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Springer Proceedings in Physics 221

Olena FesenkoLeonid Yatsenko Editors

Nanocomposites, Nanostructures, and Their Applications Selected Proceedings of the 6th International Conference Nanotechnology and Nanomaterials (NANO2018), August 27-30, 2018, Kyiv, Ukraine

Springer Proceedings in Physics

Volume 221

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Olena Fesenko • Leonid YatsenkoEditors

Nanocomposites,Nanostructures,and Their Applications

Selected Proceedings of the 6th InternationalConference Nanotechnologyand Nanomaterials (NANO2018),August 27-30, 2018, Kyiv, Ukraine

123

EditorsOlena FesenkoNational Academy of Sciences of UkraineInstitute of PhysicsKyiv, Ukraine

Leonid YatsenkoNational Academy of Sciences of UkraineInstitute of PhysicsKyiv, Ukraine

ISSN 0930-8989 ISSN 1867-4941 (electronic)Springer Proceedings in PhysicsISBN 978-3-030-17758-4 ISBN 978-3-030-17759-1 (eBook)https://doi.org/10.1007/978-3-030-17759-1

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Preface

This book highlights the most recent advances in nanoscience from leadingresearchers in Ukraine, Europe, and beyond. It features contributions from theparticipants of the sixth International Research and Practice Conference “Nan-otechnology and Nanomaterials” (NANO-2018), held in Kyiv, Ukraine, on August27–30, 2018. This event was organized jointly by the Institute of Physics ofthe National Academy of Sciences of Ukraine, Taras Shevchenko National Uni-versity of Kyiv (Ukraine), University of Tartu (Estonia), University of Turin(Italy), and Pierre and Marie Curie University (France). Internationally recognizedexperts from a wide range of universities and research institutes shared theirknowledge and key results in the areas of nanocomposites and nanomaterials,nanostructured surfaces, microscopy of nano-objects, nanooptics and nanopho-tonics, nanoplasmonics, nanochemistry, nanobiotechnology, and surface-enhancedspectroscopy.

Today, nanotechnology is becoming one of the most actively developing andpromising fields of science. Numerous nanotechnology investigations are alreadyproducing practical results that can be applied in various areas of human life fromscience and technology to medicine and pharmacology. The aim of these booksis to highlight the latest investigations from different areas of nanoscience and tostimulate new interest in this field. Volume I of this two-volume work covers suchimportant topics as nanostructured interfaces and surfaces, and nanoplasmonics.

This book is divided into two sections: Part I, Nanocomposites and Nanostruc-tures, and Part II, Applications. Sections covering Nanophotonics and Nanooptics,Nanobiotechnology, and Applications can be found in Volume II: Nanophotonics,Nanooptics, Nanobiotechnology, and Their Applications.

The papers published in these five sections fall under the broad categories ofnanomaterial preparation and characterization, nanobiotechnology, nanodevices and

v

vi Preface

quantum structures, and spectroscopy and nanooptics. We hope that both volumeswill be equally useful and interesting for young scientists or PhD students andmature scientists alike.

Kyiv, Ukraine Olena FesenkoKyiv, Ukraine Leonid Yatsenko

Contents

Part I Nanocomposites and Nanostructures

1 A Microscopic Description of Spin Dynamics in MagneticMultilayer Nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3A. M. Korostil and M. M. Krupa

2 Development of the Nano-mineral Phases at the Steel-BentoniteInterface in Time of the Evolution of Geological Repository forRadioactive Waste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29B. H. Shabalin, O. M. Lavrynenko, and O. Yu. Pavlenko

3 Development of a Controlled in Situ Process for the Formationof Porous Anodic Alumina and Al Nanomesh From ThinAluminum Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45T. Lebyedyeva, M. Skoryk, and P. Shpylovyy

4 Electrooxidation of Ethanol on Nickel-Copper MultilayerMetal Hydroxide Electrode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Antonina A. Maizelis

5 Metal Surface Engineering Based on Formation of NanoscaledPhase Protective Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69V. M. Ledovskykh, Yu. P. Vyshnevska, I. V. Brazhnyk,and S. V. Levchenko

6 Electrical Conductivity and 7Li NMR Spin-Lattice Relaxationin Amorphous, Nano- and Microcrystalline Li2O-7GeO2 . . . . . . . . . . . . . 85O. Nesterov, M. Trubitsyn, O. Petrov, M. Vogel, M. Volnianskii,M. Koptiev, S. Nedilko, and Ya. Rybak

7 Influence of Surface Ultrafine Grain Structure on CavitationErosion Damage Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Yaroslav Kyryliv, V. Kyryliv, and Nataliya Sas

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viii Contents

8 The Effect of Mechanochemical and Ultrasonic Treatmentson the Properties of Composition CeO2 –MoO3 = 1:1 . . . . . . . . . . . . . . . . . 109V. A. Zazhigalov, O. A. Diyuk, O. V. Sachuk, N. V. Diyuk,V. L. Starchevsky, Z. Sawlowicz, I. V. Bacherikova,and S. M. Shcherbakov

9 Behavior of Tempered Surface Nanocrystalline StructuresObtained by Mechanical-Pulse Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125±. Maksymiv, V. Kyryliv, O. Zvirko, and H. Nykyforchyn

10 Nano-sized Adsorbate Island Formation in AdsorptiveAnisotropic Multilayer Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Vasyl O. Kharchenko, Alina V. Dvornichenko,and Dmitrii O. Kharchenko

11 The Effect of Ultrasonic Treatment on the Physical–ChemicalProperties of the ZnO/MoO3 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153V. O. Zazhigalov, O. V. Sachuk, O. A. Diyuk, N. S. Kopachevska,V. L. Starchevskyy, and M. M. Kurmach

12 Hybrid Nanocomposites Synthesized into Stimuli ResponsiblePolymer Matrices: Synthesis and Application Prospects. . . . . . . . . . . . . . . 167Nataliya Kutsevol, Iuliia Harahuts, Oksana Nadtoka,Antonina Naumenko, and Oleg Yeshchenko

13 Preparation and Complex Study of Thick FilmsBased on Nanostructured Cu0.1Ni0.8Co0.2Mn1.9O4and Cu0.8Ni0.1Co0.2Mn1.9O4 Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187H. Klym, Yu Kostiv, and I. Hadzaman

14 Nanoscale Investigation of Porous Structurein Adsorption-Desorption Cycles in the MgO-Al2O3Ceramics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199H. Klym, A. Ingram, R. Szatanik, and I. Hadzaman

15 Structure, Morphology, and Spectroscopy Studiesof La1−xRExVO4 Nanoparticles Synthesized by Various Methods . . . 211O. V. Chukova, S. Nedilko, S. G. Nedilko, A. A. Slepets,T. A. Voitenko, M. Androulidaki, A. Papadopoulos,and E. I. Stratakis

16 Investigation of the Conditions of Synthesis of Alumo-NickelSpinel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243L. Frolova and T. Butyrina

17 IV–VIB Group Metal Boride and Carbide NanopowderCorrosion Resistance in Nickeling Electrolytes . . . . . . . . . . . . . . . . . . . . . . . . . 251Oleksandr Goretskiy, Dmytro Shakhnin, Viktor Malyshev,and Tetiana Lukashenko

Contents ix

18 Hydrodynamic and Thermodynamic Conditions for Obtaininga Nanoporous Structure of Ammonium Nitrate Granulesin Vortex Granulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257A. V. Ivaniia, A. Y. Artyukhov, and A. I. Olkhovyk

19 Nanostructured Mixed Oxide Coatings on SiluminIncorporated by Cobalt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269Ann. V. Karakurkchi, Nikolay D. Sakhnenko, Maryna V. Ved’,and Maryna V. Mayba

20 Effect of Carbon Nanofillers on Processes of StructuralRelaxation in the Polymer Matrixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293T. G. Avramenko, N. V. Khutoryanskaya, S. M. Naumenko,K. O. Ivanenko, S. Hamamda, and S. L. Revo

21 Simulation of Tunneling Conductivity and ControlledPercolation In 3D Nanotube-Insulator Composite System . . . . . . . . . . . . . 307I. Karbovnyk, Yu. Olenych, D. Chalyy, D. Lukashevych, H. Klym,and A. Stelmashchuk

22 Radiation-Stimulated Formation of Polyene Structuresin Polyethylene Nanocomposites with Multi-walled CarbonNanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323M. A. Alieksandrov, T. M. Pinchuk-Rugal, O. P. Dmytrenko,M. P. Kulish, V. V. Shlapatska, and V. M. Tkach

23 Theoretical Analysis of Metal Salt Crystallization Processon the Thermoexfoliated and Disperse Graphite Surface . . . . . . . . . . . . . . 333Luidmila Yu. Matsui, Luidmila L. Vovchenko, Iryna V. Ovsiienko,Tatiana L. Tsaregradskaya, and Galina V. Saenko

24 Modeling of Dielectric Permittivity of Polymer Compositeswith Mixed Fillers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349Luidmila L. Vovchenko, Oleg V. Lozitsky,Luidmila Yu. Matsui, Olena S. Yakovenko, Viktor V. Oliynyk,and Volodymyr V. Zagorodnii

25 Nanostructural Effects in Iron Oxide Silicate Materialsof the Earth’s Crust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367A. V. Panko, I. G. Kovzun, O. M. Nikipelova, V. A. Prokopenko,O. A. Tsyganovich, and V. O. Oliinyk

26 Two-Dimensional Ordered Crystal Structure Formed by ChainMolecules in the Pores of Solid Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387A. N. Alekseev, S. A. Alekseev, Y. F. Zabashta, K. I. Hnatiuk,R. V. Dinzhos, M. M. Lazarenko, Y. E. Grabovskii,and L. A. Bulavin

x Contents

27 Joint Electroreduction of Carbonate and Tungstate Ions asthe Base for Tungsten Carbide Nanopowders Synthesis in IonicMelts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397Oleksandr Yasko, Viktor Malyshev, Angelina Gab,and Tetiana Lukashenko

28 The Kinetics Peculiarities and the Electrolysis Regime Effecton the Morphology and Phase Composition of Fe-Co-W(Mo)Coatings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403Iryna Yu. Yermolenko, Maryna V. Ved’, and Nikolay D. Sakhnenko

29 Dispersing of Molybdenum Nanofilms at Non-metallicMaterials as a Result of Their Annealing in Vacuum. . . . . . . . . . . . . . . . . . . 425I. I. Gab, T. V. Stetsyuk, B. D. Kostyuk, O. M. Fesenko,and D. B. Shakhnin

Part II Applications

30 Effective Hamiltonians for Magnetic Ordering Within PeriodicAnderson-Hubbard Model for Quantum Dot Array . . . . . . . . . . . . . . . . . . . 441L. Didukh, Yu. Skorenkyy, O. Kramar, and Yu. Dovhopyaty

31 PET Ion-Track Membranes: Formation Features and BasicApplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461Artem Kozlovskiy, Daryn Borgekov, Inesh Kenzhina,Maxim Zdorovets, Ilya Korolkov, Egor Kaniukov, Maksim Kutuzau,and Alena Shumskaya

32 Impact of Carbon Nanotubes on HDL-Like Structures:Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481Mateusz Pabiszczak, Krzysztof Górny, Przemysław Raczyński,and Zygmunt Gburski

33 Approximation of a Simple Rectangular Lattice fora Conduction Electron in Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489L. V. Shmeleva and A. D. Suprun

34 Simulation of the Formation of a Surface Nano-Crater Underthe Action of High-Power Pulsed Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505L. V. Shmeleva, A. D. Suprun, and S. M. Yezhov

35 Ballistic Transmission of the Dirac Quasielectrons Throughthe Barrier in the 3D TÑpological Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517A. M. Korol, N. V. Medvid’, A. I. Sokolenko, and I. V. Sokolenko

36 The Perspective Synthesis Methods and Researchof Nickel Ferrites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527Iryna Ivanenko, Serhii Lesik, Ihor Astrelin, and Yurii Fedenko

Contents xi

37 Electron Irradiation of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547H. Yu. Mykhailova and M. M. Nischenko

38 Influence of Irradiation with Deuterium Ions on the MagneticProperties and Structure of Nickel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553Oleksandr Mats, Nikolay Chernyak, Oleksandr Morozov,and Volodymyr Zhurba

39 Formation of VI-B Group Metal Silicides from Molten Salts . . . . . . . . . 561Konstantin Rozhalovets, Dmytro Shakhnin, Viktor Malyshev,and Julius Schuster

40 The Structure of Reinforced Layers of the Complex Method . . . . . . . . . 569Andrew E. Stetsko, Yaroslav B. Stetsiv, and Yaryna T. Stetsko

41 Technology and the Main Technological Equipmentof the Process to Obtain N4HNO3 with Nanoporous Structure . . . . . . . 585A. E. Artyukhov and N. O. Artyukhova

42 Study of Structural Changes in a Nickel Oxide ContainingAnode Material During Reduction and Oxidation at 600 ◦C . . . . . . . . . . 595Ye. V. Kharchenko, Z. Ya. Blikharskyy, V. V. Vira, and B. D. Vasyliv

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

Contributors

A. N. Alekseev Taras Shevchenko National University of Kyiv, Physical Faculty,Kyiv, Ukraine

S. A. Alekseev Taras Shevchenko National University of Kyiv, Physical Faculty,Kyiv, Ukraine

¯. A. Alieksandrov Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

M. Androulidaki Institute of Electronic Structure & Laser (IESL) of Foundationfor Research & Technology Hellas (FORTH), Heraklion, Greece

A. E. Artyukhov Processes and Equipment of Chemical and Petroleum-RefineriesDepartment, Sumy State University, Sumy, Ukraine

A. Y. Artyukhov Processes and Equipment of Chemical and Petroleum-RefineriesDepartment, Sumy State University, Sumy, Ukraine

N. O. Artyukhova Processes and Equipment of Chemical and Petroleum-Refineries Department, Sumy State University, Sumy, Ukraine

Ihor Astrelin Department of Inorganic Substances Technology, Water Treatmentand General Chemical Engineering of National Technical University of Ukraine“Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

T. G. Avramenko Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

I. V. Bacherikova Institute for Sorption and Problems of Endoecology, NationalAcademy of Sciences of Ukraine, Kyiv, Ukraine

Z. Ya. Blikharskyy Institute of Building and Environmental Engineering, LvivPolytechnic National University, Lviv, Ukraine

Daryn Borgekov Astana branch of the Institute of Nuclear Physics, Nur-Sultan,Kazakhstan

L. N. Gumilyov Eurasian National University, Astana, Kazakhstan

xiii

xiv Contributors

I. V. Brazhnyk Gimasi SA Ukraine R&D Centre, Via Luigi Lavizzari, Mendrisio,Switzerland

L. A. Bulavin Taras Shevchenko National University of Kyiv, Physical Faculty,Kyiv, Ukraine

T. Butyrina Ukrainian State University of Chemical Technology, Dnipro, Ukraine

D. Chalyy Lviv State University of Life Safety, Lviv, Ukraine

Nikolay Chernyak National Science Center “Kharkiv Institute of Physics andTechnology”, Kharkiv, Ukraine

O. V. Chukova Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

L. Didukh Ternopil Ivan Puluj National Technical University, Ternopil, Ukraine

R. V. Dinzhos Mykolaiv V.O. Sukhomlynskyi National University, Mykolayiv,Ukraine

N. V. Diyuk Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

O. A. Diyuk Institute for Sorption and Problems of Endoecology, NationalAcademy of Sciences of Ukraine, Kyiv, Ukraine

±. P. Dmytrenko Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

Yu. Dovhopyaty Ternopil Ivan Puluj National Technical University, Ternopil,Ukraine

Alina V. Dvornichenko Sumy State University, Sumy, Ukraine

Yurii Fedenko Department of Inorganic Substances Technology, Water Treatmentand General Chemical Engineering of National Technical University of Ukraine“Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

O. M. Fesenko Institute of Physics of National Academy Sciences of Ukraine,Kyiv, Ukraine

L. Frolova Ukrainian State University of Chemical Technology, Dnipro, Ukraine

Angelina Gab Institute for Engineering & Technology, University “Ukraine”, Kyiv,Ukraine

I. I. Gab Frantsevich Institute for Problems of Materials Science of NationalAcademy Sciences of Ukraine, Kyiv, Ukraine

Zygmunt Gburski Institute of Physics, University of Silesia, Katowice, Poland

Silesian Centre of Education & Interdisciplinary Research, Chorzów, Poland

Oleksandr Goretskiy Institute for Engineering & Technology, University“Ukraine”, Kyiv, Ukraine

Contributors xv

Krzysztof Górny Institute of Physics, University of Silesia, Katowice, Poland


Y. E. Grabovskii Taras Shevchenko National University of Kyiv, Physical Depart-ment, Kyiv, Ukraine

I. Hadzaman Drohobych State Pedagogical University, Drohobych, Ukraine

S. Hamamda Laboratory of Thermodynamics and Surface Treatment of Materials,University of Frères Mentouri Constantine 1, Constantine, Algeria

Iuliia Harahuts Faculty of Chemistry, Taras Shevchenko National University ofKyiv, Kyiv, Ukraine

K. I. Hnatiuk Taras Shevchenko National University of Kyiv, Physical Faculty,Kyiv, Ukraine

A. Ingram Opole University of Technology, Opole, Poland

Iryna Ivanenko Department of Inorganic Substances Technology, Water Treatmentand General Chemical Engineering of National Technical University of Ukraine“Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

K. O. Ivanenko Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

A. V. Ivaniia Processes and Equipment of Chemical and Petroleum-RefineriesDepartment, Sumy State University, Sumy, Ukraine

Egor Kaniukov Institute of Chemistry of New Materials of the NAS of Belarus,Minsk, Belarus

Ann. V. Karakurkchi National Technical University “Kharkiv Polytechnic Insti-tute”, Kharkiv, Ukraine

I. Karbovnyk Ivan Franko National University of Lviv, Lviv, Ukraine

Inesh Kenzhina Astana branch of the Institute of Nuclear Physics, Nur-Sultan,Kazakhstan


Dmitrii O. Kharchenko Institute of Applied Physics, National Academy of Sci-ences of Ukraine, Sumy, Ukraine

Vasyl O. Kharchenko Institute of Applied Physics, National Academy of Sciencesof Ukraine, Sumy, Ukraine

Ye. V. Kharchenko Institute of Building and Environmental Engineering, LvivPolytechnic National University, Lviv, Ukraine

N. V. Khutoryanskaya Taras Shevchenko National University of Kyiv, Kyiv,Ukraine

H. Klym Lviv Polytechnic National University, Lviv, Ukraine

xvi Contributors

N. S. Kopachevska Institute for Sorption and Problems of Endoecology, NationalAcademy of Sciences of Ukraine, Kyiv, Ukraine

M. Koptiev Solid State Physics and Optoelectronics Department, Oles HoncharDnipro National University, Dnipro, Ukraine

A. M. Korol Laboratory on Quantum Theory in Linkoping, ISIR, Linkoping,Sweden

Ilya Korolkov Astana branch of the Institute of Nuclear Physics, Nur-Sultan,Kazakhstan

A. M. Korostil Institute of Magnetism of NAS of Ukraine and MES of Ukraine,Kyiv, Ukraine

Yu Kostiv Lviv Polytechnic National University, Lviv, Ukraine

B. D. Kostyuk Frantsevich Institute for Problems of Materials Science of NationalAcademy Sciences of Ukraine, Kyiv, Ukraine

I. G. Kovzun F.D.Ovcharenko Institute of Biocolloid Chemistry of NAS ofUkraine, Kyiv, Ukraine

Artem Kozlovskiy Astana branch of the Institute of Nuclear Physics, Nur-Sultan,Kazakhstan


O. Kramar Ternopil Ivan Puluj National Technical University, Ternopil, Ukraine

M. M. Krupa Institute of Magnetism of NAS of Ukraine and MES of Ukraine,Kyiv, Ukraine

¯. P. Kulish Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

M. M. Kurmach L. V. Pisarzhevskii Institute of Physical Chemistry, NationalAcademy of Sciences of Ukraine, Kyiv, Ukraine

Nataliya Kutsevol Faculty of Chemistry, Taras Shevchenko National University ofKyiv, Kyiv, Ukraine

Maksim Kutuzau Scientific-Practical Materials Research Centre, NAS of Belarus,Minsk, Belarus

V. Kyryliv Karpenko Physico-Mechanical Institute of the NAS of Ukraine, Lviv,Ukraine

Yaroslav Kyryliv Lviv State University of Life Safety, Lviv, Ukraine

O. M. Lavrynenko State Institution “Institute of Environmental Geochemistry ofNAS of Ukraine”, Kyiv, Ukraine

I.M. Frantsevych Institute of Material Science Problems of NAS of Ukraine, Kyiv,Ukraine

Contributors xvii

M. M. Lazarenko Taras Shevchenko National University of Kyiv, Physical Faculty,Kyiv, Ukraine

T. Lebyedyeva V.M. Glushkov Institute of Cybernetics, National Academy ofSciences of Ukraine, Kyiv, Ukraine

V. M. Ledovskykh National Aviation University, Kyiv, Ukraine

Serhii Lesik Department of Inorganic Substances Technology, Water Treatment andGeneral Chemical Engineering of National Technical University of Ukraine “IgorSikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

S. V. Levchenko National Aviation University, Kyiv, Ukraine

Oleg V. Lozitsky Department of Physics, Taras Shevchenko National University ofKyiv, Kyiv, Ukraine

Tetiana Lukashenko Institute for Engineering & Technology, University“Ukraine”, Kyiv, Ukraine

D. Lukashevych Lviv Polytechnic National University, Lviv, Ukraine

Antonina A. Maizelis National Technical University “Kharkiv Polytechnic Insti-tute”, Kharkiv, Ukraine

±. Maksymiv Karpenko Physico-Mechanical Institute of NAS of Ukraine, Lviv,Ukraine

Viktor Malyshev Institute for Engineering & Technology, University “Ukraine”,Kyiv, Ukraine

Oleksandr Mats National Science Center “Kharkiv Institute of Physics and Tech-nology”, Kharkiv, Ukraine

Luidmila Yu. Matsui Departments of Physics, Taras Shevchenko National Univer-sity of Kyiv, Kyiv, Ukraine

Maryna V. Mayba National Technical University “Kharkiv Polytechnic Institute”,Kharkiv, Ukraine

N. V. Medvid’ National University for Food Technologies, Kyiv, Ukraine

Oleksandr Morozov National Science Center “Kharkiv Institute of Physics andTechnology”, Kharkiv, Ukraine

H. Yu. Mykhailova G. V. Kurdyumov Institute for Metal Physics of the NAS ofUkraine, Kiev, Ukraine

Oksana Nadtoka Faculty of Chemistry, Taras Shevchenko National University ofKyiv, Kyiv, Ukraine

Antonina Naumenko Faculty of Physics, Taras Shevchenko National University ofKyiv, Kyiv, Ukraine

xviii Contributors

S. M. Naumenko Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

S. Nedilko Physics Faculty, Taras Shevchenko National University of Kyiv, Kyiv,Ukraine

S. G. Nedilko Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

O. Nesterov Solid State Physics and Optoelectronics Department, Oles HoncharDnipro National University, Dnipro, Ukraine

O. M. Nikipelova SA “Ukrainian Research Institute of Medical Rehabilitation andBalneology, Ministry of Health of Ukraine”, Odessa, Ukraine

M. M. Nischenko G. V. Kurdyumov Institute for Metal Physics of the NAS ofUkraine, Kiev, Ukraine

H. Nykyforchyn Karpenko Physico-Mechanical Institute of NAS of Ukraine, Lviv,Ukraine

Yu. Olenych Ivan Franko National University of Lviv, Lviv, Ukraine

V. O. Oliinyk F.D. Ovcharenko Institute of Biocolloid Chemistry of NAS ofUkraine, Kyiv, Ukraine

Viktor V. Oliynyk Department of Radiophysics, Electronics, and Computer Sys-tems, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

A. I. Olkhovyk Processes and Equipment of Chemical and Petroleum-RefineriesDepartment, Sumy State University, Sumy, Ukraine

Iryna V. Ovsiienko Departments of Physics, Taras Shevchenko National Universityof Kyiv, Kyiv, Ukraine

Mateusz Pabiszczak Institute of Physics, University of Silesia, Katowice, Poland


A. V. Panko F.D. Ovcharenko Institute of Biocolloid Chemistry of NAS of Ukraine,Kyiv, Ukraine

A. Papadopoulos Institute of Electronic Structure & Laser (IESL) of Foundationfor Research & Technology Hellas (FORTH), Heraklion, Greece

O. Yu. Pavlenko I.M. Frantsevych Institute of Material Science Problems of NASof Ukraine, Kyiv, Ukraine

O. Petrov Institute of Solid State Physics, Darmstadt Technical University, Darm-stadt, Germany

μ. ¯. Pinchuk-Rugal Taras Shevchenko National University of Kyiv, Kyiv,Ukraine

Contributors xix

V. A. Prokopenko F.D. Ovcharenko Institute of Biocolloid Chemistry of NAS ofUkraine, Kyiv, Ukraine

National Technical University of Ukraine “KPI”, Kyiv, Ukraine

Przemysław Raczyński Institute of Physics, University of Silesia, Katowice,Poland


S. L. Revo Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

Konstantin Rozhalovets Institute for Engineering & Technology, University“Ukraine”, Kyiv, Ukraine

Ya. Rybak Physics Faculty, Taras Shevchenko National University of Kyiv, Kyiv,Ukraine

O. V. Sachuk Institute for Sorption and Problems of Endoecology, NationalAcademy of Sciences of Ukraine, Kyiv, Ukraine

Galina V. Saenko Departments of Physics, Taras Shevchenko National Universityof Kyiv, Kyiv, Ukraine

Nikolay D. Sakhnenko National Technical University “Kharkiv Polytechnic Insti-tute”, Kharkiv, Ukraine

Nataliya Sas Stepan Gzhytskyi National University of Veterinary Medicine andBiotechnologies, Lviv, Ukraine

Z. Sawlowicz Institute of Geology, Jagiellonian University, Krakow, Poland

Julius Schuster Faculty of Chemistry, University of Vienna, Vienna, Austria

B. H. Shabalin State Institution “Institute of Environmental Geochemistry of NASof Ukraine”, Kyiv, Ukraine

D. B. Shakhnin University “Ukraine”, Kyiv, Ukraine

Dmytro Shakhnin Institute for Engineering & Technology, University “Ukraine”,Kyiv, Ukraine

S. M. Shcherbakov M.G. Kholodny Institute of Botany of the National Academyof Science of Ukraine, Kyiv, Ukraine

V. V. Shlapatska L.V. Pisarghevskiy Institute of Physical Chemistry NAS ofUkraine, Kyiv, Ukraine

L. V. Shmeleva Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

P. Shpylovyy V.M. Glushkov Institute of Cybernetics, National Academy of Sci-ences of Ukraine, Kyiv, Ukraine

Alena Shumskaya Scientific-Practical Materials Research Centre, NAS of Belarus,Minsk, Belarus

xx Contributors

Yu. Skorenkyy Ternopil Ivan Puluj National Technical University, Ternopil,Ukraine

M. Skoryk NanoMedTech LLC, Kyiv, Ukraine

A. A. Slepets Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

A. I. Sokolenko National University for Food Technologies, Kyiv, Ukraine

I. V. Sokolenko National University for Food Technologies, Kyiv, Ukraine

V. L. Starchevskyy National University «Lviv Polytechnic», Lviv, Ukraine

A. Stelmashchuk Ivan Franko National University of Lviv, Lviv, Ukraine

Yaroslav B. Stetsiv Ukrainian Academy of Printing, Lviv, Ukraine

Andrew E. Stetsko Ukrainian Academy of Printing, Lviv, Ukraine

Yaryna T. Stetsko Ivan Franko National University of Lviv, Lviv, Ukraine

T. V. Stetsyuk Frantsevich Institute for Problems of Materials Science of NationalAcademy Sciences of Ukraine, Kyiv, Ukraine

E. I. Stratakis Institute of Electronic Structure & Laser (IESL) of Foundation forResearch & Technology Hellas (FORTH), Heraklion, Greece

A. D. Suprun Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

R. Szatanik Opole University of Technology, Opole, Poland

V. M. Tkach V. Bakul Institute for Superhard Materials NAS of Ukraine, Kyiv,Ukraine

M. Trubitsyn Solid State Physics and Optoelectronics Department, Oles HoncharDnipro National University, Dnipro, Ukraine

Tatiana L. Tsaregradskaya Departments of Physics, Taras Shevchenko NationalUniversity of Kyiv, Kyiv, Ukraine

±. £. Tsyganovich F.D. Ovcharenko Institute of Biocolloid Chemistry of NAS ofUkraine, Kyiv, Ukraine

National technical University of Ukraine “KPI”, Kyiv, Ukraine

B. D. Vasyliv Karpenko Physico-Mechanical Institute of the NAS of Ukraine, Lviv,Ukraine

Maryna V. Ved’ National Technical University “Kharkiv Polytechnic Institute”,Kharkiv, Ukraine

V. V. Vira Institute of Building and Environmental Engineering, Lviv PolytechnicNational University, Lviv, Ukraine

M. Vogel Institute of Solid State Physics, Darmstadt Technical University, Darm-stadt, Germany

Contributors xxi

T. A. Voitenko Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

M. Volnianskii Solid State Physics and Optoelectronics Department, Oles HoncharDnipro National University, Dnipro, Ukraine

Luidmila L. Vovchenko Departments of Physics, Taras Shevchenko National Uni-versity of Kyiv, Kyiv, Ukraine

Yu. P. Vyshnevska National Technical University of Ukraine “Igor Sikorsky KyivPolytechnic Institute”, Kyiv, Ukraine

Institute for Renewable Energy NAS of Ukraine, Kyiv, Ukraine

Olena S. Yakovenko Department of Physics, Taras Shevchenko National Univer-sity of Kyiv, Kyiv, Ukraine

Oleksandr Yasko Institute for Engineering & Technology, University “Ukraine”,Kyiv, Ukraine

Iryna Yu. Yermolenko National Technical University “Kharkiv Polytechnic Insti-tute”, Kharkiv, Ukraine

Oleg Yeshchenko Faculty of Physics, Taras Shevchenko National University ofKyiv, Kyiv, Ukraine

S. M. Yezhov Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

Y. F. Zabashta Taras Shevchenko National University of Kyiv, Physical Faculty,Kyiv, Ukraine

Volodymyr V. Zagorodnii Department of Radiophysics, Electronics, and Com-puter Systems, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

V. A. Zazhigalov Institute for Sorption and Problems of Endoecology, NationalAcademy of Sciences of Ukraine, Kyiv, Ukraine

V. O. Zazhigalov Institute for Sorption and Problems of Endoecology, NationalAcademy of Sciences of Ukraine, Kyiv, Ukraine

Maxim Zdorovets Astana branch of the Institute of Nuclear Physics, Astana,Kazakhstan


Ural Federal University named after the first President of Russia Boris Yeltsin,Ekaterinburg, The Russian Federation

Volodymyr Zhurba National Science Center “Kharkiv Institute of Physics andTechnology”, Kharkiv, Ukraine

O. Zvirko Karpenko Physico-Mechanical Institute of NAS of Ukraine, Lviv,Ukraine

Part INanocomposites and Nanostructures

Chapter 1A Microscopic Description of SpinDynamics in Magnetic MultilayerNanostructures

A. M. Korostil and M. M. Krupa

1.1 Introduction

Stacks of alternating ferromagnetic and nonmagnetic metal layers exhibit giantmagnetoresistance (GMR), because their electrical resistance depends strongly onwhether the moments of adjacent magnetic layers are parallel or antiparallel. Thiseffect has allowed the development of new kinds of field-sensing and magneticmemory devices [1]. The cause of the GMR effect is that conduction electronsare scattered more strongly by a magnetic layer when their spins lie antiparallelto the layer’s magnetic moment than when their spins are parallel to the moment.Devices with moments in adjacent magnetic layers aligned antiparallel thus havea larger overall resistance than when the moments are aligned parallel, giving riseto GMR. At the same time, there is the converse effect: just as the orientationsof magnetic moments can affect the flow of electrons, a polarized electron currentscattering from a magnetic layer can have a reciprocal effect on the moment ofthe layer. As shown in [2, 3], an electric current passing perpendicularly through amagnetic multilayer may exert a torque on the moments of the magnetic layers. Thiseffect which is known as “spin transfer” may, at sufficiently high current densities,alter the magnetization state. It is a separate mechanism from the effects of current-induced magnetic fields. Experimentally, spin-current-induced magnetic excitationssuch as spin waves [4–9] and stable magnetic reversal [6, 7] have been observed inmultilayers, for current densities greater than 107A/cm2.

The spin-transfer effect offers the promise of new kinds of magnetic devicesand serves as a new means to excite and to probe the dynamics of magneticmoments at the nanometer scale. To controllably utilize these effects, however,

A. M. Korostil (�) · M. M. KrupaInstitute of Magnetism of NAS of Ukraine and MES of Ukraine, Kyiv, Ukraine

© Springer Nature Switzerland AG 2019O. Fesenko, L. Yatsenko (eds.), Nanocomposites, Nanostructures,and Their Applications, Springer Proceedings in Physics 221,https://doi.org/10.1007/978-3-030-17759-1_1

3

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4 A. M. Korostil and M. M. Krupa

it is necessary to achieve a better quantitative understanding of current-inducedtorques. A derivation of spin-transfer torques using a one-dimensional (1D) WKBapproximation with spin-dependent potentials presented in [3] only take intoaccount electrons which are either completely transmitted or completely reflected bythe magnetic layers. For real materials the degree to which an electron is transmittedthrough a magnetic/nonmagnetic interface depends sensitively on the matching ofthe band structures across the interface [10, 11]. It is important to incorporate suchband structure information together with the effect of multiple reflections betweenthe ferromagnetic layers, into a more quantitative theory of the torques generated byspin transfer. This could be done using the formalism [12] based on kinetic equationsfor spin currents. Instead, it can be made by employing a modified Landauer-Büttiker formalism, in which the ferromagnetic layers are modeled as generalizedspin-dependent scatterers [13]. In this case, the calculations are carried out for aquasi-one-dimensional geometry, for which formulas for the torque generated onthe magnetic layers are derived when a current is applied to the system, for eitherballistic or diffusive nonmagnetic layers.

Controlling spin flow in the mentioned magnetic nanostructures at ultrafast timescales using femtosecond laser pulses opens intriguing possibilities for spintronics[14]. These laser-induced perturbations [15, 16] stir up the extreme regime of spindynamics, which is governed by the highest energy scale associated with magneticorder: the microscopic spin exchange that controls the ordering temperature TC. Incontrast, at microwave frequencies the ferromagnetic dynamics in the bulk are welldescribed by the Landau-Lifshitz-Gilbert (LLG) phenomenology [17], which hasbeen successfully applied to the problem of the ferromagnetic resonance (FMR). Atfinite temperatures below TC, the spin Seebeck and Peltier effects [18, 19] describethe coupled spin and heat currents across interfaces in magnetic heterostructures.

Despite their different appearances, the microwave, thermal and ultrafast spindynamics are all rooted in the exchange interactions between electrons. It isthus natural to try to advance a microscopic describing of the ultrafast dynamicsbased on the established phenomena at lower energies. Although some attemptshave been made [20, 21] to extend the LLG phenomenology to describe ultrafastdemagnetization in bulk ferromagnets, no firm connection exists between theultrafast spin generation at interfaces and the microwave spin-transfer and spin-pumping effects [22] or the thermal spin Seebeck and Peltier effects.

Solving the above-mentioned problem involves unification of the energy regimesof microwave, thermal, and ultrafast spin dynamics in magnetic heterostructuresfrom a common microscopic point of view, so that the parameters that control thehigh and low energy limits of spin relaxation originate from the same electron-magnon interactions [23]. In addition to the unified framework, these uniquecontributions are the history-dependent, nonthermalized magnon distribution func-tion and the crucial role of the out-of-equilibrium spin accumulation among itinerantelectrons as the bottleneck that limits the dissipation of spin angular momentumfrom the combined electronic system.

The paper is organized as follows. In Sect. 1.2, the interconnection between thespin current and spin dynamics via the spin-dependent scattering and an accom-

1 A Microscopic Description of Spin Dynamics in Magnetic Multilayer. . . 5

panying spin torque effect in ferromagnetic (F)/normal metal (N)-based magneticmultilayer nanostructure is studied. Section 1.3 is devoted to the description of theimpact of electron-magnon spin-flop scattering on out-of-equilibrium spin dynamicsin F/N-based magnetic nanostructures.

1.2 Spin-Dependent Interface Scattering-Induced Torquesin Magnetic Multilayer Nanostructures

1.2.1 Features of Spin-Features of Spin Transfer Effect

In this section, a simple intuitive picture of the physics behind the spin-transfereffect is considered. The connection between current-induced spin-transfer torquesand the spin-dependent scattering that occurs when electrons pass through amagnetic-nonmagnetic interface can be illustrated most simply by consideringthe case of a spin-polarized current incident perpendicularly on a single thinferromagnetic layer F, as shown in Fig. 1.1.

The layer lies in the y-z plane, with its magnetic moment uniformly pointed in thez direction, and it is assumed that the current is spin-polarized in the z − x plane at anangle θ to the layer moments. The incoming electrons can therefore be consideredas a coherent linear superposition of basis states with spin in the +z direction(amplitude cosθ /2) and −zdirection (amplitude sinθ /2). At first, it is assumed thatthe layer is a perfect spin filter, so that spins aligned with the layer moments arecompletely transmitted through the layer, while spins aligned antiparallel to thelayer moment are completely reflected. For incident spins polarized at an angle θ ,the average outgoing current will have the relative weights cos2θ /2 polarized inthe +z direction and transmitted to the right and sin2θ /2 polarized in the −zdirection and reflected to the left. Consequently, both of the outgoing electron spinfluxes (transmitted and reflected) lie along the z axis, while the incoming (incident)

Fig. 1.1 Schematic of exchange torque generated by spin filter. Spin-polarized electrons areincident perpendicularly on a thin ideal ferromagnetic layer. Spin filtering removes the componentof spin angular momentum perpendicular to the layer moments from the current; this is absorbedby the moments themselves, generating an effective torque on the layer moments


electron flux has a component perpendicular to the magnetization, along the x axis,with magnitude proportional to sinθ . This x component of angular momentum mustbe absorbed by the layer in the process of filtering the spins.

Because the spin-filtering is ultimately governed by the s-d exchange interactionbetween the conduction electrons and the magnetic moments of the layer, theangular momentum is imparted to the layer moments and produces a torque on them.This exchange torque [24], which is proportional to the electron current through thelayer and to sinθ , is in the direction to align the moments with the polarization ofthe incident spin current. The symmetry of this model precludes any generation oftorque from the spin filtering of a current of unpolarized electrons. To generate theeffect, then, a second ferromagnetic layer is needed to first spin polarize the current(Fig. 1.1). In that case, spin angular momentum is transferred from one layer to theflowing electrons and then from the electrons to the second layer.

The presence of this second layer has the additional effect of allowing formultiple scattering of the electrons between the two layers, which gives rise to anexplicit asymmetry of the torque with respect to current direction. This asymmetryis an important signature which can be used to distinguish spin-transfer-inducedtorques from the torques produced by current-generated magnetic fields. To see howthe asymmetry arises, consider the ferromagnet/normal-metal/ferromagnet (F/N/F)junction shown in Fig. 1.2.

It consists of two ferromagnetic layers Fa and Fb, with moments pointing indirections ma and mb, separated by a normal metal spacer N. Normal metal leadson either side of the trilayer inject an initially unpolarized current into the system.When the current enters the sample from the left (Fig. 1.2a), electrons transmittedthrough Fa will be polarized along ma. As long as the normal metal spacer issmaller than the spin-diffusion length (100 nm for Cu), this current will remain

Fig. 1.2 Qualitative picture of asymmetry of spin-transfer torque with respect to current bias in aF/NF junction. For left-going electrons (a), initially polarized by a magnetic layer Fa, the momentsof layer Fb experience a torque so as to align them with layer Fa. The electron current reflectedfrom layer Fb, in turn, exerts a torque on layer Fa so as to antialign it with the moment of layer Fb.Subsequent reflections between the layers reduce but do not eliminate this torque. If the current isreversed (b) the overall sign of the torque is reversed, encouraging the moment of layer Fb to alignantiparallel with layer Fa


spin-polarized when it impinges on Fb and will exert a torque on the moment of Fbin a direction so as to align mb with ma.

Repeating the argument for Fb, we find that the spin of the electrons reflectedfrom layer Fb is aligned antiparallel to mb, and, hence in turn, exerts a torque onthe moment of Fa trying to align ma antiparallel with mb. Subsequent multiplereflections of electrons between Fa and Fb can serve to reduce the magnitudes ofthe initial torques, but they do not eliminate or reverse them, as the electron fluxis reduced upon each reflection. If there were no anisotropy forces in the sample,the net result would be a motion with both moments rotating in the same direction(clockwise in Fig. 1.2a), as described previously in [3]. When the current is injectedfrom the right, the directions of the torques are reversed. The flow of electrons exertsa torque on Fa trying to align its moment parallel with mb, while it exerts a torqueon Fb so as to force the moment in layer Fb antiparallel with ma.

In [3, 6], the layer Fa was taken to be much thicker than Fb, so that intralayerexchange and anisotropy forces will hold the orientation of ma fixed. In that case,one is only interested in the torque on Fb, which serves to align mb either parallelor antiparallel with the fixed moment ma depending on the current direction. Thisasymmetric current response has been employed in both a point-contact geometry[7] and in a thin-film pillar geometry [8] to switch the moments in F/N/F trilayersfrom a parallel to an antiparallel configuration by a current pulse in one direction,and then from antiparallel to parallel by a reversed current. For weakly interactinglayers, either orientation can be stable in the absence of an applied current, so thatthe resistance versus current characteristic is hysteretic, and the devices can functionas simple current-controlled memory elements.

Often, the transport properties of magnetic multilayers are described using “two-current” models [8], in which one assumes that the effects of spin-polarized currentscan be described completely in terms of incoherent currents of spin-up and spin-down electrons. Normally, only the cases of purely parallel or purely antiparallelmagnetic layers are considered, and the spin currents are conserved upon passingthrough each normal-metal–ferromagnet interface.

In the mentioned case, there can be no current-induced torque on either magneticlayer. It is important to recognize that such two-current models are not appropriateto calculate current-induced torques for the more general case of arbitrary tilt anglebetween the moments in a magnetic multilayer, as the simple example discussed inthis section demonstrates. Tilting of the spin axis is an essential point of the physics[9], and this must be described in terms of a coherent sum of spin-up and spin-down basis states. In the general case, the spin flux is not conserved upon passingthrough a magnetic layer, so that a torque is applied to each magnetic layer. As it isturned out, this is a simple consequence of different transmission amplitudes for thespin-up and spin-down components of the electron flux.


1.2.2 Passing the Electric Current Through the F/N Contact

Treating the ferromagnetic layers as perfect spin filters provides important qual-itative insights into spin transfer, but for a complete qualitative and quantitativepicture, a more general approach is required. This can be realized by introductiona scattering matrix description of the F/N/F junction which allows to deal withnonideal (magnetic and nonmagnetic) layers. It is important to relate the torqueτ b exerted on layer Fb by an unpolarized incident electron beam to the scatteringproperties of the layers. Although restriction by formulas to the F/N/F junction (seeFig. 1.3), the mentioned method is applicable for an arbitrary array of magnetic-nonmagnetic layers.

The spin flux J in the x direction (the direction of current flow) can be written inthe form [13]

J (x) = h̄2

2mIm∫dydz

(φ†(x)σφ(x)

), (1.1)

where φ(x) is a spinor wave function and σ is the vector of Pauli matrices

φ(x) =(φ↑(x)φ↓(x)

), σ =

⎛⎝σxσyσz

⎞⎠ .

Note that no local equation of conservation can be written for the spin flux,since in general, the Hamiltonian (of the itinerant electrons) does not conserve spin.Specifically, the magnetic layers can act as sources and sinks of spin flux, so thatthe spin flux on different sides of a F layer can be different. When the angle θ is 0or π (typical situation for GMR), the commutativity between the Hamiltonian andthe (electron) spins is restored (in the absence of spin-flip scattering). It should bestressed that the Hamiltonian of the full system (electrons plus local moments of the

Fig. 1.3 Schematic of the setup used for the definition of the scattering matrices of the F and Nlayers. The two layers Fb and Fa are ferromagnetic layers whose magnetic moment is oriented asshown in the bottom of the figure. The layer N is a nonmagnetic metal spacer. Amplitude of leftand right moving propagating waves are defined in fictitious ideal leads 0, 1, 2, and 3 between thelayers and between the layers and the reservoirs


ferromagnets plus the environment) does commute with the total spin. Therefore,the spin lost by the itinerant electrons has to be gained by the other parts of thesystem.

Figure 1.3 shows the F/N/F junction where (fictitious) perfect leads (labeled 0,1, 2, and 3) have been added in between the layers F and N and between the Flayers and the electron reservoirs on either side of the sample. The introduction ofthese leads allows for a description of the system using scattering matrices. In theperfect leads, the transverse degree of freedom is quantized, giving Nch propagatingmodes at the Fermi level, where Nch ∼ A/λ2F , A is the cross section area of thejunction and λF is the Fermi wave length. Expanding the electronic wave function inthese modes, we can describe the system in terms of the projection � i, j of the wavefunction onto the left (j = 1) or right (j = 2) going modes in the region i = (0, 1, 2, 3).The � ij is 2Nch-component vector, counting the Nch transverse modes and spin.The amplitudes of the wave function in two neighboring ideal leads are connectedthrough the scattering matrices S1 = Sa, S2 = SN and S3 = Sb that relate amplitudesof outgoing modes and incoming modes at the layer (see [25]) by the relation

(�i,1

�i−1,2

)= Si

(�i,2

�i−1,1

), i = (1, 2, 3) (1.2)

The scattering matrix Si is 4Nch × 4Nch unitary matrices. The generalized matrixSi is decomposed into 2Nch × 2Nch reflection and transmission matrices

Si =(ri t

′i

ti r′i

). (1.3)

Normalization is done in such a way that each mode carries unit current. Due tothe spin degree of freedom, ri =

∥∥ri,σσ ′∥∥ and σ , σ ′ = (↑, ↓), where the reflectionand transmission matrices can be written in terms of four Nch × Nch blocks: wherethe subscripts ↑, ↓ refer to spin up and down in the z-axis basis. The scatteringmatrix of the magnetic layers depends on the angle θ the moments may make withthe z axis. The matrix Si(θ ) is related to Si(0) through a rotation in spin space:

ri (θ) = Rθri(0)R−θ , r ′i (θ) = Rθr ′i (0)R−θ ,

ti (θ) = Rθ ti(0)R−θ , t ′i (θ) = Rθ t ′i (0)R−θ , (1.4)

where

Rθ =(

cos θ2 − sin θ2sin θ2 cos

θ2

)⊗ 1N. (1.5)

The nonmagnetic metallic layer will not affect the spin states, i.e.,rN ↑ ↓ = rN ↓ ↑ = 0 and rN ↑ ↑ = rN ↓ ↓. We need to keep track of the amplitudes


within the system in order to calculate the net spin flux deposited into each magneticlayer. Therefore, we define 2Nch × 2Nch matrices ij and ij (i = (0, 1, 2, 3)) sothat all the � i, j can be expressed as a function of the amplitudes incident from thetwo electrodes (regions 0 and 3):

(�i,1

�i,2

)=(i1 i1

i2 i2

)(�0,1

�3,1

)(1.6)

with the convention that 01 =32 = 1 and 32 =01 = 0. In order to calculate thetorque exercised on layer Fb for a current entering from the left, we need the matrix21, which for simplification is denoted as �(� = 21). The matrix � relates theamplitudes �21 to the incoming amplitudes�01. Putting that �32 = 0 and using(1.2), the matrix of the Eq. (1.6) can be explicitly expressed via elements of thescattering matrices Si. Consequently, the following expression can be obtained forthe matrix �:

� = 11 − rnr ′b

t ′n1

1 − ratnr ′b(1 − tnr ′b

)t ′n − tar ′n

t ′a, (1.7)

which will enter in determination of the torque on the moment of the ferromagneticlayer Fb.

If the system is connected to two unpolarized electron reservoirs on its two sides,then in equilibrium, the modes in the reservoirs are filled up to the Fermi level εF .The spin current through the system is generated when the chemical potential in theleft (right) reservoir is slightly increased by δμ3(δμ0). The spin current Ji in eachregion i = 0, 1, 2, 3 is the difference of the left going and right going contributions.In according to (1.1) and (1.6)

∂Ji

∂μ0= 1

4πRe[Trσ iR

†iR − Trσ iR†iR

], (1.8)

∂Ji

∂μ3= 1

4πRe[Trσ iL

†iL − Trσ iL†iL

]. (1.9)

Since the spin flux on both sides of Fb is different, angular momentum has beendeposited in the Fb. This creates a torque τ b = J3 − J2 on the moment of theferromagnet. Setting δμ0 = eV0 gives

∂rb

∂V0= − e

4πRe Tr2Nch

[� � �†

], � = σ − t ′′†bσ t ′b − r ′†bσ r ′b (1.10)

This equation can be simplified further if the spin-transfer effect is due entirely tospin filtering (as argued in 1.3) as opposed to spin-flip scattering of electrons fromthe magnetic layers. That is at τ b ↑ ↓ = τ b ↓ ↑ = τ a ↑ ↓(θ = 0) = τ a ↓ ↑(θ = 0) = 0,then


∂rb

∂V0= − e

4πRe TrNch

[(�↑↑�†↓↑ +�↑↓�†↓↓

) (1 − r ′b↑↑r ′†b↓↓ − r ′b↑↑r ′†b↓↓

)],

(1.11)

where off-diagonal spin-flip terms are related to spin-flip scattering both in normaland magnetic layers. There is no spin flux conservation in this system, ∂Ji/∂μ3 canbe different from ∂Ji/∂μ0 and, hence, there can be a nonzero spin flux even whenthe chemical potentials are identical in the two reservoirs.

The existence of a zero-bias spin flux and the resulting torques reflect the well-known itinerant-electron-mediated exchange interaction (also known as the RKKYinteraction) between two ferromagnetic films separated by a normal-metal spacer.This interaction can in fact be described within a scattering framework [26–29].The zero-bias torque has to be added to the finite-bias contribution (given by 1.11).Since the former is typically a factor N−1ch smaller and vanishes upon ensembleaveraging [25], the zero-bias contribution to the torque can be neglected. Then thebias-induced torque satisfies the equation ∂rb/∂V0 = ∂rb/∂V3.

1.2.3 Averaging Over the Normal Metal Layer

The torque on the moments of the ferromagnetic layers Fa and Fb not only dependson the scattering matrices Sa and Sb of these layers, but also of the scatteringmatrix SN of the normal metal layer in between. If the normal layer is disordered,τ a and τ b depend on the location of the impurities; if N is ballistic, the torquedepends sensitively on the electronic phase shift accumulated in N. In general,sample to sample fluctuations of the torque will be a factor N−1ch smaller than theaverage [25]. Hence, if Nch is large, the torque is well characterized by its average.After averaging, the zero-bias spin-transfer current, corresponding to the RKKYinteraction described above, vanishes, and only the torque caused by the electroncurrent remains. Because all effects of quantum interference in the N layer willdisappear in the process of averaging, the derived results are unchanged if thereflection and transmission matrices include processes in which the energy of theelectron changes during scattering [30].

1.2.3.1 The Torque for Disordered Normal Metal Layers

The scattering matrix of the normal layer can be written using the standard polardecomposition [25]

Sa =(U 00 V ′

)(√1 − T i√Ti√T

√1 − T

)(U ′ 00 V

), (1.12)


where U, V, U′

and V′

are 2Nch × 2Nch unitary matrices and T is a diagonal matrixcontaining the eigenvalues of tnt

†n . Since Sn is diagonal in spin space, matrices U,

U′, V and V

′are block diagonal: U(U

′)=(u(u′)

00 u

(u′))

and similar definitions for

v and v′. The outer matrices in (1.12) thus mix the modes in an ergodic way while

the central matrix contains the transmission properties of the layer, which determinethe average conductance of N.

It is necessary to average (1.11) over both the unitary matrices and T. Adiagrammatic technique for such averages has already been developed in [31] andcan be used to calculate ∂ρb/∂V0 in leading order in 1/Nch. It is a general property ofsuch averages that the fluctuations are a factor of order Nch smaller than the average.This justifies the statement above, that the ensemble-averaged torque is sufficient tocharacterize the torque exerted on a single sample.

The resulting expression for ∂ρb/∂V0 can be written in a form very similar to theone for (1.11) if one uses a notation that involves 4 × 4 matrices. Then, the averageover the transmission eigenvalues T can be obtained taking into account that theaverage of a function is the function of the average, to leading order in 1/Nch. Thusthe average over T amounts to the replacement

t̂n = gNNch

l4, r̂n =(

1 − gNNch

)l4 (1.13)

where gN is the conductance of the normal layer and l4 is the 4 × 4 unit matrix.Entering the 4 × 4 block matrix �̂ with the first and fourth rows equal to (�↑↑�↑↓ �↓↑ �↓↓) and the zero second and third rows, it can be obtained that

〈∂rb

∂V0

〉= − e

4πRe Tr4

[�̂ �†

](1.14)

where �̂ coincide with � after the formal replacements rn → r̂nn and tn → t̂n. Inthe absence of spin-flip scattering (1.14) reduces to

〈∂rb

∂V0

〉= − e

2πRe[(�̂3,1 + �̂3,4

)× TrNch

(1 − r ′b↑↑r ′†b↓↓ − t ′b↑↑t ′†b↓↓

)].

(1.15)

The same formalism can be used to calculate the conductance g of the system

using the Landauer formula. One gets 〈g〉 = (Nch/h)[t̂ ′1,1 + t̂ ′1,4 + t̂ ′4,1 + t̂ ′4,4

],

where t′

being the total matrixt ′ = t ′b�.Note that, while our theory started from a full phase coherent description

of the F/N/F trilayer, including the full 4Nch × 4Nch scattering matrices of theF/N interfaces, the final result can be formulated in terms of 2 × 4 parameters,represented by the matrices r̂a and r̂ ′b(2 × 16 parameters in the case of spin-flip


scattering). This confirms the statement that for a diffusive normal-metal spacer alleffects of quantum interferences are absent [25].

The torque is characterized by symmetry properties. Due to the conservation ofcurrent, the total torque deposited on the full system is antisymmetric with respectto current direction and the equation

∂τb/∂V0 + ∂τa/∂V0 = − [∂τb/∂V3 + +∂τa/∂V3]

must be held before averaging. The averaging results in 〈∂τ b/∂V0〉 =〈∂τ b/∂V3〉.Thus, for Nch � 1, the linear response of the torque to a small bias voltage isdescribed by the expression

τβ =〈∂τb

∂V0

〉(V0 − V3) (1.16)

In the given geometry, where Fa and Fb are in the x-z plane, the only nonzerocomponent of the torque is τxb . The torque vanishes when the moments arecompletely aligned or antialigned (all the matrices are diagonal in spin space andtherefore no x component of the spin can be found). Around these two limits, thetorque is symmetric in respect to the angle θ → − θ and π − θ → π + θ . There isno symmetry between θ and π − θ . In addition, the two layers are not equivalent,and exchanging the scattering matrices of Fa and Fb also changes the torque.

The Eq. (1.14) can be simplified in some particular cases. In the case of idealspin filter, so that majority (minority) spins are totally transmitted (reflected) byeither layer, it reduces to

〈∂rxb

∂V0

〉= − e

2π

gN sin θ

3 + cos θ = −h

4πe〈g〉 tan θ/2

2, (1.17)

where 〈g〉 = 4(e2/h)gNcos2θ /(3 + cos θ ) is the average magnetoconductance. Asexpected, for left-going electrons (V0 < 0) the torque is positive, so it acts to alignthe moment of the magnetic layer Fb toward the one of Fa.

In the considered case of weak s-d exchange coupling, i.e., when the scatteringcoefficients depend only weakly on spin, with no spin-flip scattering in theferromagnetic layers, ga and gb can be defined as the average conductance per spinof the two layers (in unit of e2/h). Then, the conductance of Fa alone is ga + δgaand ga − δga for, respectively, the majority and minority spins, which defines thespin scattering asymmetryδga. In that case, in lowest nontrivial order in δga and δgb

〈∂τxb

∂V0

〉= e

2π

sin θ

2 (1 − gb/Nch)g2Nδgaδg

2b(

gagb + gN(ga + gb − 2gagb/Nch

)2 , (1.18)


This formula shows that the torque is always nonzero for arbitrary small spinscattering asymmetry. This proves the statement that multiple reflections betweenthe F layers, fully taken into account here, cannot completely eliminate the torque.The torque is not symmetric with respect to interchanging the layer Fa andFb, incontrast to the conductance. If one changes δga to −δga, the sign of the torque isreversed. However, ∂τxb /∂V0 ∝ δg2b , so if one changes δgb to −δgb, the sign ofthe torque is unchanged. The sign of the torque on a ferromagnetic layer thereforedepends on whether the other layer is a positive or negative polarizer, but not on thesign of filtering for the layer experiencing the torque.

This is true also in the general case. The quantity g2N appears through its square.Indeed, in order for some spin to be deposited in the layer Fb, some left goingelectrons have to be reflected by Fb and exit the system from the right-hand side.Therefore, these electrons cross the normal layer at least twice and this leads tothe factorg2N . On the other hand, the conductance is linear in gN . Therefore, inorder to maximize the torque deposited per current, one has to use the cleanestpossible normal metal spacer. This statement is true in this limit of weak filtering,but not in general. Note that in the previous case (perfect spin filtering) the torqueis proportional to gN instead of the expected g2N . Indeed, in that case, once theelectron has been reflected by the layer Fb, it cannot go through Fa which works asa perfect wall for it. Therefore, current conservation implies that it goes out of thesystem through the right. For gN � Nch, the torque is actually proportional to g2Nfor arbitrary spin asymmetry (except perfect filtering), and one gets

〈∂τxb

∂V0

〉∝ g2N sin θ, gN � Nch, (1.19)

where the factor of proportionality being a complicated function of the transmissionrobabilities of the layers.

1.2.3.2 The Torque for Ballistic Normal Metal Layers

If the normal metal layers N is very clean, and the interfaces are very flat, it isreasonable to assume that the electrons propagate ballistically inside the normallayer. The different modes will not be mixed in that case, and the electron wavefunction only picks up a phase factor exp(ikiL) where L is the width of N and kiis the momentum of channel i. For a sufficiently thick normal layer (i.e., L � λF),small fluctuations of ki lead to an arbitrary change in the phase factor, and it isjustified to consider exp(ikiL) as a random phase and to average over it. This isdifferent from the case of a disordered metal spacer, where the average involvesunitary matricesu, u

′, . . . that mix the channels (see 1.25). In the case where ra↓,

ra↑, . . . are proportional to the identity matrix (i.e., the reflection amplitudes donot depend on the channel), the ballistic model reduces to the disordered model of(1.32) for gN = Nch.


The reflection matrices of N being zero, the matrix � reads

� = exp (ikiL) 11 − exp (2ikiL) rar ′b

(1.20)

Neglecting spin-flip scattering, denoting z = exp (2ikiL) , and choosingra11 = ra↑,ra22 = ra↓, . . . where ra↑, ra↓, . . . are diagonal matrices, one gets aftersome algebra

∂rxb

∂V0(z) = − ev

4πTr Re

A(z)

z|D(z)|2 sin θ, (1.21)

where A(z) and D(z) stand for

A(z) = (1 − t ′b1t ′∗b2 − r ′b1r ′∗b2)∑i=1,2

∣∣t ′ai∣∣2 (1 − zrai′r ′bi′

) (z− r∗ai′r ′∗bi

), (1.22)

where i = (1, 2) = (↑, ↓), i′ = 1 at i = 2 and i′ = 2 at i = 1,

D(z) = 1 − z [cos2 θ2(ra↑r ′b↑+ra↓r ′b↓

)+ sin2 θ2

(ra↓r ′b↑ + ra↑r ′b↓

)]+ z2ra↓ra↑r ′b↓r ′b↑.

(1.23)

A similar formula can be written for the conductance g(z):

g(z) = e2

hTr

B(z)

z|D (z)|2 , (1.24)

B(z) =∑

i,j=(1,2)

∣∣t ′ai∣∣2∣∣∣t ′aj

∣∣∣2Rij (θ)(

1 − zrai′r ′bj ′) (z− r∗ai′r ′∗bj ′

)(1.25)

where i′, j

′ = (1, 2)(1 = ↑, 2 = ↓) are determined by the relations i′ �= i, j′ �= j,Rij(z) = (δi, jcos2θ /2 + (1 − δi, j)sin2θ /2). Taking the average over the phases nowamounts to contour integration for z:〈f 〉 = (1/2π ) ∮ dzf (z)/z, where the integrationis done along the unit circle. The result is then given by the sum of the poles thatare inside the unit circle. The two poles of D(z) are outside the unit circle, while thetwo poles z1 and z2 of z2D(1/z) are inside the circle. They have the form

zi = 12�+11,22cos2 θ2 +�+21,12sin2 θ2+ 12 (−1)i

[(�−11,22

)2cos4 θ2 + 2�cos2 θ2 sin2 θ2 +

(�−11,22

)2sin4 θ2

]1/2,

(1.26)


where

�+(−)ii′,jj ′=

(rair

′bi′ ± raj r ′bj ′

), � = r ′b↓r ′b↑

(ra↑ − ra↓

)2 + ra↓ra↑(r ′2b↑+r ′2b↓

).

The averaged torque and conductance are then simply given by

〈∂τxb

∂V0

〉= ev

4π

sin θ

z1 − z2 Tr(A (z1)

D (z1)− A (z2)D (z2)

)(1.27)

and

g = e2

h

1

z1 − z2 Tr(B (z1)

D (z1)− B (z2)D (z2)

)(1.28)

In the case where all the channels are not identical, these results can begeneralized by introducing a k dependence of the different transmission-reflectionamplitudes.

1.2.4 Current-Driven Magnetic Switching

Application to current-driven magnetic switching involves calculation of torques forscattering parameters which are more appropriate for the transition metal trilayers.In this case, the torque per unit current I, τxb /I = 〈(1/g) × ∂τxb /∂V

〉. The main

features of the mentioned system are that the θ dependence of the torque is not of asimple sinθ form, and that the torque per unit current diverges at θ = 0.

The main feature of this system is that the θ dependence of the torque is not ofa simple sinθ form, and that the torque per unit current diverges at θ = 0. In theimperfect case, when one of the layer (Fb) is a nearly perfect polarizer while theother one is not, the character behavior of the torque is represented in Fig. 1.4.

As can be seen in Fig. 1.4 on the left, although the divergence at θ = πis regularized, τxb /I remains sharply peaked near θ = π . This is relevant forthe critical current needed to switch the magnetization of Fb from θ = π toθ = 0. The switching of the domains follows from a competition between the spin-transfer torques on the one hand and restoring forces from local fields, anisotropy,exchange coupling, etc. The competition between these forces has been consideredphenomenologically [7, 32] using a Landau-Lifschitz-Gilbert equation. The torquesfor θ close to 0 and π determine the critical currents to overturn a metastable parallel(antiparallel) alignment of the moment in Fa and Fb. Hence the critical currentshould be different at θ = 0 and θ = π . Features of the dependence of the derivativeof the torque (with respect to θ closely to 0) on the conductance of the normallayer gN in the same system with one perfectly polarizing F layer and one partiallypolarizing layer are represented in Fig. 1.4 on the right.


Fig. 1.4 The normalized torque as a function of θ (on the left) and its derivative as a function ofthe normalized conductance gN at θ = 0 (on the right) per unit current for the case, where Fb is anearly perfect polarizer (|tb↑|2 = 0.999, |tb↓|2 = 0.001 and Fa is not (|ta↑|2 = 0.3, |ta↑|2 = 0.01)(solid line). On the left, the case of perfect polarizers is described by the Eq. (1.17) (dashed line).On the right, the dashed line corresponds to the case, when Fa is a nearly perfect polarizer and Fbis not

Switching the two layers has a drastic effect on the torque, even at a qualitativelevel. In the case where Fa is the nearly perfect layer (dashed line), a maximum ofthe torque is found forgN /Nch � 1, i.e., in that case, a dirty metal spacer would givea higher torque (per unit of current) than a clean one.

In the abovementioned description, the scattering matrices of the ferromagneticlayers appear as free input parameters. However, it can be calculated from firstprinciple calculations for specific materials. Such an approach has been taken in[10, 11] and the results can be used to give some estimates of torques that can beexpected in realistic systems.

1.3 Out-of-Equilibrium Spin Dynamics in F/N-BasedStructures

1.3.1 Features of Spin Dynamics

The first reports on ultrafast demagnetization in Ni [33] challenged the conventionalview of low-frequency magnetization dynamics at temperatures well below TC.A multitude of mechanisms and scenarios have been suggested to explain theobserved quenching of the magnetic moment. Some advocate direct coherent spintransfer induced by the irradiating laser light as the source of demagnetization[34]. Alternative theories argue that ultrafast spin dynamics arise indirectly throughincoherent heat transfer to the electron system [35]. Recent experiments havedemonstrated that nonlocal laser irradiation also induces ultrafast demagnetization,


and atomistic modeling supports the view that heating of magnetic materials issufficient to induce ultrafast spin dynamics [16, 36].

Terahertz (THz) magnon excitations in metallic ferromagnets have recently beenproposed as an important element of ultrafast demagnetization [37, 38]. The elemen-tary interaction that describes these excitations is the electron-magnon scattering.The proposed approach is based on kinetic equations, which were used for the low-frequency spin and charge transport associated with the microwave magnetizationdynamics in heterostructures [39] and with the linear spin-caloritronic response [19,40]. One treats far-from-equilibrium spin dynamics, in which transport is dominatedby magnons and hot electrons. Electron-magnon scattering plays a critical role inthis regime. Description of this interaction is related to the transverse spin diffusionin the bulk and the spin-mixing physics, e.g., spin transfer and spin pumping [40],at the interfaces.

1.3.2 The Model of the Out-of-Equilibrium Spin Dynamics

Characteristic properties of the out-of-equilibrium ultra-fast spin dynamics aredescribed by the quantum-mechanical model bilayer system comprising ferromag-netic (F) and normal metal layers, in which the localized spins are distinct fromthe itinerant electrons at the energy scales of interest. According to the accepteddescription of relaxation in ferromagnetic metals, the loss of energy and angularmomentum from localized d electrons is mediated by the exchange interaction tothe itinerant s electrons. The spin transfer from d to s states is accompanied bythe relaxation of the s electron spins to the lattice through an incoherent spin-flipprocess caused by the spin-orbit coupling.

Ultra-fast spin dynamics in bulk ferromagnetic metal is described by the quantumkinetic equations. The F/N interfacial spin transport due to electron-magnoninteractions follows a similar essential structure, unifying the bulk and interfacialspin dynamics in magnetic heterostructures. The Hamiltonian that describes F isH = H0 + Hsd, where H0 consists of decoupled s- and d- electron energies,including the kinetic energy of the itinerant electron bath, the d-d exchange energy,dipolar interactions, and the crystalline and Zeeman fields. The s-d interaction is

Hsd = Jsd∑j

Sdj s(rj), (1.29)

where Jsd is the exchange energy and Sdj(s(rj))

is the d-electron (s-electron) spinvector (spin density) at lattice point j. The s-d interaction can be expressed in termsof bosonic and fermionic creation and annihilation operators:


Hsd =∑qkk′Vqkk′aqc

†k↑ck′↓ + H.c. (1.30)

where a†q(aq)

is the Holstein-Primakoff creation (annihilation) operator for

magnons with wave number q and c†kσ (ckσ ) is the creation (annihilation) operatorfor s electrons with momentum k and spin σ . Hsd describes how an electron flipsits spin while creating or annihilating a magnon with momentum q and spin σ . Thescattering strength is determined by the matrix element Vqkk′ .

In (1.30), terms of the forma†qaq ′c†kσ ck′σ , which describe multiple-magnon

scattering and do not contribute to a net change in magnetization along the spin-quantization axis, have been disregarded. Higher-order terms associated with theHolstein-Primakoff expansion are also disregarded. Fully addressing magnoniccorrelation effects in the ultrafast regime would require a rigorous approach, e.g.,using nonequilibrium Keldysh formalism [41]. However, when the s-d coupling(1.29) is not the dominant contribution to H, a mean-field approach and Fermi’sgolden rule were used to compute the spin transfer between the s and d subsystems.Additionally, it is assumed that all relevant energy scales are much smallerthan the Fermi energy ε = kBTF of the itinerant s electrons. In this limit, theelectronic continuum remains largely degenerate, with electron-hole pairs presentpredominantly in the vicinity of the Fermi level.

In the given bilayer system, localized spin density points in the negative zdirection at equilibrium, with saturation value S (in units of h̄). In the presenceof a magnon density nd, the longitudinal spin density becomes Sz = nd − S. Themagnons are assumed to follow a quadratic dispersion relation εq = h̄ω= ε0 + Aq2,where ε0 is the magnon gap and A parameterizes the stiffness of the ferromagnet.〈a

†qaq ′〉

= n (εq) δqq ′defines the magnon distribution function n(εq), which isrelated to the total magnon density through nd =

∫ εbε0dεqD

(εq)n(εq), where

D(εq) = (εq − ε0)1/2/(4π2A3/2) is the magnon density of states. The integral overD(εq) is cut off at an energy corresponding to the bandwidth, εb = kBTC.

Because of the s-d interaction (1.29), the itinerant s electrons have a finite spindensity at equilibrium, as it is represented in Fig. 1.5.

One of the key driving forces of the out-of-equilibrium spin dynamics is the spinaccumulation μs = δμ↑ − δμ↓. The bands for spin-up and spin-down electrons aresplit by�xc ∼ JsdSa3, where a is the lattice constant of F. By introducing a dynamicexchange splitting, we can write μs = δns/D − �xc [42], where δns is the out-of-equilibrium spin density of the s electrons, D = 2D↑D↓/(D↑ + D↓), and D↑(↓) isthe density of states for spin-up (spin-down) electrons at the Fermi level. Becausethe mean-field band splitting due to the s-d exchange vanishes when the d orbitalsare fully depolarized, δ�xc/�xc = ± nd/S, where the sign determines whether the sand d orbitals couple ferromagnetically (−) or antiferromagnetically (+).

The rate of spin transfer (per unit volume) between the s and d subsystems due toelectron-magnon spin-flop processes is determined from (1.30) by Fermi’s goldenrule [40]:


Fig. 1.5 (a) Sketch of the density of s electron states in a ferromagnetic metal with saturation spindensity S. At equilibrium, the exchange splitting �xc shifts the bands for spin-up and spin-downelectrons. (b) A laser pulse heats the s electron bath. The out-of-equilibrium spin accumulationμs = δμ↑ − δμ↓ results from two different mechanisms: electron-magnon scattering induces aspin density among the s electrons, and the meanfield exchange splitting is shifted by δ�xc by theinduced magnon density nd

Isd =εb∫

εa

dεq(εq) (εq − μs

)D(εq) [nBE

(εq − μs

)− n (εq)] , (1.31)

where (εq) parameterizes the scattering rate at energy εq. In the derivation of(1.31), it has been assumed that the kinetic energy of the itinerant electrons andthe empty states (holes) thermalize rapidly due to Coulombic scattering and thatthey are distributed according to Fermi-Dirac statistics. Correspondingly, it canbe shown that the electron-hole pairs follow the Bose-Einstein (BE) distributionfunction, nBE(εq − μs) = {exp[βs(εq − μs)] − 1}−1at the electron temperatureTs = 1/(kBβs). The number of available scattering states is influenced by the spinaccumulation μs.

In contrast to the low-energy treatment in [40], the derivation of (1.31) doesnot constrict the form of the magnonic distribution n(εq) to the thermalized BEdistribution function. When the time scale of the s-d scattering is faster thanthe typical rates associated with magnon-magnon interactions, magnons are notinternally equilibrated shortly after rapid heating of the electron bath, as alsopredicted by atomistic modeling [43]. Consequently, the occupation of the magnonstates can deviate significantly from the BE distribution on the time scale of thedemagnetization process. Such treatment differs from that, in which the excitedmagnons are assumed to be instantly thermalized with an effective spin temperatureand zero chemical potential and the thermally activated electron bath is assumed tobe unpolarized.


1.3.3 Heat Pulse-Induced Spin Dynamics

The s-d scattering rate can be phenomenologically expanded as (εq) = 0+χ (εq − ε0), where 0 (which vanishes in the simplest Stoner limit) parameterizesthe scattering rate of the long-wavelength magnons and χ (εq − ε0) ∝ q2 describesthe enhanced scattering of higher-energy magnons due to transverse spin diffusion[44]. In general, one might expect other terms of higher order in q to be present inthis expansion as well. The quantity (εq)is extrapolated up to the bandwidth εbthat should be sufficient for qualitative purposes. Neglecting any direct relaxationof magnons to the static lattice or its vibrations (i.e., phonons), ∂ tnd = Isd/h̄. Theequations of motion for the s-electron spin accumulation and the d-electron magnondistribution function are

∂tμs = −μsτs

+ ρh̄Isd , (1.32)

∂tn(εq) =

(εq)

h̄

(εq − μs

) [nBE

(εq − μs

)− n (εq)] , (1.33)

where ρ determines the feedback of the demagnetization on μs and τ s is thespin-orbit relaxation time for the s electron spin density relaxing to the lattice.Here, τ s is typically on the order of picoseconds [45] and represents the mainchannel for the dissipation of angular momentum out of the combined electronicsystem. In general, τ s also depends on the kinetic energy of the hot electrons afterlaser-pulse excitation. However, it is assumed that τ s is independent of energy.ρ = ρD + ρ� = − 1/D + �xc/S includes effects arising from both the out-of-equilibrium spin density and the dynamic exchange splitting. For ferromagnetic (−)s-d coupling, these effects add up, whereas for antiferromagnetic (+) coupling, theycompete.

At low temperatures, low-frequency excitations result in purely transverse spindynamics. In the classical picture of rigid magnetic precession, the transverserelaxation time τ 2 is determined by the longitudinal relaxation time τ 1 via therelation, 1/τ 2 = 1/(2τ 1) = αω, where α is the Gilbert damping parameter and ωis the precession frequency. Indeed, in the limit (q, Ts) → 0, ∂ tnd → (/h̄)ε0nd,which is identical to the LLG phenomenology, indicating that ε0 = h̄ω and thus0 = 2α. This result establishes the important link between the scattering rate 0in this treatment and the Gilbert damping parameter that is accessible through FMRexperiments.

In the opposite high-frequency limit, pertinent to ultrafast demagnetizationexperiments, the layer F to be in a low temperature equilibrium state before beingexcited by a THz laser pulse at t = 0, upon which the effective temperature of theitinerant electron bath instantly increases such that Ts ≥ TC. This regime is clearlybeyond the validity of the LLG phenomenology, which is designed to address thelow energy extremum of magnetization dynamics. Dissipation in the LLG equation,


Fig. 1.6 Numerical solutions of Eqs. (1.32) and (1.33) after Ts is increased from 102 to 103 K (TC)within 50 fs with a decay time of 2 ps. ε0=5 meV, A = 0.6 meV/nm2, ρ = 6 meV/nm3, τ s = 2 ps,and α∗ = 10α = 0.1. (a) The itinerant electron-hole pair distribution nBE(ε − μs) is rapidlydepleted by the spin accumulation μs that is built up via electron-magnon scattering. (b) In themagnon distribution n(εq) the high-energy magnon states are rapidly populated, whereas the low-energy states remain unaffected on short time scales. (c) Time evolution of the spin accumulationμs(t) and (d) the longitudinal spin density −Sz(t) with different decay times of Ts: 0.15, 0.5, and2 ps

including relaxation terms based on the stochastic Landau-Lifshitz-Bloch treatment[46], is subject to a simple Markovian environment without any feedback or internaldynamics. This perspective must be refined for high frequencies when no subsystemcan be viewed as a featureless reservoir for energy and angular momentum.

The nonthermalized nature of the excited magnons can be appreciated in the limitin which μs is small compared with ε0 and no magnons are excited (n(ε0 = 0)) fort < 0. After rapid heating of the itinerant electrons at t = 0, the time evolution of themagnonic distribution follows

n(εq, t

) ≈ nBE (εq, t)[1 − exp

((εq, t/h̄

)]. (1.34)

This result implies that, initially, the high-energy states are populated much fasterthan low-energy states. When μs becomes sizable, the coupled Eqs. (1.32) and(1.33) must be solved subject to a suitable Ts(t). Fig. 1.6 (a, b) presents numericalsolutions of (1.32) and

Equation (1.33) when Ts is increased from 102 to 103 K within 50 fs with a decaytime of 2 ps. By comparison, internal magnon-magnon interactions equilibrate thedistribution function on the time scale τ−1eq ∼ h̄−1εm(εm/ (kBTC))3 [40], where εmis a characteristic energy of the thermal magnon cloud. For short times, Isd (see


1.31) dominates the magnon dynamics, and the magnon population is significantlydifferent from the thermalized BE distribution.

When Ts > TC, the thermally excited electron-hole pairs are populatedin accordance with the classical Rayleigh-Jeans distribution, nBE(εq − μs)→kBTs/(εq − μs). Assuming, for simplicity, that the expansion for (εq) is validthroughout the Brillouin zone, (1.31) yields

∂tnd |t→0 = Is(0)/h̄ = (0 + 3χ (εb − ε0) /5) kBTs/h̄.

Thus, the demagnetization rate is initially proportional to the temperature ofthe electron bath but is reduced by the lack of available scattering states for highenergy magnons within the time scale of the demagnetization process. Fig. 1.6(c, d)illustrates the time evolution of the out-of-equilibrium spin accumulation μs(t) andthe longitudinal spin density −Sz(t) for different decay times of Ts.

In the ultrafast regime, the electron-magnon spin-flop scattering is governedby the effective Gilbert damping parameter α∗ = χ (εb − ε0). Experimentalinvestigations of the magnon relaxation rates on Co and Fe surfaces confirm thathigh-q magnons have significantly shorter lifetimes than low-q magnons [38]. It isreasonable to assume that the same effects are also present in the bulk. The initialrelaxation time scale in the ultrafast regime is τ i ∼ (α∗ h̄−1kBTs)−1. This generalizesthe result of [20] for the ultrafast relaxation of the longitudinal magnetization toarbitrary α∗ based on the transverse spin diffusion [44].

The notion of magnons becomes questionable when the intrinsic linewidthapproaches the magnon energy, which corresponds to α∗ ∼ 1. Staying well belowthis limit and consistent with [40], it is assumed that α∗ = 0.1. For TC = 103 K theinitial relaxation time scale τ i ∼ 102(TC/Ts) fs, which is generally consistent withthe demagnetization rates observed for ultrafas

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