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(1)1. Faculty of Computer Science, Electronics and Communications Department of Electronics. Ph.D. thesis. Magnetization and magnetoresistance dynamics of spin electronics nanodevices. Marek Frankowski. Supervisors: Prof. Tomasz Stobiecki and Dr Maciej Czapkiewicz. Kraków, July 2016.

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(3) Marek Frankowski. Magnetization and magnetoresistance dynamics of spin electronics nanodevices. AGH University of Science and Technology.

(4) I would like to express my sincere gratitude to my supervisor Prof. Tomasz Stobiecki for his patient guidance throughout my Ph.D. course and invaluable advices. This work would not be completed without his constant support and teaching of the nanomagnetism and spintronics. I am deeply grateful to my co-supervisor Dr Maciej Czapkiewicz for sharing his knowledge and experience, and introducing me into micromagnetic simulations. I would like to specially thank Dr Witold Skowronski ´ for valuable comments and insight into experimental techniques and Jakub Checi ˛ nski ´ for fruitful discussions and collaboration. I also thank all members of Magnetic Multilayers and Spin Electronics group: Monika Cecot, Sławomir ˙ Zietek, ˛ Dr Piotr Wi´sniowski, Michał Dabek, ˛ Dr Jerzy Wrona, Dr Antoni Zywczak, Wiesław Powro´znik, Dr Jarosław Kanak and Dr Zbigniew Szklarski for their help and encouragement.. Various parts of this work were supported by Polish Ministry of Science and Higher Education Diamond Grant no. DI 2011001541, NANOSPIN Grant No. PSPB-045/2010 from Switzerland through the Swiss Contribution, Harmonia-UMO-2012/04/M/ST7/00799 by National Science Center, Poland and Preludium UMO-2015/17/N/ST3/02276 from National Science Center, Poland. Numerical calculations were supported by PL-GRID infrastructure.. I dedicate this work to my beautiful wife Anna, who has been bearing with me for so long..

(5) Contents. 1. Introduction. 2. Background of the research. 3. Current-driven dynamics. 4. Towards low power consumption: voltage-driven dynamics 46. 5. Development of calculation methods, tools for data analysis and simulations. 6. Summary Bibliography. 8. 76 78. 10 29. 54.

(6) List of Figures. 2.1 Scheme of MTJ multilayer structure. 10 2.2 Real MTJ device after nanolithography process with developed contacts (top view). The MTJ pillar itself is located in the intersection of the electrodes. Courtesy of W. Skowronski. ´ 11 2.3 Simplified Julliere's model of the MTJ and electron band structure in function of energy E for P and AP states with vacuum barrier. Case T = 0K. 11 2.4 Illustration of the LLG equation quantities in the case of a single spin. 13 2.5 Toy model of the STT in-plane component origin. Electrons flow from left to right, enter a polarizing layer and exiting it spinpolarized in the direction of its magnetization. Subsequently, they enter the free layer and change the spin polarization direction by transferring net torque to its magnetization vector. 16 2.6 Illustration of the LLGS equation quantities for the single spin of the free layer. 17 2.7 Toy model of the CIMS. The dots represent the state of the system in the energy profile. The torque can be use to temporarily add energy to the system and move it over the barrier Eb to the opposite state. 17 2.8 Conceptual plot for the spin-diode effect. (a) input AC current (b) resistance changes due to excitation brought on by the input (c) mixing voltage being the product of signals (a) and (b) - the red line indicates measured DC voltage. 19 2.9 Scheme of the MTJ and experimental setup for the spin-diode effect measurement. Microwave excitation is generated by the voltage source. So-called bias T is an element dedicated to the separation of constant and alternating components of the output signal (mixing voltage). 19.

(7) Short note about the author The author has graduated with MSc diplomas in Applied Computer Science from the Faculty of Electrical Engineering, Automatics, Computer Science and Electronics (2012) and Technical Physics from the Faculty of Physics and Applied Computer Science (2014) at the AGH University of Science and Technology. He prepared his thesis during a Ph.D. course in Electronics at Faculty of Computer Science, Electronics and Telecommunications at the AGH University of Science and Technology. He is currently an assistant professor in the Department of Electronics at the same faculty. He has been and is taking part in international research projects: the National Science Center, Poland, Grant Harmonia Electric-field controlled spintronic devices and a Project supported by a grant from Switzerland through the Swiss Contribution to the enlarged European Union Nanoscale spin torque devices for spin electronics. Marek Frankowski was the principal investigator of Diamond Grant Current-induced switching of tunnel nanojunctions with perpendicular magnetic anisotropy in the presence of thermal effects 2012-2015 and is currently the principal investigator of the National Science Center, Poland, Grant Preludium Low power consumption spintronics nanodevices based on magnetic tunnel junctions - electric-field-driven magnetization and magnetoresistance dynamics. He has published 8 papers and has H-index of 4 with 16 citations excluding self-citations. He has presented 6 talks and 3 posters at various international conferences in Europe, Asia and the USA..

(8) 1 Introduction Spintronics is an innovative and rapidly progressing branch of science which utilizes information from both the charge and spin of electrons in order to introduce new types of devices to the contemporary IT industry. It combines advanced material engineering, nanotechnology, specialized measurement techniques and cutting edge computational methods. Spintronics therefore requires an interdisciplinary approach as well as wide cooperation in order to develop effectively. Consequently, however, it also brings an equally broad spectrum of opportunities for applications such as microwave nanodetectors and generators, low magnetic field sensors, HDD read heads or MRAM. All of these fit very well into the current global trends in electronics for both miniaturization and energy conservation. As the industry and huge information centers have been getting more and more into digitalization, along with society’s demand for small, smart electronics in every area of life, the urge for efficiency in power consumption has been rising dramatically. Fortunately, the seemingly impossible has become possible, and items such as nonvolatile MRAMs are now available for commercial use. Not only has the pure spin electronics industry developed but vast and reliable companies have also become involved, which has ensured that scientific research can be immediately applied, if satisfactory results can be provided. There has therefore been a run to satiate the insatiable hunger for energy efficiency in the modern world. The author of this work has taken his part in this run, investigating magnetization and magnetoresistance dynamics of MTJs, which are the core of numerous essential spintronic nanodevices. The objective of this thesis has been to utilize the micromagnetic approach, develop new calculation tools and perform simulations to model and understand complicated spintronics phenomena occurring in MTJs in view of possible applications. It cannot be done reliably without experiments, therefore the presented research remains strictly correlated with measurements of real structures, in which the author has taken a part. During his Ph.D. studies the author has written or contributed to total of 8 publications and one currently in review. This thesis fully incorporates only the articles in which the author’s input is significant, namely five works with first (main) authorship and one work. HDD - hard disk drive MRAM - magnetic random access memory. Commercial MRAMs can be bought for example at https://www.everspin.com, MRAM-related news can be found at https://www.mram-info.com Example companies in MRAM industry: NEC, Aeroflex, Avalanche Technology, Crocus Nanoelectronics, Cypress, EverSpin, Freescale, Honeywell, Hynix, Infineon, Intel, Renesas, Samsung MTJs - magnetic tunnel junctions.

(9) 9. with a single co-author. All of the papers incorporated are preceded with a description of the author’s contribution and necessary comments. This thesis has been divided into 6 chapters. After the introduction, in the second chapter, the theoretical background of all the investigated phenomena and micromagnetic method are described. The third chapter consists of three papers concerning current-induced dynamics, namely magnetization switching. The fourth chapter contains a description of the most recent and promising idea: low power consumption devices using voltage control of magnetic anisotropy for both magnetization switching and FMR. The software and calculation methods necessary to either perform or speed up simulations on supercomputing infrastructure have been developed during this Ph.D. research and are described in the fifth chapter. The thesis ends with summary and concluding notes.. FMR - ferromagnetic resonance.

(10) 2 Background of the research Although each of the incorporated articles consists of an introduction and theory essentials, due to the specialized nature of journal publications and limited space particular details and especially the basics were not included in the publications. Therefore, in this chapter all the necessary principles of the effects investigated in this thesis, as well as modelling techniques, are grouped and described. For further reading and more detailed descriptions, please refer to1 , 2 and to works cited in each of the incorporated papers (some references may overlap with citations from this chapter).. 2.1 Magnetism and magnetic effects in thin films. 1 A. Aharoni. Introduction to the Theory of Ferromagnetism, volume 109. Oxford University Press, 2000. M. Donahue J. Miltat. Handbook of magnetism and advanced magnetic materials, 2007 2. 2.1.1 MTJs and magnetoresistance Apart from numerous spin electronics systems, such as single layer bar structures, multiferroic structures or GMR devices investigated in view of possible applications3 , 4 , 5 , the most common structure is the MTJ. It is a device consisting of a multilayer system with all three dimensions typically fabricated in nanoscale. The scheme of the typical MTJ is depicted in fig. 2.1. The whole structure is commonly deposited on a thermally oxidized silicon substrate. The core of the MTJ consists of two ferromagnetic layers, each single nanometers thick, separated by an insulating layer of similar thickness. Ferromagnetic electrodes are made typically of Fe, FeCo or FeCoB alloys and an insulating barrier of MgO. Buffer layers are used for the crystallographic growth of electrodes optimization, which will be discussed in Chapter 3. The problem of proper buffer layers selection, from a material properties and crystallographic structure point of view, is still under investigation. Some of the materials commonly used are Ta, Ru, CuN. Similar materials are used for the capping layers, whose additional purpose, apart from crystallographic structure engineering, is to protect the top electrode from scratching and oxidation. As the device size is typically from hundreds down to tens of nanometers it has to be equipped with micrometer-size contacts in order to perform measurements. Buffer layers are used to make electric contact at the bottom ferromagnetic electrode while the top electrode’s contact is deposited additionally. GMR - giant magnetoresistance K.D. Sattler. Handbook of nanophysics: Functional nanomaterials. CRC Press, 2010 4 T.J Meitzler E. Bankowski V.S. Tiberkevich A.N. Slavin O.V. Prokopenko, I.N. Krivorotov. Magnonics: From fundamentals to applications. Eds. S.O. Demokritov and A.N. Slavin, . Series in Applied Physics, 125, 2013 3. S. Tumanski. ´ Thin film magnetoresistive sensors. CRC Press, 2001 5. Capping layers. Ferromagnetic electrode (free) Insulating barrier Ferromagnetic electrode (reference). Buffer layers Substrate. Figure 2.1: Scheme of MTJ multilayer structure..

(11) 11. during the nanolithography process and is typically made of aluminium or gold. The details of the nanolithography process can be found in6 . As the resistance of the contacts is typically of several tens of Ω and MTJ resistance may vary from hundreds of kΩ down to single Ω, both top and bottom electrodes have two electric contact pads which allow for four-point resistance measurements. The picture of the real MTJ with contacts taken by means of tunnelling electron microscopy is depicted in fig. 2.2. The electrons can tunnel through the thin insulating barrier from one ferromagnetic electrode to the other. However, the probability of tunnelling strongly depends on the electron spin and the layer magnetization vector orientations. The effect of the MTJ resistance alternation due to changes in magnetization directions is called TMR. It is best explained using the extreme cases of P and AP states of the MTJ as examples. A toy model of the MTJ and electron band structure in each case is depicted in fig. 2.3. A voltage U applied to the junction introduces a change in the Fermi level between the opposite electrodes. Regarding P state (fig. 2.3 (a)) the spin-up current, being the major one, is high due to plenty of unoccupied states on the Fermi level in the opposite electrode, while spin-down electrons, being in the minority, have fewer possible states to occupy in the opposite electrode and therefore the probability of tunnelling is lower in their case.. W. Skowronski. ´ Current induced magnetization switching and noise characterization of MgO based magnetic tunnel junctions. PhD thesis, 2013 6. Figure 2.2: Real MTJ device after nanolithography process with developed contacts (top view). The MTJ pillar itself is located in the intersection of the electrodes. Courtesy of W. Skowronski. ´ TMR - tunnel magnetoresistance P - parallel, AP - anti-parallel orientations of magnetization vectors in ferromagnetic electrodes with respect to each other. AP state. P state. E. E. E. eU. Density of states. In AP state the majority spin-up electrons have few unoccupied states in the opposite electrode and low tunnelling probability resulting in a low current, while the spin-down electrons bring only a small contribution to the current since they are in a minority. This. E. eU. Density of states Figure 2.3: Simplified Julliere's model of the MTJ and electron band structure in function of energy E for P and AP states with vacuum barrier. Case T = 0K..

(12) 12. results in a rise of resistance in the case of AP state in comparison to P state. The TMR ratio of the MTJ developed as a result of typical technological processes at the AGH Department of Electronics can reach up to 200%7 , which is sufficient from an applications’ point of view. However, theoretical calculations shows that it is possible to achieve even 1500% of the TMR ratio8 . The exact value of TMR in particular element depends on the spin polarization obtained in both electrodes. Approximation of the TMR dependence on spin polarizations is commonly done using Julliere's model9 : TMR ≡. R ap − R p 2p1 p2 = . Rp 1 − p1 p2. (2.1). Spin polarizations can be obtained from theoretical models such as band structure calculations10 , tight-binding models11 or estimated using experimental techniques12 , 13 . In the case of the research presented in this thesis, TMR has always been obtained from the measured resistances of P and AP states. To calculate the total junction resistance for particular magnetization vectors orientations distorted from collinear arrangement, usually a phenomenological formula is used: R = Rp +. R ap − R p (1 − cosθ ). 2. (2.2). where the value of the angle θ has to be obtained using a model of magnetization vector dynamics. Such a model is described in the following sections.. 2.1.2 General magnetization dynamics - Landau-Lifshitz-Gilbert equation Magnetism itself originates from relativistic quantum mechanical effects. However, the application of the first principles theory for MTJ in order to determine time evolution of magnetization is not possible due to its size (at least tens of cubic nanometers). For a description of spin electronics devices, a phenomenological theory of Landau and Lifshitz extended by Gilbert is used14 , 15 :. ~ ~ dm ∂m ~ eff + αm ~ ×H ~ × = −γ0 m . dt ∂t. (2.3). ~ , the normalized magnetization vector, precesses around Here m ~ eff which is the effective field, derived the equilibrium pointed by H by minimizing the local energy densities. This precession is described by the first term of the LLG equation. Due to the energy dissipation in the system the precessional motion is damped towards the equilibrium with the material-dependent internal damping factor α which is described by the second term.. J. Wrona K. Rott A. Thomas G. Reiss-S. van Dijken W. Skowronski, ´ T. Stobiecki. Interlayer exchange coupling and current induced magnetization switching in magnetic tunnel junctions with MgO wedge barrier. Journal of Applied Physics, 107(9):093917, 2010 8 A. Umerski J. Mathon. Theory of tunneling magnetoresistance of an epitaxial Fe/MgO/Fe (001) junction. Physical Review B, 63(22):220403, 2001 9 M. Julliere. Tunneling between ferromagnetic films. Physics letters A, 54(3): 225–226, 1975 R ap - resistance of the AP state, R p - resistance of the P state, p1 and p2 - spin polarization in both layers respectively 7. J.C. Slonczewski. Conductance and exchange coupling of two ferromagnets separated by a tunneling barrier. Physical Review B, 39(10):6995, 1989 11 J. Mathon. Tight-binding theory of tunneling giant magnetoresistance. Physical Review B, 56(18):11810, 1997 10. M.S. Osofsky B. Nadgorny T. Ambrose S.F. Cheng-P.R. Broussard C.T. Tanaka J. Nowak J.S. Moodera A. Barry J. M. D. Coey R.J. Soulen, J.M. Byers. Measuring the spin polarization of a metal with a superconducting point contact. science, 282(5386):85–88, 1998 13 R.J. Celotta J. Unguris, D.T. Pierce. Low-energy diffuse scattering electronspin polarization analyzer. Review of Scientific Instruments, 57(7):1314–1323, 1986 θ - angle between magnetization vectors of the free and reference layers 12. Es. Lifshitz L.D. Landau. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion, 8(153):101–114, 1935 15 T.L. Gilbert. A Lagrangian formulation of the gyromagnetic equation of the magnetization field. Physical Review, 100:1243, 1955 m γ0 = 2.21 · 105 As - the gyromagnetic factor 14. LLG - Landau-Lifshitz-Gilbert.

(13) 13. A schematic of this behavior is depicted in fig. 2.4. To be able to investigate magnetization dynamics by solving the ~ eff needs to be defined and calculated. This LLG equation, the H process is described in the next section.. 2.1.3 Energies in ferromagnetic material ~ eff In ferromagnetic material treated as a continuous medium, the H can be obtained by calculating the derivative of energy E in considered volume V: 1 ∂E . (2.4) ~ µ0 Ms V ∂m The energy density is the sum of varying internal and external contributions:. ~ eff = − H. • Zeeman energy - the energy of the dipole in the external magnetic field Hex : Z ~ ex dV. ~ ·H EZeeman = −µ0 Ms m (2.5) V. • Anisotropy energy - the energy of a magnetocrystalline anisotropy originating from spin-orbit interaction and distinguishing a particular direction in crystal structure. By using ab-initio calculations, it is possible to obtain the anisotropy energy, however in the most common case of uniaxial anisotropy a phenomenological formula is used: Z E Anisotropy = (K1 sin2 α + K2 sin4 α)dV, (2.6) V. where K1 and K2 are anisotropy constants in J/m3 and α is an angle between magnetization and the anisotropy axis. This energy term can also be used for phenomenological calculations of anisotropy energies of differing origins but featuring a uniaxial character, like surface anisotropy in ultra-thin films. A so-called shape anisotropy originates from the demagnetization field, which is described later in this section. It is worth underlining that the uniaxial or uniaxial-like anisotropy favors two magnetization vector orientations (positive and negative) along the anisotropy axis and, therefore, can create two energy minima and two stable states of magnetization orientation. • Exchange energy - the quantum effect of the exchange described in micromagnetic approach by classical Heisensberg’s Hamiltonian for neighboring spins with indexes i, j:. ~j, He f f = − ∑ Ji,j ~ Si · S. (2.7). i6= j. where the coefficients Ji,j , called exchange integrals, measure the degree of overlap of electronic wave functions. In this case abinitio calculations are not only difficult and enormously timeconsuming but also face the problem of description of non-localized. Figure 2.4: Illustration of the LLG equation quantities in the case of a single spin. Ms - saturation magnetization, µ0 is vacuum permeability.

(14) 14. electrons in metallic material. In practice, the following formula for exchange energy is broadly used: EExchange = − A. Z. V. ~ · m. . ~ ~ ~ ∂2 m ∂2 m ∂2 m + + ∂x2 ∂y2 ∂z2. . dV.. (2.8). The so-called exchange constant A is well-known for all typical materials used in spin electronics. • IEC energy - apart from the exchange interaction within a single layer, exchange coupling can occur in the system where two films are separated by a very thin non-ferromagnetic barrier. In the case of metallic spacers it is called RKKY interaction16 , 17 , 18 , named after the researchers who discovered it. In the RKKY interaction, due to the spin-orbit interactions of electrons moving through metallic spacer of area S, another energy term arises: E IEC = −. Z S. ~1·m ~ 2 dS − J1 m. Z S. ~ 2 )2 dS, J2 (~ m1 · m. (2.9). ~ 1, m ~ 2 are normalized magnetization vectors of the two where m layers in question. IEC constants J1 , J2 in J/m2 are called bilinear and biquadratic, respectively. The RKKY model allows for the calculation of the those constants which oscillate with spacer thickness between positive and negative values. The positive bilinear constant corresponds to the ferromagnetic coupling and negative to the antiferromagnetic one. Biquadratic exchange seldom occurs, however it can lead to the interesting effect of favoring ~ 1 and m ~ 2 for J2 > 0. perpendicular orientations of m It has been shown that strong IEC can also be observed in the case of an insulating spacer layer19 . This is attributed to the vacancies in the atomic structure of the MgO crystal spacer. The energy of such a coupling can be described similarly to bilinear RKKY coupling (using eq. 2.9).. IEC - interlayer exchange coupling. Ch. Kittel M.A. Ruderman. Indirect exchange coupling of nuclear magnetic moments by conduction electrons. Physical Review, 96(1):99, 1954 16. T. Kasuya. A theory of metallic ferroand antiferromagnetism on Zener’s model. Progress of Theoretical Physics, 16 (1):45–57, 1956 18 K. Yosida. Magnetic properties of CuMn alloys. Physical Review, 106(5):893, 1957 17. J.P. Velev M.Y. Zhuravlev S.S. Jaswal E.Y. Tsymbal T. Katayama, S. Yuasa. Interlayer exchange coupling in Fe/MgO/Fe magnetic tunnel junctions. Evgeny Tsymbal Publications, page 3, 2006 19. • Demagnetization energy - the energy contribution from the magnetic field generated by the ferromagnetic material. This interaction is responsible for the antiferromagnetic coupling between magnetization vectors, which can result in shape anisotropy, spin wave creation (due to the competition between demagnetization and exchange energies) or net anti-ferromagnetic coupling between layers. In general, the demagnetization field originates from Maxwell laws:. ~ = 0, ∇×H ∇ · ~B = 0.. (2.10). ~ +M ~) by substituting the definition of field induction ~B = µ0 ( H ~ = −∇U and defining the scalar potential of magnetic field H. ~ = Ms · m ~ M.

(15) 15. inside and outside the ferromagnetic material we obtain:. ~ ∇2 Uin = ∇ · M,. (2.11). ∇2 Uout = 0.. It can be shown that together with the boundary condition for the surface of the material in question defined with the normal unit vector ~n: Uin = Uout , (2.12) ∂Uout ∂Uin ~ · ~n, − =M ∂n ∂n a unique solution for potential U and therefore for field demagnetization field H always exists and is as follows:. U (~r ) = −. 1 4π. Z. ~ Demag (~r ) = − 1 H 4π. Z. Z ~ (~r‘) ~ (~r‘) ~n · M ∇‘ · M 1 dV‘ + dS‘, 4π | ~r − ~r‘ | | ~r − ~r‘ |. ~ (~r‘) ~r − ~r‘ ∇‘ · M. | ~r − ~r‘ |3. dV‘ +. 1 4π. Z. (2.13). ~ (~r‘) ~r − ~r‘ ~n · M. | ~r − ~r‘ |3 (2.14) Here ~r stands for the position in space where the field is to be calculated and ~r‘ stands for the position of the ferromagnetic material generating the field. The energy of the demagnetization is equal to: 1 EDemag = − µ0 Ms 2. Z. V. ~ Demag (~r )dV. ~ (~r ) · H m. dS‘.. (2.15). 1 2. The factor results from the fact that for any two ~ri , ~r j during integration the energy is to be taken into account twice - from ~ri in ~r j and vice versa. Additionally, in the case of ellipsoid and uniformly magnetized ferromagnetic material, the demagnetization field inside it is given by the product of the so-called demagnetization tensor N and the magnetization vector:. ~ Demag = − N Ms m ~. H. (2.16). Although eq. 2.16 was introduced for a very specific case of ellipsoid sample, good approximations of N can be obtained for more typical shapes used in spintronics discs and rectangular prisms 20 , 21 .. 2.1.4 Current-induced dynamics - Slonczewski’s spin-transfer-torque term ~ eff In the previous section all the most common contributions to H have been described, however STT effect induced by spin-polarized current flow could not be included in that way. The mechanism of the STT was introduced in 1996 simultaneously by John Slonczewski22. J.A. Osborn. Demagnetizing factors of the general ellipsoid. Physical Review, 67(11-12):351, 1945 21 A. Aharoni. Demagnetizing factors for rectangular ferromagnetic prisms. Journal of Applied Physics, 83(6):3432– 3434, 1998 20. STT - spin transfer torque. J.C. Slonczewski. Current-driven excitation of magnetic multilayers. Journal of Magnetism and Magnetic Materials, 159 (1):L1–L7, 1996 22.

(16) 16. and Luc Berger23 . As the current enters the ferromagnetic layer it becomes polarized in the direction of the layer’s magnetization vector. Spin-polarized current flows through the second layer and due to the law of conservation of angular momentum, torque of the electrons is transferred to the layer. The idea of the STT is schematically pictured in the fig. 2.5.. L. Berger. Emission of spin waves by a magnetic multilayer traversed by a current. Physical Review B, 54(13):9353, 1996 23. Figure 2.5: Toy model of the STT inplane component origin. Electrons flow from left to right, enter a polarizing layer and exiting it spin-polarized in the direction of its magnetization. Subsequently, they enter the free layer and change the spin polarization direction by transferring net torque to its magnetization vector.. The discussed effect can be included in the LLG equation modifying it into the following form:. ~ ~ dm ∂m ~ eff + αm ~ ×H ~ × = −γ0 m + dt ∂t ~ × (~ ~ × ~p, +γ0 aJ m m × ~p) + γ0 bJ m. (2.17). where ~p is the unit vector of magnetization of the polarizing layer and therefore it points at the direction of electron’s spin to be trans~ . There are two terms added: ferred to the magnetization vector m the first one, so-called in-plane STT, is parallel to the plane contain~ and ~p and collinear to the damping term. The direction of ing m the in-plane term can be either parallel or anti-parallel to the damping term depending on the current sign as the torque coefficient is defined as: aJ =. h¯ ηj, 2eµ0 Ms t. (2.18). where t is the thickness of the layer on which the STT acts, η is the effective spin current polarization, j is the current density. Therefore the DC spin polarized current can be used to counteract the damping and induce a stable precession or, with a sufficiently high current amplitude, to overcome the damping totally and switch the magnetization to the opposite state. The second term is called out-of-plane torque or field-like-torque ~ as its vector is oriented perpendicularly to the plane containing m and ~p and is collinear with the term responsible for precession around the effective field.. h¯ - Planck constant divided by 2π, e - electron charge, µ0 - vacuum permeability. DC - direct current.

(17) 17. The out-of-plane torque factor is given by the formula: bJ = b0 + b1 J + b2 J 2 , where the values of parameters b0 , b1 and b2 are obtained usually through experiment24 . The LLG equation with STT extension is schematically depicted in fig. 2.6. Typically, out-of-plane torque is much smaller than in-plane torque, however in certain circumstances it can have a noticeable effect on the dynamics. Details of such an effect are described in the following section.. M. Frankowski J. Wrona T. Stobiecki G. Reiss-K. Chalapat G.S. Paraoanu S. van Dijken W. Skowronski, ´ M. Czapkiewicz. Influence of MgO tunnel barrier thickness on spin-transfer ferromagnetic resonance and torque in magnetic tunnel junctions. Physical Review B, 87(9):094419, 2013 24. 2.1.5 Backhopping effect The so-called backhopping effect, an occurrence of random backand-forth magnetization switching between P and AP states, can have some applications25 , however it is a parasite phenomenon for the most devices, such as MRAM. It originates from the interplay of the in-plane and perpendicular components of the STT in MTJs. Although vectors of both components are perpendicular to each other, they compete indirectly as the field-like torque modifies the precession speed and, therefore, the damping term of the LLG equation by ~ . For the sake of the backhopping effect, the fieldaffecting the ∂∂tm like component needs to be comparable with the in-plane one for the current density which is to be used for the switching operation. It is possible, as the quadratic formula for the perpendicular torque has a constant current-independent component, which has an interpretation as the interlayer exchange coupling and can be used instead of introducing the coupling by its energy contribution to the effective field. Regardless of the method used, the coupling effectively shifts the torque vs. current dependence up or down depending on the coupling sign. Therefore, whether the current amplitude for which the torques have similar amplitudes and opposite signs is located in the switching regime is dictated by the effective coupling value.. Figure 2.6: Illustration of the LLGS equation quantities for the single spin of the free layer. A. Thomas. Memristor-based neural networks. Journal of Physics D: Applied Physics, 46(9):093001, 2013 25. 2.1.6 Critical current and thermal stability The current necessary for overcoming the damping and achieving switching is called the critical current. From applications’ point of view it is crucial to minimize it, as the power dissipated in the MTJ is directly proportional to the current squared. The most fundamental role of the current is to supply the energy needed to overcome the energy barrier between the magnetization stable states (in the case of MTJ-based MRAM between AP and P states) and ensure switching. The CIMS idea is usually pictured as a toy model (see fig. 2.7). However, lowering the energy barrier between states as much as possible is not desirable in real devices, as random thermal fluctuations can switch the magnetization and therefore erase the data written in MRAM cell. The quantity that describes how durable the MTJ is, where the energy barrier Eb is at the appropriate level for switching at a particular temperature T is called thermal stability and is. CIMS - current induced magnetization switching. Eb. STT. P state. AP state. Figure 2.7: Toy model of the CIMS. The dots represent the state of the system in the energy profile. The torque can be use to temporarily add energy to the system and move it over the barrier Eb to the opposite state..

(18) 18. defined as follows: ∆=. Eb . kB T. (2.19). It is assumed that thermal stability must be of at least 40 for industrial applications26 . On the other hand, during the switching operation thermal effects are desirable and additionally enhanced by the Joule-Lenz heat emitted. As thermal energy contribution to CIMS is indeterministic, the probability of successful switching strongly depends on the time period during which the device is affected by the current. The critical current Jc necessary for successful CIMS during pulse of t p length is described by the formula27 , 28 : . Jc = Jintrinsic 1 −. 1 ln ∆. . tp t0. . .. (2.20). The Jintrinsic is called the intrinsic current. The t0 is the switching attempt time, which is typically equal to 1 ns. In the experimental approach can be extrapolated from the linear dependence of Jintrinsic Jc on ln. tp t0. .. k B - Boltzmann constant Y.M. Lee F. Matsukura Y. Ohno T. Hanyu-H. Ohno S. Ikeda, J. Hayakawa. Magnetic tunnel junctions for spintronic memories and beyond. Electron Devices, IEEE Transactions on, 54(5):991– 1002, 2007 26. J.Z. Sun R.H. Koch, J.A. Katine. Time-resolved reversal of spin-transfer switching in a nanomagnet. Physical Review Letters, 92(8):088302, 2004 28 J. Wrona K. Rott A. Thomas G. Reiss-S. van Dijken W. Skowronski, ´ T. Stobiecki. Interlayer exchange coupling and current induced magnetization switching in magnetic tunnel junctions with MgO wedge barrier. Journal of Applied Physics, 107(9):093917, 2010 27. 2.1.7 Spin-diode effect In the previous sections the DC current effects have been discussed. However, there are also important effects driven by AC excitation. One of them is so-called spin-diode effect. If the magnetization vectors of both layers in the MTJ are oriented non-collinearly and an AC current is applied, then magnetization will start to oscillate at the frequency of the excitation, as the STT will drive it back-and-forth with its sign changes. Due to the TMR effect, the resistance will also oscillate at the same frequency. The voltage resulting from the mixing of the excitation current and the oscillating resistance measured in the MTJ has two components - AC and DC. Analytically, if δR is the amplitude of resistance oscillations, the input current has an amplitude of I0 and angular frequency of ω, and β is the phase shift between the current and the resistance signals, then the output voltage can be written as the sum of the time-dependent and time-independent component namely:. Vout = δRcos(ωt + β) × I0 cos(ωt) =. =. δRI0 (cosβ + cos(2ωt + β)) = VDC + VAC . 2. (2.21). If the DC component is separated, the whole system starts to work as a diode and converts AC current to DC voltage. The idea behind the spin-diode effect is depicted in fig. 2.8. A schematic of the experimental setup for the spin-diode effect measurements is depicted in fig. 2.9.. AC - alternating current.

(19) C u rre n t [m A ]. 19. 2. a ) 1. 0 -1 -2 5 0. [ Ω] ∆R. -2 5 -5 0. b ). 2 5. ∆V [ m V ]. Figure 2.8: Conceptual plot for the spin-diode effect. (a) input AC current (b) resistance changes due to excitation brought on by the input (c) mixing voltage being the product of signals (a) and (b) - the red line indicates measured DC voltage.. 0. 5 0. c ). 4 0 3 0 2 0 1 0 0 0. 1. 2. 3. 4. 5. 6. 7. 8. T im e [n s ]. The spin-diode effect can be used to detect microwave signals as the amplitude of mixing voltage is inversely proportional to the difference between the excitation and resonance frequency. Therefore it is a complimentary phenomenon to DC excitation which can be used for microwave generation.. V. Bias T. Figure 2.9: Scheme of the MTJ and experimental setup for the spin-diode effect measurement. Microwave excitation is generated by the voltage source. So-called bias T is an element dedicated to the separation of constant and alternating components of the output signal (mixing voltage).. 2.1.8 Voltage-induced dynamics by voltage controlled anisotropy Last but not least, the most novel and promising concept of the VCMA effect used for driving dynamics in MTJs will be discussed. This effect is of growing interest due to the great reduction of power needed for performing operations on MTJs. If the tunnelling barrier is thick (about 1.5 nm), the resistance of the junction, which depends exponentially on the barrier thickness, is high enough to make the current density orders of magnitude lower than in the case of current-controlled devices and therefore negligible from an STT point of view and its influence on MTJ dynamics. However, the voltage can still excite the magnetization. One of the possible explana-. VCMA - voltage controlled magnetic anisotropy.

(20) 20. tions is that the electric field applied to an ultra-thin ferromagnetic layer causes an accumulation of charge which affects the interfacial PMA by changing the occupation states of the 3d electrons at the surface of the layer29 . The general problem of anisotropy control by means of voltage is broadly discussed in various publications, and has several other possible origins have been proposed such as the voltage-induced redox reaction30 , the electromigration effect31 or modfication of Rashba spin-orbit coupling by the screening potential induced by applied electric field32 . It is also possible that some of these effects may together contribute to the VCMA measured in a device. However, alternation of the PMA will modify the effective field and therefore the equilibrium of the magnetization in the system will change. It can be used either for spin-diode rectification by AC excitation or for switching with the use of a short voltage pulse. In the case of the spin-diode effect, the small current leaking through the barrier is desired, as it mixes with the resistance and produces an output voltage, similarly to the current-driven spin-diode effect. Nevertheless, as the resistance of the MTJ used for VCMA is high, the output voltage may be of similar magnitude to STT-controlled devices. VCMA also allows for switching operations. In the case of CIMS, the pulse time was reduced to save energy and it was the effect of DC current which induced switching. However, DC voltage itself can not excite precession. It is the change of voltage and therefore the change of equilibrium in the system which induced the magnetization vector motion around the newly set effective field. If the voltage pulse is high, the precession angle can be wide sufficiently to drive the magnetization vector near to the opposite equilibrium of the base state. Then, if the voltage is turned off at the right time, the magnetization will relax in this opposite state and the switching process is complete. This is described in details in33 . This phenomenon can be used in MRAM devices. Specifics of the solution are described in a very recent patent34 .. 2.2. Micromagnetic modelling. In order to investigate magnetization dynamics in nanoscale during the course of this thesis, micromagnetic modelling methods were used and developed. Micromagnetic theory was first introduced by William Fuller Brown in 196335 . In this approach, the ferromagnetic material is assumed to be a continuous medium where local magnetization vectors dynamics is driven by the LLG equation and the equilibrium of magnetization is given by an energy minimum as presented in section 2.1.3. This is the most suitable approach for analysis and predictions in spin electronics devices with sizes typically in the range of 1 - 1000 nm36 .. PMA - perpendicular (with respect to the sample plane) magnetic anisotropy R.F. Sabirianov Z. Zhu J. Chu S.S. Jaswal E.Y. Tsymbal Ch.-G. Duan, J.P. Velev. Surface magnetoelectric effect in ferromagnetic metal films. Physical Review Letters, 101(13):137201, 2008 30 A.J. Tan P. Agrawal S. Emori H.L. Tuller-S. Van Dijken G.S.D. Beach U. Bauer, L. Yao. Magneto-ionic control of interfacial magnetism. Nature materials, 14(2):174–181, 2015 31 F. Montaigne S. Mangin S. Andrieu A. Rajanikanth, T. Hauet. Magnetic anisotropy modified by electric field in V/Fe/MgO (001)/Fe epitaxial magnetic tunnel junction. Applied Physics Letters, 103(6):062402, 2013 32 S. Zhang L. Xu. Electric field control of interface magnetic anisotropy. Journal of Applied Physics, 111(7):07C501, 2012 29. F. Bonell S. Murakami T. Shinjo Y. Suzuki Y. Shiota, T. Nozaki. Induction of coherent magnetization switching in a few atomic layers of feco using voltage pulses. Nature materials, 11(1): 39–43, 2012 34 Pedram Khalili Amiri, Kang L Wang, and Kosmas Galatsis. Voltagecontrolled magnetic anisotropy (VCMA) switch and magneto-electric memory (MERAM), April 26 2016. US Patent 9,324,403 33. W.F. Brown. Micromagnetics. Number 18. Interscience Publishers, 1963 35. R. Schäfer A. Hubert. Magnetic domains: the analysis of magnetic microstructures. Springer Science & Business Media, 2008 36.

(21) 21. 2.2.1 Space discretization The continuous medium of the modelled material has to be discretized for the sake of numerical calculations. There are two general ways to do this: the finite elements approach and finite differences approach. In the former case the investigated system is divided into parts called finite elements, and in micromagnetism these are often tetrahedrons. The solution to the problem is then found in all the nodes of the mesh created by the finite elements. To find the solution at any point in the system not located in the node, the neighboring solutions are interpolated. The most significant advantage of such an approach is the ability to model complex three-dimensional geometries. In the finite-differences method, the space is divided into cuboids called cells, each of the same size. The magnetization is assumed to be uniform within the volume of the cuboid. In this approach, the exact modelling of a three-dimensional system shape is difficult and sometimes even impossible. However, most of the typical systems for spintronic devices are either in the shape of a bar or an ellipsoid. The subsequent layers, although differ in thickness from each other, are usually flat in one direction. Furthermore, the finite differences method has several crucial advantages: • Cells can be easily ordered and identified in the rectangular-based mesh. • As the sample mesh is periodic, significant simplifications in calculations can be introduced. It is particularly important for the demagnetization field calculations which in general have to be performed between each pair of cells in the whole system. • Due to the regularity of the mesh interfaces, the interfacial cells are easy to identify and pair with corresponding cells in the opposite layer. • Periodicity of the rectangular mesh allows for the usage of periodic boundary conditions for simulations of relatively large samples (up to tens of µm). Therefore, the finite differences method has been chosen for all the research presented in this thesis. The exact calculation of the energies described in section 2.1.3. in the case of a rectangular mesh is discussed below.. 2.2.2 Numerical calculation of energies in ferromagnetic material • Zeeman energy Zeeman energy is defined in eq. 2.5. In the discrete space case the operation of integration is to be replaced by the sum: n. EZeeman = −µ0 Ms. ~ (~ri ) Vi . ~ (~ri ) · H ∑m. i =1. (2.22). Vi - volume of the discretization cell of index i, ~ri - vector pointing to the center of that cell. Note that ri is a discretized quantity..

(22) 22. The error contributing to the calculation is by definition equal to:

(23)

(24) Z

(25) ~ (~r )dV − ~ (~r ) · H e =

(26)

(27) −µ0 Ms m

(28) V. !

(29)

(30)

(31) ~ (~ri ) Vi

(32) = ~ (~ri ) · H − µ 0 Ms ∑ m

(33)

(34) i =1 n.

(35)

(36)

(37)

(38) n Z n

(39)

(40)

(41) ~ (~r )dV − µ0 Ms ∑ m ~ (~ri ) Vi

(42) = ~ (~r ) · H ~ (~ri ) · H =

(43) µ 0 Ms ∑ m

(44)

(45)

(46) i =1 V i =1 i.

(47)

(48)

(49)

(50)

(51) n Z

(52) ~ (~r ) − m ~ (~ri ) dV

(53) , ~ (~r ) · H ~ (~ri ) · H m =

(54)

(55) µ0 Ms ∑

(56)

(57)

(58) i =1 V. (2.23). i. which can be estimated using Taylor’s theorem applied to magnetization:. ~ (~r ) = m ~ (~ri ) + m   m x (~ri )    =  my (~ri )  +   mz (~ri ) . ~ (~ri ) ∂m (~r − ~ri ) + Rm (~ξ m )(~r − ~ri )2 = ∂~r ∂m x ri ) ∂x (~ ∂my ri ) ∂x (~ ∂mz ri ) ∂x (~. ∂m x ri ) ∂y (~ ∂my ri ) ∂y (~ ∂mz ri ) ∂y (~. ∂m x ri ) ∂z (~ ∂my ri ) ∂z (~ ∂mz ri ) ∂z (~. Rm (~ξ m ) and R H (~ξ H ) - can be estimated from the Lagrange form of the remainder in Taylor’s theorem, ξ is a vector pointing between ~r and ~ri. .   (~r −~ri ) + Rm (~ξ m )(| ~r −~ri |2 ), . (2.24). and similarly for field:. ~ ~ (~r ) = H ~ (~ri ) + ∂ H (~ri ) (~r − ~ri ) + R H (~ξ H )(~r − ~ri )2 = H ∂~r   Hx (~ri )    =  Hy (~ri )  +   Hz (~ri ) . ∂Hx ri ) ∂x (~ ∂Hy ri ) ∂x (~ ∂Hz (~ r i) ∂x. ∂Hx ri ) ∂y (~ ∂Hy ri ) ∂y (~ ∂Hz (~ r i) ∂y. ∂Hx ri ) ∂z (~ ∂Hy ri ) ∂z (~ ∂Hz (~ r i) ∂z. .   (~r −~ri ) + R H (~ξ H )(| ~r −~ri |2 ), . (2.25). which gives:

(59)

(60) n Z

(61) ~ (~ri ) ∂m 2

(62) ~ (~ri ) + e =

(63) µ 0 Ms ∑ m (~r − ~ri ) + Rm (~r )(~r − ~ri ) ∂~r

(64) i =1 V i ! !

(65)

(66) ~ (~ri ) ∂ H 2 ~ (~ri ) + ~ (~ri ) dV

(67)

(68) ~ (~ri ) · H (~r − ~ri ) + R H (~r )(~r − ~ri ) − m H

(69) ∂~r. (2.26). ~ (~ri ) terms cancel ~ (~ri ) · H After multiplying the brackets, the m 3 out. Terms with (~r − ~ri ) and (~r − ~ri ) are odd functions with respect to ri and the considered integral of them in a cuboid mesh.

(70) 23. cell equals zero. Therefore, only the terms with a second and fourth power remain. (~r − ~ri ) is bounded above by half of the cell longest dimension d, which in micromagnetism is typically of single nanometers, while V ∼ 1 − 100nm3 . Therefore, e is dominated by the quadratic term if the constant coefficients before both (~r − ~ri )2 and (~r − ~ri )4 are sufficiently small. These coefficients are ~ spatial derivatives which are indeed ~ and H proportional to the m small as the spatial inhomogeneities cannot be too large if the formula for exchange energy, which is the core of micromagnetism (eq. 2.8), is to be justified. Thus, finally, the error drops with the quadratic function by lowering the discretization cell dimensions: e ∼ d2 ,. (2.27). as long as the magnetization is continuous. This assumption is fulfilled even in complex magnetization distribution e.g. domain walls, vortexes or skyrmions. Therefore, the angle between adjacent cells magnetization vectors is typically used to control the simulation correctness as both physics (eq. 2.8) and numerical conditions demand to keep it low. • Anisotropy energy Similarly to Zeeman energy, anisotropy energy defined in eq. 2.6 can be numerically calculated by replacing integration with summation: n. E Anisotropy =. ∑ (K1 sin2 α + K2 sin4 α)Vi .. (2.28). i =1. After analysis analogous to the one in the case of Zeeman energy, the error can be estimated proportional to d2 . • Exchange energy Exchange energy is defined in eq. 2.7. In order to approximate the second derivatives, the three-point method can be used:. d x , dy , dz - cell edges lengths.

(71) 24. . ~ (~r ) ∂2 m = ∂x2.      rx − dx rx rx + dx       ~   r y   − 2m ~  ry  + m ~  ry  m rz rz rz d2x. . ~ (~r ) ∂2 m = ∂y2.      rx rx rx       ~   r y − d y   − 2m ~  ry  + m ~  ry + dy  m rz rz rz d2y. . ~ (~r ) ∂2 m = ∂z2.      rx rx rx       ~   r y   − 2m ~  ry  + m ~  ry  m rz − dz rz rz + dz d2z. (2.29) Replacing integration with summation and substituting eq. 2.26 into 2.7 the following formula can be obtained: n. EExchange =. ~r j points center of 6 nearest neighbour cells. 6. ~ (~ri )) (~ m(~ri )) · (~ m(~ri +~r j ) − m Vi . 2 rj i =1 j =1. ∑∑. (2.30). It can be shown that the error of such an approach is again proportional to d2 . • IEC energy Interlayer exchange coupling energy is introduced in the eq. 2.9. Similarly to previous discussions, the numerical formula can be derived and reads as follows: n. n. i =1. i =1. ~1·m ~ 2 Si − ∑ J2 (~ ~ 2 ) 2 Si , E IEC = − ∑ J1 m m1 · m. Si - surface element of the interface. (2.31). with the same arguments leading to error estimation being proportional to d2 . • Demagnetization energy Combining equations 2.15 and 2.16 and integrating by parts one can obtain the formula for demagnetization energy:. EDemag. µ M2 =− 0 s 8π. n. n. ~ ∑∑m. i =1 j =1. T. . (~ri ) . Z Z. V S.  ~r − ~r 0 T ~0 ~n (r )dSdV  m ~ (~r j ), | ~r − ~r 0 |3. (2.32). then, the introduction of the demagnetization tensor, which depends on the demagnetization cell edges lengths d x , dy , dz into the above equation leads to:. EDemag = −. µ0 Ms2 2. n. n. ~ T (~ri ) N (~ri −~r j , d x , dy , dz )~ m(~r j ). ∑ ∑ Vi m. i =1 j =1. (2.33).

(72) 25. Therefore the only remaining issue is to calculate the components of the tensor N. Due to the magnetic reciprocity theorem37 the tensor is symmetrical and therefore consists only of six independent components. Additionally, only Nxx and Nxy are nontrivial to calculate, as the other four can be obtained by simple argument permutation. Introducing ∆x, ∆y, ∆z defined as:       ri,x r j,x ∆x       (2.34)  ri,y  −  r j,y  =  ∆y  , ri,z r j,z ∆z. W.F. Brown. Micromagnetics. Number 18. Interscience Publishers, 1963 37. the final form of demagnetization tensor is as follows:. N (∆x, ∆y, ∆z, d x , dy , dz ) = . Nxx (∆x, ∆y, ∆z, d x , dy , dz )   Nxy (∆x, ∆y, ∆z, d x , dy , dz ) Nxy (∆x, ∆z, ∆y, d x , dz , dy ). Nxy (∆x, ∆y, ∆z, d x , dy , dz ) Nxx (∆y, ∆x, ∆z, dy , d x , dz ) Nxy (∆y, ∆z, ∆x, dy , dz , d x ).  Nxy (∆x, ∆z, ∆y, d x , dz , dy )  Nxy (∆y, ∆z, ∆x, dy , dz , d x )  . Nxx (∆z, ∆y, ∆x, dz , dy , d x ). (2.35). The Nxx and Nxy can be obtained by analytical calculations38 , 39 , 40 . p Defining R = x2 + y2 + z2 as well as auxiliary functions: f ( x, y, z) =. 1 2 y z − x2 arcsinh 2. . √. y x 2 + z2. 1 + z y2 − x2 arcsinh 2. − xyz arctan and. p. . z x 2 + y2. !. yz . 1 + R 2x2 − y2 − z2 (2.36) xR 6. g( x, y, z) =. xyz arcsinh. p. z x 2 + y2. !. 1 + y 3z2 − y2 arcsinh 6. 1 + x 3z2 − x2 arcsinh 6. . √. y x + z2. . 1 − y2 z arctan 2. p . x y + z2 xz yR. !. . yz 1 xy 1 1 − x2 z arctan − z3 arctan − xyR (2.37) 2 xR 6 zR 3. the formulas are as follows:. A. Aharoni M. Schabes. Magnetostatic interaction fields for a threedimensional array of ferromagnetic cubes. IEEE Transactions on Magnetics, 23(6):3882–3888, 1987 39 D.J. Dunlop A.J. Newell, W. Williams. A generalization of the demagnetizing tensor for nonuniform magnetization. Journal of Geophysical Research: Solid Earth, 98(B6):9551–9555, 1993 40 N. Hayashi H. Fukushima, Y. Nakatani. Volume average demagnetizing tensor of rectangular prisms. IEEE Transactions on Magnetics, 34(1):193–198, 1998 38.

(73) 26. Nxx (∆x, ∆y, ∆z, d x , dy , dz ) = 1 4πd x dy dz. 8 f (∆x, ∆y, ∆z) − 4. ∑. f (k) + 2. k ∈ S1. ∑. k ∈ S2. f (k) −. ∑. !. f (k) ,. k ∈ S3. (2.38). Nxy (∆x, ∆y, ∆z, d x , dy , dz ) = 1 4πd x dy dz. 8g(∆x, ∆y, ∆z) − 4. ∑. k ∈ S1. g(k) + 2. ∑. k ∈ S2. g(k) −. ∑. !. g(k) ,. k ∈ S3. (2.39). where S1 , S2 , S3 are sets of f and g corresponding to different neighboring cell groups in the rectangular mesh. Namely, the coordinates of the centers of cells neighboring the cell in question through a face (6 neighbors): S1 = {(∆x + d x , ∆y, ∆z), (∆x − d x , ∆y, ∆z), (∆x, ∆y + dy , ∆z),. (∆x, ∆y − dy , ∆z), (∆x, ∆y, ∆z + dz ), (∆x, ∆y, ∆z − dz )}, (2.40). through an edge (12 neighbors): S2 = {(∆x + d x , ∆y + dy , ∆z), (∆x − d x , ∆y + dy , ∆z),. (∆x + d x , ∆y − dy , ∆z), (∆x − d x , ∆y − dy , ∆z), (∆x, ∆y + dy , ∆z + dz ), (∆x, ∆y − dy , ∆z + dz ),. (∆x, ∆y + dy , ∆z − dz ), (∆x, ∆y − dy , ∆z − dz ),. (∆x + d x , ∆y, ∆z + dz ), (∆x − d x , ∆y, ∆z + dz ),. (∆x + d x , ∆y, ∆z − dz ), (∆x − d x , ∆y, ∆z − dz )}, (2.41). through a vertex (8 neighbors): S3 = {(∆x + d x , ∆y + dy , ∆z + dz ), (∆x − d x , ∆y + dy , ∆z + dz ),. (∆x − d x , ∆y − dy , ∆z + dz ), (∆x − d x , ∆y − dy , ∆z − dz ),. (∆x + d x , ∆y − dy , ∆z − dz ), (∆x + d x , ∆y + dy , ∆z − dz ),. (∆x + d x , ∆y − dy , ∆z + dz ), (∆x − d x , ∆y + dy , ∆z − dz ), } (2.42) and the term 8g(∆x, ∆y, ∆z) corresponds to the cell in question itself, which together results in a total of 27 components in both expressions eq. 2.37 and eq. 2.38. Demagnetization energy is certainly the most complicated energy term in ferromagnetic material. Therefore, it is the most crucial element for optimization from a calculation time point of view. The direct calculation of Hd from eq. 2.32, namely: n. Hd,i = −µ0 Ms. ∑ N (~ri −~r j , dx , dy , dz )~m j. j =1. (2.43).

(74) 27. has a computational complexity of O(n2 ). Although numerous lesser improvements are important for the minimization of time needed for Hd calculation41 , the most important is the qualitative change of complexity to O(n log(n)). This is achieved by treating the right hand side of eq. 2.42 as a discrete three dimensional ~ . Therefore, the calculations can be done convolution of N and m ~ sequences separately, multiby performing the FFT of N and m plying them and performing inverse FFT. Details of efficiency and acceleration of FFT for Hd calculations can be found in publications42 , 43 .. 2.2.3 Software and hardware features One of the most popular software tool sets which implements micromagnetic paradigms is OOMMF44 . It was developed at the Information Technology Laboratory at NIST by Michael Donahue and Donald Porter. The first version of OOMMF was published in 1998. The ever-growing popularity of the OOMMF project has been ensured as it has fulfilled the following design briefs: • public domain, • portability and flexibility, • extensibility. This has been done by using the common and well-known C++ as the core implementation language, using a module structure for the whole framework and publishing the extensions developed by the users on the project’s website to allow the whole user community to benefit. Furthermore, the OOMMF has been fully parallelized on the CPU calculations with the most crucial functionalities available for acceleration by GPU calculations45 . Thanks to all of the above together with its wide functionality and scientific usefulness, the authors of OOMMF have noted over 2040 publications in peer-reviewed journals citing thier software since 199946 . From an implementation point of view, the program has been designed as a three-level structure. On the top level the fully-separated modules interact with each other using TCP/IP protocols. The modules are: the main controller of the simulation core called oxsii in the GUI version or boxsi in the batch version, the data writer module mmArchive, GUI modules for data visualization mmDataTable (quantities averaged over whole sample as numerical outputs), mmGraph (plot of outputs from mmDataTable), mmDisp (plot of local values at a selected sample intersection). On the second level the modules’ interfaces have been designed using the tcl/tk language. Finally the calculations at the bottom level are implemented in C++. A full list of the so-called OOMMF child classes, meaning C++ classes responsible for the implementation of energy therms, the supporting classes allowing for various methods of definition of scalar and vector fields and other miscellaneous features such as demagnetization details or numerical parameters of the scheme for solving. M. Donahue J. Miltat. Handbook of magnetism and advanced magnetic materials, 2007 41. FFT - fast Fourier transform. G. Thomas Jr. Stockham. High-speed convolution and correlation. In Proceedings of the April 26-28, 1966, Spring joint computer conference, pages 229–233. ACM, 1966 43 B. Rust D. Donnelly. The fast Fourier transform for experimentalists, part I: concepts. Computing in Science and Engineering, 7(2):80–88, 2005a; and B. Rust D. Donnelly. The fast Fourier transform for experimentalists, part II: convolutions. Computing in Science and Engineering, 7(4):92–95, 2005b 44 D.G. Porter M. Donahue. OOMMF user’s guide, version 1.0, interagency report nistir 6376. National Institute of Standards and Technology, Gaithersburg, MD, 1999 OOMMF - Object Oriented Micromagnetic Framework NIST - National Institute of Standards and Technology 42. CPU - central processing unit GPU - graphical processing unit M. Hu R. Chang M. Donahue V. Lomakin-Vitaliy S. Fu, W. Cui. Finite-difference micromagnetic solvers with the object-oriented micromagnetic framework on graphics processing units. IEEE Transactions on Magnetics, 52 (4):1–9, 2016 46 Full citation list available on the OOMMF homepage: math.nist.gov/oommf/oommf_cites.htm 45. GUI - graphical user interface oxsii - OOMMF extensible solver interactive interface, boxsi - OOMMF extensible solver batch interface.

(75) 28. the LLG equation have been listed in the documentation47 . Basic OOMMF functionality encompass all the phenomena described in this chapter apart from TMR and STT which have been implemented by the author of this thesis and are described in the enclosed publications. The very first calculations presented in this thesis were initially performed on a three-node computer located in the Department of Electronics using a GUI tunneled to a personal computer. Shortly thereafter, as the investigations developed, much more computational power was necessary and the majority of the results have come from supercomputers Zeus and Prometheus both located at the Academic Computer Center Cyfronet AGH. The performing of hundreds of simulations simultaneously, in order to either check different free parameters to fit experimental data, or to obtain results for varying conditions (like spin-diode voltage for several tens of different frequencies) forced the author to use a batch mode of work as well as to develop numerous lesser tools for automated data analysis. The author of this thesis is now a co-manager of a group registered on the PL-GRID infrastructure. The group has benefited from a total of 7 computational grants, and presently has access to over one million computation hours per year.. D.G. Porter M. Donahue. OOMMF user’s guide, version 1.0, interagency report nistir 6376. National Institute of Standards and Technology, Gaithersburg, MD, 1999 47.

(76) 3 Current-driven dynamics In this chapter the results of the research on current-driven dynamics is presented. The author has been involved in both theoretical and experimental aspects of current-controlled devices. The first paper Micromagnetic model for studies on Magnetic Tunnel Junction switching dynamics, including local current density is a direct continuation of the author’s Master’s Thesis, where the initial model development was started, but no results presented in this work were included. In this publication the author wrote the manuscript, developed a local current model for OOMMF, performed all the simulations, analyzed the results and prepared schematics and graphs. In the second work Backhopping in magnetic tunnel junctions: Micromagnetic approach and experiment the author again developed initial model started in the Master Thesis (which allowed for phenomenological modelling of AP resistance dependence on voltage), wrote the manuscript, performed all the simulations, developed a synthetic anti-ferromagnet random field model for OOMMF simulations, compared experimental results with simulations and prepared graphs. In the third publication Buffer influence on magnetic dead layer, critical current, and thermal stability in magnetic tunnel junctions with perpendicular magnetic anisotropy the author wrote the manuscript, took part in critical current measurements, performed critical current and thermal stability calculations, compared and correlated results obtained by different measurement techniques, discussed overall results and prepared figures. Additionally, the author of this thesis contributed to two more papers on current-induced dynamics in spin electronics devices. In Influence of MgO tunnel barrier thickness on spin-transfer ferromagnetic resonance and torque in magnetic tunnel junctions by Witold Skowronski, ´ Maciej Czapkiewicz, Marek Frankowski, Jerzy Wrona, Tomasz Stobiecki, Günter Reiss, Khattiya Chalapat, Gheorghe S. Paraoanu and Sebastiaan van Dijken, Phys. Rev. B 87, 094419, (2013) he took part in simulations of STT-induced oscillations as a function of the IEC between the free and reference layer. The simulation results were in agreement with the experimental data and showed that the qualitative difference in spectrum, namely the occurrence of single or double resonance peak, originates from different values of IEC in the system, and therefore depends on the MgO barrier thickness..

(77) 30. In Rectification of radio-frequency current in a giant-magnetoresistance spin valve by Sławomir Zietek, ˛ Piotr Ogrodnik, Marek Frankowski, Jakub Checi ˛ nski, ´ Piotr Wi´sniowski, Witold Skowronski, ´ Jerzy Wrona, ˙ Tomasz Stobiecki, Antoni Zywczak, and Józef Barna´s, Phys. Rev. B 91, 014430, (2015) he contributed to the micromagnetic simulations of the spin-diode effect induced by inhomogeneous Oersted field which showed that a non-zero output signal can be measured only in the case of a non-zero net field in the excited layer. Additionally, he took part in the calculations of the net Oersted field in the free layer..

(78) 6 Summary As spin electronics is an interdisciplinary domain, the work presents an overview of techniques essential for effective research in this field. All the necessary background presented in chapter 2 has led towards current-induced switching which has been presented in view of memory-oriented applications in chapter 3. Initially, a complete model of STT effects was developed and tested proving that for realistic calculations of the current driven dynamics it is necessary to take into account inhomogeneities in current distribution. Then, the model was used to investigate the problem of backhopping and the coupling influence on it. Simulation results obtained are in good agreement with the analyzed experimental data. The third work takes benefits from the combination of numerous experimental techniques in order to provide a complete analysis of selected buffers influences on critical current density - a crucial parameter in view of MRAM solutions. Further optimization of power consumption resulted in the investigation of voltage-induced effects for both the switching of memory cells as well as the detection of microwave signals investigated for a wide range of fundamental MTJ parameter free layer anisotropy energy. The fifth chapter presents useful software tools allowing for or enhancing the performance of advanced simulations and results analysis of magnetization and magnetoresistance dynamics developed during this research, which are now in the public domain available for other researchers. This thesis therefore presents the most important applications of MTJ in spin electronics devices and the complete set of methods and software for the modeling and analysis of dynamics in MTJs in view of experimental results to support. Further research will be focused on the emerging possibility of using voltage to control not only anisotropy energy but also IEC1 , 2 , 3 . This effect could be used to develop a new generation of voltagetuned microwave devices, voltage-tuned magnetic field detectors or MRAMs. Micromagnetic simulations can be used to predict the efficiency of these devices before investment into experiments is made. It is also planned to produce samples and apply the model of voltage-driven dynamics in quantitative analysis of experimental data. Additionally, developing reliable model of temperature effects calibrated by noise measurements would allow for the investigation of. J. Salafranca N. Nemes E. Iborra G. Sanchez-Santolino M. Varela M.G. Hernandez J.W. Freeland M. Zhernenkov M. R. Fitzsimmons S. Okamoto S. J. Pennycook M. Bibes A. Barthelemy S.G.E. te Velthuis Z. Sefrioui C. Leon J. Santamaria F.A. Cuellar, Y.H. Liu. Reversible electric-field control of magnetization at oxide interfaces. Nature Communications, 5, 2014 2 J.-X. Zhu J. Fransson, J. Ren. Electrical and thermal control of magnetic exchange interactions. Physical Review Letters, 113(25):257201, 2014 3 Current developments in electric-field control of interfacial anisotropy and interactions in perpedndicularly magnetizaed thin films 1.

(79) 77. switching probabilities in the cases of both current and voltage control..

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