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5.0043905.pdf	J. Appl. Phys. 129, 214505 (2021); https://doi.org/10.1063/5.0043905 129, 214505 © 2021 Author(s).Modeling of magnetization dynamics and thermal magnetic moment fluctuations in nanoparticle-enhanced magnetic resonance detection Cite as: J. Appl. Phys. 129, 214505 (2021); https://doi.org/10.1063/5.0043905 Submitted: 12 January 2021 . Accepted: 11 May 2021 . Published Online: 03 June 2021 Tahmid Kaisar , Md Mahadi Rajib , Hatem ElBidweihy , Mladen Barbic , and Jayasimha Atulasimha ARTICLES YOU MAY BE INTERESTED IN Magnetism in curved geometries Journal of Applied Physics 129, 210902 (2021); https://doi.org/10.1063/5.0054025 Special optical performance from single upconverting micro/nanoparticles Journal of Applied Physics 129, 210901 (2021); https://doi.org/10.1063/5.0052876 Electromechanical coupling mechanisms at a plasma–liquid interface Journal of Applied Physics 129, 213301 (2021); https://doi.org/10.1063/5.0045088Modeling of magnetization dynamics and thermal magnetic moment fluctuations in nanoparticle- enhanced magnetic resonance detection Cite as: J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 View Online Export Citation CrossMar k Submitted: 12 January 2021 · Accepted: 11 May 2021 · Published Online: 3 June 2021 Tahmid Kaisar,1 Md Mahadi Rajib,1 Hatem ElBidweihy,2 Mladen Barbic,3 and Jayasimha Atulasimha1,a) AFFILIATIONS 1Department of Mechanical and Nuclear Engineering, Virginia Commonwealth University, Richmond, Virginia 23284, USA 2Department of Electrical and Computer Engineering, United States Naval Academy, Annapolis, Maryland 21402, USA 3NYU Langone Health, Tech4Health Institute, New York, New York 10010, USA a)Author to whom correspondence should be addressed: jatulasimha@vcu.edu ABSTRACT This study presents a systematic numerical modeling investigation of magnetization dynamics and thermal magnetic moment fluctuations of single magnetic domain nanoparticles in a configuration applicable to enhancing inductive magnetic resonance detection signal to noise ratio (SNR). Previous proposals for oriented anisotropic single magnetic domain nanoparticle amplification of magnetic flux in a magnetic reso- nance imaging (MRI) coil focused only on the coil pick-up voltage signal enhancement. In this study, the numerical evaluation of the SNR hasbeen extended by modeling the inherent thermal magnetic noise introduced into the detection coil by the insertion of such anisotropic nano-particle-filled coil core. The Landau –Lifshitz –Gilbert equation under the Stoner –Wohlfarth single magnetic domain (macrospin) assumption was utilized to simulate the magnetization dynamics due to AC drive field as well as thermal noise. These simulations are used to evaluate the nanoparticle configurations and shape effects on enhancing SNR. Finally, we explore the effect of narrow band filtering of the broadband mag-netic moment thermal fluctuation noise on the SNR. It was observed that for a particular shape of a single nanoparticle, the SNR could beincreased up to ∼8 and the choice of an appropriate number of the nanoparticles increases the SNR by several orders of magnitude and could consequently lead to the detectability of a very small field of ∼10 pT. These results could provide an impetus for relatively simple modifications to existing MRI systems for achieving enhanced detection SNR in scanners with modest polarizing magnetic fields. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0043905 I. INTRODUCTION Sensitivity enhancement in magnetic resonance detection con- tinues to be an important challenge due to the importance of nuclear magnetic resonance (NMR) and MRI in basic science, medical diagnostics, and materials characterization. 1–5Although many alternative methods of magnetic resonance detection havebeen developed over the years, the inductive coil detection of mag-netic resonance of precessing proton nuclear magnetic moments is by far the most common. 6The challenge in magnetic resonance detection stems from the low nuclear spin polarization at roomtemperature and laboratory static magnetic fields. An additionalchallenge is the fundamental requirement that the detector in amagnetic resonance experiment needs to be compatible with and immune to the large polarizing DC magnetic field while also be sufficiently sensitive to weak AC magnetic fields generated by theprecessing nuclear spins. The inductive coil, operating on the prin- ciple of Faraday ’s law of induction, satisfies this requirement, and enhancing the inductive coil detection signal to noise ratio (SNR)has been pursued through various techniques. 7–9However, an unlimited increase of the polarizing magnetic field is cost prohibi- tive, and technical challenges often inhibit the development of mobile MRI units, their access, sustainability, and size. Therefore,solutions to achieving sufficient or improved SNR in NMR induc-tive coil detection in lower magnetic fields and more accessible andcompact configurations remain highly desirable. 10 A. Signal amplification by magnetic nanoparticle-filledcoil core An idea has been put forward to increasing the magnetic field flux from the sample through the coil by filling the coil with a coreJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-1 Published under an exclusive license by AIP Publishingof oriented anisotropic single domain magnetic nanoparticles,11,12 as shown in Fig. 1 . The sample and the inductive coil detector are both in the prototypical MRI environment of a large DC polariz-ing magnetic field (H Zdc)a l o n gt h e zaxis, or a large magnetic induction BZdc=μ0HZdc,w h e r e μ0is the permeability of free space. This field generates a fractional nuclear spin polarization of protons in the sample. The appl ication of RF magnetic fields along the xaxis is subsequently used to tilt the magnetic moment of the sample away from the zaxis and generate precession of the sample magnetization around the zaxis at the proton NMR fre- quency ω0=γBZdc, where γis the proton nuclear gyromagnetic ratio. This sample moment precession around the zaxis generates a time-varying magnetic induction BXac=μ0HXacthrough the inductive coil detector of Nturns and sensing area Aalong the x axis. By Faraday ’s law of induction, an AC signal voltage Vat fre- quency ω0generated across the coil terminals is V¼N/C1A/C1ω0/C1BXac: (1) It is a well-known practice in electromagnet design and ambient inductive detectors that a soft ferromagnetic core withinthe coil significantly amplifies the magnetic flux through thecoil. 13,14The challenge, however, in the configuration of NMR detection of Fig. 1 is that the presence of the large polarizing mag- netic field along the zaxis, HZdc, would generally saturate the detec- tion coil core made of a soft ferromagnet along the zaxis and render the AC magnetic field due to proton precession along the x axis of the coil ineffective. In other words, a high polarizing mag- netic field would saturate the coil core in its own direction andleave the core ’s magnetization unresponsive to the AC magnetic field arising from proton precession. The solution proposed 11,12 was that the oriented anisotropic magnetic nanoparticles filling the coil core actually have an appreciable magnetic susceptibility along thexaxis precisely in the presence of a significant DC magneticfield along the zaxis. The pick-up coil voltage is then V¼N/C1A/C1ω0/C1μ0/C1(HXacþMXac), (2) where MXacis the magnetization component of the nanoparticle- filled coil core along the x-direction (sensing direction of the coil) due to the magnetic field HXacfrom the precessing sample nuclear spin magnetization, MXac=χRTHXac(where χRT=ΔMXac/ΔHXacis defined as reversible transverse susceptibility). Therefore, if the reversible transverse susceptibility, χRT, of the magnetic nanoparticle-filled coil core along the xaxis is significant at the large polarizing DC magnetic field HZdcalong the zaxis, the induc- tive coil signal voltage will be enhanced. Various theoretical15–17 and experimental investigations18–25of reversible transverse sus- ceptibility in oriented magnetic nanostructures indeed reveal thatits magnitude can be appreciable and, therefore, might provide aviable route for magnetic resonance signal amplification, as dia-grammatically shown in Fig. 1 . In this study, the coil signal voltage has been numerically eval- uated by modeling individual nanoparticle magnetic momentdynamics in the Stoner –Wohlfarth (SW) uniform magnetization approximation. 26More specifically, the AC nanoparticle moment along the xaxis in Fig. 1 ,mXac, has been investigated in the pres- ence of a large DC magnetic field HZdcalong the zaxis and under the driven sample AC magnetic field HXacalong the xaxis. Though artificially synthesized magnetic nanoparticles follow a lognormalsize distribution, for simplicity the total coil core of volume, V c, has been assumed to be composed of “n”number of identical oriented single domain magnetic nanoparticles, and that for each particle, the average x-component of the AC magnetic moment, mXac, equally contributes to the coherent amplification of the pick-upvoltage signal of the coil detector. Therefore, the total coil AC voltage due to the magnetic nanoparticle core contribution is V¼N/C1A/C1ω/C1μ 0/C1n/C1mXac Vc: (3) B. Noise contribution by magnetic nanoparticle-filled coil core Essential to the SNR consideration of any NMR experimental arrangement is the evaluation of the noise sources in the signal chain. In this work, the focus is specifically on the magneticnanoparticle-filled inductive coil detector since the sample noisealong with the amplifier noise and the Johnson noise contributionshave been addressed in numerous works. 27–31Any magnetic mate- rial placed inside the inductive detection coil will introduce addi- tional pick-up voltage noise due to intrinsic magnetizationfluctuations. 32These thermal fluctuations of the coil core magneti- zation along the xaxis, which were numerically modeled in detail in this work, generate a total mean squared coil noise voltage, V2/C10/C11 ¼N2/C1A2/C1ω2/C1μ2 0/C1M2 X/C10/C11 : (4) For simplicity, it is assumed that the total coil core of volume, Vc, is composed of nnumber of identical oriented single domain magnetic nanoparticles and that each particle magnetic moment, FIG. 1. Schematic diagram for enhanced NMR detection with magnetic nanoparticle-filled coil core.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-2 Published under an exclusive license by AIP Publishingm, undergoes random uncorrelated thermal fluctuation. Therefore, the total mean squared coil noise voltage due to the magnetic nanoparticle core is33,34 V2/C10/C11 ¼N2/C1A2/C1ω2/C1μ2 0/C1n/C1m2 X/C10/C11 V2 C: (5) It is to be noted that the magnetic moment fluctuation phe- nomena have previously been investigated in various spin systems,materials, and detection modalities. 35–42However, there does not appear to be a theoretical, numerical, or experimental study where thermal magnetic moment fluctuations of single domain nanopar- ticles of the configuration of Fig. 1 (where the large polarizing mag- netic field is applied perpendicular to the nanoparticle hard axisand the coil detection axis) and their contribution to the coil noise voltage have been carried out. In this study, therefore, in order to assess the signal and the noise of the configuration of the magnetic resonance coil detectorofFig. 1 , the room temperature magnetization dynamics of a single domain nanomagnet and its thermal fluctuation in the coil core has been simulated. Such single nanomagnet dynamics for the macrospin Stoner –Wohlfarth (SW) model uniform magnetization in both oblate and prolate ellipsoid geometries has been explored.The SW model assumes that the entire nanomagnet behaves like agiant classical spin. Thus, this model assumes that the spin (or magnetization) in all regions of the nanomagnet point in the same direction, i.e., different regions of a nanomagnet cannot have spinspointing in different directions. As a first approximation, thisassumption is valid for the nanomagnets we model as their dimen-sions are less than 100 nm. 43 This will explain the optimum nanoparticle orientation and bias field needed to maximize SNR of the experimental arrange-ment of Fig. 1 . This analysis has been extended to scaling proper- ties of an ensemble of nanomagnets and the effect of applying a bandpass filter to provide an estimate on the extent to which the insertion of magnetic particles in the sensing coil can enhance thelimits of detection of magnetic fields due to proton spin resonancesin MRI/NMR. C. Modeling particle magnetization dynamics in the presence of room temperature thermal noise Modeling of the single particle magnetization dynamics was per- formed by solving the Landau –Lifshitz –Gilbert (LLG) equation, 44–46 which was formulated for laboratory frame of reference, d~m dt¼/C0γ~m/C2~Heff/C0αγ[~m/C2(~m/C2~Heff)]: (6) In Eq. (6),γis the gyromagnetic ratio (m/A s), αis the Gilbert damping coefficient and ~mis the normalized magnetization vector, found by normalizing the magnetization vector ( ~M) with respect to saturation magnetization ( Ms),46 ~m¼~M Ms;m2 xþm2 yþm2 z¼1: (7)Here, mx,my,a n d mzare the normalized components of ~malong the three Cartesian coordinates. The effective field ( ~Heff) was obtained from the derivative of the total energy ( E) of the system with respect to the magnetization (~M),46,47 HQ eff¼/C01 μ0ΩdE dMQþHQ thermal , (8) where μ0is the permeability of the vacuum and Ωis the volume of the nanomagnet. The total potential energy in Eq. (8)is given by E¼Eshape anisotropy þEzeeman , (9) where Eshape anisotropy is the shape anisotropy due to the prolate or oblate shape and can be calculated from the following equation:46 Eshape anisotropy ¼μ0 2/C16/C17 Ω[NdxxM2 xþNdyyM2 yþNdzzM2 z], (10) where Nd_xx,Nd_yy,a n d Nd_zz represent the demagnetization factors along the x,y,a n d zdirections, which depend on the dimensions of the nanoparticle and follow the relation ofN d_xx+Nd_yy+Nd_zz=148,49andMx,My,a n d Mzare the compo- nents of magnetization vector ( ~M) along the three Cartesian coordinates. In Eq. (9),Ezeeman is the potential energy of nanomagnet for an external magnetic field ( ~H), given by Ezeeman¼/C0μ0Ω~H/C1~M: (11) The thermal field HQ thermalis modeled as a random field incorporated in the manner of47,50 HQ thermal(t)¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2kTα μ0MsγΩΔts /C1~G(t): (12) In Eq. (12),~G(t) is a Gaussian distribution with zero mean and unit variance in each Cartesian coordinate axis, kis the Boltzman constant, Tis the temperature, Msis the saturation mag- netization and γis the Gyromagnetic ratio. Δtis the time step used in the numerical solution of Eq. (6)and was chosen to be 100 fs. This was chosen to be small enough to ensure that all results areindependent of the time step. It is to be noted that m x,my,a n d mzare not independent and they are related by Eq. (7)and can be represented parametrically46as mx(t)¼sinθ(t)c o sf(t); my(t)¼sinθ(t)s i nf(t); mz(t)¼cosθ(t):(13) With this parametric representation, the number of variables reduces from three ( mx,my,a n d mz)t ot w o( θ,f). When Eq. (6)is written in the component form, three scalar equations are obtained of which two equations are enough to solve for θandf.B ye m p l o y i n gJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-3 Published under an exclusive license by AIP Publishingthe Euler method, the differential equations can be solved as given in Ref. 46and temporal evolutions of θandfcan be obtained, which provides the magnetization components along the three coordi- nate axes. The total angular dependence of energy can be obtained from11,12,51 E(θ)¼[Kusin2(θ)/C0μ0HdcMssin(θ)]Ω: (14) The first term of the equation represents the uniaxial anisot- ropy energy and the second term stands for Zeeman energy of the nanoparticle moment in the DC magnetic field. In the uniaxialanisotropy energy term, K u, is the anisotropy constant that has the value of51 Ku¼1 2μ0/C1(Na/C0Nc)/C1M2 s: (15) NaandNcare the demagnetization factors along the hard and easy axis of the ellipsoids, respectively, and the critical field Hcis obtained from11 Hc¼2Ku μ0Ms: (16) Table I lists the values of the material properties of the nanomagnet.II. RESULTS AND DISCUSSIONS Consider a prolate ellipsoid of nominal volume ∼5000 nm3 (principal axes 100, 10, and 10 nm) shown in Fig. 2(a) , where it has been assumed that the sample proton spin precession produces a magnetic field along the easy (long) xaxis of the nanomagnet while the DC bias field is applied along one of the hard (short)axes, viz., the zaxis. When the DC bias field is zero, there are two deep energy wells at θ= 0°, 180° as obtained from Eq. (14). When the magnetization is in one of these states, the magnetization response to an AC magnetic field along the xaxis (a simplified rep- resentation of the signal at the pick-up coil due to proton spin pre-cession of Fig. 1 ) is very small as the magnetization is in this deep potential as seen in Figs. 3(a) and 3(b) and Table II . The corre- sponding magnetization fluctuation due to room temperature thermal noise [that is modeled as a random effective magnetic field; see Eq. (8)] is also very small. As the DC field increases along the zaxis to the point H dc=Hc(the DC bias field is equal to the critical field Hc), the mean magnetization orientation is at 90°. However, the potential well at 90° [ Fig. 2(a) ] is characterized by a flat energy profile where the energy is nearly independent of the polar angle θ, around θ= 90°. This leads to a large magnetization response along the x axis to an applied AC magnetic field along the xaxis [ Figs. 3(a) and3(b) andTable II ] in the presence of a large DC magnetic field along the zaxis. Essentially, since the energy profile is flat in this configuration, the particle moment fluctuation due to room tem-perature thermal noise is also high. Nevertheless, it is found thatthe signal to noise ratio (SNR) is highest at H dc=Hc. In fact, the magnetization response and the SNR ratio are found to increase monotonically with the applied bias field up to Hc[see Table II based on the selected simulations shown in Figs. 3(a) and3(b) and all the simulations shown Fig. S1 in the supplementary material ] and then decreases as Hdc>Hc(for example, at Hdc= 1.25 Hcin Table II and Fig. S1 in the supplementary material ) due to an energy well deepening at θ= 90° for Hdc>Hc[Fig. 2(a) ]. SNR isTABLE I. Material properties of CoFe.52 Parameters Material property Saturation magnetization ( Ms) 1.6 × 106(A/m) Gilbert damping ( α) 0.05 Gyromagnetic ratio ( γ) 2.2 × 105(m/A s) FIG. 2. Energy profile for various DC bias magnetic fields ( Hdc) for (a) prolate: bias field along minor axis, (b) prolate: bias field along major axis, and (c) oblate: bias field along minor axis.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-4 Published under an exclusive license by AIP Publishingcalculated in the following manner. First, no thermal noise is included and the LLG simulation is performed to determine themagnetization response due to only the AC field from sampleproton spin precession. Then, another LLG simulation is per-formed with no signal and the magnetization fluctuation is studied due to the thermal noise only. The ratio of the rms values of the response due to the sample precessing field and that due to noise isdefined as the SNR. It is noteworthy that the AC drive magnetic field has an amplitude of 800 A/m (10 Oe or 1 mT) for all cases discussed in this work, which is much larger than the typical signal due toproton spins, which may be several orders of magnitude smaller. However, a higher drive amplitude was chosen to elicit a reasonablemagnetization response that would be easily visible in the plots andresult in reasonable SNR ratios for a single nanomagnet. In prac-tice, the number of nanomagnets placed in the detector coil could be over n∼10 12or more resulting in sub-nT sensing capability as discussed later. Furthermore, the proton resonance of 42.5 MHzoccurs at a DC field ∼1 T, which would change if the DC bias is changed. However, to keep the simulations consistent, the signal due to proton resonance is assumed as 42.5 MHz for all cases as this would not change the qualitative findings. FIG. 3. Magnetization dynamics with (a) 800 A/m, 42.5 MHz AC field with no thermal noise for single prolate nanomagnet with bias along minor axis and (b) only th ermal noise at Hdc= 0 and Hdc=Hc(large magnetic response and magnetization fluctuation due to thermal noise at Hdc=Hccompared to Hdc= 0). Magnetization dynamics with (c) 800 A/m, 42.5 MHz field with no thermal noise for single oblate nanomagnet with bias along minor axis (very large magnetic response at Hdc= 0.625 Hccompared to Hdc= 0) and (d) only thermal noise at Hdc= 0 and Hdc= 0.625 Hc. (e) SNR vs Hdc/Hcfor single prolate and single oblate nanomagnet cases and (f) zoomed version of SNR vs Hdc/Hcfor single prolate nanomagnet with bias along major axis.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-5 Published under an exclusive license by AIP PublishingNext, a prolate ellipsoid is considered as shown in Fig. 2(b) , where it is assumed that the proton spin precession produces a magnetic field along one of the hard (short) axes of the nanomag-net while a DC bias field is applied along the long (easy) axis.Initially, the magnetization points downward ( θ= 180°) and at H dc= 0 is in a deep potential well as shown in Fig. 2(b) . As the magnetic field applied along the + zdirection ( θ= 0°) increases, the energy well around θ= 180° is flattened as described by Eq. (14). Thus, for a higher field, the magnetization response increases, asdoes the magnetization fluctuation due to thermal noise as shown inTable II (detailed simulations are shown Fig. S2 in the supplementary material ). However, the shallow wells improve the magnetization response due to the drive AC magnetic field morethan the increased magnetization fluctuations due to the thermalnoise (as in the prior case) and increase the overall SNR ratio (Table II ). However, as one approaches H dc=Hc, the SNR drops signifi- c a n t l y .T h i si sb e c a u s et h ee n e r g yp r o f i l ei sf l a ta t θ=1 8 0 ° b u t e v e n small perturbations from this angle make the magnetization switchand rotate to the + zaxis ( θ= 0°) as shown in Fig. 2(b) . Once it reaches this state, the energy well profile at H dc=Hcandθ=0 ° i s a deeper than the well at Hdc= 0. The reason is that the Zeeman energy due to a field along the + zaxis makes the already deep shape anisotropy well even deeper at θ= 0° reducing both the magnetiza- tion response and the magnetization fluctuations due to thermal noise, as well as the SNR ratio. Thus, the best SNR is seen atH dc<Hc(see the high SNR at 0.875 HcinTable II )b u tc l o s et o Hc. Finally, the case of an oblate ellipsoid of nominal volume ∼5000 nm3(principal axes 40, 40, and 6 nm) is considered, similar to the volume of a prolate ellipsoid, shown in Fig. 2(c) . Again, it is assumed that the proton spin precession produces a magnetic fieldalong one of the easy (long) axes of the nanomagnet while the DCbias field is applied along the hard axis. The symmetry of the problem is such that at H dc= 0 and the magnetization is free to rotate in the x–yplane as there is no energy barrier to such rota- tion. As Hdcincreases, the magnetization is still free to move in a cone of the x–yplane at a specific angle to the zaxis that decreases with increasing Hdc, finally coinciding with it when Hdc=Hc. Thus, at a range of DC bias fields (for example, from Hdc= 0.25 Hcto Hdc= 0.75 Hc) a high SNR > 1.4 is observed when a single nanomag- net is driven by an AC magnetic field. This is due to the combina-tion of high magnetization response given the symmetry and noise limited by the presence of the DC bias field. The simulations of magnetic response to the AC magnetic field and magnetizationfluctuations due to random thermal noise are, respectively, showninFigs. 3(c) and 3(d) comparing H dc= 0 and Hdc= 0.625 Hccases with all other bias field cases shown in Fig. S3 in the supplemen- tary material . In summary, as far as the SNR is concerned, the prolate ellip- soid with DC bias magnetic field along the hard axis [ Fig. 2(a) ]i s the better choice over the prolate ellipsoid with DC bias magneticfield along the easy axis. However, the oblate geometry and config- uration shown in Fig. 2(c) produces the highest SNR as shown in Figs. 3(e) and3(f) andTable II [more than twice the highest SNR for the single prolate ellipsoid configuration in Fig. 2(a) , and more than 10 times the single prolate ellipsoid configuration of Fig. 2(b) ]. What makes this oblate configuration even more attrac- tive to detection coils in MRI/NMR applications is that the highSNR performance is seen over a large range of DC bias fields (e.g.,H dc= 0.25 HctoHdc= 0.75 Hc), making it attractive for a broad range of MRI scanner fields. This best-case nanoparticle (oblate ellipsoid at Hdc= 0.625 Hc with SNR = 1.71) was then taken and investigated if the SNR can further be improved by applying a narrow band filter aroundTABLE II. Magnetization oscillations in the single nanomagnet of different geometries for different values of DC bias magnetic field (based on simulations sh own in Fig. 3 and Figs. S1 –S3 in the supplementary material ). Boldfaced rows represent the highest SNR values for each case. CasesValue of bias field (Hdc)RMS normalized magnetization ( M/Ms) for a sinusoidal magnetic signal of 800 A/m (10 Oe) amplitudeRMS normalized magnetization (M/Ms) due to thermal noise only (no signal)SNR (defined here as ratio of columns 3 and 4) Prolate applying bias field along minor axis0 1.66 × 10−66.45 × 10−40.003 0.25Hc 2.82 × 10−40.0075 0.05 0.5Hc 0.002 0.0103 0.2885 0.75Hc 0.0128 0.0202 0.63 Hc 0.1035 0 .1464 0 .7042 1.25Hc 0.009 0.0484 0.193 Prolate applying bias field along major axis0 9.3 × 10−40.025 0.03 0.5Hc 0.0017 0.034 0.0465 0.75Hc 0.0032 0.05 0.0653 0.875 Hc 0.0063 0 .0662 0 .09 Hc 0.0018 0.025 0.075 Oblate applying bias field along minor axis0.25Hc 0.94 0.68 1.4 0.5Hc 0.843 0.582 1.51 0.625Hc 0.76 0 .45 1 .71 0.75Hc 0.645 0.46 1.44 Hc 0.0934 0.0921 0.95Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-6 Published under an exclusive license by AIP Publishing42.5 MHz. The rationale is that the magnetization response driven by the magnetic field due to proton spin precession at 1 T appliedDC field is dominant around 42.5 MHz while the magnetization fluctuations driven by thermal noise are a broad band as evidenced by the single-sided amplitude spectrum shown in Fig. 4(a) . When a bandpass filter of 42 –44 MHz was applied, the SNR improved to ∼8 as shown in Fig. 4(b) . A. Scaling of SNR with coil core nanoparticle number While the SNR with an AC magnetic field of 10 Oe (equiva- lent to 1 mT) shows a SNR ∼8 for optimal conditions, this was assuming a single nanomagnet. However, for a large number ofnanomagnets nwithin the core, the magnetization response increases as n(as more nanomagnets coherently contribute to more magnetic moment and, therefore, greater induced voltage in the coil) according to Eq. (3), while the magnetic noise would only increase as √n(as the thermal noise induced fluctuations of nano- magnets have a random phase) according to Eq. (5), leading to a SNR increase of n/√n=√n. The scaling trend of SNR obtained from Eq. (5)has been corroborated by performing simulations for 5, 10, 100, 150, and 300 nanoparticles under best-case conditions(oblate ellipsoid at H dc= 0.625 Hc). The analytically calculated trend of SNR increasing as √ntimes is compared with numerical simu- lations using the LLG formalism incorporating noise as shown in Fig. 5 . It should be noted that for higher than 150 nanomagnets,the simulated gain in SNR is smaller than the SNR expected due to √nscaling. This is possibly due to numerical issues in not main- taining random (completely uncorrelated) magnetization dynamics between different nanoparticles as the number of particles simu- lated increases beyond 100. If a square detection coil of 2 cm on a side and the pitch between nanomagnets ∼200 nm is considered, 10 billion nanomag- nets can be accommodated in a single layer of 2 cm by 2 cm dimen- sion. Additionally, as the single nanoparticle layer thickness is∼6 nm, the average distance between two such layers can be ∼25 nm. Thus, 400 000 such magnetic nanoparticle layers can be accommodated in 1 cm coil thickness. Consequently, n=4 0×1 0 14 nanomagnets can be incorporated into the sensing coil. So, the inser- tion of 40 × 1014nanomagnets in a core of 2 × 2 × 1 cm3size has ∼0.005 or 0.5% volume fraction (defined as the ratio of the volume of the nanoparticles to the volume of coil core). Since the dipolarinteraction decreases as the nanoparticle density decreases, 53,54the insertion of 40 × 1014nanomagnets constitutes a very low volume fraction with ∼5 times higher pitch than the lateral dimension and ∼4 times higher separating distance in the vertical direction than the height of the individual oblate nanomagnet. This justifies ignoringthe dipole coupling between nanoparticles. This number of nano- magnets also leads to an increase in SNR from 8 to ∼5×1 0 8.I n other words, with a SNR of 5 × 108, one could conceivably detect an AC magnetic field of 1/(108) mT, i.e., an AC magnetic field of 10 pT or better depending on the density with which nanoparticles are inserted into the NMR detection coil. However, it should be noted that nanoparticle pinning sites, inherent inhomogeneities, etc. can FIG. 4. (a) Frequency spectrum of signal + noise, before filtering. (b) Frequency spectrum of signal + noise, after filtering. Both cases for oblate nanomagnet. FIG. 5. The scaling trend of SNR as √nfornnumber of nanoparticles calculated analytically (red) from Eq. (5)and obtained from simulation (blue).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-7 Published under an exclusive license by AIP Publishingimpede magnetization dynamics55and create phase differences, thus decreasing signal enhancement. The key point is that merely filling the detection coil with a soft (high permeability) core would not help as the core would besaturated under the high DC bias fields used for MRI. However, byusing anisotropic nanoparticles of appropriate geometry that are still responsive to AC fields from proton precession in the presence of orthogonal strong DC fields, the coil detection sensitivity can beenhanced. III. CONCLUSION This numerical investigation of nanoparticle magnetization dynamics and room temperature thermal moment fluctuationsconfirm the initial zero temperature proposal for nanoparticle-based amplification of a NMR signal. Such a consideration of thethermal fluctuation allows us to predict not just idealized zero tem- perature signal amplification values but realistic room temperature SNR values. This analysis suggests specific orientations of aniso-tropic oblate ellipsoid particles can lead to SNR improvements overconventional air-filled MRI coils. Much will depend on the qualityof the particles used in the coil core: shape uniformity, quality of particles orientation within the core, smoothness of the particles and surface pinning sites (that degrade the effect of magnetizationdynamics), and uniformity of the nanoparticle aspect ratio (whichdetermines where the particle has a peak in transverse susceptibil-ity). Further consideration would have to be made of the effect of magnetic particles on the field non-uniformity within the nuclear spin sample that is being detected/imaged since such field distor-tions will broaden the sample spin resonance and will have to beaddressed in both the MRI scanner bore designs that incorporate the nanoparticles within the coils, as well as in the pulse sequences that deal with such inhomogeneous broadening. Nevertheless, theseresults provide further strong impetus for relatively simple modifi-cations to existing MRI inductive detection coils for achievingimproved SNR in scanners operating in 0.1 –2 T polarizing field range. This promise of a higher SNR would allow for shorter MRI scan time, more compact MRI systems, lower operating fields, andhigher accessibility. SUPPLEMENTARY MATERIAL In the supplementary material , figures have been provided for normalized magnetization ( M/M s) for (i) a sinusoidal magnetic signal of 800 A/m (10 Oe) amplitude and (ii) thermal noise for allcases of DC bias magnetic fields for prolate nanomagnet with biasalong minor axis and major axis as well as oblate nanomagnet with bias along minor axis. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1P. Mansfield and P. G. Morris, NMR Imaging in Biomedicine (Academic Press, London, 1982).2R. R. Ernst, G. Bodenhausen, and A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford University Press, Oxford, 1987)). 3P. T. Callaghan, Principles of Nuclear Magnetic Resonance Microscopy (Oxford University Press, Oxford, 1991). 4B. Blumich, NMR Imaging of Materials (Oxford University Press, Oxford, 2000). 5D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. 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1.551122.pdf	Network structure dependence of volume and glass transition temperature Jeffry J. Fedderly, Gilbert F. Lee, John D. Lee, Bruce Hartmann, Karel Dušek, Miroslava Dušková-Smrková, and Ján Šomvársky Citation: Journal of Rheology (1978-present) 44, 961 (2000); doi: 10.1122/1.551122 View online: http://dx.doi.org/10.1122/1.551122 View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/44/4?ver=pdfcov Published by the The Society of Rheology Articles you may be interested in Composition-dependent damping and relaxation dynamics in miscible polymer blends above glass transition temperature by anelastic spectroscopy Appl. Phys. Lett. 93, 011910 (2008); 10.1063/1.2945889 Measurement of the x-ray dose-dependent glass transition temperature of structured polymer films by x-ray diffraction J. Appl. Phys. 102, 013528 (2007); 10.1063/1.2752548 Thermal stress and glass transition of ultrathin polystyrene films Appl. Phys. Lett. 77, 2843 (2000); 10.1063/1.1322049 Conformational transition behavior around glass transition temperature J. Chem. Phys. 112, 2016 (2000); 10.1063/1.480761 Fast structural relaxation of polyvinyl alcohol below the glass-transition temperature J. Chem. Phys. 108, 10309 (1998); 10.1063/1.476492 Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55Network structure dependence of volume and glass transition temperaturea) Jeffry J. Fedderly,b)Gilbert F. Lee, John D. Lee, and Bruce Hartmann Naval Surface Warfare Center, West Bethesda, Maryland 20817-5700 Karel Dus ˇek, Miroslava Dus ˇkova´-Smrc ˇkova´, and Ja ´nSˇomva´rsky Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, 162 06 Prague, Czech Republic (Received 16 February 2000; ﬁnal revision received 21 April 2000) Synopsis A series of polyurethanes was used to determine the molar contributions of chain ends ~CE!and branch points ~BP!to free volume and glass transition temperature Tg. The polyurethanes were copolymers of diphenylmethane diisocyanate and poly ~propylene oxide !~PPO!with hydroxyl functionalities of one, two, and three. The equivalent weights of all the PPOs were equal, such thatthe chemical composition of the chain segments was essentially identical. Therefore, the onlydistinctions among polymers were differences in CE and BP concentration. Theory of branchingprocesses computer simulations were used to determine the concentration of CE due to imperfectnetwork formation. Other CE contributions were from the monofunctional PPO. Polymer volumes andT gs were correlated to CE and BP concentrations, and the contributions of these species were determined from least squares ﬁts. The molar volume and Tgcontributions were then used to determine free volume thermal expansion coefﬁcients. These values were compared to thermal expansion coefﬁcients obtained from WLF parameters ( c1,c2) obtained from the measurement of dynamic moduli as a function of temperature. © 2000 The Society of Rheology. @S0148-6055 ~00!01204-9 # I. INTRODUCTION The glass transition temperature Tgand speciﬁc volume vof an amorphous polymer network can be quantiﬁed as the summation of contributions from the chain segments,chain ends ~CEs!and branch points ~BPs!in the system. CEs increase mobility and generate volume, BPs restrict mobility and reduce the volume in their vicinity. Tradition-ally, the polymer sets used to determine CE and BP contributions to free volume andglass transition temperature have been series of homopolymers of varying molecularweight. These polymers have CE concentrations inversely proportional to their molecularweights. If the polymer can be crosslinked without the introduction of additional species,the BP concentration can be varied without affecting the structure of the chain segments.A different approach for generating polymers with variable CE and BP concentration,while keeping chain segment properties identical, was used here. Polyurethane networkswere synthesized for which the chain segments are essentially identical and the only a!Dedicated to Professor John D. Ferry. b!Author to whom correspondence should be addressed; electronic mail: FedderlyJJ@nswccd.navy.mil © 2000 by The Society of Rheology, Inc. J. Rheol. 44 ~4!, July/August 2000 961 0148-6055/2000/44 ~4!/961/12/$20.00 Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55differences are in network structure. The relevant network structure differences are the CE and BP concentrations. These concentrations were calculated and their contributions to volume and Tgwere determined. A series of ten polyurethanes was produced from reacting diphenylmethane diisocy- anate ~MDI!and poly ~propylene oxide !polyols with functionalities of one, two, or three. Through the use of polyols of identical chemical composition and equal equivalentweight, the polymers were designed to have identical chain segment composition, butwidely varying network structure. The network structures exhibit signiﬁcant differencesin properties such as the relative concentrations, molecular weights, and molecularweight distributions of elastically active network chains, dangling chains, and sol. How- ever, the volumes and T gs of amorphous copolymers such as these can be viewed as the summation of group contributions regardless of where the groups are located within thesystem. This concept of additive group contributions to polymer properties has beendeveloped extensively by Van Krevelen ~1990!. The polymers presented here have es- sentially the same chemical group composition and it is assumed that they have identical T gs except for contributions from CE and BP. The CE and BP can be viewed as addi- tional groups in the additive properties approach. The BP concentration was determinedby the amount of trifunctional polyol in the system. The determination of the CE con-centration is more complicated as a result of the use of a monofunctional component andthe lack of complete reaction. Theory of branching processes simulations were run to aidin the prediction of the CE concentration. This work is a part of our ongoing efforts tocharacterize the properties of monofunctionally modiﬁed polymer systems @Fedderly et al. ~1996!, Fedderly et al. ~1999!#. Speciﬁc volumes of the polymers were measured and a least squares ﬁt of these volumes to a model incorporating CE and BP concentrations was made. From this, molar volumes for CE and BP were determined. A similar treatment was performed for T g. Using free volume relationships, the free volume thermal expansion coefﬁcient was de- termined from the volume and Tgbehavior. Dynamic mechanical properties of the poly- mer set were also measured at several temperatures using a resonance technique. Modulifrom the various temperatures were shifted in accordance to the time–temperature super-position principle. The shift factors, plotted versus temperature, were ﬁt to the WLF equation. The WLF parameters ( c 1,c2) were used to obtain an independent determina- tion of the free volume thermal expansion coefﬁcient as well as to determine a freevolume fraction. II. EXPERIMENTAL PROCEDURES A. Sample preparation The polyurethanes synthesized for this study are divided into two sets. The ﬁrst set is comprised of typical polyurethane networks formed from a blend of difunctional ~2F!and trifunctional ~3F!poly~propylene oxide !~PPO!polyols. The 2 Fand 3Fpolyols were Poly-G 20-112 and Poly-G 30-112, respectively from Olin Chemical. Each of thesematerials has a nominal equivalent weight of 500 g/eq. The 2 Fand 3Fmaterials were blended to have speciﬁc number average functionalities F nranging from 2.1 to 3 ~nomi- nal!. The second set of polyurethanes was formed from a blend of monofunctional ~1F! and 3FPPOs. The 1 Fmaterial was a blend of UCON LB-65 ~nominally 400 g/eq !and UCON LB-135 ~nominally 700 g/eq !from Union Carbide. The two UCON materials were blended to achieve an equivalent weight of 500 g/eq. Polymers from this set hadpolyol functionalities ranging from 2.0 to 3.0. The polyol blends from both sets werereacted with a stoichiometric amount of MDI. The MDI used was Mondur ML from962 FEDDERLY ET AL. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55Bayer Chemical. This material is a blend of 4,4 8-diphenylmethane diisocyanate and 2,48-diphenylmethane diisocyanate. Other data from these samples were presented in an earlier publication and further details on the sample preparation can be found there@Fedderly et al. ~1999!#. B. Glass transition temperature Glass transition temperature measurements were made using a DuPont 910 differential scanning calorimeter ~DSC!cell with a DuPont 9900 controller. Thin ﬂat sections of approximately 10 mg were cut from the resonance bar samples. A scanning rate of10°C/min was used. Measurements were made under a nitrogen atmosphere. Glass tran-sition temperatures were determined from the inﬂection points in the DSC thermograms. C. Sol fraction Sol fractions were measured from 10 mm 320 mm specimens cut from 2 mm thick sheets. Samples were weighed dry in a dry room environment ~less than 0.25% relative humidity !then immersed in sealed jars containing nominally 40 g of 2-methoxyethyl ether as solvent. Afte r 2 d the samples were transferred to jars containing fresh solvent and kept for an additional 6 d. The samples were then removed from the solvent andallowed to dry to constant weight. The sol fraction was determined from the difference inweight. D. Speciﬁc volume The speciﬁc volume ~or density !measurements were made on bar specimens follow- ing the general procedures of ASTM D 792 ‘‘Density by Liquid Displacement’’ using octane as the liquid. Measurements were made at 23°C and at the T gof the polymer being tested. The octane density was obtained at the various temperatures using a cali-brated Pyrex bob, accounting for the thermal expansion of the bob. The low temperaturemeasurements were obtained by cooling the sample and octane separately to just belowthe desired measurement temperature. The octane was in an insulated cup and the samplein a desiccator to prevent frost from forming on the surface. The sample was quicklyimmersed in the octane and placed in the balance. As the temperature reached the desiredvalue, the immersed sample mass was recorded. E. Dynamic mechanical properties The dynamic mechanical properties were measured using a resonance technique de- veloped at this laboratory @Madigosky and Lee ~1983!#. This technique has been used in a number of studies on polymer properties @Duffyet al. ~1990!, Hartmann and Lee ~1991!#. The apparatus is based on producing resonances in a bar specimen. Typical specimen length is 10–15 cm with square lateral dimensions of 0.635 cm. Measurementsare made over 1 decade of frequency in the kHz region from 260 to 70°C at 5° intervals. By applying the time–temperature superposition principle, the raw data are shifted togenerate a reduced frequency plot ~over as many as 20 decades of frequency !at a constant reference temperature. III. RESULTS AND DISCUSSION The primary objective of this work was to determine the molar contributions of CE and BP to the polymer glass transition temperature and speciﬁc volumes. To accomplishthis, it is necessary to have a set of polymers in which the variations in CE and BP963 NETWORK STRUCTURE DEPENDENCE Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55dominate the changes in these properties. This requires that the composition of the poly- mers all be the same. This was achieved by using mono-, di-, and trifunctional PPOs ofequal equivalent weight. The use of equal equivalent weight polyols is a novel approachand is central to this work, so it is worth pointing out the signiﬁcance of this choice. Forthe monofunctional component, the equivalent weight is 503 g/eq with a functionality of1.01, thus a molecular weight of 508 g/mol. For the difunctional component, the respec-tive values are 502 g/eq, 1.99, and 999 g/mol. For the trifunctional component, therespective values are 504 g/eq, 2.97, and 1496 g/mol. Each polyol equivalent reacts withone equivalent of the diisocyanate, 125 g/eq, giving rise to one urethane group. Thus, anystoichiometric blend of these three polyols with diisocyanate will have essentially thesame urethane concentration, the same propylene oxide concentration, the same aromaticconcentration, etc. Slight differences exist in the PPOs due to the use of different alcoholinitiators in their production. To verify that this and the slight equivalent weight differ- ences have only a small effect on volume and T g, the sample set was analyzed using an additive group contribution approach similar to that described by Fedderly et al. ~1998!. It was found that the polymer volumes are constant 60.001 cm3/g and that the Tgs are constant 60.5°C. Given these small variations, it was felt that the contributions of CE and BP could be determined. The novel approach taken here can be contrasted with studies on some similar PPO polymers from mono- and trifunctional polyols reacted with diisocyanate @Randrianan- toandroet al. ~1997!#. There the triol used had a molecular weight of 720 g/mol or 240 g/eq while the monofunctional had a molecular weight of 136 g/mol or 136 g/eq, nearlya factor of two smaller than the trifunctional. Thus, the higher the fraction of triol in thosepolymers, the lower the urethane concentration. Also, the monofunctional componentused there was aromatic while the trifunctional component was an aliphatic poly ~propyl- ene oxide !. In typical systems such as this, it is difﬁcult to separate the contributions of composition and network structure to the polymer T gor other properties. Some other complicating factors include the extent of reaction, which is taken into account by aand the degree of cyclization, which has been shown by Dus ˇek~1989!to be no more than 2%–3% for the trifunctional polymer. The range of speciﬁc volumes and Tgs in the polymers presented here is quite small, but as shown above, the variations should be predominantly due to differences in network structure ~concentrations of CE and BP !. The volume and Tgdata are ﬁt to simple models that account for the concentration of these structural features. From the ﬁts of the data tothe models, material constants are determined which predict the dependence of volume andT gon CE and BP concentration. In other treatments, the CE and BP are speciﬁcally identiﬁed to consist of various numbers of repeat units or atoms along the chain @Chompff ~1971!#. In this work, there are no assumptions concerning the magnitude or the range of the CEs and BPs. Theirconcentrations simply have an overall effect on the system. The polyurethane networks used in this study and their polyol compositions are listed in Table I. The polymers are speciﬁed by polyol type and number average functionality of the polyol blend. For example, a designation of 2 F13F, 2.20 indicates a polymer made from a blend of difunctional and trifunctional PPOs having an average functionalityof 2.20. A. Chain end and branch point concentrations The BP concentration is determined strictly from stoichiometry. The BP concentration is equivalent to the 3 Fpolyol concentration in the starting materials mixture ~except for a small correction because the 3 Fmaterial does not have a functionality of exactly three !.964 FEDDERLY ET AL. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55CE concentrations are more difﬁcult to determine. For every monofunctional chain in the starting formulation one CE remains in the ﬁnal network. In addition to this, however,CEs remain from unreacted functional groups. Conversion of the functional groups is not100% and the resulting network is less than ideal. Dangling chains and sol are generated.Using the sol fraction, determined experimentally, the degree of conversion was calcu-lated as described below. Computer simulations were performed on the polymer set, using the theory of branch- ing processes. This theory is based on the generation of structures from component unitsin different reaction states @Dusˇek~1989!#. The formalism of using probability generating functions has been used to describe and transform various distributions. For details seeDusˇek~1986!. The simulation predicts the critical gel point conversion and several prop- erties including: molecular weight averages before the gel point, sol and gel fractions,and molecular averages of the sol as a function of conversion of functional groups. Thesimulation also offers information on the fraction of material in dangling chains and onconcentration and molecular weight averages of elastically active network chains. Usually, all these parameters ~e.g.,X!are calculated as a function of conversion a. Beyond the gel point, all these parameters are also a function of the so-called extinction probability v, which itself is a function of a X5C~a,v~a!!. ~1! If the determination of ais difﬁcult experimentally, one can calculate afrom the weight fraction of the sol ws, which is readily measured a5F~ws,v~a!!. ~2! The extinction probability v(a) is determined from Eq. ~3! v5F~a,v!, ~3! whereF(a,v) is obtained from the probability generating function for the number of additional bonds of a unit already connected by one bond to another unit, F(a,z), by substituting z5v@Dusˇek~1989!#. The procedure is illustrated for the case of F3 PPO triol, component a, and diisocy- anate, compound b!, where the OH and NCO groups, respectively, have the same reac- tivity and the system is stoichiometric. The sol fraction is given byTABLE I. Polymer compositions, sol fractions, and degree of conversion. Sample FnN1F ~mole!N2F ~mole!N3F ~mole! Ws a 2F13F 2.12 0.000 0.867 0.133 0.070 0.9772.20 0.000 0.786 0.214 0.040 0.9712.40 0.000 0.582 0.418 0.017 0.9602.60 0.000 0.378 0.622 0.007 0.9582.97 0.000 0.000 1.000 0.003 0.942 1F13F 1.99 0.500 0.000 0.500 0.244 0.9392.12 0.434 0.000 0.566 0.122 0.9592.20 0.393 0.000 0.607 0.087 0.9622.40 0.291 0.000 0.709 0.041 0.9632.60 0.189 0.000 0.811 0.022 0.953965 NETWORK STRUCTURE DEPENDENCE Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55ws5wa~12a1avb!31wb~12a1ava!2, ~4! wherewaandwbare weight fractions of components aandb, and ais the molar conversion of isocyanate and hydroxy groups; vaandvbare extinction probabilities determined by the equations va5~12a1avb!2, ~5! vb5~12a1ava!. ~6! The extinction probabilities can be calculated explicitly by solving Eqs. ~5!and~6!and eliminating the trivial roots va5vb21. The solution is as follows: va5~12a2!2 a4, ~7! vb5122a21a3 a3. ~8! By substituting Eqs. ~7!and~8!into Eq. ~4!and solving the equation numerically with respect to a, one gets the desired result. The sol fraction and degree of conversion values are also shown in Table I. The amount of unreacted OH and NCO groups are readily calculated from the degree ofconversion and the concentration of the starting materials. These values plus the concen-tration of monofunctional component are added to give the CE concentration. The CE and BP concentrations nEandnB, respectively, are shown in Table II. B. Speciﬁc volume and glass transition temperature The molar contributions of CE and BP to speciﬁc volume were determined using the CE and BP concentrations and the measured volumes. It is assumed here that the speciﬁc volume can be expressed as the sum of v0~the volume of an inﬁnitely long linear polymer !, positive chain end contributions, and negative branch point contributions, as shown in Eq. ~9! v5v01VEnE2VBnB, ~9!TABLE II. Chain end and branch point concentration, speciﬁc volume, and Tg. SamplenE3104 ~mol/g !nB3104 ~mol/g !vg~meas! ~cm3/g!vg~pred! ~cm3/g!v23~meas! ~cm3/g)v23~pred! ~cm3/g!Tg~meas! ~°C!Tg~pred! ~°C! 2F13F 2.12 0.733 0.970 0.8969 0.8966 0.9311 0.9311 228 227.6 2.20 0.924 1.503 0.8961 0.8962 0.9305 0.9303 227 227.1 2.40 1.274 2.690 0.8953 0.8952 0.9282 0.9285 225 225.9 2.60 1.377 3.693 0.8937 0.8941 0.9267 0.9266 225 225.9 2.97 1.845 5.193 0.8929 0.8929 0.9242 0.9245 223 223.2 1F13F 1.99 5.897 3.876 0.9001 0.8998 0.9337 0.9342 229 229.7 2.12 4.529 4.120 0.8969 0.8977 0.9314 0.9313 228 227.8 2.20 4.022 4.257 0.8969 0.8969 0.9308 0.9301 228 227.0 2.40 3.086 4.558 0.8961 0.8953 0.9276 0.9279 226 225.5 2.60 2.639 4.812 0.8945 0.8944 0.9268 0.9266 224 224.6966 FEDDERLY ET AL. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55whereVEandVBare the CE and BP molar volumes, respectively. In the present case, Eq. ~9!leads to a set of ten simultaneous equations ~for the ten polymers !in three unknowns ~v0,VE, andVB!which can be solved by the method of least squares. To achieve an accurate determination of the three parameters, it is neces-sary that there be considerable variation in the CE and BP concentrations and that they bepresent in many unique ratios. This type of diversity is greatly enhanced through the use of both conventional formulations using 2 F13Fpolyols and unique formulations using 1F13Fpolyols. This variability can be seen in Fig. 1, which shows speciﬁc volume measured at 23°C ( v23) versus CE concentration and in Fig. 2, which shows the same volumes versus BP concentration. In both ﬁgures, the intersection of the 1 F13Fand 2F13Flines is at the pure 3 Fsample point. Extending out from this point, the number FIG. 1.Speciﬁc volume at 23°C vs chain end concentration. FIG. 2.Speciﬁc volume at 23°C vs branch point concentration.967 NETWORK STRUCTURE DEPENDENCE Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55average functionality monotonically decreases. The uniqueness of the CE and BP ratios is evident from the 1 F13Fand 2F13Fdata having dependencies of opposite slope for CE concentration ~Fig. 1 !, and slopes of the same sign, but signiﬁcantly different mag- nitude for BP concentration ~Fig. 2 !. Equation ~9!was evaluated using volumes measured at Tg(vg) and at 23°C ( v23). AtTg, the values for v0,VE, andVBwere determined to be 0.8968 cm3/g, 13.0 cm3/mole, and 12.3 cm3/mole, respectively. At 23°C, the respective values were found to be 0.9318 cm3/g, 17.4 cm3/mole, and 20.3 cm3/mole. Volumes for each polymer were predicted from the model at both temperatures and are shown along withthe measured volumes in Table II. In a manner similar to speciﬁc volume, T gis assumed to be of the form shown in Eq. ~10!. It is the sum of Tg0, theTgof a linear polymer of inﬁnite molecular weight, positive BP contributions, and negative CE contributions. Tg5Tg01~TBnB2TEnE!/vg, ~10! whereTBandTEare molar Tgcontributions for BP and CE, respectively, and vgis the speciﬁc volume at Tggiven in Table II. Again, this provides a set of ten simultaneous equations in three unknowns ( Tg0,TB, andTE!. A least squares solution for Tg0was determined to be 245 K, while TBandTEwere 1.21 3104and 1.05 3104Kcm3/mole, respectively. Tgvalues were predicted for each polymer and these values along with the measured values are given in Table II. The parameters TEandTBare related to other material constants. Consider the simple case of a polymer with no branch points nB50. Then the molecular weight is twice the reciprocal of the concentration of chain ends, Mn52/nE, and Eq. ~10!reduces to the well known result @Fox and Loshaek ~1955!#for the effect of chain ends on Tg. Tg5Tg02K/Mn, ~11! whereK52TE/vg. Likewise, the TBvalues can be compared to the KXparameter which represents the contribution of branch points to Tg@Chompff ~1971!#, with the result that KX52TB. TheVEandVBparameters are also related to other material constants. Again consider the case of no branch points. The change in volume from one polymer to another withdifferent molecular weight is assumed to result from a change in free volume. Thecontribution that chain ends make to free volume fraction in polymers of ﬁnite molecularweight has been given by Ninomiya et al. ~1963!in the form f5f 01A/Mn ~12! and it follows from Eq. ~9!thatA52VE/vg. Likewise, the VBvalues can be compared to theAxparameter, which represents the contribution of a pair of branch points to volume @Chompff ~1971!#, with the result that Ax52VB. Values for the four parameters determined here are given in Table III along with the equivalent AandKvalues. These values are typical of those found for other polymers @Chompff ~1971!, Nielsen and Landel ~1990!#. It should be noted that the branch points in these systems are trifunctional. Free volume relationships show that the Tgdependence on branch points is proportional to the fractional free volume and to j22~jis the functionality of the branch point !and inversely proportional to the free volume thermal968 FEDDERLY ET AL. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55expansion coefﬁcient af@Chompff ~1971!#. Therefore the TBorKxvalues shown here would be half the value expected for a similar polymer with tetrafunctional branch points. Chompff ~1971!has shown that a reasonable approximation of af~the free volume thermal expansion coefﬁcient !is given by the ratio of AXtoKX. Since these parameters have been related to constants from Eqs. ~9!and~10!, we can write af5AX/KX5VB/TB. ~13! Using the values of the parameters at Tg,VB/TB510.231024°C21. Similarly, the ratio of the total relative volume change to change in Tgwill also provide an estimate for the expansion coefﬁcient. This is obtained from the reciprocal of the slope of a plot of Tg vsvg/vg0, shown in Fig. 3. A value of 10.3 31024°C21was obtained, in excellent agreement with the value from Eq. ~13!. Note that this value is based on the ratio of two experimentally determined values and does not depend on calculated CE or BP concen-trations. C. Dynamic mechanical measurements Free volume parameters can also be determined from dynamic moduli obtained as a function of temperature. Dynamic shear moduli were measured from 260 to 70°C at 5° intervals using the resonance apparatus described previously. Using the time– temperature superposition principle, the G 8values were shifted in log frequency space to obtain the shift factor log aT. The temperature at which G8versus frequency has theTABLE III. Volume and glass transition temperature parameters. VE ~cm3/mol!VB ~cm3/mol!TE31024 (K cm3/mol)TB31024 (K cm3/mol)A ~g/mol !AX (cm3/mol!K31024 ~K g/mol)KX31024 (K cm3/mol) Tg13.0 12.3 1.05 1.21 29.0 24.6 2.34 2.42 23°C 17.4 20.3 37.3 43.6 FIG. 3.Glass transition temperature vs relative speciﬁc volume at Tg.969 NETWORK STRUCTURE DEPENDENCE Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55steepest slope was chosen as the reference temperature T0. A plot of log aTvsTwas produced for each polymer and least-squares ﬁt to the WLF equation @Ferry ~1980!# logaT52c10~T2T0!/~c201T2T0!. ~14! Two constants c10andc20are obtained from the ﬁt for the given reference temperature. Thec1andc2parameters were redetermined using Tgas the reference temperature from the following transformations: c2g5c201Tg2T0, ~15! c1g5c10c20/c2g. ~16! Thec1gandc2gparameters are related to the free volume parameters afandfg~the fractional free volume at Tg!, using the following equations @Ferry ~1980!#: fg5B/2.303c1g, ~17! af5B/2.303c1gc2g, ~18! whereBis an empirical constant near unity. A typical plot of log aTvsTﬁt to the WLF equation is shown in Fig. 4. The WLF and free volume parameters for the polymer set arelisted in Table IV. Thef gvalues appear to be fairly randomly distributed, therefore it was not possible to correlate these value with vg. An average value of 0.043 was obtained. There was also a fair amount of scatter in the afvalues, but the average value of 8.9 31024°C21 compares very closely with the value of 10.2 31024°C21obtained from Eq. ~13!. As- suming a linear temperature dependence between Tgand 23°C, the rubbery thermal expansion coefﬁcients aLwere also determined and are also shown in Table IV. The average value is 7.5 31024°C21. Although it would be expected that the afvalues be FIG. 4.LogaTvsTplot for 1 F13F,Fn52.6, ﬁt to the WLF equation.970 FEDDERLY ET AL. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55somewhat less than those of aL, it is reasonable that the aLvalue compares very closely with the Eq. ~13!and Eq. ~18!values. IV. CONCLUSIONS A novel method was used to determine volume and Tgcontributions from chain ends and branch points in a series of polyurethane networks. The volumes and Tgso ft h e polymers, which were identical in composition but varied in CE and BP concentration,were ﬁt to simple models incorporating these elements. It was found that the opposing contributions of CE and BP were nearly identical in magnitude for both volume and T g. AtTg, the volume contribution of a CE was found to be 13.0 cm3/mole. The volume that is removed from the system by a BP was found to be 12.3 cm3/mole. At 23°C these values were 17.4 and 20.3 cm3/mole, respectively. These values represent the additional free volume associated with a chain end or lack of it by having a branch point and are not identiﬁed with any speciﬁc atoms in the vicinity of these points. For Tg, a positive contribution of 1.2°C for every 1024mole of BP per cm3was determined. The negative contribution for CE was 1.1°C for every 1024mole/cm3. The free volume thermal expansion coefﬁcient afwas determined from the volume andTgparameters and found to be approximately 10 31024°C21. Dynamic mechanical moduli of the materials were also measured as a function of temperature. From WLF ﬁts of the shift factors, the c1andc2parameters were used to calculate free volume thermal expansion coefﬁcients and the free volume fractions at Tg. The average expansion co- efﬁcient was found to be about 9 31024°C21and compares closely with the value obtained from the volume and Tgmeasurements. The free volume fraction fgwas also determined from the WLF ﬁts. A reasonable average free volume fraction of 0.043 wasdetermined. ACKNOWLEDGMENTS This work was supported by NATO Collaborative Research Grant No. CRG 970041, the CDNSWC In-house Laboratory Independent Research Program sponsored by theOfﬁce of Naval Research, and by Grant Agency of the Academy of Sciences of the CzechRepublic, Grant No. A4050808.TABLE IV. WLF parameters and thermal expansion coefﬁcients Sample c1gc2g ~°C! fg/Baf/B3104 ~°C21)aL3104 ~°C21) 2F13F 2.12 12.2 41.0 0.036 8.7 7.52.20 10.5 55.9 0.041 7.4 7.72.40 10.6 37.5 0.041 10.9 7.72.60 8.3 50.1 0.053 10.5 7.72.97 7.2 44.6 0.060 13.4 7.6 1F13F 1.99 11.6 47.9 0.037 7.8 7.12.12 11.0 62.4 0.040 6.3 7.62.20 11.0 62.5 0.040 6.3 7.42.40 10.4 44.8 0.041 9.3 7.22.60 11.0 48.3 0.040 8.2 7.7971 NETWORK STRUCTURE DEPENDENCE Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55References Chompff, A. J., ‘‘Glass Points of Polymer Networks,’’ in Polymer Networks, Structure and Mechanical Prop- erties, edited by A. J. Chompff and S. Newmann ~Plenum, New York, 1971 !, pp. 145–192. Duffy, J. V., G. F. Lee, J. D. Lee, and B. Hartmann, ‘‘Dynamic mechanical properties of poly ~tetramethylene ether!glycol polyurethanes,’’ in Sound and Vibration Damping with Polymers , ACS Symposium Series 424, edited by R. D. Corsaro and L. H. Sperling ~American Chemical Society, Washington, D.C., 1990 !, pp. 281–300. Dusˇek, K., ‘‘Network formation in curing of epoxy resins,’’ Adv. Polym. Sci. 78, 1–59 ~1986!. Dusˇek, K., ‘‘Formation and structure of networks from telechelic polymers: Theory and application to poly- urethanes,’’ in Telechelic Polymers: Synthesis and Applications , edited by E. J. Goethals ~Chemical Rubber Corp., Boca Raton, FL, 1989 !, pp. 289–315. Fedderly, J., E. Compton, and B. Hartmann, ‘‘Additive group contributions to density and glass transition temperature in polyurethanes,’’ Polym. Eng. Sci. 38, 2072–2076 ~1998!. Fedderly, J. J., G. F. Lee, D. J. Ferragut, and B. Hartmann, ‘‘Effect of Monofunctional and Trifunctional Modiﬁers on a Phase Mixed Polyurethane System,’’ Polym. Eng. Sci. 36, 1107–1113 ~1996!. Fedderly, J. J., G. F. Lee, J. D. Lee, B. Hartmann, K. Dus ˇek, J. Sˇomvarsky, and M. Smrc ˇkova´, ‘‘Multifunctional Polyurethane Network Structures,’’ Macromol. Symp. 148, 1–14 ~1999!. Ferry, J. D., Viscoelastic Properties of Polymers , 3rd ed. ~Wiley, New York, 1980 !, pp. 264–320. Fox, T. G. and S. Loshaek, ‘‘Inﬂuence of molecular weight and degree of crosslinking on the speciﬁc volume and glass temperature of polymers,’’ J. Polym. Sci. 15, 371–390 ~1955!. Hartmann, B. and G. F. Lee, ‘‘Dynamic mechanical relaxation in some polyurethanes,’’ J. Non-Cryst. Solids 131–133, 887–890 ~1991!. Madigosky, W. M. and G. F. Lee, ‘‘Improved resonance technique for materials characterization,’’ J. Acoust. Soc. Am. 73, 1374–1377 ~1983!. Nielsen, L. E. and R. F. Landel, Mechanical Properties of Polymers and Composites , 2nd ed. ~Marcel Dekker, New York, 1990 !, pp. 1–32. Ninomiya, K., J. D. Ferry, and Y. O¯yanagi, ‘‘Viscoelastic properties of polyvinyl acetate II. Creep studies of blends,’’ J. Phys. Chem. 67, 2297–2308 ~1963!. Randrianantoandro, H., T. Nicolai, D. Durand, and F. Prochazka, ‘‘Viscoelastic relaxation of polyurethane at different stages of gel formation. 1. Glass transition dynamics,’’ Macromolecules 30, 5893–5896 ~1997!. Van Krevelen, D. W., Properties of Polymers , 3rd ed. ~Elsevier, New York, 1990 !.972 FEDDERLY ET AL. Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014 22:13:55
1.4995240.pdf	Low spin wave damping in the insulating chiral magnet Cu 2OSeO3 I. Stasinopoulos , S. Weichselbaumer , A. Bauer , J. Waizner , H. Berger , S. Maendl , M. Garst , C. Pfleiderer , and D. Grundler Citation: Appl. Phys. Lett. 111, 032408 (2017); doi: 10.1063/1.4995240 View online: http://dx.doi.org/10.1063/1.4995240 View Table of Contents: http://aip.scitation.org/toc/apl/111/3 Published by the American Institute of Physics Articles you may be interested in Semitransparent anisotropic and spin Hall magnetoresistance sensor enabled by spin-orbit torque biasing Applied Physics Letters 111, 032402 (2017); 10.1063/1.4993899 Nanoconstriction spin-Hall oscillator with perpendicular magnetic anisotropy Applied Physics Letters 111, 032405 (2017); 10.1063/1.4993910 Fast vortex oscillations in a ferrimagnetic disk near the angular momentum compensation point Applied Physics Letters 111, 032401 (2017); 10.1063/1.4985577 Ultrafast imprinting of topologically protected magnetic textures via pulsed electrons Applied Physics Letters 111, 032403 (2017); 10.1063/1.4991521 Integration of antiferromagnetic Heusler compound Ru 2MnGe into spintronic devices Applied Physics Letters 111, 032406 (2017); 10.1063/1.4985179 Inversion of the domain wall propagation in synthetic ferrimagnets Applied Physics Letters 111, 022407 (2017); 10.1063/1.4993604Low spin wave damping in the insulating chiral magnet Cu 2OSeO 3 I.Stasinopoulos,1S.Weichselbaumer,1A.Bauer,2J.Waizner,3H.Berger,4S.Maendl,1 M.Garst,3,5C.Pfleiderer,2and D. Grundler6,a) 1Physik Department E10, Technische Universit €at M €unchen, D-85748 Garching, Germany 2Physik Department E51, Technische Universit €at M €unchen, D-85748 Garching, Germany 3Institute for Theoretical Physics, Universit €at zu K €oln, D-50937 K €oln, Germany 4Institut de Physique de la Matie `re Complexe, /C19Ecole Polytechnique F /C19ed/C19erale de Lausanne, 1015 Lausanne, Switzerland 5Institut f €ur Theoretische Physik, Technische Universit €at Dresden, D-01062 Dresden, Germany 6Institute of Materials (IMX) and Laboratory of Nanoscale Magnetic Materials and Magnonics (LMGN), /C19Ecole Polytechnique F /C19ed/C19erale de Lausanne (EPFL), Station 17, 1015 Lausanne, Switzerland (Received 5 May 2017; accepted 8 July 2017; published online 21 July 2017) Chiral magnets with topologically nontrivial spin order such as Skyrmions have generated enormous interest in both fundamental and applied sciences. We report broadband microwave spectroscopy performed on the insulating chiral ferrimagnet Cu 2OSeO 3. For the damping of magnetization dynamics, we ﬁnd a remarkably small Gilbert damping parameter of about 1 /C210/C04at 5 K. This value is only a factor of 4 larger than the one reported for the best insulating ferrimagnet yttrium iron garnet at room temperature. We detect a series of sharp resonances and attribute them to conﬁned spin waves in the mm-sized samples. Considering the small damping, insulating chiral magnets turnout to be promising candidates when exploring non-collinear spin structures for high frequency appli- cations. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4995240 ] The development of future devices for microwave appli- cations, spintronics, and magnonics 1–3requires materials with a low spin wave (magnon) damping. Insulating compounds are advantageous over metals for high-frequency applications as they avoid damping via spin wave scattering at free chargecarriers and eddy currents. 4,5Indeed, the ferrimagnetic insula- tor yttrium iron garnet (YIG) holds the benchmark with a Gilbert damping parameter aintr¼3/C210/C05at room tempera- ture.6,7During the last few years, chiral magnets have attracted a lot of attention in fundamental research and stimu- lated new concepts for information technology.8,9This mate- rial class hosts non-collinear spin structures such as spin helices and Skyrmions below the critical temperature Tcand critical ﬁeld Hc2.10–12Dzyaloshinskii-Moriya interaction (DMI) is present that induces both the Skyrmion lattice phase and nonreciprocal microwave characteristics.13Low damping magnets offering DMI would generate new prospects by par-ticularly combining complex spin order with long-distance magnon transport in high-frequency applications and mag- nonics. 14,15At low temperatures, they would further enrich the physics in magnon-photon cavities that call for materials with small aintrto achieve high-cooperative magnon-to-pho- ton coupling in the quantum limit.16–19 In this work, we investigate the Gilbert damping in Cu2OSeO 3, a prototypical insulator hosting Skyrmions.20–23 This material is a local-moment ferrimagnet with Tc¼58 K and magnetoelectric coupling24that gives rise to dichroism for microwaves.25–27The magnetization dynamics in Cu2OSeO 3has already been explored.13,28,29A detailed investigation on the damping which is a key quality for mag- nonics and spintronics has not yet been presented however. To evaluate aintr, we explore the ﬁeld polarized state (FP)where the two spin sublattices attain the ferrimagnetic arrangement.21Using spectra obtained by two different copla- nar waveguides (CPWs), we extract a minimum aintr¼(9.9 64.1)/C210–5at 5 K, i.e., only about four times higher than in YIG at room temperature. We resolve numerous sharp reso- nances in our spectra and attribute them to modes that are conﬁned modes across the macroscopic sample and allowed for by the low damping. Our ﬁndings substantiate the rele-vance of insulating chiral magnets for future applications in magnonics and spintronics. From single crystals of Cu 2OSeO 3, we prepared two bar-shaped samples exhibiting different crystallographic ori- entations. The samples had lateral dimensions of 2 :3/C20:4 /C20:3m m3. They were positioned on CPWs that provided us with a radiofrequency (rf) magnetic ﬁeld hinduced by a sinusoidal current applied to the signal (S) line surrounded by two ground (G) lines (Fig. 1andsupplementary material , Table SI). We used two different CPWs with either a broad30 or narrow signal line width of ws¼1m m o r 2 0 lm, respec- tively. The central long axis of the rectangular Cu 2OSeO 3 rods was positioned on the central axis of the CPWs. The static magnetic ﬁeld Hwas applied perpendicular to the sub- strate with Hkh100iandHkh111ifor samples S1 and S2, FIG. 1. Sketch of a single crystal mounted on either a broad or narrow CPW with a signal (S) line width wsof either 1 mm or 20 lm, respectively (not to scale). The rf ﬁeld his indicated. The static ﬁeld His applied perpendicular to the CPW plane.a)Electronic mail: dirk.grundler@epﬂ.ch 0003-6951/2017/111(3)/032408/5/$30.00 Published by AIP Publishing. 111, 032408-1APPLIED PHYSICS LETTERS 111, 032408 (2017) respectively. The direction of Hdeﬁned the z-direction fol- lowing the deﬁnition of Ref. 4. The rf ﬁeld component h?H provided the relevant torque for excitation. Components hk Hdid not induce precessional motion in the FP state of Cu2OSeO 3. We recorded spectra by a vector network ana- lyzer using the magnitude of the scattering parameter S12. We subtracted a background spectrum recorded at 1 T toenhance the signal-to-noise ratio (SNR) yielding the dis-played DjS 12j. In Ref. 7, Klingler et al. have investigated the damping of the insulating ferrimagnet YIG and found that Gilbert parameters aintrevaluated from both the uniform pre- cessional mode and standing spin waves conﬁned in the mac-roscopic sample provided the same values. We evaluateddamping parameters as follows (and further outlined in thesupplementary material ). 31When performing frequency- swept measurements at different ﬁelds H, the obtained line- width Dfwas considered to scale linearly with the resonance frequency fras32 Df¼2aintr/C2frþDf0; (1) with the inhomogeneous broadening Df0.I nF i g s . 2(a)–2(d) , we show spectra recorded at 5 K in the FP state of the materialusing the two different CPWs. For the same applied ﬁeld H, we observe peaks residing at higher frequency fforHkh100i compared to Hkh111i. From the resonance frequencies, we extract the cubic magnetocr ystalline anisotropy constant K¼ð /C0 0:660:1Þ/C210 3J/m3for Cu 2OSeO 3[compare supple- mentary material , Fig. S1 and Eqs. (S1)–(S3)]. The magnetic anisotropy energy is found to be extremal for h100iandh111i reﬂecting easy and hard axes, respectively. The saturation mag-netization of Cu 2OSeO 3amounted to l0Ms¼0:13 T at 5 K.22 Figure 2summarizes spectra taken with two different CPWs on the two different Cu 2OSeO 3crystals S1 and S2, exhibiting different crystallographic orientations in the ﬁeldH(further spectra are depicted in supplementary material , Fig. S2). For the broad CPW [Figs. 2(a) and2(c)], we mea- sured pronounced peaks whose linewidths were small. Weresolved small resonances below the large peaks [arrows in Fig.2(b)] that shifted with Hand exhibited an almost ﬁeld- independent frequency offset dffrom the main peaks that we will discuss later. For the narrow CPW [Figs. 2(b) and2(d)], we observed a broad peak superimposed by a series of reso-nances that shifted to higher frequencies with increasing H. The ﬁeld dependence excluded them from being noise orartifacts of the setup. Their number and relative intensities varied from sample to sample and also upon remounting the same sample in the cryostat (not shown). They disappearedwith increasing temperature Tbut the broad peak remained. It is instructive to ﬁrst follow the orthodox approach 29and analyze damping parameters from modes reﬂecting the exci- tation characteristics of the broad CPW. Second, we followRef. 7and analyze conﬁned modes. Lorentz curves (blue) were ﬁtted to the spectra recorded with the broad CPW to determine resonance frequencies andlinewidths. Note that the corresponding linewidths were larger by a factor ofﬃﬃ ﬃ 3p compared to the linewidth Dfthat is conventionally extracted from the imaginary part of the scat-tering parameters. 33The extracted linewidths Dfwere found to follow linear ﬁts based on Eq. (1)at different temperatures [supplementary material , Figs. S2 and S3(a)]. In Fig. 3(a), we depict the parameter aintrobtained from the broad CPW.34ForHkh100i[Fig. 3(a)], between 5 and 20 K, the lowest value for aintramounts to (3.7 60.4)/C210–3. This value is three times lower compared to preliminary datapresented in Ref. 29. Beyond 20 K, the damping is found to increase. For Hkh111i, we extract (0.6 60.6)/C210 –3as the smallest value. Note that these values for aintrstill contain an extrinsic contribution due to the inhomogeneity of hin the z-direction and thus represent upper bounds for Cu 2OSeO 3. For the inhomogeneous broadening Df0in Fig. 3(b), the data- sets taken with Happlied along different crystal directions are consistent and show the smallest Df0at lowest tempera- ture. Note that a CPW wider than the sample is assumed toexcite homogeneously the ferromagnetic resonance (FMR)atf FMR35transferring an in-plane wave vector k¼0 to the sample. Accordingly, we ascribe the intense resonances of Figs. 2(a) and2(c) tofFMR. Using fFMR¼6 GHz and aintr ¼3:7/C210/C03at 5 K [Fig. 3(a)], we estimate a minimum relaxation time of s¼½2paintrfr/C138/C01¼6:6 ns. In the following, we examine in detail the additional sharp resonances that we observed in spectra of Fig. 2.I n Fig. 2(a) taken with the broad CPW for Hkh100i,w e FIG. 2. Spectra DjS12j(magnitude) obtained at T ¼5 K for different Hval- ues using (a) broad and (b) narrow CPWs when Hjjh100ion sample S1. Corresponding spectra taken on sample S2 for Hjjh111iare shown in (c) and (d), respectively. Note the strong and sharp resonances in (a) and (c) when using the broad CPW that provides a much more homogeneous excitation ﬁeld h. Arrows mark resonances that have a ﬁeld-independent offset with the corresponding main peaks and are attributed to standing spin waves. An exemplary Lorentz ﬁt curve is shown in blue color in (a).FIG. 3. (a) Damping parameters aintrand (b) inhomogeneous broadening Df0 forHparallel to h100i(circle) and h111i(square). aintrandDf0are obtained from the slopes and intercepts at fr¼0, respectively, of linear ﬁts to the linewidth data (compare supplementary material , Figs. S2 and S3).032408-2 Stasinopoulos et al. Appl. Phys. Lett. 111, 032408 (2017)identify sharp resonances that exhibit a characteristic fre- quency offset dfwith the main resonance at all ﬁelds (black arrows). We illustrate this in Fig. 4(a)in that we shift spectra of Fig. 2(a) so that the positions of their main resonances overlap. The additional small resonances (arrows) in Fig.2(a)are well below the uniform mode. This is characteristic for backward volume magnetostatic spin waves (BVMSWs). Standing waves of such kind can develop if they are reﬂected at least once at the bottom and top surfaces of the sample.The resulting standing waves exhibit a wave vector k¼np=d, with order number nand sample thickness d¼0.3 mm. The BVMSW dispersion relation f(k)o fR e f . 13(compare also supplementary material , Fig. S4) provides a group velocity v g¼/C0300 km/s at k¼p=d[triangles in Fig. 4(b)]. The decay length ld¼vgsamounts to 2 mm considering s¼6:6 ns. This is about seven times larger than the relevant thickness d, thereby allowing standing spin wave modes to form across thethickness of the sample. Based on the dispersion relation of Ref. 13, we calculated the frequency splitting df¼ f FMR/C0fðnp=dÞ[open diamonds in Fig. 4(inset)] assuming n¼1a n d t¼0.4 mm for the sample width tdeﬁned in Ref. 13. Experimental values (ﬁlled symbols) agree with the calcu- lated ones (open symbols) within about 60 MHz. In the caseof the narrow CPW, which provides a broad wave vector dis-tribution, 36we observe even more sharp resonances [Figs. 2(b)and2(d)]. A set of resonances was reported previously in the ﬁeld-polarized phase of Cu 2OSeO 3.26,28,37,38Maisuradze et al. assigned secondary peaks in thin plates of Cu 2OSeO 3to different standing spin-wave modes38in agreement with our analysis outlined above. We attribute the series of sharp resonances in Figs. 2(b) and2(d) to further standing spin waves. In Figs. 5(a) and 5(b), we highlight prominent and particularly narrow reso- nances with #1, #2, and #3 recorded with the narrow CPWforHkh100iandHkh111i. We trace their frequencies f ras a function of H. They depend linearly on Hshowing that for both crystal orientations, the selected sharp peaks reﬂect dis-tinct spin excitations. From the slopes, we extract a Land /C19e factor g¼2.14 at 5 K. Consistently, this value is slightly larger than g¼2.07 reported for 30 K in Ref. 13.F r o m g¼2.14, we calculate a gyromagnetic ratio c¼gl B=/C22h¼1:88/C21011rad/ sT, where lBis the Bohr magneton of the electron. Note thatFIG. 4. Spectra of Fig. 2(a) replotted as f/C0fFMRðHÞfor different Hvalues such that all main peaks are at zero frequency and the ﬁeld-independent fre- quency splitting dfbecomes visible. The numerous oscillations seen particu- larly on the bottom curve are artefacts from the calibration routine. The inset depicts experimentally evaluated (ﬁlled circles) and theoretically predicted (diamonds) values dfusing dispersion relations for a platelet. Triangles indi- cate calculated group velocities vgatk¼p=ð0:3m m Þ. Dashed lines are guides to the eyes.FIG. 5. Resonance frequencies as a function of ﬁeld Hof selected sharp modes labelled #1 to #3 extracted from individual spectra (insets) for (a) Hkh100iand (b) Hkh111iat T¼5 K. (c) Lorentz ﬁt of a sharp mode #1 forHkh100iat 0.85 T. (d) Extracted linewidth Df as a function of reso- nance frequency fralong with the linear ﬁt performed to determine the intrinsic damping a0 intrfrom conﬁned modes. Inset: Effective damping a0 effas a function of resonance frequency fr. The red dotted lines mark the error margins of a0 intr¼ð9:964:1Þ/C210/C05.032408-3 Stasinopoulos et al. Appl. Phys. Lett. 111, 032408 (2017)the different metallic CPWs of Fig. 1vary the boundary condi- tions and thereby details of the spin wave dispersion relations in Cu 2OSeO 3. However, the frequencies covered by dispersion relations vary only over a speciﬁc regime; for, e.g., forwardvolume waves, the regime even stays the same for different boundary conditions. 4Following Klingler et al. ,7the exact mode nature and resonance frequency were not decisive whenextracting the Gilbert parameter. We now concentrate on mode #1 in Fig. 5(a) forHk h100iat 5 K that is best resolved. We ﬁt a Lorentzian line- shape as shown in Fig. 5(c) for 0.85 T and summarize the corresponding linewidths Dfin Fig. 5(d). The inset of Fig. 5(d) shows the effective damping a eff¼Df=ð2frÞevaluated directly from the linewidth as suggested in Ref. 29. We ﬁnd thataeffapproaches a value of about 3.5 /C210/C04with increas- ing frequency. This value is a factor of 10 smaller compared toaintrin Fig. 3(a) extracted from FMR peaks by means of Eq.(1). This ﬁnding is interesting as aeffmight still be enlarged by inhomogeneous broadening. To determine the intrinsic Gilbert-type damping from standing spin waves, we apply a linear ﬁt to the linewidths Dfin Fig. 5(d)atfr>10:6 GHz and obtain (9.9 64.1)/C210–5. For fr/C2010.6 GHz, the resonance amplitudes of mode #1 were small reducing the conﬁdence of the ﬁtting procedure. Furthermore, at low fre-quencies, we expect anisotropy to modify the extracted damping, similar to the results in Ref. 39. For these reasons, the two points at low f rwere left out for the linear ﬁt provid- inga0 intr¼ð9:964.1)/C210–5. We ﬁnd Dfand the damping parameters of Fig. 3to increase with T. It does not scale linearly for Hkh100i.A deviation from linear scaling was reported for YIG single crystals as well and accounted for by the conﬂuence of a low-kmagnon with a phonon or thermally excited magnon.5 We now comment on our spectra taken with the broad CPW that do not show the very small linewidth attributed to the conﬁned spin waves. The sharp mode #1 yields Df¼15:3 MHz at fr¼16:6 GHz [Fig. 5(d)]. At 5 K, the dominant peak measured at 0.55 T and fr¼15:9 GHz with the broad CPW provides however Df¼129 MHz. Dfobtained by the broad CPW is thus increased by a factor of eight. This increase is attributed to the ﬁnite distribution of wave vectors provided by the CPW. We conﬁrmed this larger value on a third sam-ple with Hkh100iand obtained (3.1 60.3)/C210 –3using the broad CPW ( supplementary material , Fig. S2). The discrep- ancy with the damping parameter extracted from the sharpmodes of Fig. 5might be due to the remaining inhomogene- ity of hover the thickness of the sample, leading to an uncer- tainty in the wave vector in the z-direction. For a standing spin wave, such an inhomogeneity does not play a role as the boundary conditions discretize k. Accordingly, Klingler et al. extracted the smallest damping parameter of 2 :7ð5Þ/C210 /C05 reported so far for the ferrimagnet YIG at room temperature when analyzing conﬁned magnetostatic modes.7The ﬁnding of Klingler et al. is consistent with the discussion in Ref. 33. From Ref. 33, one can extract that the evaluation of damping from ﬁnite-wave-vector spin waves provides a damping parameter that is either equal or somewhat larger than theparameter extracted from the uniform mode ( supplementary material ). The evaluation of Fig. 5(d) thus overestimates the parameter.To summarize, we investigated the spin dynamics in the ﬁeld-polarized phase of the insulating chiral magnet Cu 2OSeO 3. We detected numerous sharp resonances that we attribute to standing spin waves. Their effective damping parameter is small and amounts to 3 :5/C210/C04. A quantitative estimate of the intrinsic Gilbert damping parameter extracted from the conﬁned modes provides (9.9 64.1)/C210–5at 5 K. The small damping makes an insulating ferrimagnet exhibit-ing the Dzyaloshinskii-Moriya interaction a promising candi- date for exploitation of complex spin structures and related nonreciprocity in magnonics and spintronics. Seesupplementary material for further spectra, the mag- netic anisotropy constant, and linewidth evaluation. We thank S. Mayr for assistance with sample preparation. 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1.3677838.pdf	The concept and fabrication of exchange switchable trilayer of FePt/FeRh/FeCo with reduced switching field T. J. Zhou, K. Cher, J. F. Hu, Z. M. Yuan, and B. Liu Citation: J. Appl. Phys. 111, 07C116 (2012); doi: 10.1063/1.3677838 View online: http://dx.doi.org/10.1063/1.3677838 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i7 Published by the American Institute of Physics. Related Articles Reduced spin transfer torque switching current density with non-collinear polarizer layer magnetization in magnetic multilayer systems Appl. Phys. Lett. 100, 252413 (2012) A study on exchange coupled structures of Fe/NiO and NiO/Fe interfaced with n- and p-silicon substrates J. Appl. Phys. 111, 123909 (2012) Origin of magneto-optic enhancement in CoPt alloys and Co/Pt multilayers Appl. Phys. Lett. 100, 232409 (2012) Reproducible domain wall pinning by linear non-topographic features in a ferromagnetic nanowire Appl. Phys. Lett. 100, 232402 (2012) Magnetoplasma waves on the surface of a semiconductor nanotube with a superlattice Low Temp. Phys. 38, 511 (2012) Additional information on J. Appl. Phys. Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors Downloaded 23 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsThe concept and fabrication of exchange switchable trilayer of FePt/FeRh/FeCo with reduced switching field T. J. Zhou,a)K. Cher, J. F . Hu, Z. M. Yuan, and B. Liu Data Storage Institute, ASTAR (Agency for Science Technology and Research), 5 Engineering Drive 1, Singapore 117608 (Presented 31 October 2011; received 14 October 2011; accepted 21 November 2011; published online 8 March 2012) We report the concept and fabrication of exchange switchable trilayer of FePt/FeRh/FeCo with reduced switching ﬁeld for heat assisted magnetic recording (HAMR). A thin layer of FeRh is sandwiched between L10FePt and magnetically soft FeCo. At room temperature, FePt and FeCo are magnetically isolated by the antiferromagnetic FeRh layer. After the metamagnetic transitionof FeRh layer by heating, FePt and FeCo are exchange-coupled together through ferromagnetic FeRh layer. Therefore, the switching ﬁeld of FePt can be greatly reduced via exchange-spring effect. Simulation work was carried out to understand the exchange coupling strength and the FeCothickness effects on the switching ﬁeld reduction. It is found that switching ﬁeld decreases with the increase of exchange coupling strength and FeCo thickness. The trilayer ﬁlms were also successfully fabricated. A clear change of reversal mechanism from two-step to one-step switchingupon heating was observed and a 3-time switching ﬁeld reduction was demonstrated. The results show the promise of the trilayer for HAMR applications. VC2012 American Institute of Physics . [doi:10.1063/1.3677838 ] I. INTRODUCTION Heat-assisted magnetic recording (HAMR) is believed to have the potential to achieve multiple Tbit/in2recording density.1The conventional HAMR technology requires to write information at temperature close to or slight higherthan the Curie temperature, 2Tc, which is about 750 K for FePt. Such high writing temperature puts stringent require- ments on the overcoat and lubricants. Thiele et al. proposed the use of FePt/FeRh bilayer as the composite HAMR media for heat-assisted recording with reduced writing temperature (500 K or lower).3FeRh is antiferromagnetic at room tem- perature and it undergoes a metamagnetic transition to ferro- magnetic state at elevated temperatures (350–400 K).4 Therefore, this FePt/FeRh structure can provide thermal sta- bility at room temperatures while the coupling between FePt and FeRh reduces switching ﬁeld after the metamagnetic transition by slightly heating. Zhu et al. also proposed a bi- nary anisotropy media consisting a trilayer of a magnetic re- cording layer with perpendicular anisotropy, a magnetic assist layer with negative anisotropy, and a phase transitionlayer between the recording and assist layers to reduce writing temperature. 5With such structure, simulation results showed the switching ﬁeld can be reduced to a few percentage of theanisotropy ﬁeld of the recording layer. In this work, we further develop the concept and pro- pose the exchange switchable trilayer of FePt/FeRh/FeCowith a purpose of reducing both writing temperature and switching ﬁeld. In the trilayer, FeRh forms a very thin layer between FePt and FeCo and works as an exchange switchinglayer to turn on/off the coupling between FePt and FeCoupon heating/cooling. As shown in Fig. 1, at room tempera- ture, FePt and FeCo are magnetically isolated by the antifer- romagnetic FeRh layer. After the metamagnetic transition of FeRh by heating, FePt and FeCo are exchange-coupledtogether through ferromagnetic FeRh layer, and therefore the switching ﬁeld of FePt can be greatly reduced due to exchange spring effects. 6,7Here the FeCo provides a higher magnetic moment that can further reduce the switching ﬁeld compared to the FePt/FeRh bi-layer. FeCo layer also func- tions as a soft magnetic underlayer to enhance writing. Simu-lation work was carried out to study the exchange coupling strength and FeCo thickness effects on the switching ﬁeld reduction. The switching ﬁeld decreases with both exchangecoupling strength and FeCo thickness. The trilayer ﬁlms were fabricated. About 3-time switching ﬁeld reduction was experimentally demonstrated. The results show the promiseof the trilayer for HAMR applications. II. SIMULATION MODEL AND RESULTS To understand the exchange coupling strength and soft layer thickness effect on the reduction of switching ﬁeld, the FIG. 1. (Color online) Schematic representing the exchange switchable tri- layer of FePt/FeRh/FeCo before and after metamagnetic transition of FeRh.a)Author to whom correspondence should be addressed. Electronic mail:zhou_tiejun@dsi.a-star.edu.sg. 0021-8979/2012/111(7)/07C116/3/$30.00 VC2012 American Institute of Physics 111, 07C116-1JOURNAL OF APPLIED PHYSICS 111, 07C116 (2012) Downloaded 23 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsmagnetic switching of the proposed FePt/FeRh/FeCo trilayer was simulated by micro-magnetic modeling8where gyro- magnetic motion of magnetization is governed by the Lan-dau-Lifshitz-Gilbert equation given by d^m ds¼^m/C2~heff/C0a^m/C2ð^m/C2~heffÞ: (1) ^mis the magnetization unit vector and ais the damping constant. ~heffis the effective ﬁeld, which includes the anisot- ropy ﬁeld, exchange ﬁeld, external ﬁeld, thermal ﬁeld, and demagnetization ﬁeld. The following parameters are used: Anisotropy constant KFePt¼6/C2107erg/cc, KFeRh¼7/C2104 erg/cc, KFeCo¼9.5/C2104erg/cc, saturation magnetization MsFePt¼1140 emu/cc, M sFeRh¼1400 emu/cc, M sFeCo¼1900 emu/cc, and interlayer exchange coupling constant C is 0.4for the hysteresis loop calculation. The simulated hysteresis loops for the trilayer at 300 K and 473 K are shown in Fig. 2. At 300 K, a clear two-step switching is obtained. One is around zero ﬁelds correspond- ing to the switching of the soft layer. The other is at applied ﬁeld, H a, of 0.75 H k, which corresponds to the switching of hard layer. The switching ﬁeld of the trilayer is deﬁned as that of the hard layer. At 473 K, the switching of the soft layer is shifted to higher ﬁeld and that of the hard layer ismoved to lower ﬁeld—a one-step switching is observed with a much reduced switching ﬁeld of /C240.25 H k. At 300 K, FeRh is antiferromagnetic and the FePt and FeCo layers are mag-netically isolated. Therefore the trilayer exhibits two distinct switching states. FeRh is ferromagnetic at elevated tempera- tures of 473 K and the trilayer forms an exchange spring. Inexchange-spring media, magnetic reversal starts in the soft layer by forming a Neel-type domain wall when an external ﬁeld is applied. This wall propagates toward and penetratesinto the hard layer, assisting in the switching, which facili- tates a single state reversal at much lower switching ﬁelds. Figure 3reveals how the switching ﬁelds change with exchange coupling strength, C. The initial increment of C from 0.25 to 0.5 reduced the switching ﬁeld by a factor of 2, which is close to one fourth the anisotropy of FePt. Subse-quent increase of exchange coupling only yields smaller changes to the switching ﬁeld. In the exchange-spring media, the spins in the soft layer act on the magnetization of thehard layer like a (exchange) spring. The spring strength isproportional to both the exchange coupling strength and sat- uration magnetization of the soft layer. Therefore, certainexchange coupling is needed to have high enough spring strength in order to minimize the switching of the hard layer. Figure 4plots the switching ﬁeld as a function of FeCo thickness at ﬁxed exchange coupling strength of C* ¼0.5. The switching ﬁeld can be reduced to one ﬁfth the anisotropy ﬁeld of FePt at FeCo thickness of 15 nm or thicker. Suchreduction makes it possible to use the conventional perpen- dicular head to write information into the recording medium. The reduction of switching ﬁeld is mainly due to the springeffect plus the demagnetization effect from the bottom FeCo layer. The demagnetization energy is proportional to the magnetic volume of the FeCo layer, which is a function ofFeCo thickness and saturated at certain thickness. This can explain why the switching ﬁelds decrease with FeCo thick- ness and saturated at certain thickness. A theoretical analysis of the magnetization reversal pro- cess in a structure of FePt/FeRh bi-layer was conducted by Guslienko et al. to understand the underlying physics. 9It was concluded that the switching ﬁeld was related to the interlayer exchange-coupling strength and the saturation magnetization of FeRh at ferromagnetic state. For the pro-posed trilayers, it is plausible to treat the bottom two layers of FeRh and FeCo as one magnetically soft layer after FIG. 2. (Color online) Simulated hysteresis loops before and after metamag- netic transition of FeRh. FIG. 3. (Color online) Switching ﬁelds vs the exchange-coupling strength. FIG. 4. (Color online) Switching ﬁelds as a function of FeCo layerthickness.07C116-2 Zhou et al. J. Appl. Phys. 111, 07C116 (2012) Downloaded 23 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsmetamagnetic transition of FeRh. Then, for strong interlayer exchange coupling, we have Hswðstrong coupling Þ /C25Ku;FePt/C2tFePt Ms;FePt/C2tFePtþ/C22Ms;ðFeRhþFeCoÞ/C2tFeRhþFeCo(2) and for weak interlayer exchange coupling, the following applies: Hswðweak coupling Þ¼HK;FePt/C0J tFePt/C22MFeRhþFeCo /C21þJ tMs;FePt ðHK;FePt/C04p/C22Ms;FeRhþFeCo/C20/C21 ; (3) where Jis the interlayer exchange coupling and /C22MFeRhþFeCo is the average saturation magnetization of the bottom two layers of FeRh and FeCo. It is clearly shown that the switch- ing ﬁeld decreases with both the exchange coupling and theFeCo thickness as observed based on simulation. Also due to higher saturation magnetization of FeCo, the trilayer has higher potential for the reduction of switching ﬁeld com-pared with the FePt/FeRh bilayers. III. FABRICATION OF THE TRILAYERS AND CONCEPT DEMONSTRATION The trilayer was fabricated. Firstly, (002) oriented FeCo was deposited onto MgO substrate at 300/C14C. Then (001) ori- ented FeRh was grown on FeCo at 400–500/C14C. Last, (001) oriented FePt was deposited onto FeRh layer at 400–500/C14C. Due to high temperature process, 0.5 nm Ta layer was inserted between FePt and FeRh and between FeRh and FeCoto prevent the interlayer diffusion. XRD (Fig. 5) showed good (001) orientated FePt layer on top of (001) orientated FeRh and (001) orientated FeRh on top of (002) orientatedFeCo layers. Temperature-dependent dc demagnetization(DCD) curves of the trilayers were measured at different tem- perature to study temperature-dependent magnetizationswitching behavior. The measured results are shown in Fig. 6. At low temperature (250 K and 300 K), a clear two-step switching was observed. When temperature was increased to350 K and above, a single-step switching was shown. The switching ﬁeld as reduced from 4500 Oe to about 1500 Oe, which is about a 3-time reduction, after the metamagnetictransition of FeRh. IV. SUMMARY We proposed and experimentally demonstrated that the switching ﬁeld can be effectively reduced without the loss of thermal stability in exchange switchable trilayer of FePt/FeRh/FeCo. The writing temperature of the trilayer can also be reduced to the metamagnetic transition temperature of FeRh, which is about 400 K. The trilayer has higher heatingefﬁciency because only a very thin FeRh layer is needed to be heated above the metamagnetic transition temperature. Although the results presented show the promise for the tri-layer structure for HAMR applications, much work needs to be done for the improvement of magnetic properties and microstructure in order for it to be used as practical HAMRmedia. 1M. H. Kryder, E. C. Gage, T. W. McDaniel, W. A. Challener, R. E. Rott- mayer, G. Ju, Y–T. Hsia, and M. F. Erden, Proceedings of the IEEE 96, 1810 (2008). 2N. Kazantseva, D. Hinzke, R. W. Chantrell, and U. Nowak, Europhys. Lett. 86, 6 (2009). 3J. U. Thiele, S. Maat, and E. E. Fullerton, Appl. Phys. Lett. 82, 2859 (2003). 4P. H. L. Walter, J. Appl. Phys. 33, 938 (1964). 5J. G. Zhu and D. E. Laughlin, U.S. patent US2008/0180827 (31 July 2008). 6R. H. Victora and X. Shen, IEEE Trans. Magn. 41, 537 (2005). 7T. J. Zhou, B. C. Lim, and B. Liu, Appl. Phys. Lett. 94, 152505 (2009). 8C. K. Goh, Z. M. Yuan, and B. Liu, J. Appl. Phys. 105, 083920 (2009). 9K. Yu. Guslienko, O. Chubykalo-Fesenko, O. Mryasov, R. Chantrell, and D. Weller, Phys. Rev. B 70, 104405 (2004). FIG. 5. (Color online) XRD pattern of the exchange switchable trilayer of FePt/FeRh/FeCo. FIG. 6. (Color online) DCD curves of the exchange switchable trilayer of FePt/FeRh/FeCo at different temperature.07C116-3 Zhou et al. J. Appl. Phys. 111, 07C116 (2012) Downloaded 23 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
1.5042417.pdf	An analog magnon adder for all-magnonic neurons T. Brächer , and P. Pirro Citation: Journal of Applied Physics 124, 152119 (2018); doi: 10.1063/1.5042417 View online: https://doi.org/10.1063/1.5042417 View Table of Contents: http://aip.scitation.org/toc/jap/124/15 Published by the American Institute of PhysicsAn analog magnon adder for all-magnonic neurons T. Brächer and P. Pirro Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany (Received 31 May 2018; accepted 2 August 2018; published online 2 October 2018) Spin-waves are excellent data carriers with a perspective use in neuronal networks: Their lifetime gives the spin-wave system an intrinsic memory, they feature strong nonlinearity, and they can be guided and steered through extended magnonic networks. In this work, we present a magnon adder that integrates over incoming spin-wave pulses in an analog fashion. Such an adder is a linearprequel to a magnonic neuron, which would integrate over the incoming pulses until a certain non- linearity is reached. In this work, the adder is realized by a resonator in combination with a paramet- ric ampli ﬁer which is just compensating the resonator losses. Published by AIP Publishing. https://doi.org/10.1063/1.5042417 I. INTRODUCTION In certain tasks like pattern recognition, the brain outper- forms conventional CMOS-based computing schemes by far in terms of power consumption. Consequently, neuromorphiccomputing approaches aim to mimic the functionality of neurons in a network to boost computing ef ﬁciency. 1–5In the brain, stimuli are conveyed by short wave packets from oneneuron to another, where they lead to stimulation which adds up and then, ultimately, triggers a nonlinear response. Thus, it is natural to consider waves as data carriers for bio-inspiredcomputing and arti ﬁcial neuronal networks. Certain key components need to be accessible by the used kind of waves: It should be possible to convey them through extended net-works as well as to store the information carried by the waves for a certain time so that stimuli can add up. In addition, the waves should exhibit nonlinear dynamics in order to mimicthe threshold characteristics of a neuron. Among the possible waves that one can consider, spin waves, the collective exci- tation of magnetic solids, are a highly attractive candidate: 6–10 The dynamics of spin-waves and their quanta, magnons, are governed by a nonlinear equation of motion,11,12providing easy access to nonlinearity.13–15They can be guided through reprogrammable networks by using spintronics and nonlinear effects and their ﬁnite lifetime provides an intrinsic memory to the spin-wave system. In addition, their excitation energy is very low and their nanometric wavelengths at frequencies in the GHz and THz range promise a scalable and power-efﬁcient platform for neuromorphic computing. In this work, we employ micromagnetic simulations to demonstrate an analog magnon adder, which can be regardedas a pre-step to a magnon based neuron. The adder, which is sketched in Fig. 1(a), consists of two building blocks: a leaky spin-wave resonator and a parametric ampli ﬁer. 16–19 Spin waves can enter the resonator by dipolar coupling to the input.20Within it, their amplitude is added to or subtracted from the amplitude of the already accumulated amplitudes, asis sketched in Fig. 1(b). This process is, in principle, equiva- lent to the arrival of excitation pulses in the axon, where the neuron integrates over the incoming stimuli until a criticalstimulus is reached. In our scheme, the parametric ampli ﬁeracts to counteract the spin-wave losses that arise from propa- gation through the resonator and the leakage to the input and output of the resonator. We show that by working at thepoint of loss compensation, the adder can add and subtract the spin-wave amplitudes over a large range and enables to store the sum of these calculations in the resonator. II. LAYOUT AND WORKING PRINCIPLE To demonstrate the magnonic adder, we perform micro- magnetic simulations using MuMax3.21We chose dimen- sions that are compatible with the time scales and feature sizes accessible in state-of-the-art magnonic experiments. For our simulations, we assume the material parameters ofYttrium Iron Garnet (YIG), 22,23a widely used material in magnonics:6,24saturation magnetization Ms¼140 kA m/C01, exchange constant Aex¼3:5p J m/C01, and Gilbert damping parameter α¼0:0002, which represents the damping of the spin-waves mainly into the phonon system. The geometry we study consists of three w¼0:5μm wide and 40 nm thick rectangular YIG waveguides in a row [see Fig. 1(a)], similar to the general design proposed in Ref. 25. The length of the central waveguide, which acts as the resonator, is L¼20μm. The waveguide to the left of the resonator acts as input, where spin-waves are excited by a source creating a local magnetic ﬁeld. In a magnonic network, this input could be connected to an arbitrary number of other waveguides acting as individual inputs. The waveguide on the right of the reso- nator acts as output, which again could be reconnected in anetwork. In our simulations, input and output are 10 μm long and separated from the resonator by g¼75 nm wide gaps. Spin-waves can tunnel through this gap, 20which constitutes the coupling channel from the resonator to the input and output, respectively. In the present simulation, about 0 :1% of the spin-wave amplitude is tunneling through the gap. A dif-ferent gap spacing or a different magnetization con ﬁguration leads to different tunneling amplitudes, resulting in different, potentially larger losses for the resonator. 26These can be compensated by adjusting the parameters of the ampli ﬁca- tion. Toward their outer edges, the damping in the input and output is increased exponentially to mimic the transport ofJOURNAL OF APPLIED PHYSICS 124, 152119 (2018) 0021-8979/2018/124(15)/152119/5/$30.00 124, 152119-1 Published by AIP Publishing. spin-waves out into the network that would take place in a real extended system. Figure 2(a)shows the simulated spin-wave dispersion27,28 of the fundamental mode in a color-coded scale. The external ﬁeld of μ0Hext¼20 mT is applied along the long axis of the resonator. The excitation frequency f¼5:8 GHz corresponds to the excitation of spin-waves with a wave-vector of kk¼ 56 rad μm/C01(i.e., wavelength λ/C25112 nm) and the periodic excitation source is matched to excite this wave-vector reso- nantly. From the simulated spin-wave dispersion, a groupvelocity of v g¼(0:88+0:05)μmn s/C01can be extracted. This corresponds to a roundtrip time of Δt¼2/C1L=vg/C2546 ns through the resonator. During one trip, the spin-wave amplitudendecays exponentially following A p(t)¼Ap(0)/C1exp (/C0t=τ) with their lifetime τ. From this, it can be inferred that during one pass lasting Δt, the relative amplitude change is Ap(tþΔt) Ap(t)¼e/C0Δt τ: (1) As mentioned above, the dipolar coupling between the reso- nators is very weak and only a small fraction of 0 :1%of the spin-wave amplitude is actually coupled from the resonator to the input and output, respectively. Consequently, thelosses of the resonator are dominated by the propagation loss. In order to counteract these losses, we employ paramet- ric ampli ﬁcation, also known as parallel pumping. 16–19In this technique, the system is pumped at the frequency fp which equals twice the resonance frequency f. One possible driving force is Oersted ﬁelds,16,19μ0hp, where microwave photons split into pairs of magnons as indicated in Fig. 2(a). Here, we consider this mechanism, but also other, more energy ef ﬁcient realizations like the use of electric ﬁelds have been proposed.29,30In the simplest case of adiabatic parametric ampli ﬁcation,16the pumping at 2 fleads to theformation of wave pairs at fwith wave-vector +kkin order to conserve momentum, as is sketched in Fig. 2(a). Parallel pumping counteracts the damping losses with two key fea-tures: 16(1) It only couples to already existing waves and, in the absence of nonlinear saturation, leads to an exponential increase of the spin-wave amplitude following Ap(t)¼ Ap(0)/C1exp [( Vμ0/C1hp/C0τ/C01)t] if the energy per unit time V/C1 μ0hpinserted into the spin-wave system exceeds the losses given by τ/C01. Here, Vconstitutes the coupling parameter of the given spin-wave mode at ( f,kk). (2) Parallel pumping conserves the phase of the incident spin-waves. This is important to pro ﬁt from the phase of the spin-wave in encod- ing which is, for instance, vital to be able to perform subtrac- tion in the presented magnon adder. In the simulated structure, the local parametric ampli ﬁer exhibits an extent of 1μm along the resonator and it is situated in the center of the resonator. For simplicity, we only take into account the FIG. 1. (a) Sketch of the magnon adder consisting of an input, a resonator, and an output, as well as a parametric ampli ﬁer to compensate the losses in the resonator. (b) Sketch of the operation principle of the adder: Subsequent pulses with amplitude Ap(n) (indicated by the numbers) enter the resonator, which integrates over their amplitude. Periodically, a pulse leaves the resona- tor at the output. The value stored in the resonator and the value in the output are equal to the sum Sof the input amplitudes. FIG. 2. (a) Simulated spin-wave dispersion relation at a ﬁeld of μ0Hext¼ 20 mT applied along the resonator long axis and illustration of the parallel pumping process. (b) Dynamic magnetization 6 μm away from the center of the resonator as a function of time. Red: Only excitation of a single spin- wave pulse with amplitude “1”in the input. Green: Only periodic application of ampli ﬁcation pulses, no stimulus at the input. Black: Periodic application of an ampli ﬁcation pulse twice per roundtrip with an input stimulus of one pulse with amplitude “1.”The gray shaded areas indicate the position of the idler waves.152119-2 T. Brächer and P. Pirro J. Appl. Phys. 124, 152119 (2018)parallel component of the microwave ﬁeld created by a stri- pline.19,31Please note that a reduction of the ampli ﬁer size below the wavelength of the spin waves to be ampli ﬁed results in the nonadiabatic regime of paramteric ampli ﬁca- tion.16,32In this regime, two co-propagating spin waves will be created, i.e., the idler wave runs along with the signalwave. In this case, the ampli ﬁcation is not only phase- conserving but also phase-sensitive. 16,19This allows for alter- native designs of the adder or a magnonic neuron. However,for the wavelength we employ here, a very small pumping source would be required to access this regime, as the pumping source has to provide the necessary momentum. 16 This results in a very short interaction time of the spin waves with the pumping ﬁeld which leads to a large increase of the needed ampli ﬁcation ﬁelds. Similar to the operation in the brain, we assume that incident information is carried by pulses. As sketched in Fig.1(a), these pulses can arrive at the ampli ﬁer with differ- ent amplitudes and at different times. They exhibit a ﬁxed duration of 5 ns and delayed pulses are sent to the input at times which are integer multiples of the roundtrip time Δt. The ampli ﬁcation is also pulsed: tp¼5 ns long pumping pulses are applied whenever the spin-wave pulse in the reso- nator passes the ampli ﬁer, i.e., twice per roundtrip. When the net increase of the spin-wave amplitude by the pumping is equivalent to the losses, this leads to the formation of a pair of signal and idler spin-waves running back and forth in theresonator. The general act of the parametric ampli ﬁcation is shown in Fig. 2(b) for one single input pulse of amplitude “1.”In our simulations, this amplitude was arbitrarily chosen to correspond to an external excitation with a local ﬁeld amplitude of 65 μT in the input. The diagram shows the out-of-plane dynamic magnetization component m zas a func- tion of time at a point 6 μm away from the resonator center. The red curve shows the amplitude if no pumping ﬁeld is applied —the spin-waves pass the position where mzis recorded for the ﬁrst time at t¼t0/C2527 ns. They are reﬂected at the end of the resonator and pass the measure- ment position again at t/C2536 ns. Then they pass a roundtrip through the resonator and arrive again at t¼t0þΔt/C25 73 ns, and so forth. The damping of the waves can be clearly seen and it amounts to about 25 %per roundtrip. In contrast, the black curve shows the time evolution of the spin-wave amplitude if the parametric ampli ﬁcation is switched on and is just strong enough to compensate thelosses during one roundtrip. Now, the idler pulses are created, which give rise to additional pulses highlighted by the gray shaded areas. After the idler is build up and aftersome initial ﬂuctuations, the quasi-steady-state is reached and the pulses run back and forth with constant amplitude. For completeness, the green curve shows the dynamic magnetiza-tion if only the pumping pulses are applied, showing that for the presented parameters, noise creation by parametric gener- ation is negligible. 16,33 III. WORKING POINT OF THE MAGNONIC ADDER In the following, we want to elaborate the impact of the parametric ampli ﬁcation in more detail, since it plays acrucial role for the operation of the resonator as an adder or as a nonlinear device. The absolute gain per roundtrip is determined by the strength of the pumping ﬁeldμ0hp.F o r half a roundtrip and assuming that t¼0 is the point in time when the pulse enters the ampli ﬁer, we can modify Eq. (1)to ApΔt 2/C18/C19 ¼Ap(0)/C1eVμ0hp/C0τ/C01ðÞ Δtp/C1e/C0τ/C01Δt 2/C0ΔtpðÞ ¼Ap(0)/C1eVμ0hpΔtp/C0τ/C01Δt 2 ¼Ap(0)/C1e0:5/C1G0(hp), (2) with the gain G0(hp)¼2Vμ0hpΔtp/C0τ/C01Δtper roundtrip. In the following, we will consider a normalized gain factor G¼G0(hp)=G0(0)¼G0(hp)=(τ/C01/C1Δt), which is /C01 in the absence of parametric ampli ﬁcation, 0 when the parallel pumping is just compensating the losses, and which takes positive values if more energy is inserted per roundtrip than is lost by dissipation. Figure 3shows the gain factor Gas a function of the applied pumping ﬁeld, which has been extracted from a linear ﬁtt ol n [ mz(t)]/ln[Ap(t)] as is exem- plarily shown in the insets for μ0hp¼0(G¼/C01), corre- sponding to the intrinsic spin-wave decay with the lifetime τ¼155 ns and for μ0hp¼24:3m T( G¼1:2). The amplitude has hereby been integrated in time over the forward travelingsignal pulse in a time window of +4 ns, i.e., for each pulse n from t¼(t 0/C04n sþn/C1Δt)t o t¼(t0þ4nsþn/C1Δt). As can be seen from the linearity in the insets, the data show aclear exponential decay/growth, respectively. While the regime G.0 is highly interesting for neuromorphic applica- tions in general, since it provides easy access to nonlinearity,for the magnon adder, we chose the working point at G/C250. In this case, the energy inserted is just enough to com- pensate the losses and the current amplitude of the pulsewithin the resonator is preserved. Please note that due to the fact that the ampli ﬁcation is proportional to the amplitude, FIG. 3. Gain factor as a function of the applied pumping ﬁeldμ0hp. The insets show the amplitude of the spin-wave pulses Apas a function of time for the case of μ0hp¼0(G¼/C01, upper inset) and μ0hp¼24:3m T (G¼þ1:2, lower inset) on a semi-logarithmic scale. The adder is operated at the damping compensation point G¼0, marked by the dashed lines.152119-3 T. Brächer and P. Pirro J. Appl. Phys. 124, 152119 (2018)this compensation point holds for a small and a large amplitude spin-wave alike, as long as no nonlinearity sets in. In the following, we will ﬁxt h ea m p l i ﬁcation ﬁeld to μ0hp¼12:8 mT, the ﬁe l da l s ou s e di nF i g . 2(b), to stay at G/C250. IV. DEMONSTRATION OF ANALOG ADDING AND SUBTRACTING ForG¼0, the resonator losses are compensated. In this regime, a spin-wave pulse within it is cached as long as the ampli ﬁcation remains switched on and the compensated reso- nator can be used as a spin-wave adder. Since the spin-wave dynamics in the resonator are linear, the amplitude of the spin-wave pulse stored within the resonator corresponds tothe sum Sover all incident pulses. This sum Sis given by S¼P nAp(n)/C1(/C01)f(n), where Ap(n) is the amplitude of the individual pulse nandf(n) represents its phase, being either f(n)¼0 for a phase-shift of 0 or 2 πandf(n)¼1 for a phase-shift of π. A phase-shifted pulse, thus, corresponds to a negative value and allows for a subtraction. The phasecould, for instance, be given by a global reference in the magnonic network and it could be altered by reprogramma- ble, local phase shifters such as nanomagnets. For an inputamplitude A p(n) ranging from “0”to“100,”individual pulses with the respective value of Apcan be applied at the input without signi ﬁcant nonlinear effects, corresponding to excitation ﬁeld amplitudes ranging from 65 μTu pt o6 :5m T in the input. Numbers /C20100 can, therefore, be injected into the ampli ﬁer and will be summed over in an analog fashion. For larger excitation ﬁelds in the input, the spin-wave dynam- ics in the input become nonlinear, which distorts the summa- tion. Nevertheless, within the ampli ﬁer, much larger numbers can be handled, since only a fraction of the input spin-wave is coupled into the resonator by the dipolar coupling. Figure 4(a) shows the amplitude of the spin-wave pulse within the resonator in the quasi-steady-state, which corre- sponds to the sum S, as a function of the input amplitude on a double-logarithmic scale. As can be seen from the ﬁgure, the output is perfectly linearly proportional to the sum of the input amplitudes, no matter if an individual spin-wave pulse or a series of pulses are applied. This holds in the entiretested input range ranging from “1”up to at least “1500. ” The latter corresponds to the sum of 15 pulses with an indi- vidual amplitude of “100,”which are subsequently added in the resonator. The straight line is a linear ﬁt yielding a slope of 1 :029+0:004, con ﬁrming the linear relationship between input and output. From the inputs exceeding “100,”it can already be inferred from Fig. 4(a)that the spin-wave packets in the reso- nator add up in a linear fashion. For instance, the output“200”corresponds to the sum of two input pulses of value “100,”and so on. The key feature of the adder is that the device performs the summation purely analog, and the ampli-tude of the wave running back and forth in the resonator is directly proportional to the sum of the amplitudes of the input pulses. To illustrate this further, Fig. 4(b) shows the amplitude in the resonator for several combinations adding up to 10: 1 /C2input “10,”5/C2input “2,”10/C2 input “1,”aswell as the more involved pattern “1”+“3”+“0”+“3”+“0”+ “0”+“0”+“2”+“1.”As mentioned above, the fact that spin- waves carry amplitude and phase can also be used to do a subtraction. This is also shown in Fig. 4(b) by the combina- tion “20”/C0“3”/C0“0”/C0“0”/C0“3”/C0“0”/C0“0”/C0“4”/C0“0”/C0“0,” being equal to “10.”The transitionary dynamics visible in Fig. 2(b) at the beginning of the ampli ﬁcation process are also visible in Fig. 4(b): For the ﬁrst few roundtrips, the stored value decreases until it reaches the steady-state. Please note that this has no sizable impact on the adding or subtract-ing function. It should be noted that in the demonstrated regime of operation of the adder, the spin-wave dynamics stay linear.This allows one to add up individual spin-wave pulses. The device already performs similar to a neuron in the brain: Incident pulses are converted into an amplitude informationwithin the resonator and this amplitude is given by the inte- gration over the incoming signals. In the presented adder, small pulses carrying the amplitude of the sum are constantlyejected into the output [cf. Fig. 1(b)]. Toward a neuromor- phic application, the resonator could be designed in a way FIG. 4. (a) Quasi-steady-state amplitude in the resonator vs. input stimulus on a double-logarithmic scale. The straight line is a linear ﬁt yielding a slope of 1 :029+0:004 con ﬁrming the linear relationship between the input ampli- tude and the sum in the resonator. (b) Different combinations resulting in a sum of 10 within the resonator. Since the number of applied pulses as well as the time of their arrival is different in all cases, the sum of “10”is reached at different times for the different combinations.152119-4 T. Brächer and P. Pirro J. Appl. Phys. 124, 152119 (2018)that its quality factor is a function of the spin-wave ampli- tude, for instance, by a change of the dipolar coupling ef ﬁ- ciency associated with a nonlinear change of wave-vector.This way, a nonlinearity can open up the resonator once the critical stimulus is overcome. In such a way, spin-wave axons that can be conveniently integrated into extended networksbecome feasible. V. CONCLUSION To conclude, by means of micromagnetic simulations, we have demonstrated a magnon adder, where the magnon amplitude adds and subtracts in an analog fashion. The spin- wave summation is performed in a resonator, whose lossesare compensated by a parametric ampli ﬁer. This way, the amplitude is stabilized and constant in time if no mathemati- cal operation is performed. The spin-wave signal in the reso-nator is directly proportional to the time-integrated amplitude of the incoming pulses. Hereby, the phase degree of freedom of the spin-waves allows one to add spin-wave pulses in the case of constructive interference between the incoming spin- wave pulses. If a pulse is shifted by π, it will instead be sub- tracted. The presented device can perform as a magnon cache memory that can store an analog magnon sum on long time scales and, thus, constitutes the ﬁrst step toward an all- magnonic neuron. ACKNOWLEDGMENTS The authors thank B. Hillebrands and A. Chumak for their support and valuable scienti ﬁc discussion. 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1.1721128.pdf	Metrization of Phase Space and Nonlinear Servo Systems Chi Lung Kang and Gilbert H. Fett Citation: Journal of Applied Physics 24, 38 (1953); doi: 10.1063/1.1721128 View online: http://dx.doi.org/10.1063/1.1721128 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/24/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonlinear friction model for servo press simulation AIP Conf. Proc. 1567, 918 (2013); 10.1063/1.4850119 Phase space method for identification of driven nonlinear systems Chaos 19, 033121 (2009); 10.1063/1.3207836 Using phase space reconstruction to track parameter drift in a nonlinear system J. Acoust. Soc. Am. 101, 3086 (1997); 10.1121/1.418806 Tracking servo system J. Acoust. Soc. Am. 81, 1665 (1987); 10.1121/1.394771 Phase Space Hydrodynamics of Equivalent Nonlinear Systems: Experimental and Computational Observations Phys. Fluids 13, 980 (1970); 10.1063/1.1693039 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Tue, 25 Nov 2014 02:25:4338 H. L. ROBINSON eter. In both cases the experimental data are compared with calculated curves using Young's circuital form. Note that the difference between the experimental and theoretical curves is greater when the diameter is one half wavelength. The experimental data in Fig. 5 have been normalized as follows: All values of 1/10 at the center of an aperture one-half wavelength in diameter were averaged, then each reading was multiplied by a factor such that the reading at the center would have this average value. This makes it possible to compare the shapes of the two experimental curves. The relative intensities over a given curve as shown by its shape are more precise than the actual value of 1/10• The lack of agreement near the edge of the aperture indicated that results within a sixteenth-wavelength of the edge are not reliable. CONCLUSION Although Young's circuital form predicts the inten sity in the plane of apertures a few wavelengths in diam eter, it does not agree with experimental values for apertures less than a wavelength in diameter. Theo retical curves based upon it indicate neither the sharp increase in intensity near the ends of the electric diam eters nor the high intensity at the centers of apertures near one-half wavelength in diameter. ACKNOWLEDGMENT It is a pleasure to acknowledge the assistance through out this study of C. L. Andrews, who brought the prob lem to the author's attention and whose guidance was often sought. The calculations of H. S. Story, P. Pi saniello, and R. F. Tucker, Jr., have been of great value. JOURNAL OF APPLIED PHYSICS VOLUME 24, NUMBER 1 JANUARY, 1953 Metrization of Phase Space and Nonlinear Servo Systems* CHI LUNG KANGt AND GILBERT H. FETTt University of Illinois, Urbana, Illinois (Received July 29, 1952) By introducing a proper distance function, the phase space for a servomechanism is completely metrized. A new approach is developed to study servo systems directly on the basis of instantaneous performance under an arbitrary input function. A criterion for determining the effect of nonlinearity on performance is obtained. It will serve as basis for the design of nonlinear servo systems. INTRODUCTION CONTROL systems that lead to the following dif ferential equation are to be considered: e(n)+ale(n-l)+ ... + an_le(l)+ ane =G(e(n-ll, e(n-2), .. ·e, t), (1) where e stands for error and superscript in parenthesis indicate order of differentiation with respect to time. The left side of the equation is a linear equation with constant coefficients. It represents a basic system to which the actual system (which may be changed from time to time) always refers. Any nonlinearity pur posely introduced or parasitic to the basic system is lumped with the input function on the right side of the equation as function G. The existence theorem of solu tion to such a differential equation is well established.! * This paper is part of a thesis submitted by the first named author in partial fulfillment of requirements for the degree of Doctor of Philosophy in Electrical Engineering at University of Illinois. t Formerly University Fellow, University of Illinois; now with Boonton Radio Corporation, Boonton, New Jersey. t Professor of Department of Electrical Engineering, University of Illinois. 1 S. Lefschetz, Lectures on Differential Equations (Princeton University Press, Princeton, 1948), p. 23. Suppose there exists an unique solution to the differ ential equation. Then G(e(n-l), e(n-2), .. 'e, t) can be considered as another function of time, say F(t), which thereby becomes a forcing term to the basic linear system. For any system represented by an nth order differ ential equation, its states are specified by the set (e(n-!), e(n-2), .. 'e, t) in the phase space. Hence, the phase space becomes the configuration space for all the states of all the systems of nth order. By definition, e(t) = (Jdt) -(to(t), (2) where (Ji(t), (Jo(l) are the input and the output functions of the servo system, respectively. Since (Jo(t) is always continuous and (Ji(t) should be continuous almost everywhere, the trajectory of the representing point of the state of the system in error coordinates is continu ous almost everywhere. Good servo performance means that this error trajectory remains for most of the time near to the origin. Hence, at any point in the phase space, this state point of the system should tend to move back to the origin quickly. [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Tue, 25 Nov 2014 02:25:43METRIZATION OF PHASE SPACE 39 DEFINITION OF DISTANCE FUNCTION The notion of distance from the state point to the origin thus comes up. Mathematically, it really does not matter what the actual distance function is, as long as the usual hypotheses for a distance function are satisfied. Thus, in the three-dimensional case, a spherical neighborhood is equivalent to an ellipsoidal one, which incidently is what is to be adopted here. However, a properly defined neighborhood may greatly simplify the actual analysis. So the problem under investigation is to choose a logical, rational, and physi cally meaningful definition for the distance function. Transform the given differential equation (1) with its right side replaced by F(t) to normal coordinates.2 The following substitutions, with dot on top of the letters indicating differentiation with respect to time, e=el } el=e2 e~l~e7O ' lead to the vector equation, de -=Be+f(t), dt (3) (4) where e=(e1, e2, ···en),f(t)=(O,O, ···O,F(t)),andBis a constant matrix in terms of the constant coefficients in the left side of the given differential equation. Let the characteristic roots of the basic linear system be AI, A2, •• ·A2r-1, A2r, 'Y2r+l, .. ·1'70, where A2r-1, A2r are complex conjugate pair and 'Yis are real roots. It can be shown that, when they are distinct, 1 Al l-'l2 P= A13 A1n-1 1 A2r A2,2 A2r3 " n-1 1\2r 1 'Y2r+1 'Y2r+12 'Y2r+13 'Y2r+l7O-1 is a nonsingular matrix such that. P-1BP=R, 1 'Yn 1'702 'Yn3 (5) 'Ynn-1 (6) where R is a diagonal matrix and has the characteristic roots as its diagonal elements. Thus, the transformation e=Pz (7) gives dz/dt=Rz+P-1f(t), (8) 2 H. Goldstein, Classical Mechanics (Addison-Wesley Press, Cambridge, Massachusetts, 1950), p. 329. i.e., Z2r= A2rZ2r+q2r. 7OF(t) (9) Z2r+l = 'Y2r+lZ2r+l+q2r+1. "F(t) where {qi;}=P-1. The actual trajectory of the system is the result of the motion of the force free (i.e., F(t) = 0) trajectory caused by the forcing function F(t). The state point will jump from one to another force free trajectory. These trajectories never cross each other. Thereby, it is natural to derive the notion of distance from the force free case. The state point can be considered as a ma terial point, whose motion in the phase space will be characterized by a Lagrangian function leading to the same set of equations of motion, Eq. (9). This La grangian function for the force free case is i-2r B=n-2r L=[ L: Zi2+ L: Z2r+.2] i=1 8=1 2r ~n-2r +![L: Alzl+ L: 'Y2r+.2Z2r+,2J. (10) i-I.-I It is quite instructive to note that the first sum in the above expression can be looked upon as the kinetic energy of the system with the time derivatives of the coordinates considered as generalized velocities; the other sum can be considered the negative of the po tential energy corresponding to a force field propor tional to the. coordinates. The Hamiltonian function, hence the total energy, is a constant and is equal to zero. Therefore, the value of the Lagrangian function which is twice the kinetic. energy would be a measure of the swing of the energy content of the system away from the equilibrium position. It can also be easily shown that the necessary and sufficient condition that the basic system is stable is to have the value of its Lagrangian function on its force free trajectory tending to zero as time tends to infinity. This property of the Lagrangian function suggests it at once as a natural and rational distance function. However, a distance func tion has to be positive definite; so, in case the La grangian function contains oscillatory terms, the en velope to the Lagrangian function instead of the func tion itself will be used. Thus, by making use of Eq. (9) with F(t) = 0, the following definition of distance is derived from the Lagrangian function: r n-2r D(z, 0) = 2 L: A2r-1A2rZ2r-1Z2r+ L: 'Y2r+.2Z2r+.2 r-1 .=1 (11) and D(z, z')=D(z-z', 0). (12) [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Tue, 25 Nov 2014 02:25:4340 C. L. KANG AND G. H. FETT where o o S= rA2-IA2r A2r-IA2r (13) o Since Zi differs from Zi by only constants, z defines the state as well as z does. And in the space of z, which, in general, is complex, the above distance function is nothing but the ordinary norm in n-dimensional space over the field of complex number. If the matrix C = (P-I)TS(P-I) has all its characteristic roots positive (this is always the case when the basic linear system has only real characteristic roots), then D= constant will be an ellipsoidal surface. This point is of prime im portance in the present discussion and must be checked to assure that it is satisfied. The e space is therefore topologically Euclidian. And the above defined dis tance function satisfies all the hypotheses for a dis tance function and completely metrizes the phase space. NONLINEAR SERVO SYSTEMS In a servo system, this distance function can readily be used as an ordering relation defining at least par tially a preference among all the states in the phase space. To have a smaller distance from the origin is therefore a necessary condition for one state to be "better" than another state in a servo system. Since error itself, more than its time derivative, is of im portance, some auxiliary ordering relation can be set up to assure real improvement of the performance of the servo system. Now the effect of the forcing function F(t) at any .instant is to be examined. Differentiation of D gives r D= 2 L: A2r-IA2r(Z2r-IZ2r+Z2r-IZ2r) r=1 n-27 +2 L: 'Y2r+b2r+.Z2r+o. (14) 0=1 Substitution of the expressions of Eq. (9) for the z/s gives o r n-2r D= 2[L: A2r-IA2r(A2r+A2r-I)Z2r-IZ2r+ L 'Y2r+.3Z2r+02J r=l 8=1 r + 2F(t)[L: A2r-IA2r(q2r, nZ2r-l+q2r-I, nZ2r) r=1 n-2r + L: 'Y2r+.2q2r+8. nZ2r+oJ (15) 0=1 = Do+ 2F(t)K, where Do represents the rate of reduction of distance for force free case, and the other term represents the effect of the forcing function. It can be readily proved that Do is always negative for a stable basic system as can be expected. Whether the effect of the forcing func tion is favorable (i.e., to make the D more negative) or not depends on the sign of the term F(t)K. K is a linear function of the coordinates; hence, K = 0 is a plane in the phase space through the origin. The whole space is divided into two halves by the K = 0 plane, on one side of which a larger (algebraically) F(t) is preferred,. and on the other side, a smaller F(t). The functional dependence ofG(e(n-I), e(n-2), ., ·e, t) on the error and its time derivatives depends on the nature of the system, the magnitude of its parameters, its nonlinearity, etc. If at any point in the phase space, the function G(e(n-I), e(n-2), .. ·e, t), hence F(t), can be changed favorably by some modification of the system, whatever it may be, then the performance of the sys tem would be improved at that point. And the effect of any nonlinearity in a supposedly linear system can also be studied in the light of this K plane criterion. If the state of the system or at least the sign of its K value can be monitored by direct measurement, proper change can be made in the system accordingly to improve the performance. As a special case, there may be two linear systems, one faster in response and the other heavily damped. They can be switched into action alternately, as the K plane criterion permits, to improve the servo performance. In fact, this study is [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 128.114.34.22 On: Tue, 25 Nov 2014 02:25:43METRIZATION OF PHASE SPACE 41 motivated by such a heuristic attempt of switching among systems in a composite system. And a nonlinear system can naturally be considered as the result of continuous switching among linear systems. Notice that the behavior of the actual system is always ex pressed in terms of forcing with respect to a basic sys tem. This provides a simple and unique way to treat the general servo system. . It should be pointed out that the comparison of D has been made with respect to that at the particular point under consideration in the phase space. When the trajectory is changed by any modification of the sys~ tem, the basis of comparison is changed too. This makes the general study of the over-all rate of reduc tion of distance rather difficult. An investigation for the special case with the real parts of all the char acteristic roots equal will help to understand the situa tion. To insure that the rate of reduction of the abso lute value of the error is increased, extra forcing control should be used only when eK>O. (16) Thus, the general scheme of extra forcing control for a third-order system may be K>o>O el>O LlF<O el<O LlF=O, K<-o el>O LlF=O el<O LlF>O, IKI~o for all e LlF=O, where LlF stands for the extra forcing control, and 0 is introduced to give a zone about the K plane without extra forcing. This is to avoid possible instability at the origin due to the presence of inevitable delay in switching. When D has been reduced to the extent that the maximum dimension of the corresponding ellip soidal surface of constant distance is less than 0, the whole system will behave exactly as the basic linear system. It is difficult to say much about the resulting trajectory in general. While it seems hardly possible for some trajectory to remain in the LlF=O region forever, the question is to what extent will any trajectory come into some region with extra control and expose itself to it. The trajectories emerging out of the planes K = ± 0 into regions with extra forcing may be forced immedi ately back to these planes. This situation is certainly intolerable practically. A purposely designed hysteresis band (not given in the above scheme) for the switching on and off of the extra forcing around the K = ± il planes should solve this difficulty. The choice of the basic system is evidently an im portant problem in the design of such composite sys tem. To reduce the region where extra forcing is for bidden, the normal to the K plane should make a small angle with the error axis. This will likely give a system more suitable for this kind of extra control. One possible way of introducing the extra control is suggested below. An extra control box is used to feed an extra error signal H to the actual error, as is shown 90(:1) FIG. 1. A possible way of introducing extra forcing control. in Fig. 1. Thus, e'(t)=e(t)+H. (17) Let kN(p)/S(p) represent the forward transmission characteristics of the servo loop where N(p) and S(p) are polynomials in p = d/ dt. Therefore, 8o(t) = kN(p)e'(t)/ S(p) = kN(p)e(t)S(p) +kN(p)H/S(p) (18) = 8i(t) -e(t). Hence, [S(p)+kN(p)Je(t) =S(p)8 i(t)-kN(p)H. (19) The last term in the above equation is the extra control needed. Thus, a constant forcing term can be obtained by making H=ct', where p is the lowest power of p in N(p) and c is a constant. If a network with transfer function l/N(p) is available, then the extra forcing term of the form h(e) can be obtained by feeding this h(e) through such a network to give the function H. Since the extra forcing term of the form clel+c2e2 +Caea, where the c's are constants, is equivalent to a change to another linear system, it can be achieved by direct adjustment of the parameters of the basic system. But either theS(p) of the system should not be changed, or its effect on the term S(p)8i(t) should be taken into consideration. CONCLUSION To facilitate the study of a general servo system directly on the basis of performance, the phase space is metrized by defining a distance function. The defini tion adopted here seems to be quite natural and physi cally meaningful. And above all, it leads to a simple partition of the phase space and hence a simple cri terion to determine the effect of any nonlinearity, purposely introduced or parasitic, in the system. Actual design of specific systems have not been at tempted here. This work should be considered as a new approach for the study of nonlinear systems. Since this work is based on the differential equation, its application should not be limited to servo systems. It will certainly be useful in the study of nonlinear damper for vibration. BIBLIOGRAPHY (1) L. A. MacCoU, Fundamental Theory of Servomechanisms (D. Van Nostrand Company, Inc., New York, 1945). (2) C. Lanczos, The Variational Principles of Mechanics (Uni versity Press, Toronto, 1949). (3) H. Goldstein, Classical Mechanics (Addison-Wesley Press, Cambridge, Massachusetts, 1950). (4) S. Lefschetz, Lectures on Dijferential Equations (Princeton University Press, Princeton, 1946). [This article is copyrighted as indicated in the article. 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1.3396983.pdf	Gilbert damping in perpendicularly magnetized Pt/Co/Pt films investigated by all- optical pump-probe technique S. Mizukami, E. P. Sajitha, D. Watanabe, F. Wu, T. Miyazaki, H. Naganuma, M. Oogane, and Y. Ando Citation: Applied Physics Letters 96, 152502 (2010); doi: 10.1063/1.3396983 View online: http://dx.doi.org/10.1063/1.3396983 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/96/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Irreversible modification of magnetic properties of Pt/Co/Pt ultrathin films by femtosecond laser pulses J. Appl. Phys. 115, 053906 (2014); 10.1063/1.4864068 Determining the Gilbert damping in perpendicularly magnetized Pt/Co/AlOx films Appl. Phys. Lett. 102, 082405 (2013); 10.1063/1.4794538 Magneto-optical Kerr effect in perpendicularly magnetized Co/Pt films on two-dimensional colloidal crystals Appl. Phys. Lett. 95, 032502 (2009); 10.1063/1.3182689 Magnetic easy-axis switching in Pt/Co/Pt sandwiches induced by nitrogen ion beam irradiation J. Appl. Phys. 95, 8030 (2004); 10.1063/1.1712014 Giant enhancement of magneto-optical response and increase in perpendicular magnetic anisotropy of ultrathin Co/Pt(111) films upon thermal annealing J. Vac. Sci. Technol. A 17, 3045 (1999); 10.1116/1.582003 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.255.6.125 On: Thu, 11 Dec 2014 13:23:13Gilbert damping in perpendicularly magnetized Pt/Co/Pt ﬁlms investigated by all-optical pump-probe technique S. Mizukami,1,a/H20850E. P . Sajitha,1D. Watanabe,1F. Wu ,1T . Miyazaki,1H. Naganuma,2 M. Oogane,2and Y . Ando2 1WPI-Advanced Institute for Materials Research, Tohoku University, Katahira 2-1-1, Sendai 980-8577, Japan 2Department of Applied Physics, Graduate School of Engineering, Tohoku University, Aoba 6-6-05, Sendai 980-8579, Japan /H20849Received 16 February 2010; accepted 25 March 2010; published online 13 April 2010 /H20850 To investigate the correlation between perpendicular magnetic anisotropy and intrinsic Gilbert damping, time-resolved magneto-optical Kerr effect was measured in Pt /Co/H20849dCo/H20850/Pt ﬁlms. These ﬁlms showed perpendicular magnetization at dCo=1.0 nm and a perpendicular magnetic anisotropy energy Kueffthat was inversely proportional to dCo. With an analysis based on the Landau–Lifshitz– Gilbert equation, the intrinsic Gilbert damping constant /H9251was evaluated by parameter-ﬁtting of frequency and lifetime expressions to experimental data of angular variations in spin precessionfrequency and life-times. The /H9251values increased signiﬁcantly with decreasing dCobut not inversely proportional to dCo.©2010 American Institute of Physics ./H20851doi:10.1063/1.3396983 /H20852 Spin transfer torque magnetic random access memory /H20849STT-MRAM /H20850utilizing magnetic tunnel junctions /H20849MTJs /H20850is one of many candidates for next-generation nonvolatilerandom access memory. Many groups are currently develop- ing STT-MRAM, and in particular, STT-MRAM based onMTJs with perpendicularly magnetized electrodes as theseexhibit a large thermal stability factor /H9004and a very low critical current density J crequired for current-induced mag- netization switching /H20849CIMS /H20850.1,2The Jcis proportional to /H9251MsHkeffin CIMS for out-of-plane magnetization conﬁgura- tion, where the respective /H9251,Ms, and Hkeffare the Gilbert damping constant, saturation magnetization, and effectiveperpendicular magnetic anisotropy /H20849PMA /H20850ﬁeld. On the other hand, /H9004is also proportional to M sHkeff. Thus, to reduce Jc while maintaining /H9004constant, requires some intervention. One possibility is to use perpendicularly magnetized materi-als with low /H9251value. Gilbert damping originates intrinsically from a quantum mechanical electron transition mediated byspin-orbit interaction. 3Roughly speaking, /H9251is proportional to/H92642/W, where /H9264is the spin-orbit interaction energy and Wis thed-band width.4PMA also originates from spin-orbit in- teraction and broken symmetry and is also roughly propor-tional to /H92642/Win the theory.5These theories imply that Gil- bert damping tends to be stronger in materials with high-PMA and there might be a linear correlation between them.Recently, Gilbert damping in /H20851Co /Pt/H20852 Nmultilayer ﬁlms with high-PMA was investigated by the time-resolved magneto- optical Kerr effect /H20849TRMOKE /H20850, and the /H9251value increased with increasing stacking number Nwhile in contrast PMA decreased.6In addition, /H9251values were deduced from domain wall motion in Pt/Co/Pt ﬁlms and were found to be indepen-dent of Co layer thickness although PMA increased withdecreasing thickness. 7The conclusion is that the relationship between Gilbert damping and PMA is still unclear. To clarifythe Gilbert damping mechanism in materials with large-PMA, a more systematic study is required to extract the pre-cise nature of this correlation.In this paper, we report on the systematic investigation of intrinsic Gilbert damping for Pt/Co/Pt ﬁlms deduced fromangular dependence of TRMOKE and discuss its correlationwith PMA. The Pt/Co/Pt ﬁlms were deposited on naturallyoxidized Si substrate at room temperature using magnetronsputtering. The base pressure was 1 /H1100310 −7Torr and Ar pres- sure was 3 mTorr. The Pt buffer and capping layer thick-nesses were 5 nm and 2 nm, respectively, and Co layer thick-nesses d Cowere varied from 4.0 to 0.5 nm. Structural analysis was accomplished by x-ray diffraction /H20849XRD /H20850and x-ray reﬂectivity /H20849XRR /H20850. Magnetic properties were investi- gated using polar magneto-optical Kerr effect /H20849PMOKE /H20850and a superconducting quantum interference device magnetome- ter. Magnetization dynamics were investigated by x-bandferromagnetic resonance /H20849FMR /H20850and TRMOKE. Details of FMR measurements and analyses were the same as describedin a previous report. 8In the TRMOKE measurements, a stan- dard optical pump-probe setup was used with a Ti:sapphirelaser and a regenerative ampliﬁer. 9Beam wavelength and pulse width were /H11011800 nm and 100 fs, respectively, and pump beam ﬂuence was 3.8 mJ /cm2. The s-polarized probe beam, for which the intensity was much less than for thepump beam, was almost normally incident on a ﬁlm surfaceand the time variation in the magnetization was exhibited byPMOKE. TRMOKE measurements were obtained with anapplied magnetic ﬁeld Hof 4 kOe, and the angle /H9258Hbetween ﬁeld and direction normal to the ﬁlm was varied from 0° to80° using a specially designed electromagnet. From the XRD /H9258-2/H9258patterns, only a Pt /H20849111/H20850diffraction peak appeared for all ﬁlms, indicating the ﬁlms were /H20849111/H20850- textured polycrystalline. The XRR analysis showed that ac-tuald Covalues were equal to nominal values within experi- mental error, and interface roughness and/or alloying layerthickness were /H110110.4 nm. Figure 1/H20849a/H20850shows coercivity H Cas a function of dCofor Pt/Co/Pt ﬁlms. Perpendicular magneti- zation appears at dCo=1.0 nm and the coercivity exhibits a maximum value at dCo=0.8 nm, for which a PMOKE loop is shown in Fig. 1/H20849b/H20850as an example. To obtain the Hkeffvalues, we performed PMOKE measurements varying angle /H9258Hsub-a/H20850Electronic mail: mizukami@wpi-aimr.tohoku.ac.jp.APPLIED PHYSICS LETTERS 96, 152502 /H208492010 /H20850 0003-6951/2010/96 /H2084915/H20850/152502/3/$30.00 © 2010 American Institute of Physics 96, 152502-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.255.6.125 On: Thu, 11 Dec 2014 13:23:13ject to a constant applied ﬁeld of 4 kOe. Subsequently, the data were compared to theoretical data calculated numeri-cally using the expression for magnetization angle /H9258, sin 2/H9258=/H208492H/Hkeff/H20850sin/H20849/H9258H−/H9258/H20850, with the adjustable Hkeffas a ﬁt- ting parameter. The effective anisotropy energy Kueffwas evaluated from the relation Kueff=MsHkeff/2. The product KueffdCowas plotted in Fig. 1/H20849c/H20850as a function of dCo. The Kueff values were also obtained from FMR measurement for rela- tively thicker ﬁlms and are plotted with open circles; thesevalues agree with those from PMOKE. Proportionality of K ueffdCoagainst dCoindicates PMA is due to interface PMA, as reported in much of the literature.10The interface PMA energy Kswas estimated to be 0.35 erg /cm2by extrapola- tion as indicated by the solid line in Fig. 1/H20849c/H20850. The Ksvalues ranged from 0.3 to 1 in /H20849111/H20850-textured Co/Pt multilayer ﬁlms,10depending on interface quality and degree of texture, and our value is relatively lower because the buffer layer isthinner than in conventional multilayers so as to increase theKerr signal intensity. Figures 2/H20849a/H20850and2/H20849b/H20850show the representative TRMOKE measurements for Pt/Co/Pt ﬁlm with d Co=2.0 nm and 0.8 nm, respectively, measured at /H9258H=60°. MOKE signals de- crease suddenly in sub-ps time regime and subsequently ex-hibit damped oscillatory behavior for both ﬁlms that is acommon feature observed in all-optical measurements. 11Thespin precession frequency fand life-time /H9270were evaluated by ﬁtting the damped harmonic function superposed with anexponential decay function, as expressed in the formAexp/H20849−Bt/H20850+Csin/H208492 /H9266ft+/H9278/H20850exp/H20849−t//H9270/H20850, using the phase of precession /H9278and ﬁtting parameters A,B, and C, as shown with solid curves in Figs. 2/H20849a/H20850and2/H20849b/H20850. Figures 3/H20849a/H20850and3/H20849b/H20850show the /H9258Hdependence of fand 1//H9270for Pt/Co/Pt ﬁlm with dCo=2.0 nm and 0.8 nm, respec- tively. With dCo=2.0 nm, the normal direction of the ﬁlm is a magnetic hard-axis, so that fincreases with increasing /H9258H. The 1 //H9270values tend to increase with increasing fbecause Gilbert damping acts more effectively on faster spin motions,much like viscosity, as seen in Fig. 3/H20849a/H20850. Trends in fand 1 / /H9270 against /H9258Hbecome inverted in Fig. 3/H20849b/H20850because a magnetic easy-axis is perpendicular to the ﬁlm plane for dCo =0.8 nm. The experimental angular-dependence data of fand 1 //H9270were parameter-ﬁtted with expressions derived from the Landau–Lifshitz–Gilbert equation. Taking intoaccount PMA and arbitrary /H9251, these expressions are f =f0/H208811−/H208492/H9266f0/H9270/H20850−2with f0=/H20849/H9253/2/H9266/H20850/H20881H1H2//H208811+/H92512and 1 //H9270 =/H9251/H9253/H20849H1+H2/H20850//H208491+/H92512/H20850. Here, /H9253is the gyromagnetic ratio and H1=Hcos/H20849/H9258H−/H9258/H20850+Hkeffcos2/H9258and H2=Hcos/H20849/H9258H−/H9258/H20850 +Hkeffcos 2/H9258. The /H9253and/H9251values were treated as ﬁtting pa- rameters, while the Hkeffvalues were ﬁxed to those obtained from PMOKE measurements. The magnetization angle /H9258 was calculated in the same way as those in PMOKE. Thecalculated data ﬁtted well to the experimental data for bothﬁlms without invoking magnetic inhomogeneity or two-magnon scattering. This indicates that intrinsic Gilbert damp-ing is the dominant mechanism in the relaxation of magne-tization precession in these ﬁlms. Figure 4/H20849a/H20850shows the /H9251values evaluated from TRMOKE as a function of the reciprocal of dCo. The /H9251val- ues increase signiﬁcantly with decreasing dCoand are not proportional to 1 /dCo. This trend is different from the linear relationship between Kueffand 1 /dCo. The /H9251values obtained from FMR are also shown in this ﬁgure with open circles.FMR was barely measurable at d Co/H110211.0 nm because of sig- niﬁcantly large linewidths. The /H9251values from FMR show quite good agreements with those from TRMOKE for in-plane magnetized ﬁlms but these tend to deviate slightlyfrom those from TRMOKE with d Co/H110211.0 nm. The /H9251values in our ﬁlms are of the same order of magnitude as reportedvalues,6,7and the nonlinearity of /H9251against 1 /dCois similar to that observed in perpendicularly magnetized CoFeB ﬁlmsdespite a much different magnetic material.12To account for this enhanced /H9251values, the relaxation frequency G, deﬁned(a) (c) out-of-plane 01f×dCo(erg/cm2) 00.5 1.0 1.5 2.000.10.20.30.4HC(kOe) dCo(nm) 1nit) in-plane (b) 0 0.5 1.0 1.5 2.0-1Kueff dCo(nm)-4 -2 0 2 4-101MOKE (arb. u n H(kOe) FIG. 1. /H20849a/H20850The Co layer thickness dCodependence of coercivity HCfor Pt/Co/Pt ﬁlms. The curve is used as a visual guide. /H20849b/H20850A hysteresis loop for a Pt/Co /H208490.8 nm /H20850/Pt ﬁlm measured by PMOKE with applied ﬁeld perpendicu- lar to ﬁlm plane. /H20849c/H20850The effective perpendicular magnetic anisotropy energy Kueffmultiplied by dCoas a function of dCoas measured by PMOKE /H20849/L50098/H20850and ferromagnetic resonance /H20849/H17034/H20850. Solid line is a ﬁt to the experimental data. -200nal (arb. unit ) -200 0 100 200-60-40MOKE sign 0 100 200-60-40 Pump-probe delay time (ps)/g894/g258/g895 /g894/g271/g895 FIG. 2. Signals of TRMOKE measured with applied ﬁeld of 4 kOe directed at 60 deg. from the ﬁlm normal in /H20849a/H20850Pt/Co /H208492.0 nm /H20850/Pt and /H20849b/H20850Pt/ Co/H208490.8 nm /H20850/Pt ﬁlms. Solid curves are the calculated damped harmonic func- tion superposed on an exponential decay parameter-ﬁtted to the experimen-tal data.2030 2030GHz) Grad/s)2030 2030 Grad/s )(GHz)(a) 0 30 60 90010 010f(G 1/τ(G θΗ(deg.)0 30 60 90010 010 θΗ(deg.) 1/τ(Gf( (b) FIG. 3. Magnetic ﬁeld angle /H9258Hdependence of precession frequency f/H20849/H17034/H20850 and inverse precession life-time 1 //H9270/H20849/L50098/H20850for/H20849a/H20850Pt/Co /H208492.0 nm /H20850/Pt and /H20849b/H20850 Pt/Co /H208490.8 nm /H20850/Pt ﬁlms. Solid and broken curves are the calculated data of f and 1 //H9270parameter-ﬁtted to the experimental data.152502-2 Mizukami et al. Appl. Phys. Lett. 96, 152502 /H208492010 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.255.6.125 On: Thu, 11 Dec 2014 13:23:13asG=/H9251/H9253Ms, is shown in Fig. 4/H20849b/H20850as a function of 1 /dCo. The Gvalues for Pt /Ni80Fe20/H20849Py/H20850/Pt ﬁlms reported previ- ously are also shown in Fig. 4/H20849b/H20850with open triangles.8,9For dCo/H110221.0 nm, the Gvalue for Pt/Co/Pt ﬁlms seems to be proportional to 1 /dCo, and its slope evaluated by linear ﬁtting was 34 /H11003108rad /s nm, which is roughly three times larger than for Pt/Py/Pt ﬁlms /H2084913/H11003108rad /sn m /H20850. The enhanced Gilbert damping in thin Py layer in contact with a Pt layer can be caused by a spin pumping effect. The damping fre-quency is then expressible as G=G 0+/H20849/H92532/H6036/2/H9266/H20850g↑↓/dFM, us- ing the bulk relaxation frequency G0and mixing conduc- tance g↑↓.13The/H9253values for Pt/Co/Pt ﬁlms were almost the same as in Pt/Py/Pt ﬁlms, and the g↑↓is considered to be almost the same for both ﬁlms because it is approximatelyequal to the conductance of Pt layer in the diffusive transportregime. 13Thus, Gilbert damping in Pt/Co/Pt ﬁlms could be enhanced by an another mechanism, in addition of spinpumping. It is possible that Co 3 d–Pt 5 dhybridization ef- fectively decreases the bandwidth Wfor the Co atomic layer in contact with a Pt layer, 14enhancing both PMA and Gilbert damping, as mentioned earlier. However, this hybridizationmechanism seems not to explain the signiﬁcant increase in G ford Co/H110211.0 nm. This thickness regime is close to the inter-face roughness or alloying layer thickness, which might af- fect Gilbert damping but the problem remains open. The an-gular dependence of TRMOKE measurement with highmagnetic ﬁeld should be done in ﬁlms with atomically ﬂatinterface as further subject. In conclusion, Gilbert damping for perpendicularly magnetized Pt/Co/Pt ﬁlms had been investigated usingTRMOKE. The effective PMA energy was shown to linearlydependent on 1 /d Co, while /H9251andGincreased rapidly in the regime dCo/H110211.0 nm corresponding to a switch in the mag- netic easy-axis from in-plane to out-of-plane. No linear cor-relation between PMA and Gilbert damping was observed.TheGvalue deduced from /H9251for Pt/Co/Pt ﬁlms was much larger than that for Pt/Py/Pt ﬁlms, which was considered tobe due to the d-dhybridization effect. This work was partially supported by Grant for Indus- trial Technology Research /H20849NEDO /H20850and Grant-in-Aid for Sci- entiﬁc Research. 1S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E. Fullerton, Nature Mater. 5, 210 /H208492006 /H20850. 2M. Nakayama, T. Kai, N. Shimomura, M. Amano, E. Kitagawa, T. Na- gase, M. Yoshikawa, T. Kishi, S. Ikegawa, and H. Yoda, J. Appl. Phys. 103, 07A710 /H208492008 /H20850. 3V . Kambersky, Can. J. Phys. 48,2 9 0 6 /H208491970 /H20850. 4V . Kamberský, Czech. J. Phys., Sect. B 26,1 3 6 6 /H208491976 /H20850. 5P. Bruno, Physical Origins and Theoretical Models of Magnetic Aniso- tropy /H20849Ferienkurse des Forschungszentrums Jürich, Jürich, 1993 /H20850. 6A. Barman, S. Wang, O. Hellwig, A. Berger, and E. E. Fullerton, J. Appl. Phys. 101, 09D102 /H208492007 /H20850. 7P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier, J. Ferre, V . Baltz, B. Rodmacq, B. Dieny, and R. L. Stamps, Phys. Rev. Lett. 99, 217208 /H208492007 /H20850. 8S. Mizukami, Y . Ando, and T. Miyazaki, Jpn. J. Appl. Phys., Part 1 40, 580 /H208492001 /H20850. 9S. Mizukami, H. Abe, D. Watanabe, M. Oogane, Y . Ando, and T. Miyazaki, Appl. Phys. Express 1, 121301 /H208492008 /H20850. 10M. T. Johnson, P. J. H. Bloemen, F. J. A. den Broeder, and J. J. de Vries, Rep. Prog. Phys. 59, 1409 /H208491996 /H20850, and references therein. 11M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 /H208492002 /H20850. 12G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Appl. Phys. Lett. 94, 102501 /H208492009 /H20850. 13Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 /H208492002 /H20850. 14N. Nakajima, T. Koide, T. Shidara, F. Miyauchi, H. Fukutani, A. Fujimori, K. Ito, T. Katayama, M. Nyvlt, and Y . Suzuki, Phys. Rev. Lett. 81, 5229 /H208491998 /H20850.0.30.40.50.61 0.5 αdCo(nm) 4060801 0.5rad/s)dFM(nm) TRMOKE FMR(a) (b) FM = Co24 0.7 24 0.7 0 1 200.10.2α 1/dCo(nm-1)0 1 202040G(108 1/dFM(nm-1)FM = Ni80Fe20 Pt/FM( dFM)/Pt FIG. 4. Inverse thickness 1 /dCodependence of /H20849a/H20850Gilbert damping constant /H9251and /H20849b/H20850relaxation frequency Gfor Pt /Co/H20849dCo/H20850/Pt ﬁlms. The values ob- tained from the time-resolved magneto-optical Kerr effect and ferromagneticresonance are shown with the solid /H20849/L50098/H20850and open circles /H20849/H17034/H20850, respectively. The reported values of Gfor Pt /Ni 80Fe20/Pt ﬁlms are also shown with open triangles /H20849/H17005/H20850. Solid and broken lines are ﬁtted to the experimental data for 1/dFM/H110211.0 nm−1.152502-3 Mizukami et al. Appl. Phys. Lett. 96, 152502 /H208492010 /H20850 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.255.6.125 On: Thu, 11 Dec 2014 13:23:13
1.1660764.pdf	Curvature Stabilization of the Universal Instability Gilbert A. Emmert Citation: Journal of Applied Physics 42, 3530 (1971); doi: 10.1063/1.1660764 View online: http://dx.doi.org/10.1063/1.1660764 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/42/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic curvature drift instability Phys. Fluids 29, 3672 (1986); 10.1063/1.865799 Finiteβ stabilization of the universal drift instability: Revisited Phys. Fluids 25, 1821 (1982); 10.1063/1.863659 Lower hybrid density drift instability with magnetic curvature Phys. Fluids 24, 1588 (1981); 10.1063/1.863545 Effect of Magnetic Curvature on the DriftCyclotron Instability Phys. Fluids 10, 1526 (1967); 10.1063/1.1762316 Demonstration of the Minimum B Stability Theorem for the Universal Instability Phys. Fluids 8, 1004 (1965); 10.1063/1.1761313 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.111.185.72 On: Thu, 18 Dec 2014 14:43:45JOURNAL OF APPLIED PHYSICS VOLUME 42, NUMBER 9 AUGUST 1971 Curvature Stabilization of the Universal Instability Gilbert A. Emmert Department of Nuclear Engineering, University of Wisconsin, Madison, Wisconsin 53706 (Received 8 January 1971) A graphical t~chnique for de~ermining the influence of the drift velocity in a magnetic well on wave-particle resonance IS presented and applied to the universal instability. It is seen that the curvature drift velocity can strongly affect the number of resonant ions and lead to increased stabilization of the wave. The universal instability is a low-frequency electro static instability driven by a density gradient of the plasma. 1-3 It is caused by electrons whose motion parallel to B resonates with the wave. 4,5 The con ~ct~n produced by the density gradient and the E x B drift causes the resonant electrons to give energy to the wave and thus destabilize it. Compet ing stabilizing effects are electron and ion Landau damping and ion convection; these are also resonant particle effects. Neglecting magnetic field curva ture, the net electron contribution is destabilizing if the frequency w is less than the diamagnetic fre quency w* (w* = IkL v;IU~eL I, ve is the electron thermal velocity, Oe is the electron cyclotron fre quency, kL is the wave number perpendicular to B, and L is the plasma-density scale length) and is maximum when the number of resonant electrons is large, Le., 1~/k,,1 <ve, where k" is the wave number along B. The stabilizing ion contribution is small when the number of resonant ions is small L e., I wi k,,1 »v i' Thus the instability occurs ' primarily for Vi « I w Ik" I < v e' Magnetic field curvature has usually been simulated in slab models by a fictitious gravity g 3,6,7 which produces a guiding center drift g /0; g is usually chosen so that g 10 matches the drift of a thermal particle in an actual curved field. The resonance condition becomes w-kLg/O-k" v,,=O. In this model the effect of curvature is to introduce a Doppler shift of the frequency-, Wi = W -k Lg 10, and consequently a shift of the resonant velocity v , " = Wi Ik". In a magnetic well, the ion resonant veloc- ity is shifted downwards (Wi < w) which increases the ion-stabilizing contribution. For the electrons, w' > wand their destabilizing influence is decreased. From this it can be concluded that a magnetic well tends to stabilize the universal instability. In an actual curved field the resonance condition is B RESONANCE ELLIPSE f=CONSTANT A --'------'---"IL1..'-:Iv:-e---------Ll-~VIl FIG. 1. Locus of resonant electrons. 3530 w-{kL/OR) (v2+iv2)-k v =0 (1) II 1 II II ' where 0 is the cyclotron frequency and R is the radius of curvature of a field line. Unfortunately, (1) leads to intractable integrals in the dispersion relation. The slab model with gravity corresponds to replacing (v~ + i v~) by its thermal average. An alternative procedure, used by Laval et al. 8 is to replace only the v: term by its thermal average. The approximate resonance condition becomes w -(kjOR) (v~ + i <v~» -k v = O. (2) ... J) II USing (2), they found that the stabilizing influence of a magnetic well was significantly greater than that given by the slab model with gravity. They inter preted this to be due to a "Landau effect in the di rection of the drift". This paper proposes, however, that the results of Laval can still be interpreted as a "Doppler shift" of the resonant velocity, but of greater magnitude than that given by the gravity Doppler shift. We present a graphical technique for determining the validity of various resonance approx imations. Recall that the universal instability occurs primarily when the resonant ion velocity along B is much greater than Vi' The Doppler shift is proportional to v~ and thus the resonant ions experience a greater Doppler shift in a curved field than a thermal ion. Since the resonant ion contribution is proportional to exp{ -v~ Iv~) evaluated at the resonant velocity, this effect is Significant. The gravity model assumes that the resonant ions experience the same Doppler shift as the thermal ions and thus underestimates the stabilizing effect of a magnetic well. We can visualize the effect of the various resonance conditions in the following way. We rewrite (1) as V + __ II + _L_= __ + __ II ( ORk ) 2 v 2 RwO R202k2 " 2k l 2 k l 4k~' (3) B a FIG. 2. Locus of resonant ions when w/k Ii Vj « (RTe/LT//~ [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.111.185.72 On: Thu, 18 Dec 2014 14:43:45CURVATURE STABILIZATION OF THE UNIVERSAL INSTABILITY 3531 o c W/k vII II FIG. 3. Locus of resonant ions when w/kll vi» (R Te/L Ti) 1:2 which is the equation of an ellipse in the VII' V ~ plane with eccentricity 1/12 and centered at v~ = 0, VII = -nRkj2kJ.. This ellipse represents the locus of resonant particles in the VII' V J. plane. If we consider values typical for drift waves in fu sion plasmas (w-w-kJ. v;/neL, w/kll < v., and R/L > 1), we find that for the electrons, the second term on the right side of (3) dominates. Since ne < 0, the ellipse for the electrons looks like that shown in Fig. 1. A local Maxwellian distribution j is constant on circles centered about the origin and decreases as exp[ -(v~ + v~)/v;] in the radial direction. Reso nant particle effects are proportional to j, oj /ov '" and oj/ovJ. which are largest for VII' vJ. ~ve' Hence the dominant resonant particle contribution comes from the region on the ellipse nearest the origin, i. e. , near point a. At a, v II ~ w /k II and hence the curvature drift has little effect on the resonant elec tron contribution. Laval's approximation is to re place the ellipse by the two lines A and B as the locus of resonant particles; line A is insignificant and line B appears to be a reasonable approximation to the important part of the ellipse. The gravity approximation is to consider a single line very close to B; this also appears reasonable. The situation is quite different for the ions. Of the two terms on the right side of (3), either term can dominate depending on w/kll Vi compared with (RTe/ LTy/2, (assumingw-w). For l<w/kllv/ «(RTe/LT/)1/2, the second term dominates and the ellipse appears as in Fig. 2. When w/kll Vi «(RTe/ LTi)1/2, we get the ellipse shown in Fig. 3. In both cases the major part of the resonant contribution comes from points on the ellipse near a [where VII ~ w/kll in Fig. 2 and VII ~ (Rwn/kJ.)1/2 -vi(RTe/ LTi)1/2 in Fig. 3]. Laval's approximation again con sists of replacing the ellipse by the two lines A and B. The gravity model uses a single line C near w/kll which, in the case of Fig. 3, underestimates the resonant ion contribution. This graphical technique for determining the impor tant regions of the resonance ellipse can be used for other distribution functions. For example, if the ion distribution is bi-Maxwellian with TJ. > 2 T,l' the im portant region of the resonance ellipse in Fig. 3 is the top. This requires a different approximation to the resonance condition when w/kll is sufficiently large. This work is supported in part by the Atomic Energy Commission and by the Wisconsin Alumni Research Foundation. lA. B. Mikhailovskii and L. I. Rudakov, Zh. Eksperim. i Teor. Fiz. 44, 912 (1963) [Sov. Phys. JETP 17, 621 (1963) 1. 2B. B. Kadomtsev and A. V. Timofeev. Dokl. Akad. Nauk. SSSR 146, 581 (1962) [SOy. PhY. Dokl. 7, 826 (1963) 1. 3N. A. Krall and M. N. Rosenbluth, Phys. Fluids 8, 1488 (1965). 4F. C. Hoh, Phys. Fluids 8, 968 (1965). 5D. M. Meade, Phys. Fluids 12, 947 (1969). 6K. Kitao, J. Phys. Soc. Japan 26, 802 (1969). TR. Saison, Plasma Phys. 10, 927 (1968). 8G. Laval, E. K. Maschke, R. Pellat, and M. Vuillemin, Phys. Rev. Letters 19, 1309 (1967). [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 131.111.185.72 On: Thu, 18 Dec 2014 14:43:45
1.3177269.pdf	"Pitch bending and glissandi on the clarinet: Roles of the vocal\ntract and partial tone hole closur(...TRUNCATED)
1.3259398.pdf	"Inherent spin transfer torque driven switching current fluctuations in magnetic element\nwith in-pl(...TRUNCATED)

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