Magneto-crystalline anisotropy of metallic nanostructures : tight-binding and first-principles studies Dongzhe Li To cite this version: Dongzhe Li. Magneto-crystalline anisotropy of metallic nanostructures : tight-binding and first- principles studies. Physics [physics]. Université Pierre et Marie Curie - Paris VI, 2015. English. NNT : 2015PA066232. tel-01243074v2 HAL Id: tel-01243074 https://tel.archives-ouvertes.fr/tel-01243074v2 Submitted on 24 Nov 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. 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UNIVERSITE PIERRE ET MARIE CURIE ECOLE DOCTORALE DE PHYSIQUE ET CHIMIE DES MATERIAUX THESE pr´esent´ee pour obtenir le grade de : Docteur de l'Universit´ePierre et Marie Curie Sp´ecialit´e: Physique par Dongzhe Li Magneto-crystalline anisotropy of metallic nanostructures : Tight-binding and first-principles studies Soutenue le 30 Septembre 2015 en pr´esence du jury compos´ede Mairbek Chshiev Rapporteur Lionel Calmels Rapporteur Fabio Finocchi Examinateur Andres Saul Examinateur Gustav Bihlmayer Examinateur Cyrille Barreteau Directeur de th`ese Alexander Smogunov Co-directeur Table des mati`eres Contents i Abbreviationsv Introduction 1 1 Methods 5 1.1 Spin Density Functional Theory ........................ 5 1.1.1 The many-body Hamiltonian ..................... 5 1.1.2 The Kohn-Sham equation ....................... 6 1.1.3 Non-collinear magnetism ........................ 7 1.1.4 Relativistic corrections and magnetic anisotropy . 11 1.1.5 Spin-orbit coupling for a spherically-symmetric field ......... 13 1.1.6 QUANTUM ESPRESSO package ................... 13 1.2 Magnetic Tight-Binding model ......................... 14 1.2.1 Tight-Binding Hamiltonian ...................... 15 1.2.2 Local charge neutrality ......................... 16 1.2.3 Stoner model .............................. 16 1.2.4 Spin-orbit coupling ........................... 18 2 Magnetic anisotropy 19 2.1 Shape anisotropy ................................ 20 2.2 Magneto-crystalline anisotropy ......................... 20 2.2.1 Self-consistent scheme ......................... 21 2.2.2 Force theorem .............................. 21 2.2.3 Perturbation treatment ......................... 22 2.2.4 Bruno formula .............................. 23 2.3 Force Theorem : practical implementations . 23 2.3.1 Tight-binding model .......................... 23 ii TABLE DES MATIERES` 2.3.2 DFT calculations ............................ 28 3 Magneto-crystalline anisotropy of Fe and Co free-standing slabs 33 3.1 Total MCA of Fe and Co slabs ......................... 34 3.1.1 Methodology and structures ...................... 34 3.1.2 Results and discussions ......................... 35 3.2 Surface and sub-surface contributions ..................... 37 3.3 Layer-resolved MCA .............................. 38 3.4 d-orbitals-resolved MCA ............................ 38 4 MCA of free Fe and Co nanocrystals 43 4.1 Self-assembled Fe and Co nanocrystals growth ................ 44 4.1.1 STM observations ............................ 44 4.1.2 Wulff construction ........................... 46 4.2 Tight-binding model .............................. 47 4.2.1 Geometry of nanocrystals ....................... 47 4.2.2 Total MCA of truncated pyramid of different sizes .......... 48 4.2.3 Local analysis of MCA ......................... 50 4.2.4 MCA of truncated bipyramid ..................... 52 4.3 First-principles calculations .......................... 54 4.3.1 TB .vs. DFT .............................. 54 4.3.2 Real-space distribution of MCA .................... 54 5 MCA of ferromagnetic slabs and clusters supported on SrTiO3 57 5.1 Fe(Co)jSrTiO3 interfaces ............................ 58 5.1.1 Atomic structures and computational details . 58 5.1.2 Magnetic spin moment ......................... 59 5.1.3 Electronic properties .......................... 60 5.1.4 Local analysis of MCA ......................... 62 5.2 Fe and Co clusters on SrTiO3 ......................... 65 5.2.1 Atomic structures and computational details . 65 5.2.2 Magnetic spin moment ......................... 66 5.2.3 Electronic structure properties ..................... 67 5.2.4 Local analysis of MCA ......................... 69 Conclusion 73 A The 1D quantum well 75 B Length to height ratio of nanocrystals 79 C Shape anisotropy 81 C.1 Fe and Co free-standing slabs ......................... 81 C.2 Free Fe and Co nanocrystals .......................... 82 TABLE DES MATIERES` iii Publications 85 Bibliography 95 Abbreviations AMR anisotropic magnetoresistance bcc body centered cubic DFT density functional theory fcc face centered cubic FR-PPs fully-relativistic pseudopotentials FT force theorem FTgc grand-canonical force theorem GGA generalized gradient approximation hcp hexagonal close packed K-S kohn-sham LCN local charge neutrality LSDA local spin density approxiamtion MAE magnetic anisotropy energy MCA magneto-crystalline anisotropy MOKE magneto-optic kerr effect PDOS projected density of states PPs pseudopotentials QE quantum espresso QWS quantum well states SCF self-consistent field SOC spin-orbit coupling SP-STS spin-polarized scanning tunneling spectroscopy SR-PPs scalar-relativistic pseudopotentials TB tight-binding XMCD x-ray magnetic circular dichroism Introduction Although magnetism is a relatively old topic in condensed matter physics, studies of nanomagnetism have attracted large attention for its potentiial applications such as the use of magnetic units down to nanoscale for high density magnetic recording [1, 2] or nonvolatile magnetoresistive random access memory (MRAM). Magnetic anisotropy is a general phenomenon that is characterized by the change of physical properties of a ma- gnetic system with respect to the orientation of the magnetization. In particular, magnetic anisotropy energy (MAE), which is defined as the change of total energy associated to a change of the orientation of the spin moment (Fig. 1, left) is an extremely important quan- tity that is crucial for the stability of nano-scale magnetic grains. Such small nanoclusters are of great importance in view of future miniaturization of data storage devices since they have well-defined structure as well as an ability to assemble into well-ordered arrays on the substrate. However, the magnetic stability of a nanocrystal decreases proportionally to its size, therefore, one of the most challenging problems towards ultimate magnetic density storage is evidently to be able to synthesize well-ordered arrays of magnetic nanocrystals with as large magnetization and MAE as possible. That would prevent the magnetization flips due to thermal (or any other) fluctuations and increase the blocking temperature as it is sketched on Fig. 1 (left panel). There are two physical sources for MAE, both of them are due to relativistic effects : the shape anisotropy and the magneto-crystalline anisotropy (MCA). The shape anisotropy is mediated by magnetic dipole-dipole interactions and originates from the quantum-field relativistic corrections to electron-electron interaction [4]. This long range interaction de- pends on the shape of the crystal and basically favors magnetization along the most elon- gated direction of the crystal. The physical origin of MCA is the spin-orbit coupling (SOC) which can be derived theoretically from the Dirac equation [5]. Unlike shape anisotropy, the MCA is a short range effect localized around atomic cores. SOC stands at the origin of many other intriguing features in low-dimentinal magnetic materials, such as rashba ef- fect, magneto-optic Kerr effect (MOKE), tunnelling/balistic anisotropic magnetoresistance (AMR) [6{8], the chiral magneric order [9] and quantum anomolous Hall effect (QAHE) [10], to list a few. Understanding the role of the SOC in these phenomena is crucial for 2 INTRODUCTION Figure 1: Left : N´eel relaxation time τN { a mean time between two flips of the magnetic moment { with respect to the magnetic anisotropy energy MAE (over temperature), where kB is the Boltzmann constant, T is the temperature and τ0 is the attempt time (typical values of which are between 10−9 and 10−10 seconds). Right : 3D represen- tation of scanning tunneling microscopy (STM) topography of truncated pyramid- shaped Fe nanocrystals of nanometer-size grown on SrTiO3(001) substrate [3]. both fundamental interests and practical applications. A large MCA is expected in the systems with large spin moments and SOC. Large spin moments are usually found in magnetic 3d metals while the SOC is rather weak there. Heavy 4d or 5d materials, on the contrary, have larger SOC but are normally nonmagnetic. Therefore, in order to get both large magnetization and MCA, one is particularly inter- ested in the nanostructured \cheap" 3d transition metal systems or in bimetallic systems consisting of 3d elements deposited on 5d heavy element substrate. In 3d transition-metal bulk, the value of MCA per atom is extremely small (some µeV/atom) in cubic systems such as Fe, Cr and Ni due to the high symmetry reason while it is larger (∼ 65 µeV/atom) in noncubic systems such as hexagonal close packed (hcp)