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Theoretical study of magnetism,structure and chemical order intransition-metal alloy clusters

vonHerrn Junais Habeeb Mokkath

aus Kerela, India

Dissertation zur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaften (Dr. rer. nat.)

vorgelegt dem Fachbereich Mathematik und Naturwissenschaftender Universit at Kassel

Betreuer:Prof. Dr. G. M. Pastor

Disputation am 15. Februar 2012



Theoretical study of magnetism,structure and chemical order intransition-metal alloy clusters

vonHerrn Junais Habeeb Mokkath

born in Kerela, India

A Thesis submitted to the Department of Mathematics and NaturalSciences in partial fulllment of the requirements for the degree

Doctor of Natural Sciences (Dr. rer. nat.)

Theoretical physics instituteUniversity of Kassel

supervised byProf. Dr. G. M. Pastor

Examination on 15 th February 2012



To my parents, jessi and reyan

Who teacheth by the pen, Teacheth man that which he knew not.

Holy Quran.



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Abstract

Research on transition-metal nanoalloy clusters composed of a few atoms is fascinating by theirunusual properties due to the interplay among the structure, chemical order and magnetism. Suchnanoalloy clusters, can be used to construct nanometer devices for technological applications by ma-nipulating their remarkable magnetic, chemical and optical properties. Determining the nanoscopicfeatures exhibited by the magnetic alloy clusters signies the need for a systematic global and lo-cal exploration of their potential-energy surface in order to identify all the relevant energeticallylow-lying magnetic isomers.

In this thesis the sampling of the potential-energy surface has been performed by employingthe state-of-the-art spin-polarized density-functional theory in combination with graph theory andthe basin-hopping global optimization techniques. This combination is vital for a quantitativeanalysis of the quantum mechanical energetics.

The rst approach, i.e., spin-polarized density-functional theory together with the graph theorymethod, is applied to study the Fe m Rh n and Co m Pd n clusters having N = m + n ≤ 8 atoms.We carried out a thorough and systematic sampling of the potential-energy surface by takinginto account all possible initial cluster topologies, all different distributions of the two kinds of atoms within the cluster, the entire concentration range between the pure limits, and differentinitial magnetic congurations such as ferro- and anti-ferromagnetic coupling. The remarkablemagnetic properties shown by FeRh and CoPd nanoclusters are attributed to the extremely reducedcoordination number together with the charge transfer from 3 d to 4d elements.

The second approach, i.e., spin-polarized density-functional theory together with the basin-hopping method is applied to study the small Fe 6 , Fe3 Rh 3 and Rh 6 and the larger Fe 13 , Fe6 Rh 7

and Rh 13 clusters as illustrative benchmark systems. This method is able to identify the trueground-state structures of Fe 6 and Fe 3 Rh 3 which were not obtained by using the rst approach.However, both approaches predict a similar cluster for the ground-state of Rh 6 . Moreover, the

computational time taken by this approach is found to be signicantly lower than the rst approach.The ground-state structure of Fe 13 cluster is found to be an icosahedral structure, whereas Rh 13

and Fe 6 Rh 7 isomers relax into cage-like and layered-like structures, respectively. All the clustersdisplay a remarkable variety of structural and magnetic behaviors. It is observed that the isomershaving similar shape with small distortion with respect to each other can exhibit quite differentmagnetic moments. This has been interpreted as a probable artifact of spin-rotational symmetrybreaking introduced by the spin-polarized GGA. The possibility of combining the spin-polarizeddensity-functional theory with some other global optimization techniques such as minima-hoppingmethod could be the next step in this direction. This combination is expected to be an idealsampling approach having the advantage of avoiding efficiently the search over irrelevant regionsof the potential energy surface.

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Abstrakt

Ubergangsmetall-Nano-Gemisch-Cluster bestehend aus einigen wenigen Atomen stellen we-gen ihren ungew ohnlichen Eigenschaften ein faszinierendes Forschungsgebiet dar. Ihre Eigen-schaften sind zurückzufuhren auf das Wechselspiel zwischen Struktur, chemischer Ordnung undMagnetismus. Solche Nano-Gemisch-Cluster knnen durch Manipulation ihrer bemerkenswertenmagnetischen, chemischen und optischen Eigenschaften benutzt werden, um Nanometer-Bauteilefur technologische Anwendungen zu konstruieren. Die Bestimmung der nanoskopischen Besonder-heiten dieser magnetischen Gemisch-Cluster kennzeichnet die Notwendigkeit einer systematischenglobalen und lokalen Erforschung ihrer Potentialenergie-Ober¨ ache, um alle relevanten energetischniedrig gelegenen magnetischen Isomere zu identizieren.

In dieser These ist das Abtasten der Potentialenergie-Ober¨ ache mittels modernster spinpo-larisierter Dichtefunktionaltheorie in Kombination mit sowohl Graphentheorie als auch globalen

basin-hopping-Optimierungstechniken durchgef¨ uhrt worden. Diese Kombination ist entscheidendfr eine quantitative Analyse der quantenmechanischen Energetik.Der erste Ansatz, d.h. spinpolarisierte Dichtefunktionaltheorie zusammen mit der Graphen-

theorie-Methode, ist angewandt worden, um Fe m Rh n und Co m Pd n Cluster mit N = m + n ≤ 8Atomen zu untersuchen. Die Potentialenergie-Ober¨ ache wurde vollst andig und systematischabgetastet, indem alle m¨ oglichen Anfangs-Cluster-Topologien, alle verschiedenen Verteilungen derzwei Atomsorten innerhalb des Clusters, der gesamte Konzentrationsbereich zwischen den Gren-zen monoatomarer Cluster, und verschiedene magnetische Anfangskongurationen wie ferro- undantiferromagnetische Kopplung ber¨ ucksichtigt wurden. Die von FeRh und CoPd aufgezeigtenbemerkenswerten magnetischen Eigenschaften sind auf die extrem reduzierte Koordinationszahlzusammen mit dem Ladungstransfer von 3 d zu 4d Elementen zurückgefuhrt worden.

Die zweite Herangehensweise, d.h. spinpolarisierte Dichtefunktionaltheorie zusammen mit derbasin-hopping-Methode, ist gew¨ ahlt worden, um die kleinen Fe 6 , Fe 3 Rh 3 and Rh 6 und die gr oßerenFe 13 , Fe6 Rh 7 und Rh 13 Cluster als anschauliche Bezugswertsysteme zu studieren. Diese Methodeist in der Lage, die wahren Grundzustandsstrukturen von Fe 6 und Fe 3 Rh 3 zu identizieren, diemittels des ersten Ansatzes nicht gefunden worden waren. F¨ ur den Grundzustand von Rh 6 jedochsagen beide Ans atze eine ahnliche Clusterstruktur voraus. Dar¨ uber hinaus ist herausgefundenworden, dass die f ur die zweite Herangehensweise ben¨ otigte Computerzeit bedeutend geringer istals fr die erste. F ur den Grundzustand des Fe 13 Clusters ist eine Ikosaederstruktur gefundenworden, wohingegen die Rh 13 und Fe 6 Rh 7 Isomere zu k agartigen bzw. schichtartigen Struk-turen relaxieren. Alle Cluster zeigen eine bemerkenswerte Vielfalt an strukturellem und magnetis-chem Verhalten. Es ist beobachtet worden, dass Isomere mit ¨ ahnlicher Form aber geringer gegen-seitiger Verzerrung ziemlich unterschiedliche magnetische Momente aufweisen. Dieses ist als einwahrscheinliches Artefakt der Spin-Rotationssymmetrie-Brechung durch die spinpolarisierte GGAinterpretiert worden. Die M¨ oglichkeit, die spinpolarisierte Dichtefunktionaltheorie mit anderenglobalen Optimierungstechniken wie zum Beispiel der Minima-hopping-Methode zu kombinieren,

konnte der nächste Schritt in dieser Richtung sein. Es wird erwartet, dass diese Kombinationeinen idealen Abtastansatz darstellt, der den Vorteil, dass das Suchen ¨ uber irrelevante Regionender Potentialenergie-Oberche effizient vermieden wird, aufweist.

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Acknowledgments

A journey is easier when we travel together. This thesis is the result of four years of work whereby I have been accompanied and supported by many people. It is a pleasantaspect that I now have the opportunity to express my gratitude for all of them.

The rst person I would like to sincerely thank is my supervisor, Prof. Dr. G. M.Pastor for his guidance, enthusiastic and integral view on research. He gave me enoughfreedom in thinking, choosing my own problems and collaborating with other colleagueswhich helped me to bring out my best. Many thanks to Prof. Dr. Martin E. Garciafor reviewing this thesis. I would like to express my sincere gratitude to Prof. Dr. J.Dorantes D´avilla for his friendly relationship. I gratefully acknowledge Dr. Luis Dıaz foran active collaboration on basin-hopping method. He gave me motivation to learn ’python’programing language. I thank also Dr. J. L. Ch´ avez who guided my rst steps throughVASP. I thank also Dr. Pedro Ruiz Di´ az, who helped me whenever I needed his helpin Linux and VASP. Special thanks to Waldemar T¨ ows, who made my life in Germanywonderful. Special thanks to Matthieu Saubanere for reading and correcting this thesis.Thanks also to the secretary Katarina Schmidt and Andrea Wecker for their great helpsolving any situation. I thank all my friends for standing beside me all the time.

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Contents

1 Introduction 91.1 Magnetism in the bulk phase . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Magnetism in reduced dimensionality . . . . . . . . . . . . . . . . . . . . . . 121.3 Magnetic characterization methods . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.1 Stern-Gerlach (SG) experiment . . . . . . . . . . . . . . . . . . . . . 151.3.2 X-ray magnetic circular dichroism . . . . . . . . . . . . . . . . . . . 16

1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Background on electronic structure theory 192.1 The quantum many-body problem . . . . . . . . . . . . . . . . . . . . . . . 19

2.1.1 Hartree-Fock Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 The Thomas-Fermi Theory . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Density Functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . . . . . . . 23

2.2.2 The Kohn-Sham theory . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.3 Kohn-Sham equations . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Exchange-correlation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.1 Exchange and correlational functionals . . . . . . . . . . . . . . . . . 252.4 Projector augmented wave method . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 The PAW transformation operator . . . . . . . . . . . . . . . . . . . 282.4.2 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Magnetic anisotropy energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Exploring the ground-state energy surface of nanoclusters 343.1 Local Optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 Steepest Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.2 Conjugate Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.3 The Broyden Fletcher Goldfarb Shanno method . . . . . . . . . . . 36

3.2 Global Optimization schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.2.2 Simulated annealing . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.3 Basin-hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.4 Genetic algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Interplay of structure, magnetism and chemical order in small FeRhclusters 424.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2 Ab-initio relaxation of clusters . . . . . . . . . . . . . . . . . . . . . . . . . 44

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Contents

4.3 Structure and magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3.1 FeRh dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.2 FeRh trimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.3 FeRh tetramers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.4 FeRh pentamers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3.5 FeRh hexamers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.6 Exploring heptamers and octamers . . . . . . . . . . . . . . . . . . 57

4.4 Trends as a function of size and composition . . . . . . . . . . . . . . . . . . 624.4.1 Binding energy and magnetic moments . . . . . . . . . . . . . . . . . 624.4.2 Relative stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.3 Electronic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Structure and magnetism of small CoPd clusters 715.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.3.1 Dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3.2 Trimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3.3 Tetramers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.3.4 Pentamers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.3.5 Hexamers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.6 Heptamers and Octamers . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4 Size and composition trends . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.4.1 Binding energy and magnetic moments . . . . . . . . . . . . . . . . . 895.4.2 Relative stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 First principles spin-polarized basin-hopping method 936.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.1 Basin-Hopping method . . . . . . . . . . . . . . . . . . . . . . . . . . 956.2.2 Measuring the sampling efficiency . . . . . . . . . . . . . . . . . . . 96

6.3 Results for small homogeneous clusters . . . . . . . . . . . . . . . . . . . . . 986.3.1 Dominant isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3.2 Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4 Results for heterogeneous small clusters . . . . . . . . . . . . . . . . . . . . 1046.4.1 Dominant isomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.4.2 Magnetic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.5 Results for larger clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.6 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7 Composition dependent orbital magnetism 1137.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.2 Computational aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.3.1 Size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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Contents

7.3.2 Composition effects on the magnetic properties . . . . . . . . . . . . 1177.3.3 Angular dependence of the MAE . . . . . . . . . . . . . . . . . . . . 120

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8 Summary and outlook 122

Bibliography 125

List of gures 136

List of tables 139

Abbreviations and symbols 140

List of publications 142

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1Introduction

Nanoclusters composed of small number of atoms show special structural, chemical, mag-netic, and optical properties. Their properties differ qualitatively and quantitatively fromthose of atomic and the bulk limits [1]. These special features can be placed under acommon label of quantum size effects (QSE). These are due to the dramatic change inthe electronic structure of a system when its size reduced to few atoms, thus replacing thequasi-continuous density of states by a discrete energy level spectrum. Nanoclusters havelarge surface to volume ratio. This leads to the effective reduction of the mean coordi-nation number for the atoms on the cluster surface. Therefore, electronic structure of asurface atom and an interior atom within the nanocluster will be quite different. This is infact the main reason for the remarkable features shown by nanoclusters. One of the mostperplexing questions in the eld of magnetic nanoclusters is how their magnetic propertieschange in the non-scalable, reduced dimensionality regime as compared to the atomic andbulk limits. For example, it is found that nanoclusters composed of 3 d transition metal(TM) elements such as Fe N , CoN and NiN show spin moments, orbital moments, andmagnetic anisotropy energies (MAEs) that are enhanced with respect to the correspond-ing crystalline solids. Moreover, 4 d TM nanoclusters composed of elements like Rh and

Pd are known to be magnetic despite being non-magnetic in the bulk [2,4–10].The possibilities of optimizing the cluster magnetic behavior by simply tuning the sys-

tem size in the case of pure clusters have been rather disappointing, particularly concern-ing the magnetic anisotropy energy, which remains relatively small despite being ordersof magnitude larger than in solids due to the rather weak spin-orbit (SO) coupling in the3d atoms [2]. Even though magnetic nanoclusters possess sizable magnetic moments peratom, their blocking temperatures are found to be very small. From a technological per-spective the most important question is: how to realize magnetic objects which are nanoand ferro simultaneously. In other words, how we can enhance the blocking temperatureof a nanocluster in order to utilize them as recording or storage media even at room tem-perature applications. The most appropriate answer found so far to the above challenge

seems to be alloying ferromagnetic 3 d elements with highly polarizable 4 d or 5d elements.

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This so called nanoalloying is expected to materialize the long time goal of realizing thematerials having large magnetic moments and sufficiently large MAEs per atom.

Although the potential advantages of nanoalloying can be grasped straightforwardly,the problem involves a number of serious practical challenges for both theory and experi-ment. Different growth or synthesis conditions can lead to different chemical orders, whichcan be governed not just by energetic reasons but also by kinetic processes. For instance,one may have to deal with segregated clusters having either a 4 d core and a 3d outer shellor vice versa. Post-synthesis manipulations can induce different degrees of intermixing,including for example surface diffusion or disordered alloys. Moreover, the inter atomicdistances are also expected to depend strongly on size and composition. Typical TM-cluster bond-lengths are in fact 10–20% smaller than in the corresponding bulk crystals.Taking into account that itinerant magnetism is most sensitive to the local and chemical

environments of the atoms [3, 16–18], it is clear that controlling the distribution of theelements within the cluster is crucial for understanding magnetic nanoalloys. Systematictheoretical studies of binary-metal clusters are hindered by the diversity of geometricalconformations, ordered and disorder arrangements, as well as segregation tendencies thathave to be taken into account.

The above diversity of magnetic nanoclusters would make the sampling of the potentialenergy surface (PES) nearly an impossible task due to the exponentially increasing numberof local minima. This exponential growth is analogous to the famous Levinthal’s paradox,according to which a protein would never reach its native state within the lifetime of theuniverse if it would have to go through all local PES minima completely randomly [19,20].At the early stages of cluster research the most widely used approach in order to circumventthis problem was to consider educated guesses of cluster geometries. Such educated guessesmaneuvered by chemical intuition (for example: FCC, BCC, HCP, etc.) is a rst trial toexplore the magnetic behavior. However, this approach is not suitable for nanoclusters dueto the diversity of structural isomers, for example, due to strong Jahn-Teller distortionsor to the coexistence of isomers and different spin congurations in the case of magneticnanoclusters. This signals the need for a systematic, efficient, and unbiased sampling of the local and global congurational space of magnetic nanoclusters.

Concerning the solution of the many-electron problem underlying the cluster behav-ior we intent to employ the state-of-the-art DFT in the Kohn-Sham (KS) formalism [11].DFT has proven its suitability and popularity as the most powerful method in the elec-tronic structure theory. It has demonstrated to be a good compromise between accuracyand computational time. It scales favorably with the system size O (N3) compared with

Hartree-Fock (HF) and high-level correlated ab initio methods. For example, the secondorder perturbation theory by Møller and Plesset (MP2) [12] scales as O (N5). DFT hasa further advantage over HF since it is able to treat electron correlation effects. Fromthe implementation point of view, DFT is well suited for modern parallel computing andlinear-scaling techniques. Although the available computer resources are commonly one of the main considerations when choosing one particular model over the others, the dominantfactor still hinges on the physical adequacy of the chosen model for the specic propertyone is interested in. The spin-polarized version of DFT [13,14] is a powerful tool to studythe properties displayed by the magnetic materials and to bring forth results which arehaving the desired physical accuracy.

This thesis is devoted to study the unique properties exhibited by the TM magnetic

nanoclusters. The main goals are the following: i) The magnetic properties of FeRh and

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Chapter 1. Introduction

CoPd nanoclusters have been investigated as a function of size, structure, and chemicalorder. ii) We developed a novel method which combines spin-polarized DFT with thebasin-hopping (BH) global optimization method. iii) This approach has been successfullyemployed to study the properties of pure and binary magnetic nanoclusters. iv) The spinmoments, orbital moments and magnetic anisotropy energy of fcc-like FeRh clusters havingN = 13 and 19 atoms have been determined as a function of composition. A remarkablenon-monotonous dependence of the MAE is observed as a function of Fe content, i.e., upongoing from pure Fe to pure Rh. This leads to an important increase of the MAE, whichreaches about 300% at the optimal Fe concentration.

In the rest of this chapter, a brief description on the magnetic nanoalloy clusters isgiven and it is followed by a comparative study on the magnetic properties exhibitedby the materials in the bulk phase and in the reduced dimensionality such as nanowires,

nanoclusters, etc. The comparison is based on the spin moments, orbital moments, and theMAE. The main purpose of this chapter is to review the concepts and the formalism thatwill be useful for further discussion. In addition, we review two of the most widely usedmagnetic characterization methods, namely, the Stern-Gerlach (SG) cluster experimentand X-ray magnetic circular dichroism (XMCD).

1.1 Magnetism in the bulk phase

Magnetism in thermal equilibrium (not in steady states) is a quantum mechanical effectwhich cannot be explained by classical statistical mechanics. Theory shows that the par-tition function of a classical system is always independent of the vector potential eld

−→A [15]. Microscopically, the magnetism of solids originates nearly exclusively from elec-trons. Nuclear moments contribute negligibly to the magnetization, but can be important,for example, in resonance imaging. In magnetic materials, the atomic moments interactwith each other to form a variety of magnetically ordered structures. A bulk phase issaid to be ferromagnetic (FM) if all the atomic moments mutually align parallel. It isantiferromagnetic (AF) when the atomic moments of two sub-lattices of opposite direc-tion that cancel out each other. More complex non-collinear magnetic orders can also beformed depending on the interactions between the atomic magnetic moments [22]. At hightemperatures the regular magnetic order is destroyed by spin uctuations and the bulkbecomes paramagnetic [23].

One of the key parameters to quantify the magnetism in the bulk is the mean saturationmagnetization per atom (MSM). The MSM as a function of the electron lling of the dbands in TMs and their alloys can be represented by the renowned Slater-Pauling (SP)curve (see Fig. 1.1). The most remarkable feature of this plot is its triangular shapeshowing that most alloy species follow the two equal and opposite slopes. On the rightside of this plot, Co and Ni based alloys such as CoNi, NiCu, NiZn, and Co rich Fe-Co alloysare located, while the BCC Fe based alloys such as FeV, FeCr, and Fe rich FeCo alloys areon the left side. One can see from SP curve that as the total electron number decreasesfrom 28.5, where the majority and minority d-states are both lled, electron vacanciesappear only in the minority states. Therefore, magnetic moment which is dened as thedifference in the number of majority and minority electrons, increases with decreasingnumber of d-electrons at a rate of 1 µB (Bohr magneton) per electron. This leads to

a straight line on the right side with a slope of -45% degrees. In the case of FeNi, the

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1.2. Magnetism in reduced dimensionality

Figure 1.1: Mean saturation magnetization for ferromagnetic alloys as a function of totalelectron number (after Ref. [25]).

saturation moment drops sharply as the electron number decreases and approaches thephase boundary. At the peak of this curve, located at 35% Ni in Fe, a very low thermalexpansion coefficient is measured at room temperature. This alloy is called Invar, andits peculiar thermal effect is called Invar effect. The name Invar comes from the word

invariable, referring to its lack of expansion or contraction with temperature changes [24].

1.2 Magnetism in reduced dimensionalityThis section presents a compact description concerning some of the unusual magneticproperties exhibited by materials in reduced dimensionality with a special focus on nan-oclusters. In order to study the remarkable effects shown by the magnetic objects in thereduced dimensionality, we are going to analyze qualitatively the differences in electronicstructure, and the magnetic moment, in the case of Cobalt, when proceeding from onesingle atom to the bulk phase (see Fig. 1.2).

Let us consider a Co atom having seven electrons in the 3 d band and two electrons inthe 4s band. In order to satisfy the Hund’s rst rule, the spin magnetic moment shouldbe 3µB (see Fig. 1.2). Now we gradually add Co atoms one after the other to the rst Coatom. This is called bottom-up approach and this have been utilized to model many novelfunctional devices. The one by one addition of Co atoms will induce a tendency for atomsto come closer by creating bonds with each other. As the atoms combine to form solid, s–dhybridization occurs and the 3 d electrons participate in bonding. In fact, the localizationof d states near to the atomic core is the key reason for the magnetism exhibited by thetransition-metal atoms. Each Co atom provides seven electrons in the 3 d band. Thisis localized picture (Heisenberg model) and is based on the assumption that s-electronsprovide the metallic bonding and d-electrons generate the magnetic moments. This ap-proach can successfully explain the Curie-Weiss susceptibilities and the existence of spin

waves in transition metals. However, it is not really successfully to explain the saturation

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Figure 1.2: Illustration of the transition from atomic to bulk behavior in Co.

magnetization nor the magnetic moments deduced from Curie-Weiss constant [15].In the case of bulk Co, the magnetic moment per atom amounts to 1.70 µB , which is

much lower than the magnetic moment of an isolated Co atom. In fact, the appearance of non-integer value is due to the partial delocalization of the 3 d electrons. That makes themagnetic moments not entirely localized near to the atomic core. This kind of magnetismcan be explained by the Stoner model (itinerant electron model) [15, 23,26]. This modelstarts from Bloch-like electrons which are spread out through the system (localized inreciprocal k space) and the electrons are assumed to have itinerant character, formingbroad sp bands and narrow d bands. The width of the band is inversely proportional tothe degree of localization of the atomic states, and increases almost continuously whenadding one atom after the other until it reaches the bulk value. This model can successfullyexplain the fractional number of magnetic moments in transition metals, however, it givespoor description of their thermal properties.

The models mentioned above are apparently insufficient to study the unique magneticproperties exhibited by the nanoclusters, since in clusters a large number of atoms areon their surface. Consequently, there are signicant differences in the electronic structureof an atom on the surface compared to an atom inside the cluster. The reduced coor-dination number on the cluster surface will reduce the effective hybridization of orbitalsthat give rise to a prominent contribution to the total magnetic moment. We can restatethe above sentence as reducing the coordination number of a nanostructure can lead toenhanced magnetic moments . This behavior can be quantied by the Stoner criterion [37],which states that ferromagnetism occurs only when the product of the density of statesat the Fermi level D(E F ) and the material dependent exchange integral I is larger than1: D(E F )·I > 1. In the bulk only Fe, Co and Ni fulll this condition, due to their highd-density of states (DOS) at the Fermi level. A narrow d band width W d implies a largernumber of states at the Fermi level: W d 1/D (E F ). Tight binding calculations have

derived a simple relationship between W d and the local environment in nanostructures:

13



1.2. Magnetism in reduced dimensionality

Figure 1.3: Experimentally determined size dependence of the magnetic moment per atomof RhN clusters in the size range N = 9–34. Notice that the Rh 15 , Rh 16 and Rh 19 exhibitparticularly large magnetic moment. These results have been obtained by Cox et al. [51].

W d = 2√ Z ·hd , where Z is the number of nearest neighbors and hd is an average hoppingmatrix element. hd depends on the overlap of nearest neighbors d orbitals and is thereforeelement specic. It is then clear that, by reducing the effective coordination number, the dband width is reduced, with a subsequent increase of D (E F ) (for more information aboutthis concept see Ref [4]). In particular, TM elements that are non-magnetic in bulk suchas Rh, and Pd etc might become magnetic in the reduced dimensions, for example, innanowires, nanoclusters, thin lms etc. [38–50]

The rst magnetic measurements of 3 d and 4d nanoclusters as a function of the numberof constituent atoms were performed in the beginning of the 1990s by the groups of W. deHeer [30,31] and L. Bloomeld [32–34,51]. Both groups determined the average magneticmoments experimentally by using a SG magnet. In Ref. [51] magnetic moments of Rhclusters were investigated, as a function of cluster size (see Fig. 1.3). It has been foundthat, the average magnetic moment per atom increased signicantly by reducing the clustersize. In addition, the magnetic moment exhibits an oscillating behavior, which is probablydue to ferromagnetic-like or antiferomagnetic-like coupling among atomic shells inside the

cluster.First theoretical calculations on the magnetic properties of Rh clusters were performed

by Galicia [52] by using a molecular orbital approach. He computed a magnetic momentof 1 µB per atom for Rh 13 cluster with bulk-like structure. Later, Reddy, Khanna, andDunlap [53] predicted the magnetic moment for Rh 13-atom clusters with icosahedral andcubo-octahedral symmetry. They found 1.60 µB per atom for icosahedral Rh 13 and 1.46for fcc Rh13 . The rst theoretical calculations on the electronic structure and magneticproperties were performed on small Fe and Ni clusters in the 1980’s by Lee et al. [29].They found narrowing of the d bands and consequent enhancement in spin polarizationcompared to the bulk in the small Fe clusters. Shortly later, G. M. Pastor et al. [4]extended these calculations by taking different geometric structures in the size range from

2 to about 50 atoms per cluster. This study shown that Fe clusters having less than 10

14




atoms present magnetic moments of about 3 µB with decreasing values for larger clusters.These theoretical prediction where subsequently veried by the SG experiment of W. deHeer et al. [6].

1.3 Magnetic characterization methods

A concise description of two important magnetic characterization methods used to studythe behavior of nanostructures are presented in the following. The rst one concerns theworking principle of a SG device, while the second one presents the XMCD technique.

1.3.1 Stern-Gerlach (SG) experiment

Stern-Gerlach experiment [54] have been used to study the magnetic moment of clustersand established a very fundamental empirical fact: The electron has an intrinsic magneticmoment µS which adopt quantized values µB = ±e / 2mc along any axis, µB is the Bohrmagneton. A schematic diagram of the experiment is shown in the Fig. 1.4.

Figure 1.4: Schematic diagram of the Stern-Gerlach cluster beam experiment. The clusterbeam is collimated before traveling down the length of a magnet with a strongly inho-mogeneous eld along the z. (a) If the spatial components of angular momentum arequantized, the beam will be split into distinct beams along z, resulting in separate spotsat the rear detection plane. (b) If angular momentum is classically distributed in space,the beam will be evenly spread in z.

A compact description about the working principle is as follows. A beam of clustersproduced in a cluster source passes through an inhomogeneous magnetic eld along the zdirection. Due to the space quantization, the magnetic moment of clusters in the beamwill have quantized z components and will experience a quantized deecting force in thez direction. After the magnet, the beam hits a screen so that the deected distribution of clusters could be directly observed. After inspection of the detector screen, and in contrastto the atomic experiment and initial expectations, a broadening of the cluster beam proleand a shift of the distribution towards the direction of the increasing magnetic eld hasbeen observed.

The effective magnetic moment per atom µeff can then be derived by assuming a

longer superparamagnetic relaxation, namely,

15



1.3. Magnetic characterization methods

µeff = µLNµBkB T = µ coth

NµBkB T −

kB T NµB ,

1.3.1

where N is the number of atoms in the cluster, kB is the Boltzmann’s constant, T is thetemperature of the cluster ensemble, and B is the external magnetic eld.

1.3.2 X-ray magnetic circular dichroism

The XMCD technique has been experimentally developed by Sch¨ utz et al. [55]. By usingXMCD one can probe the magnitude of both orbital and spin magnetic moments of specicelements within a sample. Consequently, this versatile and powerful technique has becomewell-known in recent years. XMCD can sample specic elements of FM ordered materials,

but it is unresponsive to AF order. This is because the signal is proportional to the averagemagnetic moment. Therefore the signals from parallel and antiparallel orientated spinscancel out. A schematic diagram of the density of states is shown in Fig. 1.5, depictingthe spin-dependent absorption as a single-electron two-step process.

Figure 1.5: Schematic representation of L edge X-ray absorption of a) a nonmagneticmetal and b) a magnetic metal using right circularly polarized (RCP) and left circularlypolarized (LCP) light. The asymmetry in the spin up and spin down states give rise toXMCD signal (after Ref. [27]).

The 2 p core states are split by the spin-orbit interaction into the j = 3 / 2 and j = 1 / 2levels. Right circularly polarized (RCP) photons excites preferentially spin-up electronsat the p 3/ 2 (L3) edge because the orbital and spin angular momenta are parallel j = l + s,while left circularly polarized (LCP) photons excites preferentially spin-down electrons at

the p 1/ 2 (L2) edge because the orbital and spin angular momenta are antiparallel j = l−s.

16




Figure 1.6: Upper part: photo-absorption spectra of 7.5 nm FeCo particles deposited ona Ni (111) lm grown on W (110). The spectra are taken in remanence with circularlypolarized radiation from the 2 p core levels. The Fe and Co spectra have been enlarged bya factor 30. Lower part: the corresponding XMCD data given by the intensity differences

(after Ref. [59]).

These differences are the reason behind the L 2 ,3 XMCD spectrum. The L 3 and L2 peaksare opposite in sign. The sum of the white line intensities, denoted I L 3 and IL 2 respectively,is directly proportional to the number of d holes.

The orbital and spin moments are related to the absorption spectra by

µL = µB · Lz = 2 nh ∫ L 3 + L 2σ+ −σ− dE

∫ L 3 + L 2σ+ + σ0 + σ− dE

,

1.3.2

and

µS = 2 µB S z + 72

T z = 32

nh∫ L 3σ+ −σ− dE −2∫ L 2

σ+ −σ− dE

∫ L 3 + L 2σ+ + σ0 + σ− dE

.

1.3.3

The method measures the quantities Lz and S z + 7 T z per valence band hole nh ,where Lz and S z are the expectation values of the z components of the orbital andspin angular momenta of the atoms and T z is the expectation value of the magneticquadrupole operator. As an example, the XMCD spectra of FeCo particles deposited on

a Ni (111) lm grown on W (110) are shown in Fig. 1.6

17



1.4. Thesis outline

1.4 Thesis outlineAs discussed in the previous sections, the 3 d-4d and 3d-5d magnetic nanoalloy clusters(e.g., FeRh, CoPd etc.) exhibit special properties and offer many interesting features forfuture magnetic nanometer device applications. In this thesis, we have investigated themagnetic properties of these clusters by employing the state-of-the-art density functionaltheory (DFT) method. We have used different optimization methods such as graph theoryand basin hopping in order to study the complex energy landscape of these clusters. Someremarkable results have been observed from these studies. For instance, in the FeRh dimerwe found an important transfer of d electrons from Fe 3 d to Rh 4d orbital, which allowsthe Fe atom to develop very large spin moment, due to the larger number of d holes.

The rest of the thesis is organized as follows. Chapters 2 provides the background

information on theoretical and computational procedures we used in this thesis. For thesake of clarity we begin with Hatree-Fock wave function method. Then we focus onDFT in which framework all the calculations reported in this thesis have been performed.Different approximations to the exchange and correlation functional are discussed in moredetail: the local density approximation (LDA), the generalized gradient approximation(GGA). Chapter 3 is mainly dedicated to review the various local and global optimizationmethods used in present-day calculations. Among the local optimization methods we ndthe steepest descent (SD) and the conjugate gradient (CG) schemes, while the globaloptimization methods include the graph theory, simulated annealing (SA), basin-hopping(BH), and genetic algorithms (GA). Chapter 4 and 5 presents the main results we obtainedfor FeRh and CoPd nanoalloy clusters. As already discussed these studies were motivatedby the peculiar magnetic properties exhibited by the pure clusters of Fe, Co, Rh, and Pdand the possibilities of tailoring them through alloying. Indeed, a number of interestingnew effects are revealed by these calculations. Chapter 6 presents the results of ourcombined spin-polarized DFT and BH approach on some representative pure and alloymagnetic clusters. Chapter 7 analyzes the spin moments, orbital moments and the MAEof the FeRh clusters having 13 and 19 atoms. In this study we focus on the effect of size, and the composition on the magnetic properties. Finally, Chapter 8 summarizes ourconclusions and outlines some relevant directions for future work in this eld.

18



2Background on electronic structure theory

2.1 The quantum many-body problemAtoms, clusters and solids are composed of mutually interacting electrons and nuclei. Theprecise determination of the electronic structure is a very difficult task due to the numberof electrons involved, which renders an analytic solution impossible for systems with morethan one electron. Obviously, the complexity grows dramatically with increasing numberof electrons.

The time-independent many-body Schr¨ odinger equation can be written as

H Ψ(r 1 , r 2 ,...r N ) = E Ψ( r 1 , r 2 ,...r N ),

2.1.1

where H is the Hamiltonian, Ψ( r 1 , r 2 ,...r N ) is the many-body wavefunction and E is thetotal energy of the system [69]. The Hamiltonian in atomic units can be written as

H = − 12m zi

M

i=1

2R i −

12

N

i=1

2r i +

M

i

M

j>i

Z i Z j

|R i −R j |

−N

i=1

M

j =1

Z i

|r i −R j | +

N

i=1

N

j>i

1

|r i −r j |,

2.1.2

where M and N are total number of nuclei and electrons in the system, m z , Z and R arethe mass, charge and position of the nuclei, and r i represents the position of the electroni. The rst two terms in Eq. 2.1.2 are the kinetic energy of the nuclei and the electrons,respectively. The remaining three terms are Coulomb energies, representing the ion-ion re-pulsion, the ion-electron attraction and the electron-electron repulsion, respectively. TheSchr odinger Eq. 2.1.1 with the Hamiltonian given above is impossible to be solved ana-lytically except for selected single-electron problems. Therefore, it is unavoidable to make

approximations for both the Hamiltonian and the many-body wavefunction Ψ.

19



2.1. The quantum many-body problem

The rst major simplication is introduced by the Born-Oppenheimer (BO) approxi-mation, which allows to decouple the electronic and ionic degrees of freedom due to thelarge difference in mass between the electrons m e and the ions M I (around 103) [70].In the BO approximation the ions behave like classical particles. In other words, ionsare regarded as if they were at rest at some xed positions, while the electrons are mov-ing within a xed external potential due to the nuclei. With the BO approximation theHamiltonian can be simplied to a electronic model. In the non-relativistic approximationit is given by

H = −12

N

i=1

2r i −

N

i=1

M

j =1

Z j

|r i −R j | +

N

i=1

N

j>i

1

|r i −r j |.

2.1.3

Despite this assumption the electronic Schr¨ odinger equation, remains extremely com-plicated to solve for most of the realistic situations. The next sections give a short review of the most famous effective eld approaches, namely Hartree-Fock (HF) theory and densityfunctional theory (DFT).

2.1.1 Hartree-Fock Theory

Fock formulated an approximation [74] in order to deal with the problem associated withthe Hatree approximation by writing the wavefunction as a fully antisymmetrised productof single-orbital states:

ΨHF = 1√ N !

ψ1(r 1s1) ψ2(r 1s1) · · · ψN (r 1s1)ψ1(r 2s2) ψ2(r 2s2) · · · ψN (r 2s2)

... ...

...ψ1(r N sN ) ψ2(r N sN ) · · · ψN (r N sN )

2.1.4

ΨHF = 1√ N !

det[ψ1(r 1s1)ψ2(r 2s2) · · ·ψN (r N sN )]

2.1.5

This ansatz fullls the antisymmetry condition and known as a Slater determinant [75].The Eq. 2.1.4 has the desired full antisymmetry property, since interchanging the positionof two electrons is equivalent to interchanging the corresponding columns in the determi-nant, which changes its sign. The orbitals are subject to the orthonormality conditions

ψi (r )ψ j (r )dr = ψi|ψ j = δ ij ,

2.1.6

The HF energy can be evaluated by taking the expectation value of the Hamiltonian

with the above Slater determinant. This yields:

20



Chapter 2. Background on electronic structure theory

E HF = ΨHF

|H

|ΨHF

=N

i 12| ψi(r )|2 −

M

j =1

Z j

|r −R j||ψi(r )|2 d3 r

+12

N

i,j,i = j |ψi (r )|2

1

|r −r ′||ψi(r ′ )|2d3r d3 r ′

−12

N

i,j,i = j

δ s zi s zj ψi (r ) ψ j (r ′ ) 1

|r −r ′| ψ j (r ) ψi (r ′ ) d3 r d3 r ′

The rst term is the kinetic energy of the electrons, the second one is the electron-ioninteraction and the third one is the interaction between electrons. The last term arisesfrom the antisymmetric nature of the HF wavefunction. It vanishes when si

= s j . It isknown as the exchange energy E .

Consider a spin-up electron at position r i . Then, according to the Pauli’s exclusionprinciple, all the other spin-up electrons will be effectively moved away from the position

r i . On the other hand, the spin-down electrons will not be repelled by the Pauli principle,or more generally by the antisymmetry of the wavefunction, since they have a differentspin quantum number. In fact there is a sort of enhancement of the probability of ndingtwo electrons on the same point r i when they form a singlet spin state (antiparallel spins).Thus spin-up electron is encircled by a space which has been depleted of other spin-upelectrons. Similarly, for a spin-down electron there is a space depleted of other spin-downelectrons. This region is called the exchange hole .

In fact, the electronic motion in realistic systems are correlated beyond the predic-tions of the HF formalism. The interaction energy neglected by HF is known as thecorrelation energy E c [76].

E c = E 0 −E HF ,

2.1.7

where E 0 is the exact ground state energy [77]. Taking into account the variational prin-ciple it is clear that E c < 0.

The electronic structure calculations which neglect electron correlation are found toproduce signicant deviations from experimental results. One way to bring in the cor-relation effects is to construct linear combinations of Slater determinants correspondingto excited-state congurations. These methods are conjointly called post Hartree-Fockmethods, e.g., the conguration interaction (CI), the coupled cluster (CC) and Møller-Plesset (MP) methods. These have allowed one to include electron correlations into themulti-electron wave function. However, these implementations require a prohibitively hugecomputer time. In practice they do not allow to study systems having more than 10 elec-trons [78].

Another kind of hole (a dynamic electron density depletion) is present in the neighbor-hood of any electron, due to the electronic correlation beyond HF. This electron-depletedregion that surrounds each electron is known as the Coulomb hole or correlation hole .

The HF approach is an improvement over the Hartree theory. It is often applied asa rst approximation to oxides and crystals of small organic molecules, which have smallnumber of localized electrons. For transition metals, it is not appropriate, in particular

because of the high electron density. The HF method cannot be used for metals, as it

21



2.2. Density Functional theory

neglects the collective Coulomb screening in a completely delocalized electron system. Bythe way, one can show that HF DOS is always zero precisely at the Fermi energy. Thus,the HF approximation can never describe metallic behavior.

2.1.2 The Thomas-Fermi Theory

Thomas and Fermi (TF) introduced theory [79, 80], which can be considered as the rstDFT, since it employs electron density n(r ), rather than the many-electron wavefunction,as the fundamental unknown of the many-body problem. The total energy functional canbe written as

E T F [n(r )] = C F

n(r )5/ 3d3 r +

n(r ) vext (r ) d3 r +

12

n(r )n(r ′ )

|r

−r ′

| d3 r d3 r ′

2.1.8

The rst term is the kinetic energy of the non-interacting electrons in a homogeneouselectron gas with density n. The second term is the electrostatic interaction energy betweennuclei and electrons, where vext (r ) is the Coulomb potential

vext (r ) = −M

j =1

Z j

|r −R j |.

2.1.9

generated by the nuclei. Finally, the third term is the classical Coulomb repulsion betweenelectrons, known as the Hartree energy. The TF model is simple and appealing, but itshould not be used when quantitative predictions on realistic systems are desired. It’s main

failure comes from the incorrect approximation of the kinetic energy. Another problemis the over-simplied description of the electron-electron interactions, which are treatedclassically and hence do not take account of quantum phenomena such as the exchangeinteraction [81, 82]. In fact TF theory has been rapidly abandoned since one can showsthat it predicts that the energy of isolated atoms is always lower than the energy of any molecule. Thus the TF theory cannot predict any molecular bonding. Still, theidea of replacing the wavefunction Ψ ( r 1s1 , · · ·r N sN ) by the density of electrons n(r ) is aremarkable approach, which later on provided a major breakthrough.

2.2 Density Functional theory

Finding the approximate solution by taking advantage of modern computational powerseems not surprising nowadays. Dirac’s famous statement concerning the equations gov-erning chemistry are still authoritative today, if one takes for granted that the Schr¨ odingerequation can be directly numerically solved. DFT appears as an alternative method tothe theory of electronic structure, in which the electron density n(r ), rather than themany-body wave function Ψ, plays the key role [84]. This idea is proven to be excellent inall respects, since it represents an exact ground-state quantum theory which is alternativeto the wavefunction approach. Walter Kohn has been indeed awarded the Nobel Prizein Chemistry in 1998 for his remarkable work in this eld, shared with John Pople, whocontributed with the quantum chemistry approach. Nevertheless, although in principleexact, the theory involves an universal functional F HK [n], which is in general not known

and which must be therefore approximated.

22




DFT deals with systems of identical particles [85], basically fermions, providing a sim-ple method for describing the effects of exchange and correlation in an inhomogeneouselectron gas. The minimum of the total energy as a functional of n(r ) is the ground-stateenergy of the system, and the density that yields this minimum value is in principle theexact ground-state density n(r ). Kohn and Sham [11] showed how to replace the many-electron problem by an exactly equivalent set of self-consistent one-electron equations.Furthermore, they showed that all other ground-state properties of the system (e.g. cohe-sive energy, lattice constant, etc.) are functionals of the ground-state electron density [87].They derived eigenvalue equations from the variational approach which take a simple formand which are analogous to the HF equation. These so-called Kohn-Sham equations areactually simpler to solve than HF ones.

DFT has had a huge impact on realistic calculations of the properties of molecules

and solids, and its applications to diverse systems will continue to grow. There are state-of-the-art applications of DFT in wide variety of areas, including magnetism, catalysis,surface science, nanomaterials, biomaterials and geophysics [93]. DFT has been appliedto degenerate ground states [88], spin-polarized ground states [84,88], quantum Hall effect[94], etc. However, in some special cases DFT predicts wrong results. For example,DFT has limited accuracy in the calculation of excited states. A particularly famousexample is the underestimation of band gaps in semiconductors and insulating materials[87]. Among the failures of DFT, with the functionals known nowadays, one may mentionthe larger binding energies in LDA, inaccurate results associated with weak van der Waalsforces, the Kohn-Sham potential decays exponentially for large distances instead of 1/r ,and strongly correlated solids such as NiO and FeO are predicted as metals and not asantiferromagnetic insulators [97].

2.2.1 The Hohenberg-Kohn theorems

The formal footing of the DFT is provided by two theorems demonstrated by Hohenbergand Kohn in 1964.

Theorem I: The ground-state density n(r ) of a many-body quantum system in some external potential vext (r ) determines this potential uniquely.

This means that n(r ) determines N and vext (r ), and hence all properties of the ground-state, for example, the kinetic energy T [n(r )], the potential energy V [n(r )] = ∫ n(r )vext (r )and the electron-electron interaction energy U ee [n(r )]. The total energy E [n(r )] [99] isgiven by

E [n(r )] = V [n(r )] + T [n] + U ee [n] 2.2.1

We can group together all functionals which do not involve vext (r ) as

E [n(r )] = V [n] + F HK [n] = n(r ) vext [n] d3r + F HK [n],

2.2.2

where F HK is known as the Hohenberg-Kohn functional.Theorem II: For any trial density n(r ) it holds E 0 ≤E [n(r )], where E 0 is the ground-

state energy of the system .In other words, the minimum value of the total-energy functional E [n(r )] is the ground-

state energy of the system, and the density which yields this minimum value is the exact

ground-state density of the many-body system [99].

23



2.2. Density Functional theory

2.2.2 The Kohn-Sham theory

Although providing the theoretical foundations of DFT, the HK work of 1964 does notunravel an explicit way to solve the many-body problem. One year after, Kohn and Sham(KS) developed this important scheme in practice. The KS formulation [11] is based onmapping the full interacting electronic system onto a ctitious non-interacting system,so that the complex many-body problem can be transformed into a set of self-consistentsingle-particle equations.

2.2.3 Kohn-Sham equations

The TF idea of obtaining the ground-state kinetic energy from a non-interacting system isalso the starting point for the KS scheme. In order to assess the kinetic energy of N non-interacting particles given only their density distribution n(r ), the corresponding potentialvs [r ] yielding the exact ground-state density in the absence of interactions is introduced.The single particle Schr¨odinger equation reads

−12

2 + vs (r ) ψi(r ) = εiψi (r )

2.2.3

So that

n(r ) =N

i=1|ψi (r )|2 .

2.2.4

Since the potential vs (r) is a functional of the density, then Eqs. 2.2.3 and 2.2.4 haveto be solved self-consistently.

The KS total-energy functional for a set of occupied electronic states ψi can be writtenas

E [n(r )] = −i

ψi2ψi d3r + V ion (r ) n(r ) d3 r +

12 n(r )n(r ′ )

|r −r ′ | d3 r d3 r ′ + E xc [n(r )] + E ion (R I ),

where E ion stands for the Coulomb energy associated with interactions between the nuclei(or ions) at the positions RI and V ion is the electron-ion energy, and E xc [n(r )] is the

exchange and correlation (XC) energy.Strictly speaking only the minimum of the KS energy functional has physical meaning,corresponding to the ground-state energy of the system of electrons with the ions atpositions RI [100]. It is essential to nd out the set of wave functions ψi that minimizethe KS energy functional. These are given by the self-consistent KS equations [11], whichtake the form

−12

2 + V ion (r ) + V H (r ) + V XC (r ) ψi (r ) = εi ψi (r ).

2.2.5

As before, ψi denotes the wave function of electronic state i and εi the corresponding KSeigenvalue. The terms within the brackets can be regarded as an effective single-particle

Hamiltonian, where V H is the Hartree potential given by

24




V H (r ) =

n(r ′ )

|r −r′

|d3 r ,

2.2.6

and the exchange and correlation potential

V XC (r ) = δE xc [n(r ′ )]

δn(r )

2.2.7

is formally given by the functional derivative of the exchange and correlation energy withrespect to the density.

The KS equations must be solved self-consistently so that the occupied electronic statesrender a charge density that produces the electronic potential that is used to construct theequations. The sum of the single-particle KS eigenvalues does not give the total electronicenergy because of some kind of double counting of the effects of the electron-electroninteraction in the Hartree energy and in the exchange-correlation energy [101].

2.3 Exchange-correlation energyThe exchange and correlation (XC) potential V XC is a functional derivative of the ex-change and correlation energy with respect to the local density [see Eq. 2.2.7]. For ahomogeneous electron gas, this will only depend on the value of the electron density. Fora non-homogeneous system, the value of the exchange correlation potential at the pointr depends not only on the value of the density at r , but also on its variation close tor , where ′′ close′′ is a microscopic distance of comparable magnitude as the local Fermiwavelength or the TF screening length [78]. It can therefore be written as an expansionover the gradients to arbitrary order of the density:

V xc [n(r )] = V xc [n(r ), n(r ), ( n(r )) , ....]. 2.3.1

The exact form of the energy functional is, so far, unknown, as the inclusion of densitygradients makes the solution of the DFT equations difficult. The simplest way to obtainthis contribution is to assume that the exchange and correlation energy leads to an XCpotential at point r i that depends only on the value of the density n (r ) at r , i.e., not onits gradient. This is known as the local density approximation (LDA).

2.3.1 Exchange and correlational functionals

Some of the main XC functionals used nowadays are the following:

• The LDA which takes into account only the local aspects of the density.• The generalized gradient approximation (GGA) in which the dependence on the

gradients of the density is further added.

• The meta-GGA (MGGA), which including the dependence on the kinetic energydensity.

• The hybrid functionals, in which exchange HF-like contributions are added to thedependence on the occupied orbitals. An example of this kind is the exact exchangeapproach.

• The fully non-local functionals, in which more complex dependence on the unoccu-pied orbitals are incorporated.

25



2.3. Exchange-correlation energy

The local density approximation

LDA is a simplest approximation to quantify the exchange and correlation energy [11].In this approximation the XC energy of an electronic system is constructed by assumingthat the XC energy per electron at a point r in the electron gas εxc [n(r )] is equal to theXC energy per electron in a homogeneous electron gas having the same (homogeneous)density n(r ). Thus,

E LDAxc [n(r )] ≡ εxc [n(r )]n(r )dr

2.3.2

andδE xc [n(r )]

δn(r ) =

δ [n(r )εxc (r )]δn(r )

2.3.3

withεxc (r ) = εhom

xc [n(r )].

2.3.4

Numerical correlated calculations allow one to know εxc (r ) to a very high accuracy. Phys-ically, one expects LDA to become exact when the length scale over which n(r ) varies isvery large [84].

By construction LDA is exact for a uniform electron gas. In other words, it should bea good approximation for slowly varying densities. Surprisingly, the LDA has proven toyield very good results in many applications, even for atomic systems where the hypoth-esis slowly varying density evidently violated [100]. This is due to the fact that the LDA

shows many formal features, such as the sum rule for the exchange and correlation hole.However, the LDA also has several important failures. For example, the LDA systemat-ically underestimates the band gap in semiconductors. In the case of Ge the calculatedband gap is even negative, which erroneously indicates that Ge should be metal, and nota semiconductor [100].

The generalized gradient approximation

The next level of approximation beyond LDA is given by a number of non-local approaches.They are usually termed as generalized gradient approximations (GGAs). They take theform

E GGAXC = f (n(r , | n(r )|))dr,

2.3.5

in which f (n(r , | n(r )|)) is a suitably chosen function of n(r ) and n(r ) [84].Even though GGAs should render a systematic improvements over the LDA, it remains

a main problem that the gradients in real materials can be so large that the expansionbreaks down. By imposing conditions on the correct exchange hole given by the gradientexpansion, Perdew [89] proposed a model which leaves only a 1% error in exchange energy.This model has also been further simplied [90] to

E GGAXC = −C 1 dr 3 n4/ 3 F x (s),

2.3.6

with

26




s = |n(r )

|2kF n ,

kF = C 2n 1/ 3 .

andF x (s) = (1 + 1 .29s2 + 14 s4 + 0 .2s6)1/ 15 ,

2.3.7

C 1 and C 2 are constants. Note that the LDA corresponds to F x (s) = 1. However, severalother forms for F x (s) have been suggested [91].

The most widely used XC functionals have been proposed by Becke (B88) [115], Perdewand Wang (PW91), and Perdew, Burke, and Enzerhof (PBE) [116]. Finally, a useful

collection of explicit expressions for some GGAs can be found in the appendix of Ref. [92].

2.4 Projector augmented wave methodIn order to compute the electronic structure of a system using the DFT method, the Kohn-Sham equations have to be solved in some efficient numerical way. One of the centralissues concerning efficiency is the rather different behavior of the electronic wavefunctionat different distances from the atomic nuclei. i.e., the different behavior of the outervalance electrons and the inner core states. Indeed, the atomic wavefunctions should bemutually orthogonal. In order to comply with this condition, the valence wavefunctionsoscillate rapidly in the core region, making it very challenging to describe them precisely

without using a very large basis set. The core electrons are located very near to theatomic nucleus and therefore do not contribute to the chemical bond formation and tomagnetism. It is clear that focusing on the valence states signicantly reduces the actualcomputational time.

A widely used method to handle the core electrons is to introduce pseudopotentials bywhich nuclei and core electrons are replaced. This ideally smooth potential is constructedin order to reproduce the correct effect on the remaining valence electrons. The pseudopo-tentials are computed and tabulated once for each element. The Kohn-Sham equationsthen apply only to the valence electrons. Consequently the computational time signi-cantly reduced. However, this simplication has mainly two major disadvantages. First,all information about the true wavefunction close to the atomic nuclei is lost, renderingit impossible to determine any property which depends on the core region (e.g, electriceld gradients, hyperne parameters, etc). A second major limitation is that there is nosystematic way to construct reliable transferable pseudopotentials.

Another commonly used scheme is the all-electron method, in which all the informationabout the wavefunction is available. This method is normally connected to the frozen-core approximation, in which the core orbitals are computed and tabulated once and keptxed. This is justied by the fact that the core states do not participate in the formationof chemical bonds. One of the most important of such schemes is the augmented planewave method (APW) introduced by Slater in 1937. In this scheme the space is partitionedin two regions: a sphere one around each atom in which the wavefunction is expandedonto a local basis that is able to reproduce the fast variations, and an interstitial regionin which a plane wave basis is used. The single particle wavefunctions are of matched

at the surface of the sphere according to the usual continuity conditions. A recently

27



2.4. Projector augmented wave method

more widely spread scheme is the projector augmented wave (PAW) method developedby P. Bl¨ochl [170] as an extension to both the APW and pseudopotential methods, andwhich in fact can be retrieved by well-dened approximations [106]. The PAW methodunies all electron and pseudopotential approaches. In the following section we brieypresent this formalism, since all the calculations contained in this thesis have been withinit’s framework as implemented in the Vienna ab-initio simulation package (VASP) by G.Kresse and D. Joubert [168].

2.4.1 The PAW transformation operator

The different shape or behavior of the wavefunctions in different regions points towardsthe need for a proper partitioning of the space around the nuclei. The PAW method takes

into account this and separates the wavefunction in two parts: a partial wave expansionwithin an atom-centered sphere, and an envelope function outside. The two parts are thenmatched smoothly at the sphere edge.

We seek a linear unitary transformation ˆT which maps some computationally con-venient pseudo or smooth auxiliary wavefunctions |ψ to the physically relevant true orall-electron wavefunctions |ψ :

|ψn = ˆT |ψn ,

2.4.1

where n is a quantum-state label, including the band index, spin and k -vector index. Theground-state pseudo-wavefunction is then obtained by solving the Kohn-Sham equationsin the transformed Hilbert space

T † H T |ψn = n T † T |ψn 2.4.2

Since the true wavefunctions are smooth enough at a denite distance from the core, weanticipate that the transformation is just the unity operator beyond the augmentationcut-off and a sum of atom-centered contributions inside:

ˆT = 1 +a

ˆT a ,

2.4.3

where a is an atom index and ˆT a = ˆT a (r −R a ) = 0 for |r −R a

| > r ac . The cut-off radii

r ac is such that there is no overlap between augmentation spheres.

Within the augmentation region Ωa

, we expand the pseudo wavefunction into partialwaves φa as

|ψn =ia

cani |φa

i .

2.4.4

Similarly, for the all-electron counterpart we have

|ψn =ia

cani |φa

i

2.4.5

within Ω a . By applying Eq. (2.4.1) we obtain

|φai = (1 +

ˆ

T a

) |˜

φai

ˆ

T a

|˜

φai = |φ

ai − |

˜φ

ai ,

2.4.6

28




for all a and i, which fully determines the transformation ˆ

T in terms of the partial waves.

Hence we can express the true wavefunction as

|ψn = |ψn −ia

cani |φa

i +ia

cani |φa

i

2.4.7

with the expansion coefficients to be determined.Since ˆT is linear, the coefficients must be linear functionals of the pseudo wavefunction

|ψn , i.e., scalar productsca

ni = ˜ pai |ψn ≡P ani ,

2.4.8

where ˜ pai are some xed functions. They are known as projector functions. The notation

P ani is consistent with existing literature.Since there is zero overlap between different augmentation spheres, the one-center

expansion of a real pseudo wavefunction ∑i ˜ pai |ψn |φa

i has to be identical to |φai inside

the augmentation sphere. This is equivalent to fullling the completeness relation

i|φa

i ˜ pai | = 1

2.4.9

within Ω a which in turn implies that

˜ pai 1 | φa

i 2= δ i 1 ,i 2

2.4.10

within Ω a . In other words the pseudo functions and the partial waves are mutuallyorthonormal within the augmentation sphere.

Finally, by inserting Eq. 2.4.8 into Eq. 2.4.7 we obtain a closed form for the transfor-mation operator

ˆT =a i

(|φai − |φa

i ) ˜ pai | ,

2.4.11

which permits us to obtain the true, all-electron, Kohn-Sham wavefunction ψn (r ) = r |ψnas

ψn (r ) = ψn (r ) +a i

(φai (r ) − φa

i (r )) ˜ pai |ψn .

2.4.12

It is usually convenient to bring in the one center expansions

ψan (r ) =

i

φai (r ) ˜ pa

i |ψn

ψan (r ) =

i

φai (r ) ˜ pa

i |ψn

2.4.13

Then, the true wavefunction can be written as

ψn (r ) = ψn (r ) +a

(ψan (r ) − ψa

n (r ),

2.4.14

which split the extended-space and the atom-centered contributions. This is frequentlyemployed to obtain compact expression for various quantities in PAW. The rst term canbe evaluated on an extended grid, or on a soft basis set, while the last two terms areevaluated on ne radial grids.

In summary, the three ingredients that determine the PAW transformation are:

29



2.5. Magnetic anisotropy energy

• The partial waves φa

i , which are obtained as solutions of the Schr¨ odinger equation forthe isolated atom and utilized as an atomic basis for the all-electron wavefunctionswithin the augmentation sphere.

• The smooth pseudo partial waves φai , which match with the corresponding true

partial waves outside the augmentation sphere but are smooth continuations insidethe spheres. These are employed as atomic basis-sets for the pseudo wavefunctions.

• The smooth projector pseudo functions ˜ pai , one for each partial wave, which satisfy

the condition ˜ pai 1 |φa

i 2= δ i 1 ,i 2 inside each augmentation sphere.

2.4.2 Approximations

Up to this point the PAW method may be considered as an exact implementation of theDFT. In order to make it a practical scheme, the following three approximations are re-quired.

i ) Frozen Core The frozen-core approximation assumes that the core states are localized in theaugmentation spheres and that the core states are not modied by the formation of chemical bonds. The core Kohn-Sham states are thus chosen to be exactly the corestates of the isolated atoms:

|φcn = |ψa,core

α

2.4.15

Notice that, in contrast to Eq. 2.4.5, no projector functions need to be specied forthe core states.

ii ) Finite basis set The extended pseudo contribution ψn in Eq. 2.4.14 is measured outside the augmen-tation spheres by using an appropriate basis set or on a real-space grid. In bothcases the lack of completeness in the basis, or equivalently the nite grid-spacing,will introduce some controllable error.

iii ) Finite number of partial waves and projectors

The number of partial waves and projector functions is obviously nite. This meansthat that the completeness condition we have assumed is not strictly fullled. How-ever the approximation can be controlled by increasing the number of partial wavesand projectors, so that they form a satisfactory complete space for the expansion of the wavefunctions within the augmentation spheres.

More detailed information about the PAW method may be found in Ref. [170].

2.5 Magnetic anisotropy energyMagnetic anisotropy can be dened as the change of the total energy of a magnetic system

as a function of the orientation of the magnetization −→M with respect to the crystalline axis.

30




Figure 2.1: Denition of direction cosine. Taken from Ref. [25].

This means that there are some unique directions in space in which a magnetic materialis easy or difficult to magnetize. The former is called easy axis for −→M, whereas the latteris called hard axis for −→M. The difference in the total energy between easy and hard axismagnetization is known as magnetic anisotropy energy (MAE).

The main sources of the magnetic anisotropies are following:

• Magneto-crystalline anisotropy: The magnetization is oriented along specic crys-talline axes.

• Shape anisotropy: The magnetization is affected by the macroscopic shape of thenanoparticle.

• Induced magnetic anisotropy: Specic magnetization directions can be stabilized bytempering the sample in an external magnetic eld.

• Stress anisotropy or magnetostriction: Leading to a spontaneous deformation.

• Surface and interface anisotropy: Surfaces and interfaces often exhibit different mag-netic properties compared to the bulk due to their asymmetric environment.

As an example, we review the magnetic anisotropy exhibited by the cubic crystals. In

cubic crystals, the magneto-crystalline anisotropy energy is given by a series expansion interms of the angles between the direction of magnetization ( α 1 , α 2 , α 3) and the axes of thecube (see Fig. 2.1). In a spherical coordinate system the direction cosines are given by

α 1 = sin θcosφ, α 2 = sin θsinφ, and α 3 = cosθ,

2.5.1

with θ and φ being the polar and azimuthal angles, respectively.The energy per unit volume E MCA can be expanded phenomenologically as a power

series of the direction cosines α 1 , α 2 and α 3 :

E cubicMCA = K 0 + K 1(α 2

1α 22 + α 2

1α 23 + α 2

2α 23) + K 2α 2

1α 22α 2

3 + ...

2.5.2

The corresponding phenomenological parameters are obtained by inserting the direc-

tion cosine into Eq. (2.5.2):

31



2.5. Magnetic anisotropy energy

E 100 = K 0 ,

2.5.3

E 110 = K 0 + 14

K 1 ,

2.5.4

and

E 111 = K 0 + 13

K 1 + 127

K 2 .

2.5.5

The K n are known as the anisotropy constants. The MAE in the cubic 3 d transitionmetals is of the order of a few µeV/atom. Typical values of K 2 for bulk 3d metalsare 4.02 µeV/atom for Fe, 45 µeV/atom for Co, and -0.3 µeV/atom for Ni [117]. Onthe other hand, nanostructured materials such as monolayers, chains, clusters, or evensingle magnetic atoms deposited on non-magnetic surfaces, the MAE is of the order of meV/atom, and it can be as high as 20 meV/atom as in the case of Co/Pt(111) [50].

One of the major microscopic sources of MAE is the spin-orbit (SO) interaction. TheSO term is a relativistic correction to the Hamiltonian and is given by H SO = ζ S .Lwhich couples the spin operator S with the lattice, represented by the orbital momentumoperator L. The spin operator S is proportional to the Pauli matrices and ζ is the spin-orbit coupling constant. H SO depends on both ζ and the magnitude of the L. ζ is energydependent, whereas L strongly depends on the local environment of the atom. For example,in bulk materials the orbital moment for 3 d electrons is quenched, due to the crystal eldinteraction.

The other main contributions to the MAE is the dipolar interaction which is related tothe shape of the object under consideration. The shape anisotropy is found in materials

having non-spherical form. It has a classical macroscopic analogue: since dipoles sponta-neously align such that the north-pole of the rst one is pointing to the south pole of thesecond one, to generate a chain. Thus, in case of a chain of atoms the shape anisotropywould lead to an easy axis along the chain axis. In the case of a thin lm, the shapeanisotropy gives an in-plane easy axis.

According to the Bruno’s [118] model based on the second-order perturbation theory,the MAE is due to spin-orbit interaction (no shape anisotropy) and is proportional to theanisotropy of the orbital moment. Thus, magneto-crystalline anisotropy can be derived as

E = − ξ 4µB

∆ L

2.5.6

where ξ is the spin-orbit parameter and ∆ L is the difference in orbital moments betweenthe easy and hard axis.

Chapter 7 is devoted to the determination of MAE in nanoclusters. In fact, tight-binding method including spin-orbit coupling [2], have proved to be suitable to providethe correct trends for the MAE. Although, these methods can provide the correct trends,they do not yield accurate values for MAE. The accurate computation of MAE can befullled by employing rst principles methods, since they provide both the accurate valuesand systematic trends, even though, they are computationally very expensive to perform.In order to determine the MAE we followed some specic steps. Firstly, we conned therotation of the magnetization vector −→M with respect to the cluster frame. This allowsto restrict the calculations into a 2D coordinate system dened by the plane, where the

polar angle θ is taken as the angle between the magnetization and the easy axis direction,

32




(i) 13 atom cluster (ii) 19 atom cluster

Figure 2.2: The fcc clusters with N = 13 and 19 atoms showing the zx plane in which themagnetization angle θ is varied.

usually chosen as the z axis. Moreover, θ has been increased in the steps of 10 degreesuntil −→M completes one full rotation around the cluster.

Fig. 2.2 illustrates our procedure to determine the MAE. For θ = 0, −→M is parallel to

the z axis. When θ is gradually increased, −→M passes through the middle of a bond andalso through the center of a triangular facet. It points along a bond axis for θ = π/ 2.In the calculations reported here we employed the so called magnetic force theorem [227].This implies that we estimated the total energy difference by the corresponding differenceof the sum of the KS eigenvalues.

33



3Exploring the ground-state energy surface of nanoclusters

The potential energy surface (PES) represents the energy of a molecule or cluster as afunction of its geometry. For a molecule having N atoms, there are 3N independenttranslational degrees of freedom corresponding to each atom minus the 3 translational

degrees of freedom of the center of the mass and 3 rotational degrees of freedom aroundit. Thus, the PES has in principle 3 N −6 dimensions. The concept of PES has beenemployed in a wide variety of phenomena, from atomic clusters, over protein folding toglasses.

Finding the global minimum of the PES is one of the most difficult and importantgoals in a number of branches of physics, chemistry, and biology. For a periodic system,the global minimum renders the crystalline ground state structure of a solid, while, fora non-periodic systems such as a nanocluster it determines the ground-state geometry.The determination of the relevant isomers in the neighborhood to the ground state is alsocrucial, since the former might be observed in an experiment due to the thermodynamiceffects.

A short description of some of the most widely used techniques to explore PES and tooptimize the energy from both local and global perspectives is given below.

3.1 Local Optimization methods

Local optimization (LO) methods are employed to locate the local minima of a function.In any LO scheme, only the information about a function from the close proximity of theinitial conguration or starting point is used in order to update the conguration and thefunction value. The optimization procedure advances to whatever nearby local minimaby following the surface downhill in some way. Eventually, the LO techniques are calleddeterministic schemes. At each iteration step, local information such as the energy and

its gradients or forces are collected in order to obtain the next structure until iteratively

34



Chapter 3. Exploring the ground-state energy surface of nanoclusters

Figure 3.1: Schematic picture of the SD scheme (black arrows) and a conjugate gradient(red arrows). After Ref. [141].

the forces disappear and a local minimum has been achieved.

3.1.1 Steepest Descent

The steepest descent (SD) is the simplest of all LO methods. It strictly seeks downhill tolocate the local minimum [144]. In each iteration step, the atoms are displaced accordingto the forces acting on them

R α,k +1 = R α,k + α kF α (R α,k )

3.1.1

where α k is a technical parameter to manipulate. This method has some major disadvan-tages. If αk is too small, many iteration steps are required, the convergence speed is ratherslow, and the process can in principle go on forever. Setting a larger αk can enhance theconvergence speed. However, the system might begin to swing around the local minimum.

3.1.2 Conjugate Gradient

The CG method is introduced by Hestenes and Stiefel in 1951 [120–123,144]. This methodemploys the successive line minimizations along a search direction G α,k :

γ k = arg minα

E (R α,k + α kG α,k (R α,k ),

3.1.2

where arg min α E represents the argument α which minimizes the energy E . The atomiccoordinates are then correspondingly updated to

R α,k +1 = R α,k + α kG α,k .

3.1.3

In the rst step one has to begin along the atomic forces as rst search direction, i.e.,G α,k = F α (R α, 0). The main difference between the SD and CG methods lies in the choiceof the following search direction from current point. The SD scheme gives a criss-crossedpattern, since every new line step only takes local information into account. Instead theCG scheme does not strictly seek the PES downhill but along a search direction that issomewhat tilted or conjugate to the preceding search directions. Which is accomplishedby including a portion of the preceding search direction to the atomic forces:

G α,k = F α,k + β kG α,k − 1 .

3.1.4

35



3.2. Global Optimization schemes

(see Fig. 3.1). If the system is close to a local minima then the CG methods in principlerender the same results. However, if the system is away from the local minima, the searchdirections may become unreasonable and it is convenient often to start with a few SDsteps to bring the system near to the local minimum. One can show that for quadraticform the CG method converges in at most N line searches.

3.1.3 The Broyden Fletcher Goldfarb Shanno method

The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is sometimes called quasi-Newtonmethod. Different methods under this category differ in their updating of the inverse of the Hessian matrix H αβ . Instead of directly taking the inverse of H αβ , an iterative up-dating technique is used. The new search direction G β,i can be obtained by solving the

Newton equation

β

H αβ Gβ,i = F α (Rα,i )

3.1.5

The next structure is then obtained by performing a line minimization as in the CGscheme Eq. 3.1.2. If the PES were quadratic and the Hessian would be known accurately,the local minimum would be identied within one line search. In fact, the computationof the Hessian matrix in each iteration step can be very expensive, so that it is insteadsuccessively approximated in each iteration step, therewith being a quasi-Newton scheme.Since more information of the PES is taken into account, the BFGS method can be moreefficient than SD or even CG schemes. The best performance is achieved when the system is

near to the local minimum, i.e., where the harmonic approximation is justied. Therefore,as in the case of the CG scheme, if a LO is aimed it is important to begin with a fewSD steps to bring the system close to the local minimum. Otherwise the BFGS approachremains very efficient but the system might converge to an attractor basin that is differentfrom the one corresponding to the initial optimization.

3.2 Global Optimization schemes

Global optimization (GO) has long been an intensive research topic in many elds of science. Some of the important applications of GO are the design of integrated circuitssuch as microprocessors [129], the prediction of protein structures [126,127], and the ab-initio computation of nano-size atomic structures [125,133]. In fact GO is a non polynomial(NP) hard problem, i.e., non-deterministic and non-polynomial-time hard [124]. In otherwords there is no GO algorithm available which guarantees to locate the global minimumwithin a time that goes as some power of the number of variables.

GO schemes can be classied according to the way in which new structures are gener-ated from the initial structure or previous structures. A technique called trial move canbe used for this purpose. Every trail move generates a new structure by deforming ordistorting the initial structure in some way. The newly generated structures are acceptedor rejected according to some acceptance criterion. The total energy of the system can betaken as acceptance criterion since it is the fundamental quantity to optimize. Differentkinds of GO schemes are available at present. The most important ones will be outlined

in the following.

36




3.2.1 Graph theory

The starting cluster geometries used for the calculations reported in chapters 4 and 5 aregenerated from graph theory. Here we provide only the important aspects of this method.For more detailed information the reader refer to the Refs. [17,172,176].

The topographical bonding in the clusters can be represented by the adjacency matrixA and it is important to verify that it can be represented by a true structure in D ≤ 3dimensions. The adjacency matrix can be associated to an undirected simple graph g. Theensemble of topographical structures for cluster size N is a subset of the set G of undirectedsimple graphs having N vertices. A graph is acceptable as a cluster structure, only if a setof atomic coordinates R i with i = 1 , . . . , N exists, such that the interatomic distances R ijsatisfy the conditions R ij = R0 if the sites i and j are connected in the graph (i.e., if theadjacency matrix element Aij = 1) and Rij > R 0 otherwise (i.e., if Aij = 0). Here R0 refersto the nearest neighbor (NN) distance, which at this stage can be regarded as the unitof length, assuming for simplicity that it is the same for all clusters. The complete set of topographical structures can be generated by computing all the unidirected simple graphshaving N vertices and disregarding equivalent (isomorphic) graphs that correspond to apermutation of atomic indexes without changing the connectivity. The simplest test forgraph isomorphism consist in the comparison of the standard representation of adjacencymatrices calculated by including atomic connectivities up to third neighbors [172]. Afurther numerically more expensive test is the comparison of eigenvalue spectra of theadjacency matrices. Once the topologies are obtained the next important step is thedetermination of 3D atomic coordinates for each structure, which are consistent withthe interatomic distances and connectivity information. This is known as the distance-

geometry problem which belongs to the class of non-polynomial hard computations.The adjacency matrices for two similar cluster structures are shown in Fig. 3.2 . Theonly difference between these structures is that the atoms 1 and 6 are connected in the clus-ter structure on the bottom. The connectivity matrices for the corresponding structures,given by S1 and S2 , are of course different.

One easily nds that for N ≤ 4 all graphs are possible cluster structures. These aretetrahedron, rhombus, square, star, triangular racket and linear chain shown in Fig. 3.3[17]. However, for N ≥ 5 there are graphs, i.e., topologies, which cannot be realized inpractice. For instance, it is not possible to have ve atoms being NNs from each otherin a three dimensional space. Consequently, for N ≥5 there are less real structures thanmathematical graphs. The total number of graphs (structures) is 21 (20), 112 (104), and853 (647) for N = 5 , 6 and 7, respectively [17].

As discussed earlier, in order to perform reliable structure calculations for clusters oneneeds to consider a large, most possibly complete and unbiased set of initial structures. Inthe case of mixed clusters, a thorough geometry optimization must include not only therepresentative cluster geometries or topologies, but also all relevant chemical orders. Thusall distributions of the dissimilar atoms for any given size and composition must also betaken into account. The different distributions of atoms for representative compositionsare illustrated in the Fig. 3.4, where a square pyramid structure is considered with oneand two impurity atoms.

37




S1 =

0 1 1 1 1 01 0 1 0 1 11 1 0 1 0 11 0 1 0 1 11 1 0 1 0 10 1 1 1 1 0

=

S2 =

0 1 1 1 1 11 0 1 0 1 11 1 0 1 0 1

1 0 1 0 1 11 1 0 1 0 11 1 1 1 1 0

=

Figure 3.2: Clusters 1 and 2 with their corresponding adjacency matrices S 1 and S2 ,respectively. The clusters are similar except that the atoms 1 and 6 are connected in thecluster 2.

tetrahedron rhombus square star triangularracket

linear chain

Figure 3.3: Structures for the cluster size N = 4.

=

=

Figure 3.4: The illustration of the different chemical orders (also known as homotops) of single- and double-impurity clusters with N = 5 atoms having square pyramid topology.

38




3.2.2 Simulated annealing

The simulated annealing (SA) method is one of the most commonly used GO schemes.It was rst proposed in 1983 by Kirkpatrick [129, 130] and is based on the MetropolisMonte-Carlo (MC) scheme, by including a variable temperature to simulate the annealingprocedure of a physical system. The working principle of a typical SA method is thefollowing. Starting from a random structure with a total energy E , a new structure isgenerated by randomly displacing the atoms, leading to a change of the total energy ∆ E .If the energy has decreased, i.e., ∆ E < 0, the new structure is accepted and used as startingpoint for the following iteration. If the energy has increased, however, the structure is notrejected unconditionally. It is accepted with a probability of P (∆ E ) exp(−∆ E/k B T ).According to the acceptance criterion of the Metropolis algorithm one generates a canonicalensemble of atomic congurations at a given temperature T .

At strictly zero temperature, only structures that lower the energy would be accepted.At rst sight this might seem a good choice since one intends to push the system towardsthe ground state. However, this would be very inefficient, since the system is most likely toget stuck to a particular local minimum and never reach the true global minimum. A nitetemperature assists the system to jump out of a local minimum by following controlleduphill steps. Once the system arrives at a high temperature, an annealing schedule canbe employed. This procedure successively cools down the system, and facilitates reach-ing thermal equilibrium. Coupled to the temperature are the random displacements of the individual atoms ∆ R α which usually adopt a Gaussian distribution in the classicalsimulated annealing scheme [131]:

p(∆ R α ) exp −(∆ R α )2

/T

3.2.1

With decreasing temperature, the step width is hereby reduced,

T T 0

log(1 + t).

3.2.2

thus ideally bringing the system towards the optimal ground state geometry.

3.2.3 Basin-hopping

The calculations reported in chapter 7 are performed by using the BH method [133–135].Rather than sampling the original PES E

R i

, the method explores the transformed PES

E R i given by

E R = min E R ,

3.2.3

where min indicates a local structural relaxation.In this scheme the energy associated to a specic point R i on PES is that of the local

minimum obtained starting at this point Ri and following a local optimization. This isperformed, for example with the CG or BFGS methods. Thus, the BH scheme maps thereal PES onto a set of interpenetrating staircases with plateaus corresponding to eachbasins of attraction. A schematic view of the BH scheme is shown in Fig. 3.5.

The BH method has been derived in order to get rid of the unfavorable transition states

or high barriers on the PES. Thus, it allows for fast and efficient interbasin transitions.

39




Figure 3.5: Illustration of principle of the basin-hopping method. Depicted is a modelenergy surface together with its transformed landscape. The green arrow indicates a trialmove performed on a local minimum, being followed by a local structural relaxation (redarrows). See Ref. [222].

In contrast to the original SA scheme, Markov steps that initially result in an increaseof energy are accepted much more probably, since the subsequent LO allows the systemto relax into the corresponding local minimum which has usually a much lower energy.Although, the shapes of the original and the transformed PES are different, the nature of the local minima is not modied, since the Hamiltonian itself is not changed. A comparisonbetween molecular-dynamics and BH generated minima is thus straightforward.

In 1999, Wales et al. located global minima of Lennard-Jones (LJ) clusters having upto 110 atoms by using BH scheme. The LJ 38 is a well-known example of a double-funnelPES [137], consisting of an icosahedral funnel with a large free energy (large congurationenergy of the corresponding attraction barrier) and a fcc funnel with a comparably smallerfree energy but which actually leads to the global minimum. The original PES renders onlya small overlap in the canonical occupation probabilities with respect to the temperature,so that there is a very high possibility that the system gets trapped in the wrong funnel.Transforming the PES like in the BH scheme, however, results in a broadening of the

overlap region, which considerably helps for the system to jump out of the wrong funnel[137,138]. In fact, the BH scheme not only facilitates the interbasin transitions by removingthe often high barriers but also renders the transition between different funnels easierbecause of the barrier less stochastic-dynamics.

3.2.4 Genetic algorithms

The basic idea behind genetic algorithms (GA) is based on the Darwinian theory of naturalselection [139, 140]. New cluster structures are generated from two candidate structuresselected from a population of cluster structures, namely the parents , which are then mated to create a child .

One important condition for a good mating process is that the child should preserve

40




some of the structural properties of the parents. A common procedure for that is to chopthe parent structures by a plane that is randomly aligned, and to cross the resulting halfsfollowed by a relaxation to generate a child (see Fig. 3.6). In this way structural motifsfrom different points on the PES are then joined together to form the child rather thanperforming a pure local search like in the BH scheme.

Figure 3.6: Mating between two parent structures generating a child. After the mating,the new child is locally relaxed (after Ref. [141]).

One interesting question here is what kind of features a child should have in order tosubstitute a parent from the population. Deaven and Ho [139] formulated two arguments.The rst one is that in order to substitute a parent, a child should have lower energy.The second argument is that a child that is expected to replace a parent must be differentfrom all the other members of the population, since this will facilitates the population toremain diverse.

41



4Interplay of structure, magnetism and chemicalorder in small FeRh clusters

The present chapter is devoted to the study of the structural, electronic and magneticproperties of small Fe m Rh n clusters having N = m + n ≤8 atoms in the framework of ageneralized-gradient approximation to density-functional theory. The correlation between

structure, chemical order, and magnetic behavior is analyzed as a function of size andcomposition. For N = m + n ≤ 6 a thorough sampling of all cluster topologies has beenperformed, while for N = 7 and 8 only a few representative topologies are considered. Inall cases the entire concentration range is systematically investigated. All the clusters showferromagnetic-like order in the optimized structures. As a result, the average magneticmoment per atom µN increases monotonously with Fe content, in an almost linear wayover a wide range of concentrations. A remarkable enhancement of the local Fe momentsbeyond 3 µB is observed as result of Rh doping. This is a consequence of the increase in thenumber of Fe d holes, due to charge transfer from Fe to Rh, combined with the extremelyreduced local coordination. The Rh local moments, which are important already in thepure clusters ( N ≤ 8) are not signicantly enhanced by Fe doping. However, the overallstability of magnetism, as measured by the energy gained upon spin polarization, increaseswhen Rh is replaced by Fe. The composition dependence of the electronic structure andthe inuence of spin-orbit interactions on the cluster stability are discussed.

4.1 Introduction

Alloying elements with complementary qualities in order to tailor their physical behaviorfor specic technological purposes has been a major route in material development sincethe antiquity. Cluster research is no exception to this trend. After decades of system-atic studies of the size and structural dependence of the most wide variety of propertiesof monoelement particles, the interest has actually been moving progressively over the

past years towards investigations on nite-size binary alloys [150]. The magnetism of

42



Chapter 4. Interplay of structure, magnetism and chemical order in small FeRh clusters

transition-metal (TM) clusters opens numerous possibilities and challenges in this con-text [1,151–164]. For example, one would like to understand how to modify the magneticcharacteristics of clusters, in particular the saturation magnetization and the magneticanisotropy energy (MAE), as it has been done in solids. This would indeed allow oneto design new nanostructured materials from a microscopic perspective. Nevertheless, italso true that controlling composition, system size, and magnetic behavior sets seriousdifficulties for both experiment and theory.

Pure TM clusters such as Fe N , CoN and NiN show spin moments, orbital moments, andMAEs that are enhanced with respect to the corresponding periodic solids [2,4–9]. Still,the possibilities of optimizing the cluster magnetic behavior by simply tuning the systemsize have been rather disappointing, particularly concerning the MAE, which remainsrelatively small —despite being orders of magnitude larger than in solids [2]— due to the

weakness of the spin-orbit (SO) coupling in the 3 d atoms. This is one of the motivationsfor alloying 3d TMs with 4 d and 5d elements which, being heavier, are subject to strongerSO interactions. In this context it is useful to recall that large nanoparticles and threedimensional solids of these elements are non-magnetic. However, at very small sizes the 4 dand 5d clusters often develop a nite spontaneous low-temperature magnetization, due tothe reduction of local coordination and the resulting d-band narrowing [3,52,53,165,230].The rst experimental observation of this important nite-size effect has been made byCox et al. by performing Stern-Gerlach-deection measurements on Rh N clusters. In thiswork the average magnetic moments per atom µN = 0 .15–0.80µB have been experimentallydetermined for N ≤30–50 atoms [230]. In view of these contrasting features one expectsthat 3 d-4d and 3d-5d alloy clusters should show very interesting structural, electronic andmagnetic behaviors.

The purpose of this chapter is the investigation of the ground-state properties of thesmall FeRh clusters in the framework of Hohenberg-Kohn-Sham’s density functional the-ory [11]. Besides the general interest of the problem from the perspective of 3 d-4d nano-magnetism, these clusters are particularly appealing because of the remarkable phase dia-gram of FeRh bulk alloys (see Fig. 4.1) [166]. In the case of Fe 50 Rh 50 the magnetic orderat normal pressure and low temperatures is antiferromagnetic (AF). As the temperatureincreases this so-called α ′′ phase undergoes a rst order transition to a ferromagnetic (FM)state, the α ′ phase, which is accompanied by a change in lattice parameter. The corre-sponding transition temperature T α

′ α ′′

c increases rapidly with increasing external pressureP, eventually displacing the FM α ′ phase completely for P ≥ 7 GPa ( T α

′ α ′′

c 290K forFe50 Rh 50 at normal pressure). Moreover, T α

′ α ′′

c decreases very rapidly with decreasing Rh

content (see Fig. 4.1). At low pressures the FM α ′ phase undergoes a FM to paramag-netic (PM) transition at ( T C 670K ) [166]. In addition, the properties of α-FeRh bulkalloys have been the subject of rst principles and model theoretical investigations [167].These show that the relative stability of the FM and AF solutions depends strongly onthe interatomic distances. Such remarkable condensed-matter effects enhance the appealof small FeRh particles as specic example of 3 d-4d nanoscale alloy. Investigations of their magnetic properties as a function of size, composition, and structure are thereforeof fundamental importance.

The remainder of this chapter is organized as follows. In section. 4.2 the main detailsof the computational procedure are presented. The results of our calculations for FeRhclusters having N ≤8 atoms are reported in sections 4.3 and 4.4. First, we focus on the

interplay between structure, chemical order and magnetism in the most stable geometries

43



4.2. Ab-initio relaxation of clusters

Figure 4.1: The phase diagram of FeRh bulk alloy (after Ref. [166]).

for different cluster sizes. Second, we analyze the concentration dependence of the cohesiveenergy, the local and average magnetic moments, the spin-polarized electronic structure,relative stability and magnetic stabilization energy. Finally, we conclude in section. 4.5with a summary of the main trends and a short outlook to future extensions.

4.2 Ab-initio relaxation of clusters

The calculations reported in this chapter have been performed in the framework of Hohenberg-Kohn-Sham’s density functional theory, [11] as implemented in the Vienna ab initio sim-ulation package (VASP) [168]. The exchange and correlation energy is described by us-ing both the spin-polarized local density approximation (LDA) and Perdew and Wang’sgeneralized-gradient approximation (GGA) [224]. As discussed in the section. 2.2 theVASP solves the spin-polarized Kohn-Sham equations in an augmented plane-wave basisset, taking into account the core electrons within the projector augmented wave (PAW)method [170]. See section. 2.4 where the PAW method has extensively been described.This efficient frozen-core all-electron approach allows to incorporate the proper nodes of the Kohn-Sham orbitals in the core region and the resulting effects on the electronic struc-ture, total energy and interatomic forces. The 4 s and 3d orbitals of Fe, and the 5 s and 4dorbitals of Rh are treated as valence states. The wave functions are expanded in a planewave basis set with the kinetic-energy cut-off E max = 268 eV. In order to improve theconvergence of the solution of the self-consistent KS equations the discrete energy levelsare broadened by using a Gaussian smearing σ = 0 .02 eV. A number of tests have beenperformed in order to assess the numerical accuracy of the calculations. Increasing thecut-off energy E max = 268 eV and supercell size a = 12 A to Emax = 500 eV and a =22 A in Rh 4 increases the computation time by a factor 4–7. This yields total energy

differences of 1.75 meV and 0.25 meV, respectively. In the above calculations the changes

44




in average bond length (bond angle) amounts to 10 − 3 A (10− 4 degrees). These differenceare not signicant for our physical conclusions. In fact, typical isomerization energies inthese clusters are an order of magnitude larger, i.e., of the order of 10–30 meV. We alsofound that the total energy is nearly independent of the choice of the smearing parame-ter σ, provided it is not too large ( σ ≤ 0.05 eV). Values from σ = 0.01 to 0.1 eV havebeen checked. Therefore, we judge that our set of standard parameters (E max = 268 eV,supercell size a from 10 to 22 A, and σ = 0.02 eV) offers a sufficiently good accuracyat a reasonable computational costs. The PAW sphere radii for Fe and Rh are 1 .302 Aand 1.402 A, respectively. A simple cubic supercell is considered with the usual periodicboundary conditions. The linear size of the cell is a = 10–22 A, so that any pair of imagesof the clusters are well separated and the interaction between them is negligible. Since weare interested in nite systems, the reciprocal space summations are restricted to the Γ

point.For smaller clusters ( N ≤ 6) all the possible structures have been taken as startingpoints of our structural relaxations. See section. 3.2.1 where the cluster generation methodfor the present calculations has been described in detail. Out of this large number of different initial congurations the unconstrained relaxations using VASP lead to onlya few geometries, which can be regarded as stable or metastable isomers. For largerclusters ( N = 7 and 8) we do not aim at performing a full global optimization. Ourpurpose here is to explore the interplay between magnetism and chemical order as afunction of composition for a few topologies that are representative of open and close-packed structures. Taking into account our results for smaller sizes, and the availableinformation on the structure of pure Fe N , RhN , CoN , and Pd N clusters, we have restrictedthe set of starting topologies for the unconstrained relaxation of heptamers and octamers tothe following: bicapped trigonal bipyramid, capped octahedra, and pentagonal bipyramidfor N = 7, and tricapped trigonal bipyramid, bicapped octahedra, capped pentagonalbipyramid and cube for N = 8. Although, the choice of topologies for N = 7 and 8is quite restricted, it includes compact as well as more open structures. Therefore, it isexpected to shed light on the dependence of the magnetic properties on the chemical orderand composition.

The dependence on concentration is investigated systematically for each topology of Fem Rh n by varying m and for each size N = m + n ≤ 8, including the pure Fe N andRh N limits. Moreover, we take into account all possible non-equivalent distributions of the m Fe and n Rh atoms within the cluster. In this way, any a priori assumption on thechemical order is avoided. Obviously, such an exhaustive combinatorial search increasinglycomplicates the computational task as we increase the cluster size, and as we move awayfrom pure clusters towards alloys with equal concentrations. Finally, in order to performthe actual density-functional calculations we set for simplicity all NN distances in thestarting cluster geometry equal to the Fe bulk value [173] R0 = 2 .48 A. Subsequently, afully unconstrained geometry optimization is performed from rst principles by using theVASP [168]. The atomic positions are fully relaxed by means of conjugate gradient orquasi-Newtonian methods, without imposing any symmetry constraints, until all the forcecomponents are smaller than a threshold of 5 meV/ A. The convergence criteria are setto 10− 5 eV/ A for the energy gradient and 5 ×10− 4 A for the atomic displacements [174].The same procedure applies to all considered clusters regardless of composition, chemicalorder, or total magnetic moment. Notice that the diversity of geometrical structures and

atomic arrangements often yields many local minima on the ground-state energy surface,

45



4.3. Structure and magnetism

which complicates signicantly the location of the lowest-energy conguration.Lattice structure and magnetic behavior are intimately related in TMs, particularly in

weak ferromagnets such as Fe and its alloys. On the one side, the optimum structure andchemical order depend on the actual magnetic state of the cluster as given by the averagemagnetic moment per atom µN and the magnetic order. On the other side, the magneticbehavior is known to be different for different structures and concentrations. Therefore,in order to rigorously determine the ground-state magnetic properties of FeRh clusters,we have varied systematically the value of the total spin polarization of the cluster S z byperforming xed spin-moment (FSM) calculations in the whole physically relevant range.Let us recall that S z = ( ν ↑ −ν ↓)/ 2 where ν ↑(ν ↓) represents the number of electrons in themajority (minority) states. In practice we start from the non-magnetic state ( S min

z = 0)and increase S z until the local spin moments are fully saturated, i.e., until the Fe mo-

ments in the PAW sphere reach µFe 4µB and the Rh moments µRh 2.5µB (typically,S max

z 3N/ 2). The above described global geometry optimizations are performed inde-pendently for each value of S z . These FSM study provides a wealth of information onthe isomerization energies, the spin-excitation energies, and their interplay. These areparticularly interesting for a subtle magnetic alloy such as FeRh. In the present chapterwe focus on the ground-state properties by determining for each considered Fe m Rh n themost stable structural and magnetic conguration corresponding to energy minimum as afunction of S z and of the atomic positions [171].

Once the optimization with respect to structural and magnetic degrees of freedom isachieved, we derive the binding energy per atom E B = [mE (Fe)+ nE (Rh) −E (Fe m Rh n )]/N in the usual way by referring the total energy E to the corresponding energy of m Fe and

n Rh isolated atoms. Moreover, for each stationary point of the total energy surface (i.e.,for each relaxed structure having a nearly vanishing | E | we determine the vibrationalfrequencies from the diagonalization of the dynamical matrix. The latter is calculatedfrom nite differences of the analytic gradients of the total energy. In this way we canrule out saddle points to which the local optimization procedure happens to converge onsome occasions. Only congurations which correspond to true minima are discussed inthe following. Finally, a number of electronic and magnetic properties —for example, themagnetic energy ∆ E m = E (S z = 0) −E (S z ), the local magnetic moments µi integratedwithin the Wigner-Seitz (WS) or Bader atomic cells of atom i, [175, 176] and the spinpolarized density of electronic states (DOS) ρσ (ε)— are derived from the self-consistentspin-polarized density and Kohn-Sham spectrum.

4.3 Structure and magnetism

In this section we discuss the ground-state structure, chemical order, binding energy, andmagnetic moments of Fe m Rh n clusters having N = m + n ≤8 atoms. The main emphasisis here on understanding how the various electronic, structural and magnetic propertiesdepend on the chemical composition of the alloy. First, each cluster size N is analyzedseparately, since a strong dependence on N is expected in the small size, non-scalableregime. Comparisons between the various N are stressed by means of cross-referencesbetween different subsections. In addition the main trends as a function of size and

concentration are summarized in section. 4.4.

46




Table 4.1: Structural, electronic and magnetic properties of FeRh dimers. Results aregiven for the binding energy E B (in eV), the magnetic stabilization energy ∆ E m = E (S z =0) −E (S z ) (in eV), the average interatomic distance dαβ (in A) between atoms α and β (α, β = Fe or Rh), the average spin moment per atom µN = 2 S z /N (in µB ), the local spinmoment µα (in µB ) at the Fe or Rh atoms, and the vibrational frequency ν 0 (in cm− 1).

Cluster Struct. E B ∆ E m dαβ µN µFe µRh ν 0

Fe2 1.35 0.77 1.98 3.00 2.82 288

FeRh 1.95 0.24 2.07 2.50 3.34 1.33 359

Rh 2 1.65 0.00 2.21 2.00 1.83 224

4.3.1 FeRh dimers

Despite being the simplest possible systems, dimers allow to infer very useful trends on therelative strength, charge transfers and magnetic order in the various types of bonds whichare found in FeRh alloy clusters. The results summarized in Table 4.1 show that the FeRhbond yields the highest cohesive energy, followed by the Rh 2 bond, the Fe 2 bond being theweakest. The particular strength of the heterogeneous bond is conrmed by the fact thatthe corresponding vibrational frequency is the highest. The bond length, however, followsthe trend of the atomic radius which, being larger for Rh, gives dRhRh > d FeRh > d FeFe .Quantitatively, the binding energy per atom E GGA

B = 1 .35 eV obtained for Fe 2 withinthe GGA is smaller than the LDA result E LDAB = 2 .25 eV [178] although it still remainslarger than the experimental value E expt

B = 0 .65 eV reported in Ref. [180]. The calculatedvibrational frequency ν 0(Fe 2) = 288 cm − 1 is consistent with previous experimental results[ν 0(Fe 2) = 299 .6 cm− 1 from Ref. [180] and ν 0(Fe 2) = 300 ±15 cm− 1 from Ref. [181]]. Ourresult for E B and µN of Rh2 coincide with previous GGA calculations by B. V. Reddy et al. [182]. These are however larger than the experimental values E expt

B (Rh 2) = 1 .46 eVderived from Knudsen diffusion [185], E expt

B (Rh 2) = 0 .70±0.15 eV derived from resonanceRaman in Ar matrices [186] and E expt

B (Rh 2) = 1 .203 eV derived from the resonant two-photon ionization [187]. The calculated vibrational frequency of ν 0(Rh 2)GGA = 224 cm − 1

should be compared with the experimental value ν 0(Rh 2)expt = 283 .9 cm− 1 reported inRef. [186].

The stability of magnetism, as measured by the difference in the total energy ∆ E m of the non-magnetic ( S z = 0) and optimal magnetic solutions, is largest for Fe 2 and smallestfor Rh2 . The same trend holds for the average magnetic moment per atom which decreaseslinearly from µ2 = 3 µB to 2µB as one goes from Fe2 , to FeRh, to Rh 2 . These averagemagnetic moments per atom correspond to a full polarization of all d electrons in the WSspheres: ν d 7 for Fe and ν d 8 for Rh, where ν d stands for the number of valenced electrons of the corresponding atom. The local magnetic moments µα (α ≡ Fe andRh), are obtained by integrating the spin density within the PAW spheres which have theradius rPAW (Fe) = 1 .3 A for Fe and rPAW (Rh) = 1 .4 A for Rh. In the pure dimers, thelocal moments µFe = 2 .82µB and µRh = 1 .83µB are close to the respective total momentper atom µ2 = 3µB and 2µB , which indicates that the spin-density m(r ) = n ↑(r ) −n↓(r )

is quite localized around the atoms. Actually, the differences between µα and µN give

47




a measure of the small spill-off effect in m(r ). Taking this into account, the results forµα in the FeRh dimer seem quite remarkable. Here the Fe local moment is signicantlyenhanced with respect to the Fe 2 or Fe-atom value, while the Rh moment is reduced bya similar amount (∆ µF e = 0.52 µB and ∆ µRh = -0.50 µB , see Table 4.1). This is mainlythe consequence of a transfer of d electrons from Fe to Rh, which allows the Fe atom todevelop a larger spin moment, due to the larger number of d holes. This occurs at theexpense of the moment at the Rh atom, which has less d holes to polarize. An integrationof the electronic density in the Bader cells [175] shows that 0 .33 electrons are transferredfrom the Fe to the Rh atom in FeRh. This behavior is qualitatively in agreement with thehigher Pauling electronegativity χ of the Rh atom ( χ Fe = 1.83 and χ Rh = 2.28) [188].

4.3.2 FeRh trimers

Having established the properties of dimers, we turn now to the trimers. The results fortrimers are summarized in Table 4.2 and Fig. 4.2. We considered both triangle and linearshapes as the initial structures for the optimization. As expected, the lowest energy isomersare found to be triangles for all compositions. According to our calculations the groundstate of Rh 3 is an equilateral triangle (D 3h ) with E B = 2 .31 eV, bond length d = 2 .37 Aand average magnetic moment µ3 = 1µB . The local magnetic moments µα = 0 .93µB inthe WS cells align parallel to each other and are almost as large as µ3 . These results areconsistent with those reported in previous GGA studies of Rh 3 (E B = 2 .35 eV, d = 2 .45 Aand µ3 = 1 µB ) [182]. A single Fe substitution yields an isosceles FeRh 2 with an elongatedbase composed of the two Rh atoms. We notice that the bond-length dRhRh = 2 .57 A islarger than in Rh 3 . The linear isomer of the form Rh-Fe-Rh, i.e., with only FeRh bonds,lies 0.4 eV above the optimal structure. It is the only true local minimum among thelinear FeRh trimers. The other linear structures (Rh-Rh-Fe, Fe-Rh-Fe, and Fe-Fe-Rh) areall found to be saddle points connecting triangular minima of the potential energy surface(PES). Further Fe substitution yields a isosceles Fe 2Rh in which the FeFe bond is theshortest. One observes, as in the dimers, that the interatomic distances follow the trendsin the atomic radii. Finally, for Fe 3 , the calculated lowest-energy structure is a Jahn-Tellerdistorted isosceles triangle with two longer bonds ( d12 = d13 = 2 .30 A) and a shorter one(d23 = 2 .07 A). The calculated average magnetic moment of Fe 3 is µ3 = 3 .33µB . Theseresults coincide with previous GGA studies [177] predicting d12 = d13 = 2 .33 A andd23 = 2 .09 A. In contrast, LDA calculations [178] yield an equilateral Fe 3 with averagemagnetic moment µ3 = 2 .66µB and d = 2 .10 A. By using the spin-polarized LDA, we alsoobtain an equilateral triangle similar to the one reported in Ref. [178]. In contrast, inthe GGA one nds that the equilateral triangle (D 3h ) is unstable with respect to a Jahn-Teller distortion. The isosceles shape of Fe 3 can therefore be interpreted as a consequenceof exchange and correlation effects. Moreover, we have analyzed the GGA Kohn-Shamspectrum in the equilateral structure and found a high degeneracy at the Fermi energy,which is consistent with the interpretation that the distortion is triggered by a Jahn-Tellereffect.

Concerning the composition dependence of E B one observes a non-monotonous be-havior as for N = 2, which indicates that the FeRh bonds are the strongest. The lowestvibrational frequency follows a similar trend, despite the larger mass of Rh. Notice thatFeRh 2 is somewhat more stable than Fe 2Rh, since the bonds between Rh atoms are in

general stronger than between Fe atoms. Finally, one may also notice that the energy

48




Table 4.2: Structural, electronic and magnetic properties of FeRh trimers. Results aregiven for the binding energy per atom E B (in eV), the magnetic stabilization energy peratom ∆ E m = [E (S z =0) −E (S z )]/N (in eV), the average interatomic distance dαβ (in A)ordered from top to bottom as dFeFe , dFeRh and dRhRh , the average spin moment per atomµN = 2S z /N (in µB ) , the local spin moment µα (in µB ) at the Fe or Rh atoms. Theasterisks refer to the rst exited isomers. See also Fig. 4.2.

Cluster E B ∆ E m dαβ µN µFe µRh

Fe3 1.80 0.69 2.22 3.33 2.99

Fe2Rh 2.24 0.32 2.25 3.00 3.35 1.212.35

Fe2Rh* 1.64 0.48 2.13 3.00 3.30 1.352.29

FeRh 2 2.45 0.05 2.21 2.00 3.27 1.182.57

FeRh 2* 1.99 0.49 2.20 2.00 3.30 1.17

Rh 3 2.31 0.02 2.37 1.00 0.93

gain ∆ E m associated to magnetism only plays a quantitative role in the relative stabilityof triangular and linear FeRh 2 . ∆ E m is actually larger for the linear chain than for thetriangle. Therefore, the later remains the most stable structure even in the non-magneticcase, although with somewhat different bond lengths.

The average magnetic moment per atom µ3 amounts to 1 µB for Rh3 . In the alloysit increases monotonously with Fe doping, reaching µ3 = 10 / 3µB for Fe3 . The localmagnetic moments µα always show a FM-like coupling. They are all identical in Rh 3 ,which is consistent with the C 3 point-group symmetry. In the pure clusters µα is alwaysclose to µ3 . This indicates that the spin polarization is dominated by electrons occupyinglocalized states and that spill-off contributions are not important. For example, in thecase of Fe3 , one nds µ1 = 3 .23µB and µ2 = µ3 = 2 .87µB , the latter corresponding to thepair of atoms forming the shorter bond. On the other side, the average local momentsµRh = 0 .93µB in Rh3 should be compared with µ(Rh 3) = 1 µB . As soon as FeRh bonds arepresent, for mixed compositions, the local Fe moments are enhanced beyond 3 µB . This

is mainly due to a charge transfer from Fe to Rh, leading to an increase in the numberof Fe d holes as already observed in the dimer. Quantitatively, the local µFe and µRh inmixed trimers are similar, though somewhat smaller than the corresponding values in theFeRh dimer. Notice, moreover, the enhancement of the Rh local moments in Fe 2Rh andFeRh 2 as compared to pure Rh 3 . This reects the importance of the proximity of Fe onthe magnetic behavior of the Rh atoms.

4.3.3 FeRh tetramers

The most stable FeRh tetramers are all tetrahedra and the rst low-lying isomers arerhombi (see Fig. 4.2). The distribution of the atoms within the optimal topology does

not play a role since all sites are equivalent in a tetrahedron. In the case of Rh 4 we

49




Fe 3 Fe 2 Rh Fe 2 Rh* FeRh 2 FeRh 2 * Rh 3

Fe 4 Fe 3 Rh Fe 3 Rh* Fe 2 Rh 2 Fe 2 Rh 2 * FeRh 3 FeRh 3 *

Rh 4

Figure 4.2: The optimal structure for the ground-state and lowest energy isomers (indi-cated by an asterisk) of trimer and tetramer FeRh clusters.

obtain a nonmagnetic undistorted tetrahedron having E B = 2 .75 eV and bond lengthd = 2 .45 A. The closest isomer is found to be a bent rhombus with an average bond lengthd = 2 .35 A. Similar results have been obtained in previous studies on Rh clusters [183].Notice, however, that Bae et al. [190] have obtained a bend rhombus as the ground-statestructure for Rh 4 also by using VASP. This discrepancy is likely to be a consequence of the different choice of the pseudopotential and cutoff energy E

max. In our calculations we

considered the PAW method and E max = 268 eV, while in Ref. [190] one used ultrasoftpseudopotentials and E max = 205.5 eV.

The binding energy of the alloys shows a characteristic non-monotonous dependenceon concentration, which was also found in smaller clusters. In fact Fe 2Rh 2 and FeRh 3are the most stable tetramers with E B = 2 .74 eV and E B = 2 .76 eV, respectively. Thisconrms that the FeRh bonds are somewhat stronger than others. It is worth noting thatthese trends are not altered qualitatively if magnetism is neglected, i.e., if one considersE B for S z = 0. In addition, it is interesting to follow how E B changes from Rh 4 to Fe 4 .The stability of the clusters can be qualitatively related to the number of homogeneousand heterogeneous bonds by counting them for each of the clusters shown in Fig. 4.2. Forinstance, FeRh 3 , which is the most stable composition, has 3 FeRh and 3 RhRh bonds.Replacing a Rh by an Fe to obtain Fe 2Rh 2 implies replacing 2 RhRh bonds by a strongerFeRh and a weaker FeFe bond. Therefore, E B does not change signicantly. The fact thatE B depends weekly on composition for Rh rich tetramers shows that FeRh and RhRhbonds are comparably strong in these clusters.

Concerning the magnetic moments one observes a approximately linear dependence of µN as a function of Fe content. In general, the substitution of a Rh by and Fe atom resultsin an increase of the total moment 2 S z by 3 or 4µB , or equivalently, ∆ µN = (0 .75–1)µB(see Table 4.3). The magnetic order is always FM-like. In the alloys the local momentsµFe show the above mentioned enhancement, which is due to a Fe-to-Rh d-electron chargetransfer that increases the number of d holes and allows for the development of µFe 3.2–3.4µB . In addition, the presence of Fe in Fe m Rh n enhances the Rh local moments as

compared to pure Rh 4 .

50




Table 4.3: Structural, electronic and magnetic properties of FeRh tetramers as in Table4.2. See also Fig. 4.2.


Fe4 2.21 0.35 2.28 3.50 3.08

Fe3Rh 2.49 0.58 2.34 3.00 3.18 1.032.40

Fe3Rh* 2.43 0.90 2.44 2.75 3.08 1.072.22

Fe2Rh 2 2.74 0.37 2.52 2.50 3.39 1.032.312.72

Fe2Rh 2* 2.67 0.49 2.28 2.50 3.33 1.202.57

FeRh 3 2.76 0.21 2.30 1.75 3.25 1.122.60

FeRh 3* 2.61 0.47 2.34 1.75 3.34 1.08

Rh 4 2.75 0.00 2.45 0.00 0.00

4.3.4 FeRh pentamers

In Table 4.4 and Fig. 4.3 the results for FeRh pentamers are summarized. Although allpossible cluster topologies (20 structures) were considered as starting geometries for eachcomposition, only the most highly coordinated trigonal bipyramid (TBP) and the squarepyramid (SP) are found to be most stable geometries. The low coordinated structurestransform into compact structures after the relaxation. Except for Rh 5 , which optimalstructure is a SP, all the other FeRh pentamers have the TBP as ground-state geometry.The trend in the composition dependence of the binding energy E B of pentamers conrmsthe behavior we started to observe for N = 4. Indeed, in the Fe-rich limit E B increasesrapidly with increasing Rh content, as the weakest FeFe bonds are replaced by FeRh bonds.Later on, near 50% concentration and in the Rh-rich limit, the composition dependence isweak since FeRh and RhRh bonds are comparably strong (see Table 4.4). In particular for

Rh-rich compositions, replacing Fe by Rh atoms no longer results in weaker binding. Inother words, FeRh bonds are no longer primarily preferred. This is possibly a consequenceof the increasing coordination number, which enhances the role of electron delocalizationand band formation, thus favoring the larger Rh hybridizations.

The calculated optimal structure of Rh 5 , a square pyramid, coincides qualitatively withprevious DFT calculations [182]. Nevertheless, we obtain a binding energy that is 0.07 eVper atom lower than in Ref. [182]. Substituting one Rh atom by Fe yields FeRh 4 andchanges the optimal cluster topology to the more compact TBP. The SP remains a localminimum of the ground-state energy surface, which is only 3 meV per atom less stable thanthe optimal TBP geometry. The average magnetic moment µ(FeRh 4) = 1 .2µB is enhancedwith respect to Rh 5 due to the contribution of a large Fe local moment µFe = 3 .31µB .

Notice that the Rh moments are no longer enhanced as in the smaller FeRh N − 1 but

51




Table 4.4: Structural, electronic and magnetic properties of FeRh pentamers as in Table4.2. See also Fig. 4.2.


Fe5 2.51 1.00 2.41 3.20 2.93

Fe4Rh 2.76 1.01 2.30 3.00 3.09 1.062.47

Fe4Rh* 2.67 0.75 2.39 3.00 3.02 1.032.18

Fe3Rh 2 2.96 0.92 2.37 2.40 3.13 0.752.39

Fe3Rh 2* 2.85 0.66 2.39 2.40 3.05 0.972.402.71

Fe2Rh 3 3.06 0.55 2.35 2.20 3.36 1.082.71

Fe2Rh 3* 3.03 0.53 2.36 2.20 3.28 1.122.612.73

FeRh 4 3.01 0.33 2.39 1.20 3.31 0.572.51

FeRh 4* 3.00 0.25 2.36 0.80 3.20 0.152.52

Rh 5 3.03 0.70 2.48 1.00 0.95

signicantly reduced: µRh = 0 .62µB for the apex atoms and µRh = 0 .52µB for the Rhatoms sharing a triangle with the Fe. This is of course related to the fact that the ground-state S z is relatively low. The effect is even stronger in the case of the SP isomer of FeRh 4 .Here we nd two Rh moments µRh = 0 .43µB that couple parallel to the Fe moment, onevery small Rh local moment µRh = 0 .05µB , and an antiparallel moment µRh = −0.48µB .This explains the reduced average total moment µ5 = 0 .8µB and the very small average

Rh moment µRh = 0 .15µB found in SP isomer of FeRh 4 . The present example illustratesthe subtle competition between cluster structure and magnetism in 3 d-4d nanoalloys.Further increase in the Fe content does not change the topology of the optimal struc-

ture. Moreover, we start to see that for nearly equal concentrations of Fe and Rh (i.e.,Fe2Rh 3 and Fe 3Rh 2) the low-lying isomers are the result of changes on the chemical order,i.e., changes in the distribution of the Fe and Rh atoms within the cluster, rather than theresult of changes in the cluster topology. The most stable conguration corresponds to thecase where the 3 Rh atoms (in Fe 2Rh 3) or the 3 Fe atoms (in Fe 3Rh 2) are all NNs of eachother (see Fig. 4.3). This is understandable from a single-particle perspective, since theband energy is lower when orbitals having nearly the same energy levels are hybridized.In addition, the most stable congurations maximize rst the number of FeRh NN pairs,

followed by the number of RhRh pairs. The optimal Fe 3Rh 2 structure has 6 FeRh and 3

52




FeFe NN pairs, while the rst isomer has 5 FeRh, 3 FeFe, and 1 RhRh NN pairs. Theoptimal Fe 2Rh 3 structure has 6 FeRh and 3 RhRh NN pairs, while the rst isomer has 6FeRh, 2 RhRh, and 1 FeFe NN pairs. Finally, in the Fe-rich limit, for example in Fe 4Rh,the lowest-energy structure remains a TBP but the closest isomer corresponds to the SP,which has a different topology, rather than a different position of the Rh atom in the TBP.

The trends in the magnetic properties are dominated by the Fe content. As for smallerclusters the average magnetic moment per atom µN increases monotonously with increas-ing Fe concentration. This holds for all optimal structures and in most of the rst excitedisomers. In fact the latter show in general the same µN as the optimal structure. Theonly exception is FeRh 4 , which is also the only case where an antiparallel alignment of Rhlocal moments is found. In all other investigated cases the magnetic order was found tobe FM-like. The local Fe moments show the usual enhancement with respect to pure Fe N ,

due to an increase in the number of Fe d-holes. This effect is stronger for Rh-rich clusters,since the larger the number of Rh atoms is, the stronger is the FeRh charge transfer (seeFig. 4.3). In contrast, the substitution of Rh by Fe does not always enhances the Rhlocal moments, as we observed systematically for smaller sizes. Finally, it is interesting toobserve that the different chemical orders found in the low lying isomers of Fe 2Rh 3 andFe3Rh 2 correspond to different local magnetic moments. The environment dependenceof µα follows in general the well-known trend of higher spin polarization at the lowestcoordinated sites.

4.3.5 FeRh hexamers

In Table 4.5 and Fig. 4.3 the results for FeRh hexameters are summarized. For each com-position all possible cluster topologies (63 different graphs [17,172]) and all non-equivalentdistributions of Fe and Rh atoms were taken into account as initial guess for the ab initiooptimization of the cluster geometry. As in previous cases, all relevant values of the totalmagnetic moment 2 S z are scanned. Despite the diversity of starting topologies most lowcoordinated structures relax into compact ones in the course of the unconstrained relax-ations. In the end, the square bipyramid (SBP), in general somehow slightly distorted,yields the lowest-energy regardless of composition. Other more open structures appears,however, as the rst exited isomers (see Fig. 4.3).

The binding energy per atom E B shows a similar composition dependence as for pen-tamers. For Fe-rich clusters E B increases steadily with increasing Rh content, by about0.2 eV each time a Rh replaces an Fe (see Table 4.5). Qualitatively, this conrms that thebonding between Fe and Rh is stronger than between Fe atoms. However, for nearly equalconcentrations and in the Rh-rich clusters (Fe m Rh 6− m with m ≤ 2) E B becomes almostindependent of m. This seems to be the result of a compensation of bonding and magneticcontributions. In fact, on the one side, the magnetic energy ∆ E m continues to decreasewith increasing Rh content, by about 0 .1–0.2 eV per Rh substitution, even for high Rhcontent. And on the other side, this is compensated by an increase of the bonding energywith increasing number of Rh atoms.

In the case of Rh 6 an octahedron with an average moment µ6 = 1µB and average bondlength d = 2 .54 A yields the lowest energy. The rst isomer, a trigonal biprism (TBP),lies only 28 meV above the optimum, showing a somewhat shorter average bond lengthd = 2 .46 A and a higher average moment µ6 = 1 .67µB . These results are consistent with

previous DFT calculations [189]. A single Fe substitution enhances the average moment to

53




Fe 5 Fe 4 Rh Fe 4 Rh* Fe 3 Rh 2 Fe 3 Rh 2 *

Fe 2 Rh 3 Fe 2 Rh 3 * FeRh 4 FeRh 4 * Rh 5


Fe 3 Rh 3 Fe 3 Rh 3 * Fe 2 Rh 4 Fe 2 Rh 4 * FeRh 5

FeRh 5 * Rh 6

Figure 4.3: Lowest energy isomers of FeRh pentamers and hexamers as in Fig. 4.2.

54




Fe 6

ξ = 0.010Fe 5 Rh

ξ = 0.017Fe 4 Rh 2

ξ = 0.02Fe 3 Rh 3

ξ = 0.04

Fe 2 Rh 4

ξ = 0.006FeRh 5

ξ = 0.007Rh 6

ξ = 0.009

Figure 4.4: Constant magnetization density plots ρ↑(r ) −ρ↓(r ) = ξ and local moments(in µB ) for the ground-state structures of FeRh hexamers. The value of the constant

magnetization density ξ is given in µB /˚A

3

.

a capped trigonal bipyramid (CTBP) having a short FeFe NN bond. This structure liesonly 0.11 eV higher in energy and has the same total moment as the optimal geometry.Replacing a further Fe atom yields Fe 3Rh 3 , whose optimal structure is an octahedron.Here we nd two isosceles open Fe 3 and Rh 3 triangles that form a π/ 2 angle with respectto each other (see Fig. 4.3). Out of the 12 NN pairs in the Fe 3Rh 3 octahedron, 8 areFeRh and only 4 are homogeneous (2 FeFe and 2 RhRh). The local Fe magnetic momentsare similar to the other clusters but the Rh moments are somewhat smaller in average(µRh = 1 .14µB ). The rst excited isomer of Fe 3Rh 3 is a CTBP that lies 25 meV per atom

above the ground state. The lowest-energy structure found for Fe 4Rh 2 is an octahedron,while a distorted CTBP is an isomer lying 0 .14 eV per atom above. In the former the Rhatoms are far apart occupying the apical positions, whereas in the latter they are NNs.The situation is thus similar to what we nd in Fe 3Rh 2 . For low Rh or Fe concentrationsthe atoms are distributed in order to favor the FeRh bonds rather than homogeneous NNpairs between the atoms in the minority. The octahedron and a distorted CTBP remainthe two most stable structures as one further reduces the Rh content (see Table 4.5 forFe5Rh and Fe 6).

Concerning the magnetic properties one observes qualitatively similar trends as in thesmaller clusters. The average magnetic moment per atom µN increases monotonously withFe content. Accordingly, the energy gain ∆ E m associated to magnetism also increases with

the number of Fe atoms. There are in general very little differences in µN between the

56




optimal structure and the rst low-lying isomer. The largest part of the spin polarization(about 90%) can be traced back to the local d magnetic moments with the PAW sphere of the atoms. As expected, the s and p spin polarizations are almost negligible in comparisonto the d-orbital contributions. A signicant increase of the Fe moments is observed uponRh doping, which result from the larger number of available Fe d holes and the low coor-dination number. Moreover, the Rh moments in Fe m Rh n are stabilized by the proximityof the Fe atoms. In the alloy hexameters the values of µRh are larger than in the pureRh 6 . However, this is not a general trend, since the magnetic moments in small Rh n areoften quite important due to the extremely reduced coordination numbers. Fig. 4.4 showsplots of constant magnetization density ρ↑(r ) −ρ↓(r ) = ξ for the optimal structure of thedifferent hexameters. Only the positive values of the magnetization density ξ are consid-ered since all local moment are ferromagnetically aligned. In addition the local moment

µi for Fe and Rh atoms are indicated. Notice the strongly environment dependent.

4.3.6 Exploring heptamers and octamers

For Fe m Rh n clusters having m + n = N ≥7 we did not attempt to perform a systematicsampling of initial topologies for further unconstrained structural relaxation, as was donefor the smaller sizes. Only a few compact and open starting structures are considered.For N = 7, this includes the bicapped trigonal bipyramid (BCTBP), capped octahedron(CO) and pentagonal bipyramid (PBP), while for N = 8, they are the tricapped trigonalbipyramid (TCTBP), bicapped octahedron (BCO), capped pentagonal bipyramid (CPBP)and cube (C). This choice is motivated by previous results for pure clusters and by thetrend to compact geometries observed for smaller sizes N

≤6. Although far from exhaus-

tive, the considered geometries allow to explore various relevant growth patterns with areasonable computational effort. Certainly, a more complete study would be necessary inorder to draw denitive conclusions about the optimal topologies. For each composition,all possible distributions of the Fe and Rh atoms within the cluster, as well as all rele-vant values of the total magnetization S z are taken into account (from the non-magneticstate to saturation). Therefore, the trends on the interplay between chemical order andmagnetic behavior remain rigorous within the framework of the sampled topologies.

The results for N = 7 are summarized in Table 4.6 and Fig. 4.5. As in smaller Fe m Rh nthe binding energy per atom increases rst with increasing Rh content and becomes es-sentially independent of composition in the Rh-rich limit (5 ≤ n ≤ 7). For pure Rh 7 thePBP is the most stable structure among the considered starting geometries. This result isconsistent with some earlier DFT studies [183]. However it contrasts with the calculationsby Wang et al. , [189] who used the GGA functional of Ref. [224] (PW91) and obtained acapped octahedron, and with the calculations of Bae et al. , [190] who found a prism plusan atom on a square face. According to our results, these structures are, respectively 20and 4 meV per atom higher in energy than the PBP. In the case of a prism plus an atomon the square face, the energy difference with the ground state seems too small to be ableto draw denitive conclusions.

FeRh heptamers with high Rh concentrations also favor a PBP topology. In FeRh 6the Fe atom occupies an apex site, while in Fe 2Rh 5 and Fe 3Rh 4 the Fe atoms belong tothe pentagonal ring. Notice that the distances between the two apex atoms in Rh 7 andbetween the Fe and Rh apex atoms in FeRh 6 are relatively short. In Fe 2Rh 5 and Fe 3Rh 4

the Fe atoms are as far as possible from each other and the distance between the Rh apex

57





Fe 4 Rh 3 Fe 4 Rh 3 * Fe 3 Rh 4 Fe 3 Rh 4 * Fe 2 Rh 5

Fe 2 Rh 5 * FeRh 6 FeRh 6 * Rh 7 Fe 8

Fe 7 Rh Fe 7 Rh* Fe 6 Rh 2 Fe 6 Rh 2 * Fe 5 Rh 3

Fe 5 Rh 3 * Fe 4 Rh 4 Fe 4 Rh 4 * Fe 3 Rh 5 Fe 3 Rh 5 *

Fe 2 Rh 6 Fe 2 Rh 6 * FeRh 7 FeRh 7 * Rh 8

Figure 4.5: Lowest energy isomers of FeRh heptamers and octamers.

59




one observes a tendency of the Fe atoms to group in sub-clusters, bringing the Rh atomsto outer positions, so that the number of FeRh bonds is largest. Concerning the shape of the Fe-rich heptamers, one observes important deformations of the pentagonal bipyramid(D5h symmetry) which are similar to the distortions found in pure Fe 7 [191,192]. Whilethe precise origin of the symmetry lowering is difficult to establish in the alloys, it isreasonable to expect that it is similar to the case of pure Fe 7 . According to Ref. [192],the deformations found in Fe 7 are due to the presence of degenerate electronic states inthe undistorted PBP structure. In order to verify this hypothesis we have analyzed theKohn-Sham spectrum of the symmetric structure (D 5h symmetry) and found that it ishighly degenerate at εF . In contrast the spectrum of the distorted structure has a bandgap about 0.4 eV at εF . This suggests that the distortions in the Fe-rich clusters can beinterpreted as a Jahn-Teller effect.

In Table 4.7 and Fig. 4.5 results for FeRh octamers are reported. The general trendsconcerning the composition dependence of the binding energy, chemical order, as well asthe average and local magnetic moments are very similar to smaller clusters. The moststable structure that we obtain for Fe 8 is a BCO having E B = 3 .03 eV, an average magneticmoment µ8 = 3µB , and a relatively short average bond-length d = 2 .42 A (see Table 4.7).A similar structure is also found in previous spin-polarized LDA calculations, where E B =4.12 eV and µ8 = 3 µB were obtained [193]. We have repeated these calculation for the BCOstructure with our computational parameters and atomic reference energies and foundE B = 3 .51 eV. The discrepancies between LDA and GGA results reect the importanceof exchange and correlation to the binding energy. In the other extreme, for pure Rh 8 ,the structure that we nd with the considered starting topologies is a regular cube havingE B = 3 .59 eV, an average magnetic moment µ

8 = 1 .5µB , and all bond lengths equal to

2.40 A. These results are in good agreement with previous calculations by Bae et al. [196].It is interesting to observe that the substitution of a single Rh atom by Fe in FeRh 7results in a compact topology, which is more stable than the relatively open (relaxed)cube-like structures derived from pure Rh 8 . The same trend holds for higher Fe content(i.e., Fe m Rh 8− m with m ≥ 1). The dominant structure for non-vanishing Fe content isa BCO with slight distortions. Only for Fe 5Rh 3 we nd a different topology, namely, adistorted CPBP. The typical isomerization energies between the BCO and the TCTBPare ∆ E iso = 10–30 meV per atom. The average magnetic moments in the lowest lyingisomers are either the same or very similar.

The magnetic properties of heptamers and octamers follow qualitatively the behaviorobserved in smaller clusters. In most cases the average magnetic moment per atom µN

and the magnetic energy ∆ E m increase with Fe concentration. The only exception is thepure Rh heptamer, for which µ7 is somewhat larger than in FeRh 6 . This is not due toAF-like coupling between Fe impurity moment and the remaining Rh atoms but ratherto a reduction of the Rh local moments in FeRh 6 (µRh

i 1.61–1.63µB in Rh7 , whileµRh

i 1.25–1.30µB in FeRh 6). Remarkably, the Rh local moments in Rh 7 are the largestamong all the heptamers. They amount to 87% of the total moment, which stresses theimportance of the local d-electron contributions. Also in Rh 8 one nds quite large localmoments, which are actually larger than the Rh moments in most Fe doped clusters. Thisshows that for these sizes the Fe atoms do not necessarily increase the Rh moments bysimple proximity effects (see Tables 4.6 and 4.7). Nevertheless, a different behavior isexpected for larger N , where pure Rh clusters are no longer magnetic on their own. The

local Fe moments are strongly enhanced with respect to pure Fe N (µFe 2.8µB in Fe7

60




Table 4.7: Structural, electronic and magnetic properties of FeRh octamers as obtainedfrom a restricted representative sampling of cluster topologies (see text).


Fe8 3.03 0.76 2.42 3.00 2.77Fe7Rh 3.19 0.73 2.43 2.87 2.88 1.04

2.47Fe7Rh* 3.16 1.08 2.48 2.88 2.86 1.11

2.40Fe6Rh 2 3.32 0.66 2.43 2.75 3.18 1.04

2.47Fe6Rh 2* 3.29 0.54 2.45 2.75 3.22 1.10

2.52Fe5Rh 3 3.42 0.69 2.43 2.63 3.15 1.10

2.57Fe5Rh 3* 3.41 0.84 2.44 2.63 3.13 1.15

2.512.72

Fe4Rh 4 3.47 0.53 2.45 2.50 3.20 1.252.422.70

Fe4Rh 4* 3.45 0.84 2.51 2.50 3.24 1.232.482.60

Fe3Rh 5 3.54 0.42 2.77 2.37 3.36 1.342.392.65

Fe3Rh 5* 3.50 0.45 2.77 2.40 3.36 1.342.452.56

Fe2Rh 6 3.53 0.32 2.70 2.00 3.25 1.302.592.57

Fe2Rh 6* 3.51 0.47 2.40 2.00 3.28 1.252.59

FeRh 7 3.49 0.23 2.45 1.62 3.27 1.182.57

FeRh 7* 3.47 0.24 2.48 1.37 3.30 1.042.45

Rh 8 3.59 0.09 2.40 1.50 1.33

or Fe8) reaching values up to 3 .36µB , particularly when the Fe atoms are in a Rh richenvironment. As in the smaller clusters, this is a consequence of a charge transfer fromthe Fe to the Rh atoms, which increases the number of polarizable Fe d-holes. Notice thatsome kind of interaction between the Fe atoms seems to favor this effect, since the largestµFe are found for clusters having 2 or 3 Fe atoms rather than for the single Fe impurity.

Large Fe moments are also found in bulk FeRh alloys [161,194].

61



4.4. Trends as a function of size and composition

To conclude this section it is interesting to compare the cluster results with availableexperiments and calculations for macroscopic alloys [161,194,195]. Band structure calcu-lations for the periodic Fe 0.5Rh 0.5 alloy having a CsCl structure yield an antiferromagnetic(AF) ground state, which is more stable than the ferromagnetic solution [195]. This isqualitatively in agreement with experiments showing AF order when the Rh concentrationis above or equal to 50% [194]. In contrast our results for small clusters show a FM-likeorder for all Rh concentrations, even for the pure Rh clusters. This is a consequence of the reduction of local coordination number and the associated effective d-band narrowing,which renders the Stoner criterion far easier to satisfy, and which tends to stabilize thehigh-spin states with respect to the low-spin AF states. In fact, even in the bulk cal-culations on FeRh, the energies of the AF and FM states are not very different, and acoexistence of both solutions is found over a wide range of volumes [195]. Moreover, exper-

iment shows an AF to FM transition with increasing temperature, which is accompaniedby an enhanced thermal expansion [194]. Recent ab initio calculations have revealed theimportance of competing FM and AF exchange interactions in stoichiometric α-FeRh [161].Moreover, neutron diffraction experiments [194] on Fe 1− x Rh x for 0.35 < x < 0.5 and cal-culations [161] for x = 0 .5 show that the Fe moments µFe are signicantly enhanced withrespect to µFe in pure α -Fe, particularly in the FM state where it reaches values of about3.2µB [161,194]. These bulk results are remarkably similar to the trends found in Fe m Rh nclusters over a wide range of compositions. As in the clusters, the induced Rh momentsµRh play an important role in the stability of the FM phase. Bulk experiments [194] onFe1− x Rh x yield µRh 1µB for 0.35 < x < 0.5 which is comparable to, though somewhatsmaller than the present cluster results.

4.4 Trends as a function of size and compositionThe main purpose of this section is to focus on the dependence of the electronic andmagnetic properties of Fe m Rh n clusters as a function of size and composition for N =m + n ≤8.

4.4.1 Binding energy and magnetic moments

In Fig. 4.6 the binding energy per atom E B is given as a function of the number of Featoms m. Besides the expected monotonic increase of E B with increasing N , an interestingconcentration dependence is observed. For very small sizes ( N

≤ 4) E B is maximal for

m = 1 or 2, despite the fact that E B is always larger for pure Rh than pure Fe clusters.This indicates that in these cases the bonding resulting from FeRh pairs is stronger thanRhRh bonds. Only for m ≥ N −1, when the number of weaker FeFe bonds dominates,one observes that E B decreases with increasing m. For larger sizes ( N ≥5) the strengthof RhRh and FeRh bonds becomes very similar, so that the maximum in E B is replacedby a range of Fe concentrations x = m/N 0.5 where E B depends very weakly on m.

In Fig. 4.7 the average magnetic moments µN of Fem Rh n are shown as a functionof m for N ≤ 8. As already discussed in previous sections, µN increases monotonously,with the number of Fe atoms. This is an expected consequence of the larger Fe localmoments and the underlying FM-like magnetic order. The average slope of the curvestends to increase with decreasing N , since the change in concentration per Fe substitution

is more important the smaller the size is. The typical increase in µN per Fe substitution is

62




0 1 2 3 4 5 6 7 8m

1.0

1.5

2.0

2.5

3.0

3.5

B i n d i n g

E n e r g y

[ e V / a t o m ]

Fem

Rhn

N = 2

3

45

6 7

8

Figure 4.6: Binding energy per atom E B of Fem Rh n clusters as a function of the numberof Fe atoms. The lines connecting the points for each N = m + n are a guide to the eye.

about (1 /N )µB per Fe substitution. Notice, moreover, the enhancement of the magneticmoments of the pure clusters in particular for Fe N (m = N ), which go well beyond 3µB ,the value corresponding to a saturated d-band in the d7s1 conguration. In contrast, themoments of pure Rh N are far from saturated except for N = 2 and 7 (see Fig. 4.7 form = 0). In this context it is important to recall that a thorough global optimization, forexample, by considering a large number of initial topologies, could affect the quantitativevalues of the magnetic moments for N = 7 and 8.

The local magnetic moments in the PAW sphere of the Fe and Rh atoms providefurther insight on the interplay between 3 d and 4d magnetism in Fe m Rh n . In Fig. 4.8µFe and µRh are shown as a function of m for N = 6–8. The Fe moments are essentiallygiven by the saturated d-orbital contribution. For pure Fe clusters the actual values of µFe within the PAW sphere are somewhat lower than 3 µB due to a partial spill-off of thespin-polarized density. Notice that the Fe moments increase as we replace Fe by Rh atomsshowing some weak oscillations as a function of m. The increase is rather weak for a single

Rh impurity in Fe N − 1Rh but becomes stronger reaching a more or less constant value assoon as the cluster contains 2 or more Rh ( m ≤ N −2, see Fig. 4.8). This effect canbe traced back to a d electron charge transfer from Fe to Rh which, together with theextremely low coordination number, which yields a full polarization of the larger numberof Fe d holes. On the other side the Rh moments are not saturated and therefore aremore sensitive to size, structure and composition. The values of µRh are in the range of 1–1.5µB showing some oscillations as a function of m. No systematic enhancement of µRhwith increasing Fe content is observed. This behavior could be related to charge transferseffects leading to changes in the number of Rh d electrons as a function of m.

The magnetic stabilization energy of FeRh clusters is shown in Fig. 4.9. One observesthat the pure Fe clusters show highest ∆ E m with a largest value of 1 eV for Fe 6 , while

most of the pure Rh clusters show small ∆ E m values. For smaller cluster sizes ( N ≤ 5)

63




0 1 2 3 4 5 6 7 8m

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

µ Ν [ µ

Β ]

Fem

Rhn

N = 2

3 4 5 67 8

Figure 4.7: Total magnetic moment per atom µN of Fem Rh n clusters as a function of number of Fe atoms. The symbols corresponding to each size are the same as in Fig. 4.6.The lines connecting the points for each N = m + n are a guide to the eye.

0 1 2 3 4 5 6 7 8

1.0

1.5

2.0

2.5

3.0

3.5

µ α

[ µ Β

] N = 6N = 7N = 8

Fe

Rh

m

Figure 4.8: Local magnetic moment µα at the Fe and Rh atoms as a function of thenumber Fe atoms m.

the variation in ∆ E m as a function of the number of Fe atoms m display a somewhat non-monotonous behavior. This is due to the fact that the change in ∆ E m is more signicantthe smaller the cluster is. The similar behavior are been already found in the bindingenergy per atom and average magnetic moment (see Figs. 4.6 and 4.7).

It is interesting to analyze the role played by magnetism in dening the cluster structure

64




tions with the same cluster size N and also among the different N . Among the octamers,FeRh 7 and Fe 3Rh 5 clusters display highest and lowest stability, respectively. This can be justied by analyzing Figs. 4.5 and 4.11 for the number of different bonds and differentaverage bond length, respectively. For the optimal FeRh 7 cluster, there are 6 FeRh bonds(with average bond length of 2.45 A) and 12 RhRh bonds (with average bond length of 2.57A). Notice that this cluster has no FeFe bond which we have shown to be weakestamong the dimers (see Table. 4.1). In the case of Fe 3Rh 5 cluster, there are 11 FeRhbonds (with average bond length of 2.39 A), 6 RhRh bonds (with average bond length of 2.65A) and 3 FeFe bonds (with average bond length of 2.77 A). It can be argued that theabsence of FeFe bond provides higher relative stability to the FeRh 7 cluster compared tothe Fe 3Rh 5 cluster where 3 FeFe bonds are present.

1 2 3 4m

-0.15

-0.10

-0.05

0.00

0.05

0.10

∆ 2

E ( 5 ) [ e V ]

1 2 3 4 5m

-0.15

-0.10

-0.05

0.00

∆ 2

E ( 6 ) [ e V ]

1 2 3 4 5 6m

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

∆ 2

E ( 7 ) [ e V ]

1 2 3 4 5 6 7m

-0.10

-0.05

0.00

0.05

0.10

0.15

∆ 2

E ( 8 ) [ e V ]

Figure 4.10: Relative stability of Fe m Rh n clusters as a function of composition for differentN = m + n, ∆ 2E(N , m) = E(N , m +1 )+E(N , m −1) −2E(N , m) is given as a function of number of Fe atoms. The symbols corresponding to each size are the same as in Fig. 4.6.The lines connecting the points for each N = m + n are a guide to the eye.

66




1 2 3 4m

2.3

2.4

2.5

2.6

2.7

2.8

d ( Å )

1 2 3 4 5m

2.3

2.4

2.5

2.6

2.7

2.8

d ( Å )

1 2 3 4 5 6m

2.3

2.4

2.5

2.6

2.72.8

d ( Å )

1 2 3 4 5 6 7m

2.3

2.4

2.5

2.6

2.72.8

d ( Å )

N = 5 N = 6

N = 7 N = 8

Figure 4.11: Bond lengths for FeFe (cross), FeRh (circle) and RhRh (square) pairs as afunction of number of Fe atoms in Fe m Rh n clusters having N = 5–8 atoms. The linesconnecting the points for each N = m + n are a guide to the eye.

4.4.3 Electronic structure

In the previous sections the structure and spin moments of FeRh alloy clusters have beendiscussed as a function of 3 d/4 d concentration. Although these properties are intimatelyrelated to the size and composition dependence of electronic structure, it is in generalvery difficult to achieve a physical transparent correlation between global and microscopicbehaviors. Nevertheless, it is very interesting to analyze, at least for some representative

examples, how the electronic structure depends on the composition of magnetic nanoalloys.Fig. 4.12 shows the spin-polarized d-electron density of states (DOS) of representativeFeRh octamers having the relaxed structures illustrated in Fig. 4.5. Results for pure Fe 8and Rh 8 are also shown for the sake of comparison. The spin-polarized DOS can be givenby

N σ (ε) = 1N

k

δ (ε −εkσ )

4.4.2

represents the number of electronic states per unit energy and per atom having spin σ andenergy ε. The eigenenergies εkσ are the eigenvalues of the Kohn-Sham equations.

In all the clusters, the dominant peaks in the relevant energy range near εF correspond

either to the Fe-3 d or to the Rh-4 d states. The valence spectrum is largely dominated by

67




these d-electron contributions. In fact the total DOS and the d-projected DOS are difficultto tell apart.

-50

0

50

-50

0

50

D e n s i t y o

f s t a

t e s [ e V - 1 ]

-5 -4 -3 -2 -1 0 1 2ε − ε

F [eV]

-50

0

50

-5 -4 -3 -2 -1 0 1 2ε − ε

F [eV]

(a) (b)

(c) (d)

(e) (f)

Rh8

Fe2Rh6

Rh8

Fe4Rh4

Fe6Rh2 Fe8

Cube Bicapped octahedron

Figure 4.12: Electronic density of states (DOS) of FeRh octamers. Results are givenfor the total (solid), the Fe-projected (dotted), and the Rh-projected (dashed) d-electronDOS. Positive (negative) values correspond to majority (minority) spin. A Lorentzianwidth λ = 0 .02 eV has been used to broaden the discrete energy levels. The consideredstructures are the optimal ones illustrated in Fig. 4.5.

First of all, let us consider the DOS of the pure clusters. Our results for Rh 8 witha cube structure are similar to those of previous studies [196]. They show the dominantd-electron contribution near εF , with the characteristic ferromagnetic exchange splittingbetween the minority and majority spin states. In Fig. 4.12 we also included the DOS forRh 8 with a BCO structure, since it allows us to illustrate the differences in the electronicstructure of compact and open geometries. Moreover, the DOS of pure Rh 8 with BCOstructure is very useful in order to demonstrate the dependence of DOS on Fe content, sincethe structures of Fe m Rh 8− m with m ≥1 are similar to the BCO. Both Fe 8 and Rh 8 showrelatively narrow d-bands which dominate the single-particle energy spectrum in the range

−5eV ≤ε−εF ≤3eV. The spin polarization of the DOS clearly reects the ferromagnetic

68




order in the cluster. Putting aside the exchange splitting, the peak structure in the up anddown DOS ρσ (ε) are comparable. There are even qualitative similarities between the twoelements. However, looking in more detail, one observes that the effective d-band widthin Fe8 (about 4 eV) is somewhat smaller than in Rh 8 (about 5 eV). Moreover, in Rh 8the DOS at εF is non-vanishing for both spin directions and the nite-size gaps are verysmall (see Fig. 4.12). In contrast, the majority d-DOS is fully occupied in Fe 8 , with thehighest majority state lying about 0.5 eV below εF . In addition there is an appreciablegap (about 0.1 eV) in the corresponding minority spectrum. These qualitative differencesare of course consistent with the fact that Fe 8 is a strong ferromagnet with saturatedmoments, while Rh 8 should be regarded as a weak unsaturated ferromagnet.

The trends as a function of concentration reect the crossover between the previouscontrasting behaviors. For low Fe concentration (e.g., Fe 2Rh 6) we still nd states with

both spin directions close to εF . The magnetic moments are not saturated, although theFermi energy tends to approach the top of the majority band. Moreover, the majority-spinstates close to εF have dominantly Rh character. Small Fe doping does not reduce thed-band width signicantly. Notice the rather important change in the shape of the DOS inFe2Rh 6 as compared to the DOS in Rh 8 . This is a consequence of the change in topologyfrom cubic to bicapped octahedron.

For equal concentrations (Fe 4Rh 4) the rst signs of d-band narrowing and enhancedexchange splitting start to become apparent. The spin-up states (majority band) whichin Fe2Rh 6 contribute to the DOS at εF now move to lower energies (0.3 eV below εF ) sothat the majority band is saturated. Only spin-down (minority) states are found aroundεF , although there is a signicant gap in ρ↓(ε) (see Fig. 4.12). In the majority band Rhdominates over Fe at the higher energies (closer to εF ), while Fe dominates in the bottomof the band. In the minority band the participation of Rh (Fe) is stronger (weaker) belowεF and weaker (stronger) above εF . This is consistent with the fact that the Rh localmoments are smaller than the Fe moments.

Finally, in the Fe rich limit (e.g., Fe 6Rh 2), the majority-band width becomes as narrowas in Fe 8 , while the minority band is still comparable to Rh 8 . The exchange splitting islarge, the majority band saturated and only minority states are found close to εF . Asin Fe8 , ρ↓(ε) shows a clear gap at εF (see Fig. 4.12). However, the Rh contribution tothe minority states below εF remains above average despite the relative small Rh content.The Fe contribution largely dominates the unoccupied minority-spin DOS, in agreementwith the larger local Fe moments.

4.5 Summary

The structural, electronic and magnetic properties of small Fe m Rh n clusters having N =m + n ≤8 atoms have been investigated systematically in the framework of a generalizedgradient approximation to density-functional theory. For very small sizes ( N ≤4 atoms)the binding energy E B shows a non-monotonous dependence on concentration, whichimplies that the FeRh bonds are stronger than the homogeneous ones. However, for largersizes the FeRh and RhRh bond strengths become comparable, so that E B depends weaklyon concentration for high Rh content.

The magnetic order of the clusters having the most stable structures is found to be

FM-like. Moreover, the average magnetic moment per atom µN increases monotonously,

69



5Structure and magnetism of small CoPd clusters

In the present chapter we discuss the electronic and magnetic properties of Co m Pd n clus-ters having N = m + n ≤8. Our study shown that the optimized cluster structures havea clear tendency to maximize the number of nearest-neighbor CoCo pairs. The magneticorder is found to be ferromagnetic-like for all the ground-state topologies. However, AF-like order has also been found in some rst exited isomers. The binding energies and

average magnetic moments show an approximately linear increase with Co concentration.This is consistent with the trends found in the solid where E B is larger in Co than in Pd[E B (Co-bulk) = 4.39 eV/atom and E B (Pd-bulk) = 3.89 eV/atom]. The maximal localspin polarization for Co and Pd atoms are found in the equiatomic or nearly equiatomiccompositions (e.g., Co 3Pd 3 , Co3Pd 4 and Co 4Pd 4). A signicant enhancement of the Colocal moments is found as a result of Pd doping. This is a consequence of the increase inthe number of Co d holes, due to Co to Pd charge transfer, combined with the extremelyreduced local coordination numbers.

5.1 Introduction

The central goal in the eld of nanoalloys is to explore, understand, and characterize therich variety of alloy properties at the nanoscale as a function of size and composition [150].By using different experimental and theoretical tools, now it is known that combiningtwo metallic elements to form nanoalloys leads to an even higher degree of novelty andcomplexity [1, 2,4–7,65,151,157–159,164]. The elements considered in the present study,Co and Pd, have very interesting properties in the low dimensional non-scalable regime. Coclusters show enhanced magnetic moments compared to the bulk [65, 158, 159], whereasPd clusters display a nite magnetic moment despite being non-magnetic in the solid[10,183,205,211]. The magnetic properties of free standing Co N clusters were investigatedvia Stern-Gerlach molecular beam deection experiment by Bloomeld et al. in the sizerange N = 20–200 (see Fig. 5.1 as well as Ref. [5]) and by de Heer et al. for N = 30–300

Ref. [6]. These studies shown that in the temperature range of 77-300 K, the Co N clusters

71



5.1. Introduction

Figure 5.1: The average magnetic moment per atom of Co N clusters as a function of thecluster size N (after Ref. [5]).

show high-eld deections which are characteristic of supermagnetic behavior. Thereare also many theoretical results available for Co clusters. Castro et al. [200] performedall-electron density-functional calculations using both the local density and generalizedgradient approximations. However, the size of the clusters was limited to only up to veatoms. Later on, Lopez et al. [202] studied CoN clusters (4 ≤N≤ 60), by performinggeometry optimization by using an evolutive algorithm based on a many-body Guptapotential [203]. The magnetic properties of the cluster structures were studied by a spdtight-binding method.

The magnetism of small Pd clusters is still a subject of debate. Photoemission exper-iments [205] predicted a Ni-like spin arrangement in Pd N clusters having N ≤ 6 and aPt-like non-magnetic behavior for N ≤ 15. DC susceptibility measurements [206] foundmagnetic moments of 0.23 ±0.19µB /atom in huge Pd clusters with diameters in the rangeof 50–70 A. Recent experiment [204] by using gas-evaporative method in a high purityAr gas atmosphere has revealed a magnetic moment of 0.75

±0.31µB /atom for ne Pd

particles. A large number of theoretical studies have been devoted to the study of the Pdclusters. The DFT calculations by M. Moseler et al. [10] have shown that both neutraland anionic Pd N clusters having 2 ≤ N ≤7 and N = 13 atoms are magnetic. Moreover,there are some theoretical studies on CoPd lms. J. Dorantes-D´ avila et al. [209] carriedout tight-binding calculations by depositing Co 1D chains on Pd(110) surface. They foundthat small magnetic moments are induced ( µP d = 0 .21 −0.29)µB for Pd atoms nearestneighbor of Co. In view of these interesting behaviors one expects that CoPd clustersshould show very interesting structural, electronic and magnetic properties.

The remainder of this chapter is organized as follows. In section. 5.2 the main detailsof the theoretical background and computational procedure are presented. The resultsof our calculation for CoPd clusters having N ≤ 8 atoms are reported in Secs. 5.3 and

5.4. Here, we focus on the interplay between structure, chemical order and magnetism in

72



Chapter 5. Structure and magnetism of small CoPd clusters

the most stable geometries for different cluster sizes. We also analyze the concentrationdependence of the cohesive energy, the local and average magnetic moments, the relativestability and the magnetic stabilization energy. Finally, we conclude in section. 5.5 witha summary of main trends.

5.2 Computational details

The structural, electronic and magnetic properties of small Co m Pd n clusters having N =m + n ≤ 8 atoms has been studied using DFT and the VASP [11,168] as in the chapter4. Electronic exchange and correlation are described in the generalized gradient approx-imation, by means of the functional proposed by Perdew and Wang (PW91) [170]. Asdiscussed in Sec. 2.2 the VASP solves the spin-polarized Kohn-Sham equations in an aug-mented plane-wave basis set, taking into account the core electrons within the projectoraugmented wave (PAW) method [170]. The PAW method has extensively been describedin section. 2.4. The 4 s and 3d orbitals of Co, and the 5 s and 4d orbitals of Pd are treated asvalence states. The PAW sphere radii for Co and Pd are 1 .302 A and 1.434 A, respectively.

The validity of the present choice of computational parameters has been veried. Anumber of tests have been performed in order to assess the numerical accuracy of thecalculations. For example, increasing the cutoff and supercell size E max = 268 eV, a =12A to 500 eV, a = 22A in Co4 yields a total energy differences of only 1.0 meV/atom and0.2 meV/atom, respectively. This implies an increase in the computation time by a factors

of 3–7. In these calculations above calculations we obtain changes in the average bondlength (bond angle) of only 10 − 3 A (10− 4 degrees). Such difference are not signicantfor our physical conclusions. In fact typical isomerization energies in CoPd clusters arean order of magnitude larger, of the order of 10–50 meV/atom. We also found that thetotal energy is essentially independent of the choice of the smearing parameter values σ,provided it is sufficiently small ( σ ≤ 0.05 eV). Values of σ between 0.01 to 0.1 eV havebeen explicitly tested. We conclude that set of standard parameters E max = 268 eV,supercell size a from 10 to 22 A and smearing parameter 0.02 eV offer a good accuracy ata reasonable computational costs.

The rest of the computational procedure employed in this chapter is analogous to theone used in the chapter 4 in the case of FeRh. This concerns in particular the thorough

sampling of cluster topologies, total magnetic moments S z , and all possible chemical orders.Further methodological details may be found in Sec. 4.2.

5.3 Results and discussion

In this section we discuss the structure, chemical order, binding energy and magneticmoments of Co m Pd n clusters having N = m + n ≤ 8 atoms. Here the most importantelectronic and magnetic properties are analyzed for the different cluster sizes N and forall compositions. The general trends as a function of size and concentration are discussed

in Sec. 5.4.

73




Table 5.2: Structural, electronic and magnetic properties of CoPd trimers as in Table 5.1.The average interatomic distance dαβ (in A) ordered from top to bottom as dCoCo , dCoPd

and dPdPd . The most stable structures are illustrated in Fig 5.3.

Cluster PGS E B ∆ E m dαβ µN µCo µPd

Co3 C2v 1 .87 0.57 2.16 2.00 1.94

Co2Pd C 2v 1.84 0.47 2.01 1.33 1.95 0.062.44

Co2Pd* D ∞ 1.66 0.61 2.03 2.00 2.32 0.122.28

CoPd 2 C2v 1.70 0.19 2.31 1.00 2.25 0.332.62

CoPd 2* D∞ 1.41 0.34 2.23 1.44 2.46 0.64

Pd 3 D3h 1.25 0.10 2.52 0.67 0.63

5.3.2 Trimers

Table 5.2, Fig. 5.2 and Fig. 5.3 summarize all the important results obtained for trimers.The lowest energy isomers are found to be the triangle for all the compositions. For Co 3 ,

the ground-state structure is an isosceles triangle with E B = 1.87 eV, dCoCo = 2.16 ˚A, andµ2 = 2µB . The highest occupied molecular orbital (HOMO) is degenerate, and partially

occupied (see Fig. 5.2). These results are in agreement with the all-electron (AE) DFTcalculations by M. Castro et al. [200]. The linear Co3 is found to be 0.43 eV/atom lessstable than the ground-state. The lowest energy state for Pd 3 is an equilateral triangle withµ3 = 0.67 µB . This result is similar to the previous DFT study reported by T. Futschek et al. [183]. The DOS of Pd3 shows that the HOMO is degenerate and partially occupied.Therefore a Jahn-Teller distortion could have been expected. However, our calculation donot display such an instability. In fact, Valerio and Toulhaot [213] have shown that Pd 3is stable against Jahn-Teller distortions at the GGA level, but unstable when a hybridfunctional combining DFT and exact exchange is used or if the calculations are performedat the Hartree-Fock plus conguration interaction (HF+CI) level. For Co 2Pd, the lowestenergy state is an isosceles triangle with E B = 1.84 eV, and µ3 = 1 .33µB . The localmoments are coupled ferromagnetically. A signicant enhancement in the local momentscould be expected as in the CoPd dimer. Surprisingly, there is no such an enhancement inthe µCo , which is essentially the same as in Co 3 . Moreover, the Pd has nearly vanishingmoment. The lack of enhancement of µCo is likely to be related to the rather contractedCoCo bond length dCoCo = 2 .01 A (see Table 5.2). We found that the linear Pd-Co-Co (Co-Pd-Co) structures are 0.18 (0.48) eV/atom less stable than the the ground-statestructure. The lowest energy state of CoPd 2 is also an isosceles triangle (C 2v) with E B =1.7 eV and µ3 = 1 µB . Unlike the ground-state Co 2Pd, the CoPd 2 cluster shows a largeenhancement in the µCo = 2.25 µB . This is due to the combined effect of larger CoCointeratomic distance ( dCoCo = 2.31A) and to the signicant charge transfer from Co to

Pd. While, µP d in CoPd 2 is rather reduced compared to Pd 3 local moment.

75



5.3. Results and discussion

-4 -2 0 2 4 6

-40

-20

0

20

40

D e n s i t y o

f s t a

t e s

[ e V - 1 ]

-4 -2 0 2 4 6-40

-20

0

20

40

D e n s i t y o

f s t a

t e s

[ e V - 1 ]

-4 -2 0 2 4 6ε − ε

F [eV]

-40

-20

0

20

40

D e n s i t y o

f s t a

t e s [ e V

- 1 ]

-4 -2 0 2 4 6ε − ε

F [eV]

-80

-40

0

40

80

D e n s i t y o

f s t a

t e s [ e V

- 1 ]

Co 3 Co 2Pd 1

Co1Pd

2 Pd

3

Figure 5.2: Electronic density of states (DOS) of CoPd trimers. Results are given for thetotal DOS. Positive (negative) values correspond to majority (minority) spin. A Lorentzianwidth λ = 0 .02 eV has been used to broaden the discrete energy levels.

Co 3 Co2 Pd Co 2 Pd* CoPd 2 CoPd 2 *

Pd 3 Co4 Co3 Pd Co 3 Pd* Co 2 Pd 2

Co 2 Pd 2 * CoPd 3 CoPd 3 * Pd 4

Figure 5.3: Lowest energy isomers of CoPd trimers and tetramers. The asteriks indicaterst exited isomers.

76




Table 5.3: Structural, electronic and magnetic properties of CoPd tetramers as in Ta-ble 5.2.

Cluster PGS E B ∆ E m dαβ µN µCo µPd

Co4 S4 2.37 0.72 2.43 2.50 2.19

Co3Pd D 2h 2.18 0.52 2.18 1.75 2.10 0.322.40

Co3Pd* S 4 2.13 0.53 2.45 1.75 2.06 0.262.48

Co2Pd 2 D2h 1.874 0.44 2.39 1.50 2.30 0.432.62

Co2Pd 2* S4 1.87 0.18 2.24 -0.01 -0.02 0.012.472.63

CoPd 3 S4 1.94 0.41 2.37 0.75 2.18 0.242.71

CoPd 3* D2h 1.83 0.42 2.36 0.75 2.29 0.242.63

Pd 4 Td 1.67 0.09 2.61 0.50 0.47

5.3.3 TetramersTable 5.3 and Fig. 5.3 summarize the relevant geometric and magnetic information ob-tained for the ground-state tetramers. The lowest energy structures are tetrahedra for m= 0, 1 and 4, and rhombi for m = 2 and 3. Co 4 has a Jahn-Teller distorted ground-statewith µ4 = 2 .5µB . In this cluster there are two kinds of bond lengths: two pairs have equalshort bond lengths d = 2 .14 A, while the third pair has a much larger d = 2 .72 A. Theshorter bond lengths should enhance the binding since the localized 3 d electrons can alsotake part in the bonding. The rst excited isomer is a rhombi, which is 0.11 eV less stablethan the ground-state structure. Our results for Co 4 are similar to the previous DFTcalculations by S. Datta et al. [198]. So far, there are, to our knowledge, no experimentalresult available on the neutral tetramers. However, Yoshida et al. [201] predicted a tetra-

hedron with a bond length of 2 .25 ±0.2A as the optimal structure for Co−4 anion. The

most stable structure obtained for Pd 4 is a tetrahedron with d = 2 .61 A and µ4 = 0 .50µB .This result is in good agreement with previous DFT studies [10, 183,207]. The HOMO(minority spin) is degenerate and partially occupied by only one electron (see Fig. 5.4).The corresponding DOS is similar to the one reported by V. Kumar et al. [207].

Let us move further to Co 3Pd. The optimal structure is a rhombus and the rst exitedisomer is a tetrahedron lying 41 meV/atom above the ground-state. The rhombus is alsofound to be the optimal structure for Co 2Pd 2 . This case is an ideal example to study theinuence of chemical order on the binding energy and magnetism. Notice that E B hasbeen reduced in the ground-state of Co 2Pd 2 as compared to the ground-state of Co 3Pdand CoPd 3 . This is probably due to the presence of a Pd dimer, which is weakest among

the dimers (see Fig. 5.3). The local moments µCo and µPd show somewhat enhanced

77




-4 -2 0 2 4 6-60-40-20

02040

60

D e n s i t y o

f s t a

t e s [ e

V - 1 ]

-4 -2 0 2 4 6-60

-40-20

0204060

D

e n s i t y o

f s t a

t e s

[ e V

- 1 ]

-4 -2 0 2 4 6ε − ε

F[eV]

-60-40-20

0204060

D e n s i t y o

f s t a

t e s

[ e V

- 1 ]

-4 -2 0 2 4 6ε − ε

F[eV]

-60-40-20

0204060

D e n s i t y o

f s t a

t e s

[ e V

- 1 ]

-4 -2 0 2 4 6-60

-40-20

0204060

D

e n s i t y o

f s t a

t e s

[ e V

- 1 ]

Co 4

Co3Pd

1 Co

2Pd

2

Co1Pd

3 Pd

4

Figure 5.4: Electronic density of states (DOS) of CoPd tetramers.

values 2.30µB and 0.43µB , respectively. This is because Co atoms are farther away andweakly bonded. Consequently, Co d states will be more localized and provide a signicantenhancement to the resulting local spin polarization. The rst isomer is a tetrahedron andwhich is only 4 meV/atom less stable. Here the magnetic order is found to be AF-like. Thelocal moments are distributed as µCo1 = -1.73µB , µCo2 = 1.71µB , µPd1 = 0.03µB , µPd2= -0.02 µB . The small but still signicant energy difference of 4 meV/atom between FM-and AF-like magnetic order pointing towards a strong competition between the differentmagnetic isomers to become the ground-state structure. In the rich Pd limit (CoPd 3), theoptimal structure is a distorted tetrahedron and the rst excited isomer is a rhombus. Oneobserves that ground-state and rst excited isomer have the same µ4 = 0 .75µB and µPd= 0.24 µB , but the latter shows 0.11 µB enhancement for the µCo compared to the former(see Table 5.3).

5.3.4 Pentamers

The results are tabulated in Table 5.4, and the surface of constant magnetization den-sity is shown in Fig. 5.5. The optimal structures are trigonal bipyramids (TBP) for allcomposition expect for Co 4Pd, where a square pyramid (SP) is obtained. For Co 5 , theground-state structure is a trigonal bipyramid (D 3h ) having E B = 2.65 eV, µ5 = 2.60µB and d = 2 .41 A. In this structure there are two types of bond-lengths: all the sides of

upper and lower triangular pyramids have the same small length d = 2.18 ˚A, while the NN

78




Co5

ξ = 0.02Co4 Pd

ξ = 0.015Co3 Pd 2

ξ = 0.006

Co2 Pd 3

ξ = 0.007CoPd 4

ξ = 0.009Pd 5

ξ = 0.008

Figure 5.5: Constant magnetization density plots ρ↑(r ) −ρ↓(r ) = ξ and local moments(in µB ) for the ground-state structures of CoPd pentamers. The value of the constantmagnetization density ξ is given in µB / A3 .

5.3.5 Hexamers

The results for hexamers are summarized in Table 5.5 and Figs. 5.6, 5.7 and 5.8. The

optimal structures for Co 6 , CoPd 5 and Pd 6 clusters is an octahedra and capped-TBP(CTBP) are found for the rest of the CoPd hexamers.The optimal Co 6 has E B = 3 .03 eV and µ6 = 2.33 µB . This result is in good agreement

with those reported previous DFT calculations by S. Datta et al. [198]. The pentagonalbyramid (PBP) lies 1.7 eV higher in energy. However, photoelectron spectroscopic studies[201] predicted a PBP with bond length 2 .75±0.1 A to be the most probable structure forthe Co −

6 anion cluster. On the other extreme, a perfect octahedron (O h ) with E B = 1 .95eV, µ6 = 0 .33µB and d = 2 .65 A is found optimal for Pd 6 . The length of the bonds inthe middle plane is d = 2.60A, while we nd d = 2.69A for the bonds connecting themiddle plane and the apex positions. A non-magnetic octahedron with the length of thebond forming the middle plane d = 2.65A and the length of the bonds involving the cap

position being d = 2.67˚A is nearly degenerate and is found to be only 3 meV less stable. It

80




Co 5 Co4 Pd Co 4 Pd* Co 3 Pd 2 Co3 Pd 2 * Co2 Pd 3

Co 2 Pd 3 * CoPd 4 CoPd 4 * Pd 5 Co6

Co 5 Pd Co 5 Pd* Co 4 Pd 2 Co4 Pd 2 *

Co 3 Pd 3 Co3 Pd 3 * Co 2 Pd 4 Co2 Pd 4 * CoPd 5

CoPd 5 * Pd 6

Figure 5.6: Lowest energy isomers of CoPd pentamers and hexamers. The asteriks indicaterst exited isomers.

is interesting to observe that the cluster structures having a similar shape display differenttotal magnetic moments. This is probably an artifact of broken spin-rotational symmetry.The optimal CTBP structure is 46 meV less stable than the ground-state. These resultscoincide with those reported previous DFT calculations by T. Futschek et al. [183] andKumar et al. [207]. The optimal Co5Pd is a CTBP structure. The Co atoms form aTBP structure by pushing Pd atom to occupy the outer position. The average magneticmoment µ6 is reduced by 0.50µB compared to the pure Co 6 . The rst exited isomer isa capped square pyramid (CSP), with Co atoms forming the square pyramid structureand Pd atom to occupy the cap position. The optimal Co 4Pd 2 is a CTBP with Co atoms

sub-clustered to form a tetrahedron, and Pd atoms are capped without forming a dimer.

81




Table 5.5: Structural, electronic and magnetic properties of CoPd hexamers as in Ta-ble 5.2.

Cluster E B ∆ E m dαβ µN µCo µPd

Co6 3.03 0.77 2.27 2.33 2.08

Co5Pd 2.73 0.55 2.30 1.83 1.99 0.312.46

Co5Pd* 2.73 0.57 2.27 1.83 2.00 0.252.49

Co4Pd 2 2 .63 0.47 2.30 1.67 2.12 0.312.47

Co4Pd 2* 2.56 0.46 2.27 1.67 2.07 0.452.452.62

Co3Pd 3 2 .44 0.38 2.24 1.50 2.24 0.432.482.70

Co3Pd 3* 2.42 0.36 2.24 1.16 2.05 0.122.502.60

Co2Pd 4 2 .31 0.27 2.22 1.00 2.18 0.322.492.68

Co2Pd 4* 2.26 0.22 2.08 1.00 2.23 0.1962.502.68

CoPd 5 2.15 0.15 2.43 0.50 2.13 0.132.70

CoPd 5* 2.13 0.14 2.46 0.50 2.24 0.192.712.71

Pd 6 1.95 0.10 2.65 0.32 0.32

A CSP structure with a PdPd bond is the rst exited isomer which is 67 meV less stablein energy. For the equiatomic composition (i.e., Co 3Pd 3) a CTBP structure is found tobe optimal. Here Co atoms are grouped together to form a triangle (see Fig. 5.6). Thesegregation of similar atoms in a cluster is important since orbitals with same symmetrycan hybridize, so that cluster can reduce its total energy. The rst excited isomer is adouble tetrahedron with Co atoms making a triangle. This is 15 meV/atom less stable. Inthe Pd rich Co 2Pd 4 cluster, a CTBP structure with a relatively shorter CoCo bond dCoCo= 2.22 A is the optimal. The rst excited isomer is an octahedron, which is 47 meV lessstable. Finally, the optimal CoPd 5 is an octahedron and the rst exited isomer is a CTBP

structure that is 20 meV less stable.

82




-50

0

50

100

-4 -2 0 2 4

-50

0

50

100

-4 -2 0 2 4

D e n s i t y o

f s t a

t e s

[ e V - 1 ]

Co6 Co4Pd2

Co2Pd

4Pd6

ε − εF [eV] ε − ε

F[eV]

TotalCoPd

(a) (b)

(c) (d)

Figure 5.7: Electronic density of states (DOS) of hexamers. Results are given for the total(solid black) DOS, Co-projected (narrow red) and Pd-projected (thick blue) d-electronDOS. Positive (negative) values correspond to majority (minority) spin.

The binding energy per atom shows a similar trend as in the smaller clusters. In thecase of average magnetic moment per atom µN an interesting behavior has been observed.The average magnetic moment µ6 exhibits a similar effect found in the case of pentamers,which depends on the Co content level: µ6 shows a reduction of 0.5 µB as one goes fromCo5Pd to Co 2Pd 4 over CoPd 5 , while this reduction is only around 0.16 µB as one movesfrom Co4Pd 2 to Pd 6 over Co3Pd 3 .

In the previous sections, the total DOS has been plotted to show the variation of theHOMO-LUMO gap with concentration and size. It is also very interesting to analyze thesite projected local density of states (LDOS), at least for some representative examples.To this aim, we plot in Fig. 5.7 the spin polarized d-electron LDOS of representative CoPdhexameters having the relaxed structures illustrated in Fig 5.6. The DOS for pure Co 6 andPd 6 is also shown for the sake of comparison. In all the clusters, the dominant peaks in therelevant energy range (i.e., the occupied valence orbitals and the unoccupied ones near εF )correspond either to the Co-3 d or to the Pd-4 d states. It is found that the contributionsfrom s and p states in the DOS is negligible when compared to the d band contribution.

The Co 6 cluster shows a relatively narrow d-band which dominates the single-particleenergy spectrum in the range −4.3 eV ≤ ε −εF ≤ 1.15 eV. The majority d-DOS is

fully occupied with highest majority state lying about 1.62 eV below εF . In addition

83




Co6

ξ = 0.020Co5 Pd

ξ = 0.010Co4 Pd 2

ξ = 0.007

Co3 Pd 3

ξ = 0.006Co2 Pd 4

ξ = 0.001CoPd 5

ξ = 0.003

Figure 5.8: Constant magnetization density plots ρ↑(r ) −ρ↓(r ) = ξ and local moments(in µB ) for the ground-state structures of CoPd hexamers. The value of the constantmagnetization density ξ is given in µB / A3 .

there is an appreciable gap (about 0.26 eV) in the corresponding minority spectrum. Thespin polarized DOS clearly reects the FM-like order in the Co 6 cluster. For low Pdconcentration (e.g., Co 4Pd 2), the magnetic moments are not saturated. Only spin down

(minority) states are found around εF . The Fermi level εF lies on the top of the minorityband (see Fig. 5.7). In the majority band Pd dominates over Co at the higher energies(near to εF ) while Co dominates in the bottom of the band. In the minority band theparticipation of Pd (Co) is stronger (weaker) below εF and weaker (stronger) above εF ,which is consistent with the fact that the Pd local moments are smaller than the Comoments.

Finally for the Pd rich limit (e.g., Co 2Pd 4), the majority states are almost saturatedand only minority states are found close to εF . The Co contribution is signicant justbelow and above εF . The Co contribution largely dominates the unoccupied minority-spin DOS, in agreement with the larger local Co moments. The DOS of Co 6 is similar to

previous DFT calculation by Kumar et al. [207].

84




Co 7 Co6 Pd Co 6 Pd* Co 5 Pd 2 Co5 Pd 2 *

Co 4 Pd 3 Co4 Pd 3 * Co3 Pd 4 Co3 Pd 4 * Co2 Pd 5

Co 2 Pd 5 * CoPd 6 CoPd 6 * Pd 7 Co8

Co 7 Pd Co 7 Pd* Co 6 Pd 2 Co6 Pd 2 * Co5 Pd 3

Co 5 Pd 3 * Co4 Pd 4 Co4 Pd 4 * Co 3 Pd 5 Co3 Pd 5 *

Co 2 Pd 6 Co2 Pd 6 * CoPd 7 CoPd 7 * Pd 8

Figure 5.9: Lowest energy isomers of CoPd heptamers and octamers. Note that only veryfew topologies have been considered as starting congurations of geometry relaxation.

85




Table 5.6: Structural, electronic and magnetic properties of CoPd heptamers as obtainedfrom a restricted representative sampling of cluster topologies.


Co7 3.04 0.68 2.28 2.14 1.96

Co6Pd 3.05 0.60 2.29 2.00 2.08 0.282.48

Co6Pd* 2.94 0.61 2.30 2.00 2.06 0.3622.54Co5Pd 2 2.88 0.51 2.28 1.86 2.15 0.43

2.48Co5Pd 2* 2.83 0.50 2.28 1.86 2.16 0.42

2.492.68

Co4Pd 3 2.71 0.45 2.26 1.43 2.08 0.372.492.73

Co4Pd 3* 2.70 0.43 2.39 1.71 2.25 0.50

2.46Co3Pd 4 2.50 0.30 2.30 1.29 2.16 0.432.502.72

Co3Pd 4* 2.48 0.31 2.34 1.29 2.21 0.402.482.70

Co2Pd 5 2.39 0.26 2.22 1.14 2.24 0.502.522.66

Co2Pd 5* 2.37 0.23 2.26 0.86 2.18 0.27

2.462.70CoPd 6 2.21 0.14 2.42 0.71 2.31 0.36

2.70CoPd 6* 2.19 0.13 2.48 0.43 2.25 0.15

2.73

Pd 7 1.99 0.04 2.69 0.29 0.28

87



5.4. Size and composition trends

Table 5.7: Structural, electronic and magnetic properties of CoPd octamers as obtainedfrom a restricted representative sampling of cluster topologies.


Co8 3.17 0.70 2.31 2.00 1.88

Co7Pd 3.11 0.57 2.30 1.88 1.95 0.272.51

Co7Pd* 3.08 0.56 2.32 1.75 1.91 0.302.48

Co6Pd 2 3.05 0.53 2.31 1.75 2.06 0.292.49

Co6Pd 2* 3.00 0.53 2.30 1.75 2.03 0.322.53

Co5Pd 3 2.93 0.49 2.27 1.63 2.13 0.422.48

Co5Pd 3* 2.91 0.47 2.33 1.63 2.13 0.412.49

Co4Pd 4 2.76 0.41 2.36 1.50 2.23 0.492.46

Co4Pd 4* 2.75 0.41 2.31 1.50 2.23 0.472.472.73

Co3Pd 5 2.62 0.28 2.29 1.13 2.19 0.362.482.73

Co3Pd 5* 2.59 0.29 2.28 1.13 2.19 0.262.472.75

Co2Pd 6 2.47 0.21 2.34 1.00 2.26 0.432.452.71

Co2Pd 6* 2.46 0.20 2.34 1.00 2.26 0.432.442.72

CoPd 7 2.27 0.12 2.42 0.63 2.31 0.342.69

CoPd 7* 2.25 0.12 2.47 0.63 2.33 0.322.74

Pd 8 2.07 0.59 2.68 0.25 0.25

88




following we present and discuss results for the binding energy, average and local spinmoments, relative stability and the magnetic stabilization energy.

5.4.1 Binding energy and magnetic moments

In order to summarize the important trends, we present in Fig. 5.10 the binding energyper atom E B as a function of the number of Co atoms m. For most of the sizes studied,

0 1 2 3 4 5 6 7 8m

0.5

1

1.5

2

2.5

3

B i n d i n g E n e r g y

[ e V / a t o m ]

Com

PdnN = 2

3

45

6 78

Figure 5.10: Binding energy per atom E B of Com Pd n clusters as a function of the numberof Co atoms. The lines connecting the points for different total number of atoms N = m+ nare a guide to the eye.

the slope of E B as a function of m is maximal for single Co doping ( m = 1) and decreasingfor m > 1. There is an almost linear increase in the E B with increasing Co content (seeFig 5.10). Notice the small dip for Co 2Pd 2 in Fig. 5.10. This can be explained as follows:Co2Pd 2 is a 2D rhombus in which two Co atoms are well separated by a Pd dimer. Thepresence of weaker Pd dimer is the key reason for the reduced binding energy for Co 2Pd 2 .

In Fig. 5.12 the average magnetic moments µN of Com Pd n is shown as a function of m. As already discussed in the previous sections, µN increases almost linearly with theCo content. This is expected due to the larger Co local moments and the underlying FM-like order. Among the pure clusters, Co 5 and Pd 2 show the highest magnetic moments.Notice, the enhancement of the magnetic moments of the pure clusters in particular forCoN , which go well beyond 2.5µB (e.g., in Co4 and Co 5). In general, the slope of the curvestend to decrease with increasing N , since the change in concentration per Co substitutionis more important the smaller the size is.

The magnetic stabilization energy ∆ E m is shown in Fig. 5.11. One observes that ∆ E mincreases approximately linearly with the Co concentration except for CoPd 7 . The pureCo clusters show the highest ∆ E m . This is understandable since Co is a FM-3 d element.

In contrast, Pd clusters show low ∆ E m , since Pd is a non-magnetic element.

89




5.4.2 Relative stability

1 2 3 4m

-0.05

0.00

∆ 2

E ( 5 ) [ e V ]

1 2 3 4 5m

-0.10

-0.05

0.00

0.05

∆ 2

E ( 6 ) [ e V ]

1 2 3 4 5 6m

1 2 3 4 5 6 7m

-0.05

0.00

∆ 2

E ( 8 ) [ e V

]

0.20

0.00

-0.18

0.10

∆ 2

E ( 7 ) [ e V

]0.03

0.15

0.10

Figure 5.13: The relative stability ∆ 2E (N, m ) = E (N, m +1)+ E (N, m −1)−2E (N, m ) of Com Pd n clusters as a function of the number of Co atoms. The symbols corresponding toeach size are the same as in Fig. 5.10. The lines connecting the points for each N = m + nare a guide to the eye.

1 2 3 4m

2

2.2

2.4

2.6

2.8

d ( Å )

1 2 3 4 5m

2

2.2

2.4

2.6

2.8

d ( Å )

1 2 3 4 5 6 7m

2

2.2

2.4

2.6

2.8

d ( Å )

1 2 3 4 5 6 7m

2

2.2

2.4

2.6

2.8

d ( Å )

N = 5N = 6

N = 7

N = 8

Figure 5.14: Bond lengths for CoCo (cross), Co Pd (circle) and PdPd (square) pairs asa function of number of Co atoms in Co m Pd n clusters having N = 5–8 atoms. The linesconnecting the points for each N = m + n are a guide to the eye.

91



5.5. Summary

Like FeRh clusters, the CoPd clusters also show signicant variation in the relativestability among the different cluster sizes N having different compositions. For instance,for N = 6, Co 5Pd shows the highest stability while Co 4Pd 2 has lowest stability. This canbe justied by analyzing the number of different bonds and average bond length, as wasdone in the Sec. 4.4.2 in the case of FeRh clusters. In Co 5Pd, there are 9 CoCo bonds(average bond length 2.30 A) and 3 CoPd (average bond length 2.46 A). While in Co 4Pd 2 ,there are 5 CoCo bonds (average bond length 2.30 A) and 6 CoPd bonds (average bondlength 2.47 A) present. Let us recall that the CoCo bond is the strongest, as it was shownin the case of the dimers (see Table. 5.1).

5.5 Summary

The structural, electronic and magnetic properties of small Co m Pd n clusters having N =m + n ≤8 atoms have been investigated systematically in the framework of a generalizedgradient approximation to density-functional theory. Geometry optimization yields bothcompact 3D and 2D open topologies. The magnetic order of the clusters is found to beFM-ike, at least in the most stable structures. The average magnetic moment per atomµN increases approximately linearly with the increasing Co content. Accordingly, theenergy gain ∆ E m associated to magnetism also increases with the number of Co atoms.The maximal spin polarization for Co and Pd atom is found in the equiatomic or nearlyequiatomic compositions (Co 3Pd 3 , Co3Pd 4 and Co 4Pd 4). The s and p spin polarizationsare almost negligible in general. A remarkable enhancement of the local Co momentsis observed as result of Pd doping. This is due to the enhancement in the number of

Co d holes, due to Co to Pd charge transfer, combined with the extremely reduced localcoordination.The spin-orbit coupling (SOC) is not included in the present calculations. Several

authors have discussed the inuence of SOC for a variety of 3 d, 4d, and 5d transition-metalclusters [219–221]. These investigations have predicted that for clusters of 3 d and 4d TMsthe SOC was found much smaller whereas for 5 d TMs like Pt SOC can be strong enoughso that it can change the energetic ordering of structural isomers. We have performedsome representative calculations by taking into account spin-orbit coupling (SOC) in orderto explore their effect on the ground-state structure, chemical order and spin moments.For example in Co 6 , Co3Pd 3 and Pd 6 we nd that the changes in the ground-state energyresulting from SO interaction are typically of the order of 0.2 eV for the whole cluster. Thisis often comparable to or larger than the energy differences between the low-lying isomers.However, the SO energies are very similar for different structures, so that the ground-statestructures remain essentially the same as in the scalar relativistic (SR) calculations. Thechanges in the bond lengths and in the spin moments resulting from SOC are also verysmall (e.g., |µSOC −µSR | 0.04µB and |dSOC

ij −dSRij | 0.001A in Co4Pd 4). As a result,

the conclusions drawn from our SR calculations on the relative stability and local spinmoments seem to be unaffected by the spin-orbit contributions.

92



6First principles spin-polarized basin-hoppingmethod

6.1 Introduction

The reported results in the chapters 5 and 6 have been obtained by employing those struc-tures which are derived from graph theory method [17]. However, it should be emphasizedthat performing calculations by considering all the possible initial cluster structures, allthe different magnetic congurations and all the possible different distribution of two kindsof atoms is nearly impossible (or impracticable) beyond the hexamer clusters ( N = 6) dueto the prohibitively huge computational time. This was the main reason for consideringonly a small set of initial cluster topologies for cluster sizes N = 7 and 8 in chapters 5 and6. On the other hand, we would like to elucidate whether the ground-state structures weidentied in the last chapters represent the true ground-states, at least in the case of FeRhclusters. The overall situation signals the need for an efficient global optimization method,and by employing this we could x the inherent problems associated with the graph theorymethod. For this purpose, we chose and employed the basin-hopping (BH) global opti-mization method. Then we developed a scheme which combines the BH method with thespin-polarized DFT code VASP. As was discussed in Sec. 3.2.3, the BH method has alreadyproven its applicability and suitability to optimize smaller and larger clusters [19,20,222].In this work we applied the proposed scheme to study both small (Fe 6 , Fe3Rh 3 and Rh 6)and larger (Fe 13 , Fe6Rh 7 and Rh 13 ) clusters. Having selected some clusters with differentsizes and compositions, the next step should be the choice of suitable technical parametersin order to optimize the performance of the BH calculations. In the past, an investigationhave been reported by combining DFT with the BH scheme [222]. However, that studywas based on the Si and Cu clusters, which are obviously non-magnetic. To the best of ourknowledge, this is the rst time one proposes to combine the spin-polarized DFT with aBH scheme. Although one easily understand the potential advantages of proposed scheme,

it poses a number of serious computational challenges. We begin our computations on ho-

93



6.1. Introduction

Figure 6.1: Schematic diagram of potential energy surface (PES) of a nanoalloy cluster.

mogeneous clusters and then follow with the heterogeneous ones. In order to implementthe proposed scheme, one should consider different initial cluster structures and variousstarting magnetic congurations. The different magnetic congurations such as FM andAF-like are introduced by assigning random initial magnetic moment to the BH samplingruns. Obviously, the proposed scheme would take more computational time in the caseof mixed clusters. This is because one has to consider different homotops for a speciccomposition. This has been done by exchanging or swapping the dissimilar atoms within acluster on-the-y. Adding the complexities mentioned above makes the computation veryexpensive. This indeed shows the need for optimizing the efficiency of the BH runs.

A widely used approach to measure the sampling efficiency of a global optimizationcalculation is the mean rst encounter (MFE), which is dened as the number of BH stepsuntil the algorithm locates the ground-state structure for the rst time. The MFE is foundto be an ideal sampling efficiency indicator . In order to obtain useful or meaningful re-sults, the sampling should be performed by monitoring on-the-y analyzable performanceindicators that allow to modify an ongoing run. However, there is no previous informationwhich illustrates how to build move parameters that do not depend upon system-specicintuition in the case of spin-polarized clusters. The inherent stochastic nature of the BHscheme can be another source of difficulties in the proposed scheme. Any investigation

which judges the credibility of BH technical parameters or on-the-y analyzable perfor-mance indicators must involve an averaging over a fairly large number of BH samplingruns.

Studies have been reported in the past, which combine the BH scheme with numericallyundemanding model potentials such as Sutton-Chen [223] and Lennard-Jones [19,20]. Butthese model potentials are unsuitable to sample the quantum-mechanical PESs of magneticTM nanoclusters. In fact, the basic motivation behind this work is to illustrate an efficientframework for a systematic performance analysis of rst-principles spin-polarized BH sam-pling runs and investigate the unique magnetic properties exhibited by TM nanoclusters.This method should considerably reduce the computational cost of the spin-polarized BHruns required for the averaging process and help to obtain statistically promising perfor-

mance results.

94



Chapter 6. First principles spin-polarized basin-hopping method

6.2 Computational aspectsThe calculations reported in this chapter have been performed in the framework of Hohenberg-Kohn-Sham’s density functional theory, [11] as implemented in the Vienna ab initio simu-lation package (VASP) [168]. The detailed information concerning this method has beenexplained in the Sec. 4.2.

6.2.1 Basin-Hopping method

In this section, all the relevant parameters that govern the performance of the BH samplingrun are explained. The detailed information regarding this method has been discussed inthe Sec. 3.2.3. Three key points in a BH sampling method deserve special attention: i)the trial move method to generate new cluster structures, ii) the acceptance criteria, andiii) the entropy driven dissociations. These are discussed in the following.

The trial move method

The BH trail moves class can be classied into single and collective particle moves . Thisis schematically shown in the Fig. 6.2. The studies carried out by R. Gehrke [222] madea systematic comparison on the performance between these move classes and reportedthat they does not produce any signicant difference in the performance. Therefore, weemployed collective particle moves with uniform distribution of atoms to perform all thecalculations reported in this chapter.

Figure 6.2: Single and collective particle move classes.

In the collective random move all the atoms in the cluster are randomly displacedwithin a predened range of distance. In the coordinate space, the position of an atom iwithin a cluster is represented by R i = (x i , yi , zi ) Randomly displacing an atom meansthat changing the current (x i , yi , zi ) position into a new position and the correspondingdisplacement vector of an atom i can be written as

∆ R i = d x ±∆ x, y ±∆ y, z ±∆ z.

6.2.1

The move distance d controls the extend of the jumps in the congurational space andthe resulting algorithmic performance. The real advantage of the proposed scheme is thatit can be implemented to study any atomic cluster (for instance, metallic, semiconductor,

etc.) without a prior knowledge about the properties owned by the specic clusters. The

95




Figure 6.3: Pictorial illustration of successful, unsuccessful and high-energy trial moves inthe BH scheme. The horizontal red line indicates the energy-window acceptance criterionfor BH sampling. After Ref. [222].

outcome of trail moves can be divided into three categories: i) unsuccessful trail moves,ii) high-energy trail moves, and iii) successful trail moves. These are shown schematicallyin Fig. 6.3.

Unsuccessful trail moves

A trial move is considered to be unsuccessful (see Eq. 6.2.2) if the initial cluster, wherethe trial move has been started, and the optimized cluster where the optimization endsup are the same. In this case no new information concerning the isomers is obtained. Thefraction of unsuccessful moves α uns . up to the n th trial move can is given by

α uns = β uns

n ,

6.2.2

where β uns is the number of unsuccessful moves up to that point.

High-energy trail moves

A Markov step is rejected (see Eq. 6.2.3) if the move generates a new relaxed clusterhaving a total energy difference ∆E that is above the BH energy window imposed by theacceptance criterion . The fraction of moves rejected in this way is given by

α hE = β hE

n ,

6.2.3

where β hE is the corresponding number of rejected moves up to that point.

Successful trail moves

A Markov step is considered to be successful (see Eq. 6.2.4) once the move and subsequentrelaxation brings the cluster into local minimum within the energy window that is differentfrom the starting structure. The fraction of successful moves is given by

α suc = ns

n = 1 −β uns −β hE

6.2.4

97



6.3. Results for small homogeneous clusters

6.3 Results for small homogeneous clusters

This section is devoted to discuss the performance and the results of the proposed schemein the case of homogeneous small clusters. The main emphasis is here on identifyingthe ground-state structures and other low-lying isomers, and discuss their energetic andmagnetic properties.

6.3.1 Dominant isomers

As pointed out in the introduction of this chapter, we have varied the move distance bystarting from d = 0.10a up to a maximum value d = 1.0 a, in terms of the smallest bondlength a within the specic cluster. In order to make the discussion compact and moreinformative, we provide here only the results for three different move distances d = 0.15a,0.26a and 0.40a. It would be however noted that many other move distances (for instance,0.10a, 0.20a, 0.60a, etc) have been considered.

0, 0.00, 20 1, 0.02, 20 2, 0.18, 20 3, 0.20, 20 4, 1.18, 6

5, 1.20, 12 6, 1.22, 12 7, 1.24, 14 8, 1.25, 12 9, 1.34, 8

10, 1.38, 10 11, 1.41, 14 12, 1.74, 12

Figure 6.4: Identied isomers of Fe 6 cluster in the energy range up to 1.74 eV above theground-state. The isomers are ordered by increasing energy. In the subcaption the rstentry refers to the isomer number, the second entry to the energy relative to the ground-state energy in eV, and third entry to the the cluster magnetic moment in µB . Bondlengths are indicated in A.

98




This choice is justied since the BH algorithm run with very small move distance (sayd = 0.10 a) ends up most of the time in the same structure as the starting one. In contrastBH runs using a too large move distance (say d = 0.60a) mostly produce loosely connectedopen cluster structures having a much higher energy than the ground-state structure (seeFig. 6.6). We present in Fig. 6.7 the binding energy of isomers starting from the ground-state structure , the probability of nding a given isomer is given by the number of timesthat the low-energy isomers were located, and the cluster magnetic moment along the BHruns obtained for Fe 6 and Rh 6 . It should be emphasized that each spin-polarized BH runfor a particular move distance is composed of several hundred steps and was performeduntil the probability plots for all the relevant isomers were converged. We have explicitlychecked convergence of the probability by taking Fe 3Rh 3 cluster as a representative testsystem. The progress towards convergence is schematically shown in Fig. 6.5. The plotshows the probability over consecutive sampling periods containing 50 calculations each.One may observes that some isomers show up as dominant. The main reason for theexistence of dominant isomers is that their corresponding basins of attraction on the PESis large and thus hit by the trial moves many times. In other words, the existence of dominant isomers is because of the reduced system size and the resulting small number of low-lying isomers .

50 100 150 200

BH step n

00.10.20.30.40.50.6

P r o

b a

b i l i t y

# 1# 2# 3# 4> # 4

Figure 6.5: Normalized probabilities for the lowest-energy isomers of Fe 3Rh 3 as shown inthe Fig. 6.10. The probabilities over consecutive sampling periods containing 50 moveseach are shown by using collective-particle moves and uniformly distributed move distancesaround the average values d = 0.26 a. The probabilities for all isomers higher in energythan isomer # 4 are bundled into one entry labeled > # 4.

The following analysis provides more information concerning the dominant isomers.Fig. 6.7 demonstrates the results obtained for Fe 6 and Rh 6 cluster using three different

move distances. First we discuss the results for Fe 6 with and without window acceptance

99




0 10 20 30 40 50 60 70 80 90 100

-36.0

-35.5

-35.0

-34.5

-34.0

E n e r g y

( e V / c l u s t e r )

-36.4

-36.2

-36.0

-35.8

-35.6

-35.4

-35.2

d = 0.15a

Number of BH steps (n)0 10 20 30 40 50 60 70 80 90 100

-36.0

-35.0

-34.0

-33.0

-32.0

-31.0

-30.0

E n e r g y

( e V / c l u s t e r )

-36.4

-36.2

-36.0

-35.8

-35.6

-35.4

-35.2

Number of BH steps (n)

d = 0.26a

0 10 20 30 40 50 60 70

-36.0

-35.0

-34.0

-33.0

-32.0

E n e r g y

( e V / c l u s t e r )

-36.4

-36.2

-36

-35.8

-35.6

-35.4

-35.2


d = 0.40a

0 10 20 30 40 50 60 70 80

-36

-35

-34

-33

-32

-31

-30

E n e r g y

( e V / c l u s t e r )

-36.4

-36.2

-36.0

-35.8

-35.6

-35.4

d = 0.60a


Figure 6.6: The total energy of Fe 6 as a function of the number of BH steps n for differentmove distances d = 0.15a, 0.26a, 0.40a and 0.60a.

100




-2.8-2.7-2.6-2.6-2.5

E B

[ e V / a t o m

]d = 0.15ad = 0.26ad = 0.40a

00.20.40.60.8

P r o b a b i l i t y

0 2 4 6 8 10 12 14 16 18Isomer number

8

12

1620

T o t a l m o m e n

t

Fe 6 NAC

(a)-3.2

-3.1

-3

E B

[ e V / a t o m

]

00.20.40.60.8


0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30Isomer number

0246810

T o

t a l m o m e n

t

Rh6

NAC

(c)

-2.7

-2.6

E B

[ e V / a t o m

]

00.2

0.40.60.8

P r o

b a b i l i t y

0 2 4 6 8Isomer number

12

16

20

T o t a l m o m e n t

Fe 6 WAC

(b)-3.2

-3.1

-3

E B

[ e V / a t o m ]

0

0.1

0.2


0 2 4 6 8 10 12 14 16 18 20 22 24 26Isomer number

0369

T o t a l m o m e n t

Rh 6 WAC

(d)

Figure 6.7: The binding energy per atom (in eV), the probability that trial moves generatethe lowest energy isomers, and the total magnetic moment are plotted as a function of isomer number for Fe 6 and Rh 6 cluster. NAC refers to the calculations using No Accep-

tance Criterion and WAC refers to With Acceptance Criterion. The obtained isomers arenumbered with decreasing stability, with isomer 0 refers to the ground-state structure.The gure contains all the isomers found for collective moves with uniform distributionof move distances d = 0.15 a, 0.26a and 0.40a, where a refers to the shortest NN distancewithin the cluster. The geometric structures of obtained isomers for Fe 6 and Rh 6 are givenin Fig. 6.4 and in Fig. 6.8, respectively.

101




0,0.000,6 1,0.020,0 2,0.023,6 3,0.054,6 4,0.162,4 5,0.179,2

6,0.217,10 7,0.359,10 8,0.363,6 9,0.367,6 10,0.386,8

11,0.434,10 12,0.435,4 13,0.469,10 14,0.472,8 15,0.508,4

16,0.545,6 17,0.555,8 18,0.567,0 19,0.664,2

20,0.684,8 21,0.698,0 22,0.743,4 23,0.771,0

24,0.799,6 25,0.841,2 26,0.903,8 27,0.947,4 28,1.045,2

29,1.119,2 30,1.390,0

Figure 6.8: Calculated isomers of Rh 6 cluster in the energy range up to 1.39 eV above theground-state energy. The entries below the illustration have the same meaning as in the

Fig. 6.4

102




criterion of 1eV. See plots (a) and (b) in the Fig. 6.7.

One observes that only a small set of isomers which are close to the ground-statestructure (labeled as isomer number 0) are much more often sampled than other isomers.Notice that, more than 2/3 of all executed moves in the BH runs ended up in the ground-state isomer, regardless of the actual move distance and the window acceptance criterionemployed. The plots (a) and (b) in the Fig. 6.7 display some differences too. One cansee that for the same move distances the calculations without an energy window (with anenergy window) generate 9 (4) isomers. All the isomers are having compact geometries(see Fig. 6.4). The results for Rh 6 [see plots (c) and (d) in the Fig. 6.7] also display theexistence of the dominant isomers as in the case of Fe 6 cluster.

One can see that, some of the Rh 6 isomers relax into the open structures higher up

in energy compared to the ground-state structure. However, the possibility for a moveto end up in an open structure is very small. They are not often identied in the BHsampling run. Overall, these results show that in the case of small homogeneous clustersit is sufficient to carryout the calculations without using a window acceptance criterion .As pointed out earlier, without a window acceptance criterion, the system can jump outof the dominant isomers and is able to visit different rare local minima.

6.3.2 Magnetic properties

Fe6 and particularly Rh 6 clusters display remarkable structural and the magnetic diversity.In fact, the results for Rh 6 reveal a unique relation between the magnetism and the clusterstructure. It was been shown in chapter 4 that the reduced system dimensionality andthe subsequent d-band narrowing helps the Rh clusters to fulll the Stoner criteria of ferromagnetism, despite being non-magnetic in the bulk. And even a small distortionin the cluster structure could produce quite different magnetic ordering (see Fig. 6.8).For instance, the shape of the ground-state structure and the rst isomer of Rh 6 clusterare octahedra. However, their cluster magnetic moments 6 µB and 0µB , respectively arequite different. It is also interesting to see that isomers having the same cluster magneticmoments adopt different geometries. See for instance, the 5 th and 8 th isomers in theFig. 6.4 and ground-state, 2 nd , and 24 th isomer in the Fig. 6.8. The rst example of AF-

like coupling of the local moments is found in the 4th

isomer of Fe6 clusters, where twolocal moments couple antiferromagnetically with the rest of the local moments and thetotal cluster moment is reduced to 6 µB (see Fig. 6.4). In the case of Rh 6 clusters, AF-likecoupling is observed in the rst isomer. The associated total cluster moment is zero.

It is interesting to compare the present results for Fe 6 and Rh 6 with those reportedin chapter 4. One can see that the true ground-state structure for Fe 6 cluster is the onereported in the present chapter, namely a distorted octahedra. The perfect octahedrareported in chapter 4 (see Fig 4.3) is only a low-lying isomer. The corresponding energydifference is 0.04 eV/atom. This shows the signicance of the BH method, since it caneffectively take into account even very small cluster distortions, and thereby identifyingthe true ground-state structure. The conclusion drawn in chapter 4 concerning the local

magnetic moments are also affected by the distortions found in the present BH calculation.

103



6.4. Results for heterogeneous small clusters

6.4.2 Magnetic properties

The magnetism of the mixed clusters display remarkable diversity since not only the clusterstructure but also the different distribution of dissimilar atoms within the cluster matters.Fig. 6.10 shows the obtained isomers of Fe 3Rh 3 cluster. The ground-state structure isfound to be an octahedra, which coincide with the one reported in chapter 4 by usinggraph theory method in which two isosceles open Fe 3 and Rh 3 triangles form a π/ 2 angle

0,0.00,15 1,0.06,17 2,0.15,15 3,0.21,15 4,0.31,13

5,0.33,5 6,0.35,13 7,0.39,17 8,0.41,13 9,0.43,13

10,0.44,15 11,0.46,15 12,0.49,15 13,0.50,15 14,0.57,11

15,0.58,15 16,0.59,15 17,0.62,15 18,0.63,5

19,0.64,5 20,0.65,15 21,0.67,15

Figure 6.10: The optimal structures of Fe 3Rh 3 cluster. The entries below each structureshave the same meaning as in the Fig. 6.4

106



6.5. Results for larger clusters

energy than ground-state. The diversity of the initial starting cluster structures is alsoan essential prerequisite for a successful BH global optimization. Therefore, we chose thedifferent structures such as hcp, icosahedra and fcc. The chemical order of the mixedcluster during the calculations has been changed on the y.

The calculations reveal the remarkable structural and magnetic diversity among theisomers of Fe13 , Rh 13 and Fe 6Rh 7 . In the case of Fe 13 most of the isomers relax to highlysymmetric structures. The magnetism displayed by the cluster structures having similarshapes is analogous to what has been observed in the small clusters. In fact, differentsolutions of the Kohn-Shams equations corresponding to the different values of the totalspin moment may be found which yield very similar structures after relaxation. Forexample, we nd icosahedral isomers having small distortions with respect to each other,which correspond to different magnetic moments. For instance, the structures labeled as

0 (44µB ), 1 (42µB ), 2 (40µB ), and 3 (38µB ), see Fig. 6.11. This is surely an artifactof the broken spin rotational symmetry of the spin-polarized GGA approximation, onlythe ground state is relevant. Although most of the isomers relax into the icosahedralshape, some of the isomers optimize into different shapes. For instance, we obtained theisomer 4 as a distorted hcp structure with a cluster magnetic moment of 40 µB that is 0.64eV above the the ground-state structure. We also found a cage-like structure (isomer 5)with a cluster magnetic moment of 40 µB that is 0.75 eV less stable than the ground-statestructure. The isomer number 6 deserve special attention since it presents an AF-likecoupling among the local magnetic moments. Fe atom located on one of the pentagonalrings (-3.11 µB ) couples anti-ferromagnetically to the rest of the local moments. Takinginto account that the structure is similar to the ground-state, it is likely that this situationis a artifact of broken spin-symmetry.

The next step is to compare the present results for Fe 13 with those reported in previousstudies [at least the results of the ground-state and the rst isomer]. The DFT studiesreported in Ref. [229] predicted that the ground-state structure is an icosahedra with acluster magnetic moment of 44 µB , which coincides with our results. The rst isomerin Ref. [229] is also an icosahedra with S z = 34µB . We also found the rst isomer is anicosahedra, but with a cluster magnetic moment of 40 µB . In fact, for Sz = 40 µB we shouldhave obtained the same energy and the structures for S z = 44µB , i.e., the ground-state.

Unlike Fe 13 , Rh13 does not relax into highly symmetric structures but into a cage-like and layered structures (see Fig. 6.12). The ground-state is a cage-like structure witha total cluster magnetic moment S z = 13µB in which all local moments are coupledferromagnetically. The DFT studies reported in Ref. [190] also yield a cage-like ground-state structure but with a magnetic moment S z = 17µB . We actually nd that thestructure given in the Ref. [190] as the optimal one is our rst isomer, which lies 0.09 eVabove of ground-state. The isomer number 1 shows the highest total moment S z = 17µBwith all the local moments above 1 µB and coupled ferromagnetically. The isomer number9 is the rst AF-like coupled cluster and having a total cluster moment S z = 1µB . Infact, 5 out of 13 local moments coupled anti-ferromagnetically with the rest of the localmoments. Finally, we comment on the results for Fe 6Rh 7 cluster. In this case most of theisomers adopt the cage-like and layered structures as in Rh 13 . There are rich structuraland magnetic diversity due to the presence of two kind of atoms. In fact, there are 20homotops in an energy interval of 0.61 eV. It is interesting to analyze the diversity in thestructures having same cluster magnetic moment. Out of 19 isomers 13 have a cluster

magnetic moment S z = 27µB . They display different shapes and the different distribution

108




0,0.00,44 1,0.09,42 2,0.18,40 3,0.34,38

4,0.64,40 5,0.75,40 6,0.91,34 7,1.37,28

8,1.51,36 9,1.56,8 10,2.28,24 11,2.42,2

12,2.53,16 13,2.99,30 14,3.08,0 15,3.13,4

Figure 6.11: Obtained isomers of Fe 13 cluster in the energy range up to 3.13 eV above theground-state energy. The entries below the structures start with isomer number followedby energy of the isomer relative to the ground-state, and last entry is the total clustermoment.

109



6.5. Results for larger clusters

0,0.00,13 1,0.09,17 2,0.13,5 3,0.13,13

4,0.16,11 5,0.16,15 6,0.21,13 7,0.22,15.6

8,0.23,15 9,0.26,1 10,0.27,15 11,0.28,11

12,0.29,13 13,0.33,15 14,0.35,13 15,0.37,17

16,0.40,9 17,0.41,15 18,0.42,7 19,0.43,5

Figure 6.12: Obtained isomers of Rh 13 cluster in the energy range up to 0.43 eV abovethe ground-state energy.

110




0,0.00,27 1,0.024,27 2,0.03,27 3,0.09,27

4,0.17,27 5,0.19,27 6,0.23,27 7,0.24,15

8,0.27,25 9,0.34,17 10,0.36,27 11,0.38,17

12,0.41,17 13,0.42,27 14,0.44,27 15,0.49,17

16,0.53,27 17,0.56,27 18,0.57,27 19,0.61,17

Figure 6.13: Low-lying isomers of Fe 6Rh 7 in the energy range up to 0.61 eV above theground-state energy.

111



7Composition dependent orbital magnetism

7.1 Introduction

This chapter is devoted to study the orbital magnetism and MAE in the FeRh clusters asa function of size and composition. Atomic magnetism is characterized by Hund’s rules,which predicts maximum orbital angular moment L compatible with maximum spin multi-plicity S , while in TM solids electron delocalization and subsequent band formation resultin an almost complete quenching of L . A similar kind of quenching or reduction is alsoobserved in the MAE. In fact, MAE in the cubic 3 d transition metals is the order of onlya few µeV/atom. The purpose of the present investigations of the orbital magnetism andMAE in the alloy TM clusters is to reveal novel composition and size-dependent featuresthat are important both from a fundamental standpoint and in view of applications of thecluster-based magnetic nanometer devices.

7.2 Computational aspects

The calculations reported in this work have been performed in the framework of Hohenberg-Kohn-Sham’s density functional theory, [11] as implemented in the Vienna ab initio sim-ulation package (VASP) [168]. See Sec. 2.2 where the DFT method has been described insome detail. The wave functions are expanded in a plane wave basis set with the kineticenergy cut-off E max = 700 eV. A simple cubic supercell a is considered having a linearsize of 20 A for 13 atom clusters and 25 A for 19 atom clusters, with the usual periodicboundary conditions. In this way, pairs of images of the clusters are well separated andthe interaction between them can be neglected.

Magnetic anisotropy and orbital moments are relativistic effects depending on thestrength of the spin-orbit coupling. Spin-orbit coupling has been implemented in VASPby Kresse and Lebacq [225]. The MAE is dened as the variation of the total energy

of a system as a function of magnetization direction ( −→M) with respect to the crystalline

113



Chapter 7. Composition dependent orbital magnetism

(i) Fe 11 Rh 2 (ii) Fe 9 Rh 4 (iii) Fe 8 Rh 5 (iv) Fe 6 Rh 7

(v) Fe 4 Rh 9 (vi) Fe 3 Rh 10 (vii) Fe 2 Rh 11

Figure 7.1: The distribution of Fe and Rh atoms in the FeRh clusters having N = 13atoms. The green (blue) color refers to Fe (Rh) atom.

in the bulk. However, small Rh clusters show nite spin polarization [51]. Interestingly,the Rh dimer shows enhanced moments µS = 1.93 µB and µL = 0.91µB [231].

The results for µS and µL in pure Fe and Rh clusters shall be compared with those of extreme limits mentioned in the last paragraph (see Tables. 7.1 and 7.2, and Fig. 7.3). Weobserve a remarkable enhancement in magnetic moments for both Fe and Rh pure clustersas compared to their bulk moments. One can see that the magnetic enhancements aremore appealing in pure Rh N . For instance, µS = 1.461µB and µL = 0.336µB for Rh13 ,while, µS = 0.578µB and µL = 0.075 µB for Rh19 . In the case of pure Fe N , we nd thatFe19 shows enhanced magnetic moments compared to Fe 13 . This trend is opposite to thecase of pure Rh N since Rh13 exhibits enhanced magnetic moments compared to Rh 19 (seeFig. 7.3). It should be stressed that the above conclusions are based on our optimal fccstructures. However, we cannot completely exclude the possibility of identifying clusterstructures that are more stable for the clusters sizes and compositions considered here.

Fig. 7.4 demonstrates that the relative change in the MAE due to the size changeis more important for Rh clusters compared to Fe clusters. In the past, several authorspointed out that one can relate the MAE with the anisotropy of other spin-orbit inducedproperties. The most prominent example for this is the relation of the MAE and theanisotropy of the orbital angular momentum, that was considered by Bruno [118] on thebasis of second-order perturbation theory. According to Bruno’s model the MAE can bederived as

E = − ξ 4µB

∆ L

7.3.1

where ξ is the spin-orbit parameter and ∆ L are the difference in orbital moment

between the easy and the hard axis. Eq. 7.3.1 shows that MAE depends on the spin-orbit

115




Table 7.1: Electronic and magnetic properties of FeRh clusters having N = 13 atoms.Results are given for the average spin moment per atom µS = 2S z /N (in µB ), averageorbital moment per atom µL (in µB ), the ratio between orbital and spin moment µL

µS ,

magnetic anisotropy energy (MAE) in meV per cluster, the easy axis direction ( δ ), and γ is the difference in total energy per atom between calculations with and without SOI (inmeV).

Cluster xF e µS µL µS F e µLF e

µL

µS MAE δ γ

µS Rh µLRh

Fe13 1.00 2.769 0.056 2.597 0.023 2.73 x 6.6

Fe11 Rh 2 0.84 2.769 0.072 2.856 0.062 0.028 2.21 z 11.10.835 0.066

Fe9Rh 4 0 .69 2.615 0.097 3.031 0.082 0.041 7.80 z 16.30.899 0.130

Fe8Rh 5 0.61 2.231 0.096 2.966 0.101 0.045 5.07 xz 19.40.780 0.106

Fe6Rh 7 0 .53 2.231 0.094 3.209 0.091 0.046 7.93 z 25.21.002 0.105

Fe4Rh 9 0.31 2.077 0.103 3.355 0.065 0.055 14.95 xz 29.71.216 0.119

Fe3Rh 10 0.23 1.692 0.113 2.959 0.068 0.073 6.24 x 34.01.125 0.127

Fe2Rh 11 0.15 1.615 0.161 3.146 0.102 0.109 28.6 xz 39.81.166 0.171

Rh 13 0.00 1.461 0.336 1.304 0.257 15.6 xz 45.1

parameter and the magnitude of the orbital moment. The obtained values of magnetic

moments and MAE can be now related by using the Eq. 7.3.1. Rh, as a 4 d element, isheavier than Fe. We may expect that Rh can display larger orbital moments and spin-orbitenergy compared to Fe provided that it is magnetic. As we shall see the calculations showthat this is not always the case.

7.3.2 Composition effects on the magnetic properties

This section presents the magnetic properties as a function of the composition by increas-ing the Fe content level systematically. Computing the magnetic properties by takinginto account all possible homotops may be possible for a cluster having small number of atoms (for instance N ≤6). However this procedure is impracticable for the cluster sizes

considered here. Therefore, in order to reduce the computing time, we placed the atoms

117



7.3. Results and Discussion

0 0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

S p i n

m o m e n t

[ µ Β

/ a t o m ]

1319

0 0.2 0.4 0.6 0.8 1 0

0.1

0.2

0.3

O r b i t a l m o m e n t

[ µ Β

/ a t o m ]

Fe concentration (m)

(a) (b)

Figure 7.3: Spin and orbital moments in N = 13 and 19 FeRh clusters as a function of Fe

concentration.

0 0.2 0.4 0.6 0.8 1Fe concentration (m)

0

510

15

20

25

30

M A E ( m e V

/ c l u s t e r )1319

Figure 7.4: Magnetic anisotropy energy in FeRh clusters having N = 13 and 19 atoms asa function of Fe concentration.

118




Table 7.2: Electronic and magnetic properties of FeRh clusters having N = 19 atoms asin Table 7.1.

Cluster xF e µS µL µS F e µLF e

µL

µS MAE δ γ

µS Rh µLRh

Fe19 1.00 2.947 0.101 2.797 0.038 0.064 xz 6.3

Fe17 Rh 2 0.89 2.736 0.056 2.816 0.062 0.022 4.18 z 9.40.642 0.006

Fe15 Rh 4 0.78 2.485 0.066 2.931 0.071 0.027 0.251 z 12.60.814 0.052

Fe13 Rh 6 0.68 2.316 0.091 2.924 0.104 0.041 2.85 xz 15.70.713 0.062

Fe10 Rh 9 0.52 2.263 0.067 3.164 0.065 0.032 7.22 z 21.90.911 0.069

Fe9Rh 10 0.47 1.578 0.038 2.897 0.075 0.025 5.89 x 24.60.244 0.004

Fe6Rh 13 0.31 1.631 0.078 3.168 0.109 0.050 0.235 z 30.90.833 0.064

Fe4Rh 15 0.21 0.377 0.023 1.587 0.090 0.063 3.23 x 35.8

0.036 0.005Fe2Rh 17 0.10 0.772 0.023 2.908 0.121 0.031 24.89 z 37.9

0.494 0.012Rh 19 0.00 0.578 0.075 0.568 0.132 6.27 xz 42.6

of the same kind on the symmetric positions of the cluster, specially for low Fe or Rhconcentration (see Figs. 7.1 and 7.2).

For the 13 atom cluster, µS increases almost linearly with the Fe content level, whereasµL decreases. The decrease in the µL curve (0.336µB

→ 0.161µB ) is found for the lowest

Fe concentration (Rh 13 → Fe2Rh 11 ). In contrast in the 19 atom clusters, an oscillatorybehavior in the µS and µL are observed. The plot for µS in the Fig. 7.3 shows a diparound Fe 4Rh 15 composition. This reduction in the spin moment is partly due to theantiferromagnetic-like coupling among the Fe and Rh moments and also due to the smallaverage moment contribution from the Fe atoms. The Fe spin moments are distributedas µ1 = 2.962, µ2 = 2.972, µ3 = 0.075, µ4 = 0.341µB . The strong oscillations in the µLvalues for 19 atoms clusters shows that orbital moments are more sensitive to the localenvironment within the cluster.

We found that the µS contribution to the total magnetic moment is fairly insensitiveto the direction of magnetization, while there is a large anisotropy in the orbital momentsdepending on the magnetization direction. In the pure clusters, the easy axis is parallel to a

direction which is having largest orbital moment. While for some specic composition (for

119



7.3. Results and Discussion

instance, Fe 2Rh 17 ), it is observed that the cluster axis having the largest Lz are not easyaxis. A similar effect has been found in FeCo clusters having uni-axial symmetry [235].

Fig. 7.4 presents the variation of MAE as a function of composition. One can observethat the MAE displays a strong oscillation for both sizes and the maxima of MAE is forlow but non-zero Fe concentration. It eventually approaches very small values for the pureFe clusters. The quantitative analysis of Fig. 7.4 even reveals some remarkable results.A remarkable non-monotonous dependence of the MAE is observed as a function of Fecontent, i.e., upon going from pure Fe to pure Rh. This leads to an important increase of the MAE, which reaches about 300% at the optimal Fe concentration. This offers multiplepossibilities of tailoring the magneto-anisotropic behavior in nanoalloys [197].

7.3.3 Angular dependence of the MAE

-0.06-0.04-0.02

0

0

2

4

0

0.1

0.2

-2

-1

0

1

0246

-6-4-20

0

0.1

0.2

-3-2-10

0 30 60 9 0 120 150 1800

10

20

0 30 60 9 0 120 150 180-6-4-20 E

n e r g y d i f f e r e n c e ( m e V

/ c l u s t e r )

θ (degrees) (degrees)θ

Fe19 Fe17Rh2

Fe15Rh4

Fe13Rh6

Fe9Rh10

Fe10Rh9

Fe6Rh13

Fe4Rh15

Rh19Fe2Rh17

Figure 7.5: The energy difference ∆ E = E (θ)−E (0) in meV/cluster vs θ of FeRh 19 atomclusters. The magnetization direction θ varies in the zx plane in steps of 10 degrees.

120



8Summary and outlook

The principal motivation of this thesis was to enrich the fundamental understanding of the structural and magnetic properties exhibited by the TM nanoalloy clusters in view of applications in cluster-based magnetic nanometer devices. The results presented in thisthesis open the way to several near future investigations and developments in this eld.Specic conclusions concerning each chapter have already been mentioned at the end of the corresponding chapters. For this reason the present nal chapter is quite compact. Inthe following, I briey discuss how the thesis has been progressed.

Chapter 1 through 3 are devoted to present the essential background material for thecalculations performed in this thesis, while chapters 4 through 7 demonstrate the results.In the rst part, i.e., in chapters 4 and 5, we combined the state-of-the-art Hohenberg-Kohn Sham’s DFT with a global optimization technique based on a graph theory method.This method has been used to perform a thorough and systematic study on the interplaybetween cluster structure, magnetism and the chemical order in the Fe m Rh n and Co m Pd nnanoclusters having N = m + n ≤ 8 atoms. For N = m + n ≤ 6 a thorough samplingof all cluster topologies has been performed. We would like to emphasis that this kindof study is rather unique. For N = 7 and 8 only a few representative topologies wereconsidered including both open and compact structures. Choosing a small set of repre-sentative topologies for N = 7 and 8 is justied by the fact that the number of isomers(number of local minima on the PES) increases exponentially with the cluster size N .Indeed, the computational complexity increases even more rapidly in the case of binaryclusters, since one has to take into account all possible homotops. For all the clustersthe entire concentration range is systematically investigated, and the different initial mag-netic congurations such as ferro- and anti-ferromagnetic coupling are considered. Theresults in the case of FeRh clusters are the following: an increase of the average magneticmoment ( µN ) and magnetic stabilization energy (∆ E m ) with increasing Fe concentration,the presence of small differences in the average magnetic moment ( µN ) between low-lyingisomers, the dominant role of the d-electron spin polarization within the PAW spheres,

the enhancement of the Fe moments upon Rh doping, and a general tendency to max-

122



Chapter 8. Summary and outlook

imize the number of mixed bonds. We have adopted a similar approach in the case of Com Pd n clusters having N = m + n ≤ 8 atoms. The main results in this case are thefollowing: The optimized cluster structures have a tendency to maximize the number of nearest-neighbor CoCo pairs. An increase of µN and ∆ E m is observed with increasing Coconcentration. The magnetic order is ferromagnetic-like (FM) for all ground-state struc-tures. However, an antiferromagnetic-like (AF) order has been obtained in some of therst exited isomers. The maximal local spin polarization for Co and Pd atoms are foundin the equiatomic compositions (Co 2Pd 2 , Co3Pd 3 and Co 4Pd 4). We found that takinginto account spin-orbit (SO) interactions in FeRh and CoPd clusters does not alter theground-state structures found by using the scalar relativistic (SR) calculations. FeRh andCoPd clusters are expected to develop a variety of further interesting behaviors, whichstill remain to be explored. For instance, larger FeRh cluster should show a more complex

dependence of the magnetic order as a function of concentration. In particular for largeRh content one should observe a transition from FM-like to AF-like order with increas-ing cluster size, in agreement with the AF phase found in solids for more than 50% Rhconcentration. Moreover, the metamagnetic transition observed in bulk FeRh alloys alsoputs forward the possibility of similar interesting phenomena in nanoalloys as a functionof temperature.

In chapter 6 we have developed a DFT based spin-polarized basing-hopping algorithm.This methodological approach has been applied to study the structural and magnetic prop-erties of pure and alloy TM nanoclusters. This method is found to be very impressive.For instance, in the case of pure clusters, we obtained several Jahn-Teller distorted mag-netic isomers of the same basic structural motif which, being similar, would have been

most likely missed by using the graph or topographical scheme employed in the rst partof this thesis. A similar situation has also been encountered in the case of mixed clus-ters, where we identied several Jahn-Teller distorted magnetic isomers having the samecomposition and a similar distributions of the two kinds of atoms. We have discussedextensively the technicalities used for choosing the ideal move parameters for the opti-mizations. Moreover, we have implemented a window acceptance criterion . Moreover, inthe case of mixed clusters we swap or exchange the positions of the dissimilar atoms onthe y. We noticed that this method greatly enhances the performance of our calcula-tions by signicantly reducing the CPU time. For the small clusters (e.g. Fe 6 , Rh6 andFe3Rh 3) the main result is the presence of dominant (or frequently visited) isomers. Wefound that this is an intrinsic feature of the basin-hopping method. This is interpretedas a consequence of the reduced system size and the resulting small number of low-lyingisomers. The dominant isomers govern the overall computational demand of the samplingand are therefore the relevant isomers for the performance analysis. In the case of largerclusters (e.g. Fe 13 , Fe6Rh 7 and Rh 13 ) we applied a similar computational procedure asthe one used in the case of small clusters. In fact, both pure and mixed clusters displayremarkable structural and magnetic diversity. We found that isomers having similar shapebut with a small distortion among each other can exhibit often quite different magneticmoments. This has been interpreted as a probable artifact of the spin-rotational symmetrybreaking introduced by the spin-polarized LDA or GGA. In the case of Fe 6Rh 7 cluster, animplementation consisting of small move distances of about 0.15 a (where a is the lengthof the shortest bond) in combination with swapping of Fe and Rh atoms, could identifyall the relevant isomers including the ground-state. The ground-state structure of Fe 13

cluster is found to be an icosahedral structure, whereas Rh 13 and Fe 6Rh 7 isomers relax

123



into cage-like and layered-like structures, respectively. The possibility of combining thespin-polarized density-functional theory with some other global optimization techniquessuch as minima-hopping method could be the next step in this direction. This combina-tion is expected to be an ideal sampling approach by having the advantage of efficientlyavoiding search over irrelevant regions of the potential energy surface.

In chapter 7 we investigated the composition dependence of orbital magnetism andmagnetic anisotropy energy in FeRh nanoclusters. A remarkable non-monotonous depen-dence of the MAE is observed as a function of Fe content, i.e., upon going from pure Fe topure Rh. This leads to an important increase of the MAE, which reaches about 3 times thevalue for pure clusters at the optimal Fe concentration. This offers multiple possibilitiesof tailoring the magneto-anisotropic behavior in nanoalloys. In conclusion, it is our hopeto see the experimental verication of the reported results in this thesis.

124



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135



List of Figures

4.10 Relative stability of Fe m Rh n clusters as a function of composition. . . . . . 664.11 Bond lengths for FeFe (cross), FeRh (circle) and RhRh (square) pairs as a

function of number of Fe atoms in Fe m Rh n clusters having N = 5–8 atoms. 674.12 Electronic density of states (DOS) of FeRh octamers . . . . . . . . . . . . . 68

5.1 The average magnetic moment per atom of Co N clusters as a function of the cluster size N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2 Electronic density of states (DOS) of CoPd trimers . . . . . . . . . . . . . . 765.3 Lowest energy isomers of CoPd trimers and tetramers. The asteriks indicate

rst exited isomers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4 Electronic density of states (DOS) of CoPd tetramers. . . . . . . . . . . . 785.5 Constant magnetization density plots ρ↑(r ) −ρ↓(r ) = ξ and local moments

(in µB ) for the ground-state structures of CoPd hexameters. . . . . . . . . . 805.6 Lowest energy isomers of CoPd pentamers and hexamers. The asteriksindicate rst exited isomers. . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.7 Electronic density of states (DOS) of hexamers . . . . . . . . . . . . . . . . 835.8 Constant magnetization density plots ρ↑(r ) −ρ↓(r ) = ξ and local moments

(in µB ) for the ground-state structures of CoPd hexamers. The value of theconstant magnetization density ξ is given in µB / A3 . . . . . . . . . . . . . . 84

5.9 Lowest energy isomers of CoPd heptamers and octamers. Note that onlyvery few topologies have been considered as starting congurations of ge-ometry relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.10 Binding energy per atom E B of Com Pd n clusters as a function of the numberof Co atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.11 Magnetic stabilization energy of Co Pd m Pd n clusters as a function of num-ber of Co atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.12 Total magnetic moment per atom µN of Com Pd n clusters as a function of number of Co atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.13 The relative stability of Co m Pd n clusters as a function of the number of Coatoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.14 Bond lengths for CoCo (cross), CoPd (circle) and PdPd (square) pairs as afunction of number of Co atoms in Co m Pd n clusters . . . . . . . . . . . . . 91

6.3 Illustration of successful, unsuccessful and high-energy trial moves in theBH scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.4 Identied isomers of Fe 6 cluster in the energy range up to 1.74 eV abovethe ground-state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.5 Normalized probabilities for the lowest-energy isomers of Fe 3Rh 3 . . . . . . 996.6 The total energy of Fe 6 as a function of the number of BH steps n . . . . . 1006.7 The binding energy per atom (in eV), the probability that trial moves gen-

erate the lowest energy isomers, and the total magnetic moment are plottedas a function of isomer number for Fe 6 and Rh 6 cluster . . . . . . . . . . . . 101

6.8 Calculated isomers of Rh 6 cluster in the energy range up to 1.39 eV abovethe ground-state energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.9 Binding energy per atom (in eV), the probability with which trial movesgenerate the lowest energy isomers, and the total magnetic moment as a

function of isomer number for Fe 3Rh 3 . . . . . . . . . . . . . . . . . . . . . . 105

137



List of Figures

6.10 The optimal structures of Fe 3Rh 3 cluster. The entries below each structureshave the same meaning as in the Fig. 6.4 . . . . . . . . . . . . . . . . . . . 106

6.11 Obtained isomers of Fe 13 cluster in the energy range up to 3.13 eV abovethe ground-state energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.12 Obtained isomers of Rh 13 cluster in the energy range up to 0.43 eV abovethe ground-state energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.13 Low-lying isomers of Fe6Rh 7 in the energy range up to 0.61 eV above theground-state energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.1 The distribution of Fe and Rh atoms in the FeRh clusters having N = 13atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.2 The distribution of Fe and Rh atoms in the FeRh clusters having N = 19

atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.3 Spin and orbital moments in N = 13 and 19 FeRh clusters as a function of Fe concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.4 Magnetic anisotropy energy in FeRh clusters having N = 13 and 19 atomsas a function of Fe concentration. . . . . . . . . . . . . . . . . . . . . . . . 118

7.5 The energy difference ∆ E = E (θ) −E (0) in meV/cluster vs θ of FeRh 19atom clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

138



List of Tables

4.1 Structural, electronic and magnetic properties of FeRh dimers . . . . . . . . 474.2 Structural, electronic and magnetic properties of FeRh trimers . . . . . . . 494.3 Structural, electronic and magnetic properties of FeRh tetramers . . . . . . 514.4 Structural, electronic and magnetic properties of FeRh pentamers . . . . . . 524.5 Structural, electronic and magnetic properties of FeRh hexamers . . . . . . 554.6 Structural, electronic and magnetic properties of FeRh heptamers . . . . . . 584.7 Structural, electronic and magnetic properties of FeRh octamers . . . . . . 61

5.1 Structural, electronic and magnetic properties of CoPd dimers . . . . . . . 745.2 Structural, electronic and magnetic properties of CoPd trimers . . . . . . . 755.3 Structural, electronic and magnetic properties of CoPd tetramers . . . . . . 775.4 Structural, electronic and magnetic properties of CoPd pentamers . . . . . 795.5 Structural, electronic and magnetic properties of CoPd hexamers . . . . . . 825.6 Structural, electronic and magnetic properties of CoPd heptamers . . . . . 875.7 Structural, electronic and magnetic properties of CoPd octamers . . . . . . 88

7.1 Electronic and magnetic properties of FeRh clusters having N = 13 atoms . 1177.2 Electronic and magnetic properties of FeRh clusters having N = 19 atoms . 119

139



Abbreviations and symbols

µB Bohr Magneton

A Amstrong

APW Augmented plane waveBCC Body centered cubic

BCO Bicapped octahedra

BFGS Broyden Fletcher Goldfarb Shanno

BH Basin Hopping

BO Born Oppenheimer

CC Coupled Cluster

CG Conjugate gradient

CI Conguration Interaction

CO Capped octahedra

CSP Capped square pyramid

DFT Density functional theory

DOS Density of states

FCC Face centered cubic

GA Genetic Algorithm

GGA Generalized gradient approximation

GO Global optimization

HCP Hexagonal close packed

HF Hartree-Fock

HOMO Highest occupied molecular orbital

KS Kohn-Sham

140



LDA Local density approximation

LDOS Local density of states

LO Local optimization

LUMO Lowest unoccupied molecular orbital

MAE Magnetic anisotropy energy

MC Monte Carlo

MFE Mean rst encounter

MP Møller PlessetNm Number of BH moves

PAW Pro jector augmented wave

PBP Pentagonal bipyramid

QSE Quantum size effects

SA Simulated annealing

SD Steepest descent

SG Stern-GerlachSOC Spin orbit coupling

SP Slater Pauling

SP Square pyramid

TF Thomas and Fermi

TM Transition metal

XC exchange-correlation

XMCD X-ray magnetic circular dichroism

141



Ver¨offentlichungen - List of publications

• First-principles study of structural, magnetic and electronic properties of small Fe-Rh alloy clusters , Junais Habeeb Mokkath and G. M. Pastor, Phys. Rev. B 85,054407 (2012) and arXiv:1201.5971 Materials Science (cond-mat.mtrl-sci)

• First-principles study of magnetism, structure and chemical order in small FeRh alloy clusters , Junais Habeeb Mokkath and G. M. Pastor, arXiv:1110.2669 Atomicand Molecular Clusters (physics.atm-clus)

• Magnetism, structure, and chemical order in small CoPd alloy clusters , JunaisHabeeb Mokkath, and G. M. Pastor. (To be submitted to JPCC)

• First principles spin-polarized basin-hopping method , L. Diaz, Junais Habeeb Mokkath,and G. M. Pastor. (To be sumbitted to PRB)

In addition, results of this research were presented at the following conferences:

• 15th WIEN2k-WORKSHOP international conference (March 26-29, 2008), Vienna

dissertationjunaishabeebmokkath.pdf

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