Faculty of Physics and Applied Computer Science
Doctoral thesis
Jakub Ch¦ci«ski
Modeling of magnetization dynamics
in spintronic oscillators
Supervisor: prof. dr hab. Tomasz Stobiecki Assistant supervisor: dr in». Marek Frankowski
Declaration of the author of this dissertation:
Aware of legal responsibility for making untrue statements I hereby declare that I have written this dissertation myself and all the contents of the dissertation have been obtained by legal means.
Declaration of the thesis Supervisor: This dissertation is ready to be reviewed.
I would like to express my sincere gratitude to my supervisor prof. Tomasz Stobiecki as well as to my assistant supervisor dr Marek Frankowski for their patient guidance, numerous fruitful discussions and for introducing me into the world of spintronics and nanomagnetism. Without them, this thesis would never be completed.
I am deeply indebted to dr Piotr Wi±niowski for the enormous amount of work he put in performing magnetic tunnel junction measurements and discussions about model development.
I would like to thank all members of the Magnetic Multilayers and Spin Electronics Group, including dr Maciej Czapkiewicz, dr Jarosªaw Kanak, dr Witold Skowro«ski, dr Zbigniew Szk-larski, dr Sªawomir Zi¦tek, M.Sc. Monika Cecot, M.Sc. Michaª D¡bek, M.Sc. Krzysztof Grochot, M.Sc. Stanisªaw azarski, M.Sc. Wiesªaw Powro¹nik and M.Sc. Piotr Rzeszut for their encour-agement and accompanying me in my journey to complete this thesis.
I am particularly grateful to prof. Johan Åkerman for giving me an opportunity to stay at the University of Gothenburg and for his helpful comments on neuromorphic computations performed by spintronic devices.
I would also like to thank the members of Applied Spintronics, Magnonics and Nanomagnet-ism group at the University of Gothenburg, including (but not limited to) dr Mykola Dvornik and M.Sc. Mohammad Zahedinejad, for providing me with ideas and helping me feel welcome in a foreign country. I have only one regret about my stay in Sweden - that it lasted too short.
Various parts of the presented thesis were supported by Polish National Science Center grants: Preludium 2015/17/N/ST7/03749, Ph.D. scholarship Etiuda no. 2017/24/T/ST3/00009 and the scholarship under Marian Smoluchowski Krakow Research Consortium KNOW pro-gram. Numerical calculations were supported by the PL-Grid infrastructure.
This work is dedicated to my loving family, as without them I would not have the commit-ment and courage to pursue it.
The aim of science is to seek the simplest explanations of complex facts. We are apt to fall into the error of thinking that the facts are simple because simplicity is the goal of our quest. The guiding motto in the life of every natural philosopher should be, "Seek simplicity and distrust it."
Alfred N. Whitehead
But thought's the slave of life, and life time's fool; And time, that takes survey of all the world, Must have a stop.
Contents
1 Introduction 10
2 Oscillatory systems in spintronics 12
2.1 Spin-torque oscillators . . . 12
2.2 Dynamical phenomena in spin-torque oscillators . . . 15
2.2.1 Injection locking . . . 16
2.2.2 Mutual locking . . . 17
2.3 Spin Hall oscillators . . . 18
2.4 Antiferromagnetic oscillators . . . 19
3 Numerical methods and algorithms 24 3.1 Numerical approach to micromagnetism . . . 24
3.1.1 Space discretization . . . 24
3.1.2 Time discretization . . . 25
3.2 Neuromorphic computations based on spintronic oscillators . . . 25
3.2.1 Associative memory . . . 26
3.2.2 Edge detection . . . 26
3.3 Linear regression and cross-validation . . . 28
3.3.1 Linear regression . . . 28
3.3.2 Cross-validation . . . 29
4 Magnetization dynamics in ferromagnetic oscillators 31 4.1 [P1] Spin-Torque Oscillator Read Head in Inhomogeneous Magnetic Fields . . . 32
4.2 [P2] Simultaneous readout of two adjacent bit tracks with a spin-torque oscillator 38 4.3 Neuromorphic computations based on spin Hall nano-oscillator chains . . . 43
5 Magnetization dynamics in antiferromagnetic oscillators 46 5.1 [P3] Antiferromagnetic nano-oscillator in external magnetic elds . . . 47
6 Simulation and modeling tools 54 6.1 [P4] MAGE (M- le/Mif Automatic GEnerator): A graphical interface tool for automatic generation of Object Oriented Micromagnetic Framework con gura-tion les and Matlab scripts for results analysis . . . 55
6.2 [P5] Magnetic noise prediction and evaluation in tunneling magnetoresistance sensors . . . 67
Abstract
The presented thesis describes a contribution into the theory of magnetization dynamics in spintronic oscillators. Together, ve articles, which were previously published in scienti c journ-als from the JCR (Journal Citation Reports) list, are incorporated into this manuscript. In the theoretical introduction, the necessary background for physical systems such as spin-torque oscillators, spin Hall oscillators and antiferromagnetic oscillators was presented. The most important approaches to the theoretical description of these devices, as well as a few selected dynamical phenomena arising from their interactions with external signals, were dis-cussed. Apart from physical foundations, the numerical and statistical methods utilized in the research described in the thesis were also brie y introduced.
The further parts of the thesis focused on the results of the conducted research concerning the magnetization dynamics in ferromagnetic oscillators, the magnetization dynamics in anti-ferromagnetic oscillators and the created simulation and modeling tools. It was shown that a spin-torque oscillator can be utilized as a magnetic eld reader and therefore as a part of a hard disk read head. The quantitative in uence of the external eld inhomogeneity on the functioning of such a device was described. A method allowing for simultaneous detection of two separate magnetic elds while retaining a nearly constant signal-to-noise ratio was proposed, by utiliz-ing the concept of a hard disk read head based on a spin-torque oscillator. Opportunities for performing neuromorphic computations using a network of interconnected spin Hall oscillators were explored and an example implementations of associative memory as well as edge detection algorithms were demonstrated. The behavior of antiferromagnetic oscillators in the presence of an external magnetic eld was analyzed, including both the general solution of the dynamics equation and a number of speci c but highly typical cases. It was shown that introducing an external magnetic eld may lead to qualitatively new phenomena in the oscillator dynamics, such as e.g. the presence of a material-dependent phase shift in response to a sinusoidal excit-ation or the possibility of resonant increase in the amplitude of observed oscillexcit-ations.
An open-source software, which facilitates the preparation of micromagnetic simulation con- guration les and later in the analysis of the data, was developed to assist the conducted research. The program design and typical usage scenarios were demonstrated, and an example test simulation of a magnetic tunnel junction was described as an illustration for the future user. Additionally, a phenomenological method for assessing thermal noise levels in magnetores-istance tunnel junctions based on only a few easily measurable quantities was proposed and compared with the experimental results. The obtained results indicated that resistance, volume and eld sensitivity alone are su cient to predict both the white noise and the 1/f-type noise power in a typical magnetic tunnel junction sensor for any given measurement current.
Streszczenie
Przedstawiona rozprawa doktorska opisuje wkªad autora do teorii opisuj¡cej dynamik¦ mag-netyzacji w oscylatorach spintronicznych. Rozprawa zawiera ª¡cznie pi¦¢ artykuªów, które zostaªy uprzednio opublikowane w czasopismach indeksowanych w bazie JCR (Journal Citation Re-ports).
We wprowadzeniu teoretycznym zaprezentowano zyczne podstawy dziaªania takich systemów jak oscylatory oparte na zjawisku tzw. spin-transfer torque, oscylatory oparte na spinowym efekcie Halla oraz oscylatory antyferromagnetyczne. Ka»dy z tych ukªadów zostaª opisany od strony teoretycznej, w szczególno±ci w kontek±cie zjawisk dynamicznych, takich jak oddzi-aªywanie z sygnaªami zewn¦trznymi. Dodatkowo, opisano metody numeryczne i statystyczne wykorzystane w badaniach uwzgl¦dnionych w rozprawie.
W dalszej cz¦±ci rozprawy zawarte s¡ wyniki przeprowadzonych bada« dotycz¡cych dynamiki os-cylatorów ferromagnetycznych, dynamiki osos-cylatorów antyferromagnetycznych oraz wytworzo-nych narz¦dzi modelowania i symulacji. Wykazano, »e oscylator oparty na zjawisku spin-transfer torque mo»e sªu»y¢ jako czujnik pola magnetycznego i tym samym stanowi¢ cz¦±¢ skªad-ow¡ gªowicy odczytowej dysku twardego. Zbadany zostaª ilo±ciowy wpªyw niejednorodno±ci zewn¦trznego pola na dziaªanie takiego urz¡dzenia. Zaproponowano metod¦ jednoczesnego odczytu dwóch niezale»nych pól magnetycznych przy zachowaniu wysokiego stosunku sygnaªu do szumu, mo»liw¡ dzi¦ki wykorzystaniu gªowicy opartej o oscylator typu spin-transfer torque. Przetestowano mo»liwo±ci realizacji oblicze« neuromor cznych przy u»yciu sieci poª¡czonych ze sob¡ oscylatorów opartych na spinowym efekcie Halla, demonstruj¡c przykªadow¡ imple-mentacj¦ tzw. pami¦ci stowarzyszonej oraz algorytmu detekcji kraw¦dzi. Przeanalizowano zachow-anie oscylatorów antyferromagnetycznych w kontek±cie ich interakcji z zewn¦trznym polem magnetycznym, zarówno na poziomie ogólnego rozwi¡zania równania dynamiki, jak i wybra-nych typowych przypadków szczególwybra-nych. Wykazano, »e wprowadzenie zewn¦trznego pola mag-netycznego mo»e prowadzi¢ do jako±ciowo nowych zjawisk w dynamice oscylatora, takich jak m.in. obecno±¢ przesuni¦cia fazowego zale»nego od wªa±ciwo±ci materiaªowych w odpowiedzi na sygnaª sinusoidalny lub mo»liwo±¢ rezonansowego wzmocnienia amplitudy obserwowanych oscylacji.
Narz¦dzie typu open-source, uªatwiaj¡ce przygotowywanie plików kon guracyjnych symulacji mikromagnetycznych oraz pó¹niejsz¡ analiz¦ danych, zostaªo opracowane w ramach przygo-towania do przeprowadzenia opisanych w pracy bada«. Na potrzeby przyszªego u»ytkownika udokumentowana zostaªa konstrukcja programu i typowe przypadki u»ycia, wraz z przykªad-ow¡ symulacj¡ magnetycznego zª¡cza tunelowego. Dodatkowo, przedstawiono fenomenologiczn¡ metod¦ szacowania poziomów szumu termicznego w magnetorezystywnych zª¡czach tunelow-ych, która wykorzystuje jedynie kilka ªatwo mierzalnych wielko±ci pomiarowych. Otrzymane rezultaty, porównane z wynikami eksperymentów, wskazuj¡, »e informacja o rezystancji, ob-j¦to±ci i czuªo±ci polowej jest wystarczaj¡ca w celu przewidzenia zarówno mocy szumu bi-aªego, jak i mocy szumu typu 1/f w typowym czujniku opartym o magnetyczne zª¡cze tunelowe mierzonym przy pomocy pr¡du elektrycznego o dowolnej amplitudzie.
Short note about the author
The author graduated from AGH University of Science and Technology, Faculty of Physics and Applied Computer Science in 2015, with M.Sc. degree in Technical Physics. He prepared his thesis as a Ph.D. student in Physics at AGH University of Science and Technology, Faculty of Physics and Applied Computer Science and as a visiting Ph.D student at Gothenburg Univer-sity, Faculty of Science.
The author has been the principal investigator in the National Science Center grant Prelu-dium Analysis of the dynamics of microwave nanooscillator driven by spin-polarized current in non-uniform time-dependent magnetic eld. He has also been an investigator in the follow-ing grants: National Science Center grant Harmonia Electric- eld controlled spintronic devices, Ministry of Science and Higher Education Diamond Grant Current-induced switching of tun-nel nanojunctions with perpendicular magnetic anisotropy in the presence of thermal eects, National Science Center grant Maestro Oligo - atomic metal/metal-oxide superlattices - novel materials with tunable magnetic and electric properties, National Science Center grant Opus SpinOrbitronics, The National Centre for Research and Development grant Lider Microwave spin electronic nano devices. He has received the doctoral scholarship from National Science Center Etiuda program and the Ministry of Science and Higher Education scholarship for Ph.D. students.
The author has presented four posters and ve oral presentations at international conferences in Europe, America and Asia. Overall, he has contributed into thirteen articles published in international journals from the JCR list. He has been the rst and corresponding author in ve of them, which are incorporated into this thesis and which have been published in the follow-ing journals: IEEE Transactions on Magnetics, Journal of Magnetism and Magnetic Materials, Physical Review B, Computer Physics Communications and Sensors. Additionally, the author has served as a peer reviewer in the area of spintronic oscillator simulations.
List of incorporated publications
[P1] J. Ch¦ci«ski, M. Frankowski, T. Stobiecki, Spin-torque oscillator read head in inhomo-geneous magnetic elds, c 2016 IEEE. Reprinted, with permission, from IEEE Transactions on Magnetics, Vol. 53, no. 4, April 2017. All simulations and all calculations presented in this work were performed solely by the author of this thesis, while both him and his supervisors participated in the results analysis and manuscript preparation.
[P2] J. Ch¦ci«ski, M. Frankowski, T. Stobiecki, Simultaneous readout of two adjacent bit tracks with a spin-torque oscillator, Journal of Magnetism and Magnetic Materials 446, p. 4953, January 2018. All simulations and all calculations presented in this work were per-formed solely by the author of this thesis, while both him and his supervisors participated in the results analysis and manuscript preparation.
[P3] J. Ch¦ci«ski, M. Frankowski, T. Stobiecki, Antiferromagnetic nano-oscillator in external magnetic elds, Physical Review B 96, 174438, November 2017. All simulations and all calcula-tions presented in this work were performed solely by the author of this thesis, while both him and his supervisors participated in the results analysis and manuscript preparation.
[P4] J. Ch¦ci«ski, M. Frankowski, MAGE (M- le/Mif Automatic GEnerator): A graphical interface tool for automatic generation of Object Oriented Micromagnetic Framework con gur-ation les and Matlab scripts for results analysis, Computer Physics Communiccon gur-ations 207, p. 487498, October 2016. The author of this thesis wrote the source code for the mif automatic generation module and performed the relevant tests concerning this part of the described soft-ware. Both authors participated in overall design, results analysis and manuscript preparation. [P5] J. Ch¦ci«ski, P. Wi±niowski, M. Frankowski, T. Stobiecki, Magnetic noise prediction and evaluation in tunneling magnetoresistance sensors, Sensors 2018, 18(9), 3055. All simulations and all calculations presented in this work were performed solely by the author of this thesis. The experimental part of the work was performed by dr Piotr Wi±niowski from Department of Electronics at AGH University of Science and Technology. All authors participated in the overall design, results analysis and manuscript preparation.
1 Introduction
In the year 1897, an Irish physicist sir Joseph Larmor described the precessional movement of a charged particle around the direction of an applied magnetic eld [1]. He showed that the frequency of this precession is constant in time and linearly proportional to the eld amplitude, with the proportionality constant later named the gyromagnetic ratio. The implications of the Larmor's calculations did not become entirely clear until the advent of quantum mechanics in the 1920s and 1930s, which brought a whole new level of understanding into the research on microscopic structure of solid state materials. In 1935, Soviet scientists Lev Landau and Evgeny Lifshitz analyzed the behavior of a ferromagnetic crystal placed in an external magnetic eld and derived their famous equation for magnetization changes in time [2]. In their approach, the conservative movement of the magnetization around the eld direction was accompanied by an additional phenomenological term corresponding to energy losses. This framework was then reworked into a more mathematically consistent form by an American physicist Thomas L. Gilbert in 1955 [3, 4]. Nowadays, it is customary to refer to the rst formulation as the Landau-Lifshitz equation and to the second formulation as the Landau-Lifshitz-Gilbert equa-tion; however, in the limit of small damping they both lead to the same physical predictions. The equation discussed by Landau, Lifshitz and Gilbert described a damped precession move-ment, in which the magnetization of a ferromagnetic material oscillates with a certain frequency and an amplitude that is inevitably vanishing as the time progresses. As such, it could not have been the basis for a reliable oscillator device. This situation changed in the late 1990s, when another American physicist, John C. Slonczewski, described a curious phenomenon occurring in thin ferromagnetic lms subjected to a ow of a spin-polarized current: a transfer of spin momentum from the current to the layer magnetization, which resulted in appearance of the so-called spin-transfer torque acting on the latter [5]. The researchers in the eld of spintron-ics, a new and rapidly developing domain of science investigating spin-related phenomena in the context of electronic devices, were quick to point out to the possible applications of this mechanism, most notably emission of spin waves [6, 7], layer magnetization switching [8, 9] and sustained microwave oscillations of magnetization [10] that could be easily read electrically thanks to magnetoresistance eects. The last notion led to the development of a new class of devices, known as spin-torque oscillators, which were capable of emitting electric signal in GHz regime. Later, dierent types of spintronic oscillators, based on the spin Hall eect instead of directly utilizing the spin-transfer torque, were also proposed [11, 12].
Today, investigations of oscillatory systems are one of the main focus points in the spintronics eld. The pan-European network Spintronic Factory, which gathers over 80 academic and indus-trial partners from the European Union and associated countries, considers microwave devices to be one of the four most important directions of technological development in spintronics, next to magnetic memories, magnetic eld sensors and spin logic devices [13]. The objective of the research presented in this thesis was to provide a contribution to this eort that could combine, in the ideal case, theoretical insights and technological application perspectives. Three dierent subtypes of spintronic oscillators were considered: classical spin-torque oscillators, spin Hall oscillators and antiferromagnetic oscillators, which are all described in details in the next chapter. The scope of the undertaken research varied in accordance with the maturity of the re-spective state-of-the-art for each subtype. The spin-torque oscillators, which have been around for about 15 years, are already a well-established class of devices with robust experimental realizations, comprehensive theoretical descriptions and multiple application proposals being available. Therefore, the investigations presented here were focused on selected application de-tails and possible challenges that may arise in the future as the technology moves towards large
scale utilization. The spin Hall oscillators, which are signi cantly younger than their spin-torque counterparts, have very promising properties but are comparatively less well-established. In this thesis, they are described in the context of performing neuromorphic computations, which is an idea that has drawn a lot of attention by itself and may establish a new major direction for the development of information technology. Finally, in the case of antiferromagnetic os-cillators, a very novel idea which has entered the spintronics discourse only very recently in 2016 and 2017, the research presented here is concerned mostly with the basic physical funda-ments that still have to be mapped out before a serious discussion about applications can begin. The works incorporated in this dissertation were grouped into three main topics: modeling of magnetization dynamics in ferromagnetic oscillators, modeling of magnetization dynamics in antiferromagnetic oscillators and simulation and modeling tools, with each of these topics constituting one chapter of the thesis. In order to provide the described research with necessary context, two additional chapters are included after this introduction, which contain theoretical introductions into the relevant physical phenomena and into the numerical methods which have been utilized. The thesis is concluded with a brief summary and a list of bibliographic positions.
2 Oscillatory systems in spintronics
The objective of this chapter is to present selected elements of spintronic oscillators theory. Spin-torque oscillators are discussed rst, as they provide a useful reference point for a description of dynamical phenomena that are crucial from applications point of view and, furthermore, of a dierent class of ferromagnetic oscillators based on the spin Hall eect. At the end of the chapter, the newly proposed setup idea for antiferromagnetic oscillators is also brie y presented.
2.1 Spin-torque oscillators
Spin-torque oscillators are the oldest and the most thoroughly researched type of spintronic oscillators, with their origin dating back to the era following Slonczewski's investigations on spin-transfer torque in the second half of 1990s. In the year 2003, a group of physicists from Yale University demonstrated electrical detection of a microwave-frequency signal coming from a nanomagnet subjected to ow of a spin-polarized current [10]. Their ndings very quickly became replicated and further developed, leading to a great number of experimental and the-oretical insights into the physical nature of that new class of spintronic systems [1418]. The basic conceptual scheme of a spin-torque oscillator is presented in the gure 1 (a). The core of the device consists of two thin (few nanometers of thickness) ferromagnetic layers: the so-called reference layer, which acts as a polarizer and a reference for electric readout purposes, and the so-called free layer, where the oscillations occur. The reference and the free layer are separated by a non-ferromagnetic spacer layer, which is typically also not thicker than a few nanometers. The magnetization of the reference layer should be signi cantly stier than the magnetization of the free layer, which in real nanodevices is usually achieved either by ma-nipulating material parameters such as layer thickness or by applying an additional localized bias eld. If this condition is ful lled, the magnetization equilibrium direction in the reference layer will at the same time determine the spin polarization direction ~p for the electric current owing through the system. The spin-transfer torque eect will then be able to counteract the natural energy losses originating from magnetic damping, sustaining auto-oscillations around the equilibrium direction in the free layer. Finally, the presence of magnetoresistance (giant magnetoresistance or tunnel magnetoresistance) eects will modify the resistance of the device based on the free layer magnetization position at a given moment, thus enabling electric readout.
m p electrons equilibrium direction free layer reference layer energy losses nanopillar nanocontact insulator material electrons electrons
(a)
(b)
FM FM S FM FM SFigure 1: (a) A conceptual scheme of a spin-torque oscillator. (b) Comparison of two spin-torque oscillator geometries: nanopillar and nanocontact. Here, FM denotes a ferromagnetic material and S a non-ferromagnetic spacer material.
in gure 1 (b): a nanopillar geometry and a nanocontact geometry. In the former, a magnetic tunnel junction stack is utilized as a basis for spatially con ned oscillations, while in the latter the ferromagnetic layer are in principle continuous and it is only the electric current which is localized by means of a point contact injection. In both cases, the diameter of the localized oscillatory part should never be larger than few hundreds of nanometers at most, in order to ensure that the current density is high enough to sustain stable oscillations, which together with the very small vertical size places spin-torque oscillators rmly in the nanotechnology's scope of interest.
The magnetization dynamics in the ferromagnetic layer is governed by the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation, which can be written as: [25]
d ~m
dt =−γ(~m× µ0 ~
H)− γα(~m× (~m× µ0H))~ − γaJ ( ~m× (~m× ~p)) + γaJβ ( ~m× ~p) , (1)
where ~m = M~
MS is the normalized magnetization vector of the free layer, ~p is the normalized
magnetization vector of the reference layer, γ is the electron gyromagnetic ratio equal to ap-proximately 28.025 GHz/T, ~H stands for the strength of the eective eld which determines the magnetization equilibrium position, α is a dimensionless damping parameter and β is a secondary term describing torque. The in uence of the polarized current exercising spin-transfer torque on the magnetization vector depends on the variable aJ, which is proportional
to the current density, and , which accounts for angle dependence of the spin-transfer torque: aJ = ¯ hJ eMSd , (2) = pJΛ 2 Λ2+ 1 + (Λ2− 1)~m· ~p, (3)
with ¯h being the reduced Planck constant, J the current density, e the elementary charge, MS
the saturation magnetization of the free layer, d the free layer thickness, pJ the spin
polariz-ation of the current, Λ the spacer layer parameter proposed by Slonczewski. As can be seen in equation 1, the general expression for spin-transfer torque includes both the dominant term along ~m×(~m × ~p) direction (the so-called "damping-like torque") and the secondary term along
~
m× ~p direction (the so-called " eld-like torque") which is multiplied by the coe cient β < 1. In many practical cases, Λ is set to 1 (thus eliminating explicit dependence of the torque on angle between free layer and reference layer magnetizations in Eq. 3) and β is assumed to be equal either to zero [1921] or to the damping parameter α [22, 23].
The eective eld strength ~H is an especially important parameter of Eq.1 as it determines the direction along which the oscillations occur, as well as their frequency. In the general case, ~H is de ned using gradient of the magnetic energy density F :
~
H = −∂F µ0MS· ∂ ~m
. (4)
There are numerous possible contributions to the energy density of the system F [24], including ferromagnetic exchange energy, demagnetization eld, magnetostatic interactions, anisotropy
1 it is necessary to use numerical techniques. A very prevalent example of this approach is a micromagnetic simulation [25, 26], where the respective energy terms are expressed as clas-sical continuous vector elds which are later subject to discretization in accordance with the nite dierence (or, less commonly, nite elements) method. OOMMF [27] and MuMax3 [28] are among the most popular software tools for micromagnetic simulations, utilizing parallel computing techniques and enabling for eective simulations of several dierent physical eects. In this thesis, OOMMF was used to perform the simulations described in the incorporated works P1, P2 and P5, while MuMax3 was used to perform the spin Hall oscillator simulations described at the end of chapter 4.
If the spatial variability of energy contributions in the system is negligible, it is often pos-sible to provide a more direct description of spintronic oscillators. In the case of spin-torque oscillators, a commonly utilized approach is based on the universal model of auto-oscillator, originally proposed by Slavin and Tiberkevich in 2008 [29, 30]. In this model, three dierential equations describing the time evolution of each of the magnetization components are replaced by one dierential equation for a complex value representing a generalized state of the oscil-lator. This is possible because the magnetization in a realistic ferromagnetic oscillator has an approximately constant magnitude equal to saturation magnetization MS, so that ~m =
~ M MS
can be the normalized magnetization vector which eectively has only two degrees of freedom. Without losing generality, one can denote the magnetization equilibrium direction as z axis, which leads to the following de nition of the complex variable c:
c = √mx− imy 2 + 2mz
=√P e−iφ, (5) where mx, my and mz are the components of the magnetization vector ~m. The second
repres-entation of c, based on complex number amplitude and phase, explicitly highlights the presence of two degrees of freedom in the system and gives both of them a clear physical interpretation: P corresponds to the oscillation power and φ to the oscillation phase, so that dφdt is the os-cillation frequency. The general auto-oscillator equation, based on the complex representation introduced here, now takes the form:
dc
dt + iωc + Γ+c− Γ−c = 0, (6) where ω is the oscillator frequency, Γ+ is the abstract representation of energy losses in the
system, Γ− is the abstract representation of energy gained due to the presence of the powering
spin-transfer torque (note the plus and minus sign convention - historically, Γ− was originally
named as "negative damping" in the system, as opposed to the real physical "positive" damping corresponding to energy losses and Γ+).
The exact expressions for ω, Γ+ and Γ−, which together determine the dynamics of the
os-cillator in the model, can all in principle be derived from equation 1 for any particular set of parameters de ning the problem. In practice, however, these calculations can become very tedi-ous (or even impossible to perform analytically), especially given the large variety of physically possible oscillator con gurations: for example, free layer magnetization direction pointing in the sample plane or out of plane, reference layer magnetization direction being parallel or or-thogonal to the free layer preferred direction, symmetry-breaking physical eects in the system being present or not can all aect the outcome profoundly. Fortunately, for most physically interesting systems it is su cient to approximate these parameters by their Taylor series
ex-pansions with respect to the oscillation power P = |c|2:
ω≈ ω0+ N P,
Γ+≈ ΓG(1 + QP ),
Γ−≈ σJJ (1− P ),
(7)
where ω0, N, ΓG, Q, σJ constitute a new set of parameters controlling the dynamic behavior
of the system. In this context, they should be understood primarily as mathematical expansion coe cients for quantities introduced in the equation 6. This level of generality, allowing one to describe multiple dierent types of oscillators within one uni ed approach, is one of the biggest advantages of the universal auto-oscillator model.
From equations 5 - 7, it is possible to derive the explicit equations for oscillation power and phase: dP dt = 2P · Re 1 c dc dt = 2P (−ΓG(1 + QP ) + σJJ (1− P )), (8) dφ dt =−Im 1 c dc dt = ω0+ N P, (9)
where Re and Im stand, respectively, for real and imaginary part of a complex term. As a result, the solution describing stationary oscillations with constant power P0 are described by:
P0 =
σJJ− ΓG
σJJ + Q
, (10)
ω = ω0+ N P0. (11)
From the rst expression it is clear that, since the power of actual oscillations can only be positive, for any system with given σJ and ΓG there exists a certain threshold current density
J = ΓG/σJ below which no energetically stable precession can occur. The second equation
determines the frequency of the oscillator, which is dependent on power. The parameter N is often called the non-linearity of an oscillator, since the direct dependence between oscillation frequency and oscillation amplitude is a deviation from the simple linear (harmonic) oscillator framework. Since for realistic physical values the power P0 always increases with applied
cur-rent (as can be seen from computing the derivative dP0/dJ), the non-linearity also determines
the way the oscillator frequency is going to change with current: a positive non-linearity will lead to frequencies increasing with current amplitude while a negative non-linearity will lead to frequencies decreasing with current amplitude.
2.2 Dynamical phenomena in spin-torque oscillators
As active auto-oscillatory systems, spin torque oscillators exhibit a number of interesting dy-namical properties that can be very useful from the application point of view. In this section, two dynamical phenomena are brie y described: the so-called injection locking, which occurs when a spin-torque oscillator interacts with an external alternating signal, and the mutual locking of multiple spin-torque oscillators.
2.2.1 Injection locking
The universal model of auto-oscillator provides a simple method of considering an external signal in uence. The equation 6 can be generalized [30] to:
dc
dt + iωc + Γ+c− Γ−c = Fee
−iωet, (12)
where Fe and ωe represent the amplitude and the frequency of an external excitation,
respect-ively. Physically, such an excitation usually is achieved by means of an external magnetic eld, a radio-frequency current owing through the junction or a spin wave. Nonetheless, the equa-tion 12 represents a general form of the excitaequa-tion, so that dierent physical realizaequa-tions will aect only the value of the parameter Fe. Based on equations 8 and 9, the explicit formulas for
oscillator power and phase now take the form [30]: dP dt =−2P (Γ+− Γ−) + 2Fe √ P cos(φ− ωet), (13) dφ dt = ω− Fe √ Psin(φ− ωet), (14) where it is assumed that there is no initial mismatch in phase between the oscillator and the external signal, and that the non-linearity is small enough for ω to remain approximately constant. The resultant system of two equations can be futher simpli ed if one assumes that the solution is stationary with approximately constant power P ≈ P0, which physically corresponds
to the limit of the external signal amplitude being small. If this is the case, then the equation 14 can be rewritten as:
d(φ− ωet) dt = ω− Fe √ P0 sin(φ− ωet)− ωe. (15)
The expression φ − ωet in the equation above can be interpreted as relative phase between the
oscillator and the external signal. This relative phase can be constant (i.e., both frequencies can match each other exactly) only if the following condition is ful lled:
|sin(φ − ωet)| = √ P0 Fe (ω− ωe) ≤ 1, (16) |ω − ωe| ≤ |F e| √ P0 . (17)
In other words, the spin-torque oscillator frequency can match the external excitation frequency ωe only within a limited interval of frequencies, usually called the phase-locking bandwith of
this oscillator. Such a behavior is in stark contrast with "passive" oscillatory systems, which always oscillate with the same frequency as the external excitation (for example, a ferromag-netic resonance of the free layer).
If the excitation frequency falls outside of the phase-locking bandwith, the spin-torque oscillator frequency will be equal neither to the excitation frequency nor to the free-running frequency ω (i.e, the one that would be observed without any external signals), but it will represent a sort of compromise between these two. It can be shown [30] that in such a case the resultant oscillator frequency ωosc will be equal to:
ωosc= ωe± s (ωe− ω)2− F2 e P0 , (18)
where the sign of the square root depends on whether the excitation frequency is higher or lower than the free-running frequency.
It should be noted that the analysis presented in this section was concerned only with a simpli- ed case of an oscillator with negligible non-linearity and no initial phase mismatch relative to the excitation. In the general case, both of these assumptions can be dropped, leading to more tedious calculations but qualitatively similar results [30].
The ability to lock a spin-torque oscillator to an external signal is of great importance for the applications in microwave signal generation or detection, since it provides a practically realizable way of increasing the oscillation power as well as their resistance to thermal noise. Because of that, exhaustive experimental and theoretical investigations of this phenomenon have been conducted since it was rst reported in the middle 2000s [17, 3133]. The work P1, which is incorporated into this thesis, utilizes a method of reading a rapidly changing magnetic eld signal based on the injection locking of a spin-torque oscillator, in a parametric locking setup where Fe is proportional to c∗ . The same readout method also provided a motivation
for the research presented in the incorporated work P2, where the same oscillatory system was investigated in a dierent set of external conditions.
2.2.2 Mutual locking
Apart from locking to an external signal, a spin-torque oscillator can also lock to another oscillator, or even to an array of co-dependent oscillators. Considering two oscillators with dynamics described by equation 6 as c1(t) and c2(t), the relevant system of equations can be
written after [30] as:
dc1
dt + iω1c1+ Γ+,1c1− Γ−,1c1 = Fec2(t− τ) dc2
dt + iω2c2+ Γ+,2c2− Γ−,2c2 = Fec1(t− τ),
(19) where Fe represents the complex symmetrical coupling amplitude and τ represents the time
delay between the coupling oscillators. If the time delay is su ciently small to ignore the eect of the coupling on the oscillator amplitude during the time delay interval, which is usually the case for physical oscillators, c(t − τ) can be expressed as:
c(t− τ) = c(t)eiωoscτ, (20)
where ωosc stands for the frequency of the considered oscillator.
The general solution of the system of equations 19 depends on various parameters. However, the most practically interesting case is the coupling of oscillators which have (slightly) dierent frequency and are otherwise physically identical, i.e., Γ+,1 = Γ+,2 = Γ+ and Γ−,1 = Γ−,2 = Γ−.
In the limit of small power deviations around stationary state and small non-linearity, the explicit system of equations for oscillator phases can be obtained as:
dφ1
dt = ω1− Im[Fec2(t− τ)/c1] = ω1− Im[Fe]cos(ω2τ − ∆φ) − Re[Fe]sin(ω2τ − ∆φ) dφ2
dt = ω2− Im[Fec1(t− τ)/c2] = ω2 − Im[Fe]cos(ω1τ + ∆φ)− Re[Fe]sin(ω1τ + ∆φ),
(21) where ∆φ = φ1− φ2. The coupling condition can be again determined by demanding that the
ω1 − ω2 2
≤ |Im[Fe]sin(¯ωτ )− Re[Fe]cos(¯ωτ )| , (23)
ω1− ω2 2 ≤ |Fe| · |cos(Arg[Fe] + ¯ωτ )| , (24)
with ¯ω = (ω1+ ω2)/2. It can be seen that the width of the mutual locking regime depends not
only on the coupling strength, but also on the relation between its complex phase and ¯ωτ. In particular, setting the complex number argument Arg[F e] to e.g. ¯ωτ + π/2 would suppress the mutual coupling entirely, regardless of how large |Fe| is. Physically, such a situation occurs for
example in the case of locking mediated by spin waves in the nanocontact architecture, where for certain distances between nanocontacts the locking bandwith drops to zero. As far as the frequency of the system is concerned, it is interesting to point out that it usually does not correspond to the frequency average ¯ω, but is determined by a more complicated expression following the solution of 21 instead [30]. Finally, it should be noted that, although the result presented here is easiest to derive for an oscillator with small non-linearity, it can be once again generalized to include the frequency dependence on power [30].
Similarly to injection locking, mutual locking is a phenomenon that has drawn the researchers' attention for a long time, since its usage may show to be the key to wider application of spin-torque oscillators [16, 3436]. In this thesis, at the end of chapter 4, mutual locking of spin Hall oscillators (which are discussed in the next section) is described as a basis for a neuromorphic computing scheme.
2.3 Spin Hall oscillators
Spin Hall oscillators are another type of ferromagnetic oscillators, which utilizes the spin Hall eect instead of direct spin-transfer torque in order to supply energy to the device. The pos-sibility of conversion between an electrical current and a spin current was originally predicted as early as in the 1970s [37, 38], but it has drawn particular attention in the 21st century as the spintronics became more developed, since it allows for a reliable and energetically e -cient way of controlling spintronic devices. This conversion can occur in heavy metal materials such as tungsten, tantalum or platinum due to spin-orbit coupling eects and it results in a pure spin current owing in the direction perpendicular to the original direction of the electric current ow [39]. If the spin current encounters an interface between the heavy metal and a ferromagnet located next to it, the interaction between the spin angular momentum and the local magnetization will lead to a non-zero torque acting on the latter, in a manner analogous to the spin-transfer torque exercised by the ow of a spin-polarized electric current[40, 41]. From the perspective of the magnetization behavior in the ferromagnetic material, the torque originating from the spin Hall eect is functionally identical to the spin-transfer torque. Both of them can excite the same type of dynamics, described by the LLGS equation (1), including stationary oscillations as one of the possible classes of solutions. As a result, both the micro-magnetic simulations approach and the auto-oscillator model described in the previous section can be applied directly to the description of spin Hall oscillators, which further emphasizes the generality of these methods. Similarly, the dynamic phenomena derived from the general auto-oscillator model such as injection or mutual locking will also occur in the case of spin Hall oscillators interacting with external signal or with each other, respectively.
m electric current spin current equilibrium direction ferromagnetic layer heavy metal energy losses
(a)
(b)
1000 2000 3000 4000 0 500 1000 1500 2000 x [nm] y [n m] 0 30 60 90 120 150 180 current density [MA/cm2]Figure 2: (a) A conceptual scheme of a nanopillar-based spin Hall oscillator. (b) Simulated current density in an example array of eight nanoconstriction-based spin Hall oscillators. high current densities in order to sustain auto-oscillation modes. One common approach is to utilize a ferromagnetic nanopillar placed on top of a heavy metal layer (see gure 2 (a)) [41, 42]. This setup can be modi ed in order to include a non-ferromagnetic barrier and then another ferromagnetic layer on the top of the original one, mimicking the structure shown in gure 1 (a) and allowing for giant or tunneling magnetoresistance readout mechanism. Alternatively, a continuous heavy metal/ferromagnet lm can be patterned to form one or more nanoconstric-tions, where the current density is much higher than in the surrounding area (see gure 2 (b)) [36, 43]. The auto-oscillation modes are then clearly localized and can be observed not only electrically, but also by using optical methods [42]. Compared to the nanopillar geometry, the nanoconstriction geometry provides better basis for investigating mutual synchronization phe-nomena, but suers from lower output power since it has to rely on relatively small increase in magnetoresistance value for electrical readout. A number of other spin Hall oscillator geometries have also been reported in the literature, including a nanogap between two electrodes inject-ing current into a heavy metal/ferromagnet layer [12] and an elongated nanowire geometry [44]. Spin Hall oscillators are an intensively researched class of spintronic devices, especially since their nanofabrication process is less complex compared to traditional spin-torque oscillators, which brings them closer to the perspective of wide-scale applications. In this thesis, at the end of chapter 4, a novel scheme for low-energy neuromorphic computations based on a nanocon-striction spin Hall oscillator array is investigated.
2.4 Antiferromagnetic oscillators
So far, the discussion in this chapter was concerned with ferromagnetic materials, where the exchange interaction favors a parallel alignment of adjacent magnetic moments in the material. During the last few years, it has been suggested that spintronics-based oscillators can also be realized in an entirely dierent class of materials, which display antiferromagnetic properties in-stead: the exchange interaction favors an anti-parallel alignment of adjacent magnetic moments. This section provides a brief introduction into the dynamics of antiferromagnetic oscillators and their predicted properties.
For most practical purposes related to oscillation analysis, the internal state of a ferromag-netic material could be described with a single physical quantity, namely the magnetization
to be introduced in order to describe the collective behavior of the magnetic moments [45]. This happens because the magnetic moments inside an antiferromagnet are usually ordered in sub-lattices which, although strongly coupled, have to be described by separate physical quantities. In the simplest but still practically useful case, there exist two of those sublattices and their magnetic moments compensate each other (no net macroscopic magnetization) if no external eld is present. Figure 3 illustrates such a system with two vectors ~M1 and ~M2 representing
magnetizations of each sublattice. For dynamics description, it is convenient to work with the net magnetization ~M = ~M1 + ~M2 and the so-called Néel vector ~L = ~M1 − ~M2 instead. In
the equilibrium state, the net magnetization is equal to zero, but if an external excitation (for example a spin current or a strong magnetic eld) occurs, ~M can have a non-zero value.
antiferromagnet
excitation
M
1M
M
2L
Figure 3: Néel vector and internal magnetization in an example antiferromagnet model. Just like in the case of ferromagnets, in order to achieve a robust oscillatory system in an antiferromagnet, it is necessary to have some sort of stationary excitation that will provide an energy ow into the system and thus counteract the natural energy losses. In the commonly proposed design, a constant spin current is utilized for that purpose. The dynamics of an anti-ferromagnetic material in the presence of such a stimulus was rst analyzed by Gomonay and Loktev in 2008 [45], who proposed the Lagrange function of the system:
L = 2γ2χM2 ˙ ~ L 2 − γMχ2 ˙ ~ L× ~L· ~H+ χ 8M2 ~ L× ~H 2 , (25) where it is assumed that the Néel vector has no signi cant spatial variability across the sample and there is no anisotropy present in the system, χ is the static susceptibility of an antiferro-magnet, M is the magnetization value of a single sublattice (| ~M1| = | ~M2| = M in gure 3),
~
H is the external magnetic eld. After introducing spherical coordinates with azimuthal angle φ and polar angle θ, the relevant Lagrange equations determining the dynamics of the system can be determined: d dt δL δ ˙φ − δL δφ = δR δ ˙φ, (26) d dt δL δ ˙θ − δL δθ = δR δ ˙θ, (27) with R representing the external excitation, which for spin current takes the form [45]:
R = −αAF 2γM ˙ ~ L 2 + βs ~ L×L~˙ · ~p, (28) where αAF is the damping coe cient, βs is the parameter determining the amplitude of
direction.
The physical problem of describing the antiferromagnet dynamics in the presence of a spin current will be now solved for a simple case of external magnetic eld having amplitude H and pointing in the opposite direction as the vector ~p (chosen as -z direction in Cartesian coordin-ates). The rst step is to express L as well as R in terms of φ and θ explicitly, assuming that |~L| ≈ 2M: L =φ˙ 2 sin2θ +θ˙ 22χ γ2 − ˙φ 4χ γ Hsin 2θ + χ 2H 2sin2θ, (29) R = −φ˙ 2 sin2θ +θ˙ 22M α AF γ + 4 ˙φβsM 2sin2θ. (30)
By inserting the expressions above into equations 26 and 27, the equations of motion for φ, θ can be obtained: ¨ φsin2θ + ˙φ ˙θsin2θ +1 2 ˙ θγHsin2θ +γM αAF χ ˙ φsin2θ = γ ˙Hsin2θ + γ 2β sM2 χ sin 2θ, (31) ¨ θ + γM αAF χ ˙ θ− 1 2 ˙ φ2sin2θ + ˙φHsin2θ = γ 2H2 8 sin2θ. (32) The obtained set of equations looks relatively complicated. However, it can be noted that constant θ equal to π/2 satis es equation 32 and corresponds to a stable solution where the vector ~L lies in the xy plane (plane perpendicular to the direction of ~p and ~H). The equation 31 takes then the following form:
¨ φ +γM αAF χ ˙ φ− γ ˙H = γ 2M2 χ βs. (33) It is convenient now to introduce ωex = γM/χ, τ = −γMβs and rewrite the result as:
¨ φ ωex + αAF · ˙φ − γ ˙H ωex + τ = 0. (34) In the case of no external magnetic eld present, this equation leads to a straightforward solution ˙φ = −τ/αAF, θ = π/2. Physically, it corresponds to the magnetizations of both
an-tiferromagnetic sublattices oscillating in the xy plane of the sample with constant frequency. What is more, it can be noted that, since the expression on the left-hand side of 34 depends only on eld derivative and not on eld value, this particular solution will remain valid for any constant value of H.
If H is not constant, the solution of equation 34 takes the form: ˙ φ =− τ αAF + γe−λt Z t −∞ dH dt0 e λt0 dt0, (35) where λ = ωexαAF determines how quickly the system is able to relax after being excited by a
antiferromagnet
spin
current
heavy metal
spin
current
spin Hall effect
inverse
spin Hall effect
THz oscillations
Figure 4: Basic scheme of an antiferromagnetic oscillator.
theoretical exercise due to lack of easy method to measure the antiferromagnetic oscillation output. The rst proposal for such a robust method was made by Khymyn et al. in 2017 [49]. This was possible thanks to considering an additional term in the dynamics equation, which corresponded to an easy-plane (xy-plane) anisotropy axis. It was shown that such a term leads to a modi ed expression for ˙φ:
˙ φ→ ˙φ + γHeωex 4pα2 AFω2ex+ 4τ2/α2AF cos 2τ αAF · t , (36)
where He stands for the easy-plane anisotropy eld strength. The fact that it is the
oscilla-tion frequency ˙φ and not just the oscillaoscilla-tion phase φ which changes sinusoidally with time is crucial from the application point of view, as it enables for easy detection of the emitted signal by means of inverse spin-Hall eect. The basic scheme of the device considered can be seen in gure 4. The powering electric current ows through a heavy metal layer, where it is converted to spin current due to the spin Hall eect. This spin current ows to the adjacent antiferromagnetic layer (which can be electrically insulating), where it excites oscillations. The magnetization rotation creates another spin current, which ows back to the heavy metal layer by means of a mechanism known as spin pumping [50, 51]. Finally, the inverse spin Hall eect in the heavy metal layer converts the output spin current to the electric signal. It can be shown that this electric signal is linearly proportional to the ˙φ term in the dynamics equation solution [49], which means that the additional term shown in 36 is crucial for obtaining a non-constant oscillatory result.
The antiferromagnetic oscillators constitute a qualitatively new class of spintronic oscillators, with a great number of potentially bene cial properties. Most importantly, their characteristic frequencies are in the range of hundreds of GHz or single THz (as denoted in gure 4), which is dierent from ferromagnetic oscillators that rarely exceed several GHz in their operating frequency. Technologically, frequencies around 1 THz order of magnitude are very interesting because of their multiple potential applications for data processing, communication, biologically safe security scans and spectroscopy [52]. At the same time, relatively few physical mechanisms can be used to generate signals with such frequencies, as the corresponding wavelengths tend to be too small for most eects belonging to classical physics realm and at the same time too big for most quantum-mechanical eects. Among these few known mechanisms, antiferromagnetic
oscillators are predicted to compare favorably in terms of oscillation amplitude and operating temperature, and at the same time to display very large quality factors [49]. Therefore, their physical realization could become a breakthrough in the terahertz technology in the nearest future.
The original scheme by Khymyn et al. [49] considered only systems with no external eld being present, i.e. H equal to zero. The work P3, which is incorporated into this thesis, gen-eralized their proposed solution in order to include cases with non-zero eld derivative and to investigate various eects of an interaction between a realistic antiferromagnetic oscillator and an external magnetic eld.
3 Numerical methods and algorithms
The previous chapter was focusing on physical phenomena underlying the operation principles of dierent types of spintronic oscillators, in order to supply the reader with a brief introduction into the topic of the thesis. This chapter instead focuses on discussing algorithms and methods, i.e., the numerical and statistical learning methods that were used throughout conducting the research described in the thesis. Three main issues are provided with a brief introduction: the problem of space as well as time discretization in the computational micromagnetism, the linear regression and cross-validation methods and the emerging neuromorphic computations paradigm in the context of oscillatory devices.
3.1 Numerical approach to micromagnetism
In the previous chapter, it was brie y mentioned that micromagnetic simulations are one of the most general and most widespread approaches to modeling of spintronic oscillators based on ferromagnets. As such, they were utilized in incorporated works P1, P2 and provided a direct motivation for the incorporated work P4 in this thesis. The basic idea of a micromagnetic simulation is very simple and amounts to LLGS equation 1 being solved numerically on a mesh grid of a given size. The popular software tools for these simulations, such as OOMMF [27] and MuMax3 [28] are capable of performing that operation in an e cient way for a wide range of physical parameters. However, a few comments regarding the applications and limitations of the micromagnetic method can still be formulated.
3.1.1 Space discretization
From the physical point of view, micromagnetism is a theory of a continuous medium. This property of the micromagnetic theory can be a source of confusion sometimes, since it seems to clash both with the underlying physical reality of a ferromagnetic material, which can be understood as a lattice of adjacent discrete spins, and with the common paradigm of run-ning simulations on a discretized grid. In fact, however, it is perfectly consistent with both of these notions: micromagnetism arises as classical eld theory which approximates quantum-mechanical reality for length scales signi cantly larger than the atomic structure [24], but which in practice usually has to be re-discretized into a simulation grid in order to allow for a numer-ical calculation.
The spatial discretization can have a number of dierent forms, depending on the numerical method used. In the nite dierences method, the grid is constituted by a regular (usually rect-angular) mesh, while in the nite elements method there is no clear constraint on the geometry, as long as the investigated system is unambiguously divided into a set of smaller elements. Understandably, the latter approach allows for greater exibility, which can be especially im-portant in the case of modeling of three-dimensional structures. However, this exibility comes at a computational price [53], and in the context of micromagnetic simulations of spintronic devices it is often not needed, as majority of the considered geometries are at or nearly at systems with relatively regular shapes. Therefore, most state-of-the-art micromagnetic solvers, including OOMMF and MuMax3, rely on a nite dierences method with a rectangular mesh. When choosing the simulation cell size, it is important to consider that too large simulation cells (typically > 10 nm) may lead to physically incorrect results as too coarse grid may fail to capture the details of the magnetization structure [54]. On the other hand, too small simulation cells tend to be computationally expensive and thus should generally be avoided. In the
ex-treme case, where the simulation cell size approaches atomic lattice constant (typically fractions of a single nm), the obtained results may have no physical interpretation, even though they are mathematically correct solutions of the micromagnetic problem [55]. This happens because the micromagnetic approximation itself has physical limits, which are not explicitly present in the numerical scheme. The micromagnetic simulations presented in this thesis utilized lateral cell sizes ranging from 2nm to 4nm, which ensured reasonable computation times as well as physically reliable results.
3.1.2 Time discretization
In order to determine the oscillation dynamics numerically, the evolution of the system gov-erned by equation 1 has also to be discretized into a nite sequence of timesteps, which are then handled by an iterative method for dierential equation solution, usually coming from the Runge-Kutta family. This procedure represents yet another layer of approximation of the mi-cromagnetic problem, which can be a source of errors when not handled correctly. Similarly to spatial discretization, time discretization presents an optimization problem where accuracy has to be considered against computational cost. Too large timesteps may not only fail to capture the details of the magnetization evolution, but lead to numerical instabilities as well. On the other hand, too short timesteps directly increase the numerical simulation time. In practice, the optimal timestep size depends on the speci cation of the problem, especially on the en-ergy landscape which determines the eective eld amplitude and, in turn, the characteristic frequency of the magnetization in the system. For eld amplitudes typically encountered in ferromagnetic oscillator simulations, the optimal timestep is usually of the order of magnitude of 0.1 ps, which is the value used in all micromagnetic simulations presented in this thesis, apart from the simulations presented at the end of chapter 4 that were relying on automatic timestep choice by the dierential equation solver instead. Finally, it is interesting to note that the characteristic timescale of exchange interaction in a ferromagnetic material, which in the micromagnetic approximation is assumed to happen instantaneously, is of the order of magnitude of approximately 0.01 ps [56].
3.2 Neuromorphic computations based on spintronic oscillators
Neuromorphic computations are a relatively new paradigm within computer science, which seeks to overcome the limitations of the more traditional von Neumann architecture by mim-icking the structural features of neural systems that are present in living organisms. As the hardware solutions based on CMOS are approaching their anticipated physical limits, neur-omorphic computations are expected by many to provide a technological breakthrough, by enabling large scale calculations for an extremely small energy cost [57, 58]. One commonly cited example is the human brain, which operates at energy scales several orders of magnitude lower than what would be required from a silicon computer in order to perform an equivalent task. The unique architecture of the brain makes it also relatively fault-tolerant and extremely e cient at processing large data sets. At the same time, the biological material which nerve cells are built of has some very clear disadvantages: compared to silicon hardware, living organism neurons are orders of magnitude slower, have less potential for building high-density structures and can only work in a very limited range of temperatures. Because of that, the main idea behind neuromorphic computations paradigm is to combine the architectural solutions from the brain with the materials provided by modern nanotechnology and solid state physics, in the hope of achieving the best of both worlds.
tronic devices have drawn a lot of attention due to their resilience, exibility and relatively high technology readiness level [57, 58]. In particular, spintronic oscillators seem interesting in this context due to multiple physical properties that make them similar to biological neurons, such as signi cant non-linearity and natural ability to interact with each other [58, 59]. In this section, a brief overview of the proposed solutions for neuromorphic computations based on spintronic oscillators are provided. Additionally, at the end of chapter 4, a micromagnetically simulated demonstration of some of those solutions are described as a contribution to this thesis.
3.2.1 Associative memory
One of the most ubiquitous ideas to realize neuromorphic computations using a network of spintronic oscillators is based on the associative memory scheme. The general principle behind associative memory is straightforward and can be described as a direct comparison of a given vector of features ~u = [u1, u2, ..., uN]against a xed dictionary of size M consisting of constant
vectors ~v1, ~v2, ..., ~vM, where ~vk = [vk
1, v2k, ..., vNk]. The comparison result should be the dictionary
vector characterized by the lowest distance to ~u, usually de ned based on the Euclidean metric d(~u, ~v) = pP(ui− vi)2. Despite being a relatively simple concept, the associative memory
lies at the core of many crucial applications such as image recognition, speech recognition, contextual searches or detection of spatiotemporal events, which makes its eective realization extremely important from the technological point of view [60].
In the last few years, a number of architectures for implementation of associative memory using spintronic oscillators have been proposed[6062]. The most typical realization is the so-called frequency-shift keying, where oscillators form an interconnected network and where patterns are represented by dierences in their frequencies. The number of utilized oscillators should be equal to the number of features N and the frequency of i-th oscillator should be given by:
ωi = ω0+ ∆ω(ui − vi), (37)
where ω0 is the base frequency common for all oscillators and ∆ω is the frequency deviation
that depends on the dierence between tested and dictionary vector elements ui− vi. As the
oscillators couple with each other, the output quantity (usually represented by the total power of the signal generated by the network) behaves in a non-linear way, strongly increasing if the vectors ~u and ~vk are similar to each other. The associative memory can be then easily
imple-mented by constructing a system consisting of multiple oscillator networks, each representing one dictionary vector ~vk, and a winner-takes-all circuit that identi es the maximum degree of
match by choosing the network with the biggest output power, as illustrated in the gure 5. 3.2.2 Edge detection
Another application that is commonly proposed in the context of oscillator-based neuromorphic computations is edge detection, which is the core operation of modern image processing work-pipes and thus of utmost importance for elds such as robotics or computer vision. In a sense, the edge detection algorithm discussed here is a natural development of the associative memory algorithm presented in the previous subsection. Instead of comparing the tested image with a series of constant dictionary images, it is possible to compare a given image fragment with a so-called mask or a lter function, which is constructed speci cally in order to capture edge-like visual features. In general, the choice of a mask depends on many dierent factors, as multiple approaches to that problem have been developed in the eld of computer science.
oscN osc2 osc1 .. . oscN osc2 osc1 .. . oscN osc2 osc1 .. . . . .
winner-takes-all
u v
vs.
1u v
vs.
2u v
vs.
M . . .P
1P
2P
MFigure 5: Illustration of the associative memory architecture. Each of the M oscillator networks compares the input vector ~u with a given dictionary vector ~vk, resulting in an output power
Pk. The winner of the comparison is then determined by choosing the network with the biggest
output power.
Regardless of the speci c mask choice, the oscillator-based algorithm is able to compare them with a small fragment of the analyzed image in order to detect edge-like features. One imple-mentation that has been proposed in the literature is to use the oscillator array to nd an approximate Euclidean distance between the image fragment and the mask, and then to calcu-late their convolution value as a function of that distance [62, 63]. This approach mimicks the traditional way of detecting edges in digital image processing, which is also based on calculat-ing convolutions. However, it may not be very practical, since it requires both input vectors to be normalized and, additionally, relies on the assumption that the power of the system scales with the frequency dierence in a strictly parabolic manner. In realistic spintronic oscillator devices, this assumption is unlikely to always be ful lled, since oscillators tend to synchronize with some non-zero relative phase, which causes deviations from the parabolic regime for larger arrays. An alternative approach would be to replace the convolution value directly with the degree of match, providing an estimate of how much a given fragment of the image resembles an ideal edge structure. This way results in an approximation that is more coarse, but less reliant on the assumptions about power-frequency dierence scaling, which can be non-trivial to determine for realistic arrays. Additionally, the circuit architecture is simpli ed signi cantly because there is no longer need to normalize the image fragment input, which decreases the power consumption by the factor of two. Finally, it is interesting to note here that another, independent algorithm to detect edges with a network of coupled oscillators has also been pro-posed, utilizing visual saliency methods which measure the dierence between a single pixel and its immediate surroundings [64]. However, its usage is limited to 2D oscillator arrays, not allowing for easier 1D oscillator chain implementations.
3.3 Linear regression and cross-validation
Linear regression and cross-validation are two statistical analysis techniques that were utilized in this thesis. Their role was fundamental in constructing the model described in the work P5, which allows for prediction of magnetic noise levels in ferromagnetic layers and which was calibrated using experimental data. This section presents a brief description of these two techniques.
3.3.1 Linear regression
Linear regression is one of the most fundamental and widespread methods of statistical predic-tion. In its general form, an input vector XT = (X
1, X2, ..., Xp), which contains a set of data
variables, is used to forecast a single real output value Y [65, 66]: Y ≈ f(X) = β0+
p
X
j=1
βjXj, (38)
where f(X) represents the regression model and β coe cients are to be identi ed. In statist-ical analysis of real-life problems, it is usually assumed that the actual outcome is going to be in uenced by some inherently random error , which makes it impossible to predict Y with full accuracy given that Y = f(X) + . The random variable is often assumed to have a Gaussian distribution, but this assumption is not necessary for the regression approach to be valid as long as the expected value of equals zero.
It is important to note that the equation 38 can be very easily generalized by using simple transformations of the original variables such as Xk = log(X1), Xk =
√
X1, Xk = X12 etc.
These generalizations are still considered examples of a linear regression procedure, even if some or all of the Xj variables are explicitly non-linear functions of the original input variables.
In other words, the term 'linear' here applies to the function f(X), not to the individual vari-ables Xj.
To obtain the coe cients β, the least-squares method is usually used. The residual sum of squares, de ned as:
RSS(β) =
N
X
i=1
(yi− f(~xi))2, (39)
should be minimized by the choice of β. Instead of a general input X and a general response Y, concrete pairs of data points (~xi, yi) are now considered. These pairs can be also expressed
as a single input matrix ˆX, where each row is a separate input vector ~xi, and a single output
vector ˆY , where each element is a separate input yi (for consistency, the matrix notation will
be maintained here even though ˆY always has only one column). In this notation, the residual sum of squares has the following form:
RSS( ˆβ) = ˆ Y − ˆX ˆβ T ˆ Y − ˆX ˆβ , (40)
with ˆβ denoting the vector of β coe cients. If all columns of ˆX are linearly independent (which is usually the case for real-life data), the minimization condition for 40 can be found from the derivative: ˆ XT Yˆ − ˆX ˆβ= 0, (41) ˆ β =XˆTXˆ −1 ˆ XTY .ˆ (42)