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Spin-wave damping at spin-orientation phase transitions V. G. Baryakhtar and A. G. Danielevich Citation: Low Temperature Physics 32, 768 (2006); doi: 10.1063/1.2219498 View online: http://dx.doi.org/10.1063/1.2219498 View Table of Contents: http://scitation.aip.org/content/aip/journal/ltp/32/8?ver=pdfcov Published by the AIP Publishing 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51Spin-wave damping at spin-orientation phase transitions V. G. Baryakhtara/H20850and A. G. Danielevich Institute of Magnetism of the National Academy of Sciences of Ukraine, pr. Vernadskogo 36b, Kiev 03142, Ukraine /H20849Submitted February 6, 2006; revised March 21, 2006 /H20850 Fiz. Nizk. Temp. 32, 1010–1023 /H20849August-September 2006 /H20850 The spectra or spin waves and their damping in the vicinity of spin-orientation phase transitions are investigated. It is shown that if the ground state is degenerate and the degeneracy is charac-terized by a continuous degeneracy parameter, then the spin-wave damping is not described by aLandau-Lifshitz-Gilbert relaxation term. The dissipative function is constructed for ferromagneticcrystals of different symmetry. The cases of finite and zero longitudinal magnetic susceptibilityare considered. A method of finding the relaxation term in the Landau-Lifshitz equation for crys-tals of different symmetry is given. The spin-wave spectra and damping are calculated for ferro-magnets with uniaxial and tetragonal symmetry. It is shown that the relaxation process has atwo-step character. In the first step an equilibrium value of the magnetization is established onaccount of the exchange interaction. In the second step the spin-wave amplitude is damped asthe magnetization variable precesses around its equilibrium value. The details of the Bogolyubovmethod of quasi-averages in application to the case of ferromagnets with a spontaneously degen-erate vacuum are discussed. © 2006 American Institute of Physics ./H20851DOI: 10.1063/1.2219498 /H20852 I. INTRODUCTION Spin-orientation phase transitions are of interest because their Ginzburg number Gi/H112701, and, accordingly, the Landau theory of second-order phase transitions is applicable to thistype of transition. As we know, such phase transitions occurat values of the parameter H,Tfor which the spin-wave ac- tivation frequency /H92750/H20849H,T/H20850goes to zero. These phase tran- sitions are widely studied both theoretically and experimen- tally. The vanishing of the spin-wave activation naturally raises the question of the behavior of the spin-wave damping/H9003at the phase transition point. If the spin-wave damping /H9003 /HS110050 at the phase transition point, it would mean that addi- tional conditions are imposed on the application of Landautheory to spin-orientation phase transitions. In other words,the condition /H92750=0,/H9003/HS110050 is the condition of high viscosity of the medium, and it should /H20849and, in a number of solids, does /H20850lead to transitions that are smeared in /H20849H,T/H20850, which are neither first-order nor second-order phase transitions. As we know, the basic equation for describing the static, dynamic, and relaxation properties of ferromagnets is theLandau-Lifshitz equation, which the eponymous authorsfound in 1935: 1 /H11509M /H11509t=−/H9253/H20851M,He/H20852+R, /H208491/H20850 where He=−/H9254W//H9254M,W/H20853M,/H11509M//H11509xi/H20854is the quasiequilibrium thermodynamic potential of the ferromagnet, and the nota-tion /H20851…,…/H20852is used for the vector /H20849cross /H20850product. An impor- tant element of the general Landau-Lifshitz theory was theconstruction of the potential W/H20853M, /H11509M//H11509xi/H20854from invariants of the magnetization with respect to the symmetry group ofthe crystal. The dissipative function was not considered in Ref. 1, and the relaxation termR L=/H9261 M02/H20851M,/H20851M,He/H20852/H20852 /H20849 2/H20850 in the equation of motion for the magnetization was pro- posed on the basis of arguments that it should describe theapproach of the magnetization vector toward the effectivemagnetic field H e. Much later, in his unpublished dissertation,2Gilbert constructed the dissipative function cor- responding to Landau-Lifshitz relaxation for the case of aferromagnet and proposed to write the relaxation term interms of the time derivative of the magnetization: R G=/H9261G M0/H20851M˙,M/H20852. /H208493/H20850 In this formula M˙=/H11509M//H11509t. The relaxation terms RLandRGagree to within a con- stant factor. More precisely, the equations of motion /H11509M /H11509t=−/H9253/H20851M,He/H20852+/H9261 M2/H20851M,/H20851M,He/H20852/H20852 and /H11509M /H11509t=−/H9253G/H20851M,He/H20852+/H9261G M/H20851M,M˙/H20852/H20849 4/H20850 agree if /H9253G=/H9253/H208511+/H20849/H9261//H9253M/H208502/H20852;/H9261G=/H20849/H9261//H9253M/H20850. Both Landau and Lifshitz and Gilbert used the model of a ferromagnet with a magnetization of constant absolutemagnitude. In other words, the longitudinal susceptibility ofthe ferromagnet was assumed to be zero. In spite of the vec-tor equation of motion, the Landau-Lifshitz-Gilbert relax-ation term is contained by a single relaxation constant, cor-responding to an isotropic medium. It was noted back in thepaper by Landau and Lifshitz that the relaxation term /H208492/H20850 derives from the spin-spin and spin-orbit interactions /H20849see also Ref. 3 and Sec. III of the present paper /H20850.LOW TEMPERATURE PHYSICS VOLUME 32, NUMBER 8-9 AUGUST-SEPTEMBER 2006 1063-777X/2006/32 /H208498-9/H20850/11/$26.00 © 2006 American Institute of Physics 768 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51The first paper in which a generalization of the Landau- Lifshitz-Gilbert relaxation term to the case of the exchangeinteraction was attempted is that of Kamberskii, 4who pro- posed an additional term /H20851see Eq. /H2084937/H20850of Ref. 4 /H20852: RK,ex=/H9261K M0/H20851/H9004M˙,M/H20852. /H208495/H20850 Thus the total relaxation term has the form R=/H9261G M0/H20851M˙,M/H20852+/H9261K M0/H20851/H9004M˙,M/H20852. /H208496/H20850 It was later pointed out3that the Landau-Lifshitz-Gilbert relaxation term gives qualitatively incorrect results for one ofthe possible magnetic phases, namely, for the easy-planephase of a uniaxial ferromagnet. An examination of the ex-pression for the relaxation term shows that it completely ig-nores the symmetry of the magnetic material, and that leadsto the contradiction. Meanwhile, after the papers by Landauand Lifshitz 1,5and Kittel6,7on the domain structure of ferro- magnets, it became clear that the symmetry of ferromagnetsplays a decisive role in the formation of their domain struc-ture and other properties. Landau also pointed out the deci-sive role of the change of symmetry at second-order phasetransitions. 8 In Ref. 3 a general method of constructing the relaxation term in the equation of motion of the magnetic moment isgiven, and it is shown how to take the symmetry of thecrystal into account in the construction of the dissipativefunction. In that same paper it is proposed to take the longi-tudinal susceptibility into account, and the relaxation of themagnetic moment in magnitude was described in the sim-plest case of a crystal with magnetic anisotropy of the easy-axis type. Theoretical considerations and also experimental re- search on spin-orientation phase transitions 9have prompted us to address the issue of spin-wave damping near phasetransitions of this kind. To avoid the contradictions that havearisen we examine the question of constructing the dissipa-tive function of ferromagnets, and for the example ofuniaxial and tetragonal ferromagnets we calculate the disper-sion relations with damping taken into account in two mod-els: in the Landau-Lifshitz form, and in the form proposed inRef. 3. II. DISSIPATIVE FUNCTION OF A FERROMAGNET To construct the dissipative function of a ferromagnet we use the Landau-Lifshitz equation /H208491/H20850to find the variation of its energy with time: d dt/H20885Wdv=/H20885/H9254W /H9254M/H11509M /H11509tdv=−/H20885HRdv=−2/H20885qdv, /H208497/H20850 where qis the density of the dissipative function. It follows thatqis equal to /H20851note: HRis the scalar /H20849dot/H20850product /H20852 q=1 2HR. /H208498/H20850Before embarking on construction of the dissipative function, we recall the following circumstances. A nonequi-librium state of a ferromagnet is determined by specifying amagnetization distribution: M=M/H20849r,t/H20850. /H208499/H20850 The equilibrium state is determined from the condition H e/H20849r/H20850=−/H9254W /H9254M=0 . /H2084910/H20850 Obviously, in states that are close to the ground state in energy the effective field is small: /H20841He/H20849r,t/H20850/H20841/H11270M0, /H2084911/H20850 where M0is the equilibrium value of the magnetization. As the parameter characterizing the quasi-equilibrium state we choose not M/H20849r,t/H20850but the effective magnetic field He/H20849r,t/H20850. This field is more convenient than the magnetization because it is small for all actual nonequilibrium states. In such an approach the relaxation term Rshould be considered as a functional of He/H20849r,t/H20850. For states close to the ground state we can expand R/H20853He/H20854in a power series in He/H20849r,t/H20850and keep only the linear terms of the expansion: R/H20853H/H20854=/H9261ˆ1He−/H92612,ik/H115092He /H11509xi/H11509xk. /H2084912/H20850 In writing this formula we have taken into account that re- laxation is absent in the ground state /H20849He=0/H20850. The first term in the formula describes processes responsible for relaxation of spatially uniform magnetization to its equilibrium direc-tion. The second term describes relaxation of nonuniformdistributions H e/H20849r,t/H20850/H20851and, of course, magnetization distribu- tions M/H20849r,t/H20850/H20852to the equilibrium distribution. This type of relaxation is due mainly to the exchange interaction. For this reason /H9261ˆ2is a tensor only with respect to the coordinate variables. Substituting expression /H2084912/H20850into formula /H208497/H20850, we obtain the final form of the expression for the dissipative-functiondensity q: q=1 2/H92611,ikHe,iHe,k+1 2/H11509He /H11509xk/H92612,ik/H11509He /H11509xi. /H2084913/H20850 The second term in this formula is invariant under uni- form rotations in spin space. For comparison, the dissipative function corresponding to the Landau-Lifshitz relaxation term is qLL=/H9261 2/H20851/H20849HeM/H208502/M2−He2/H20852. /H2084913a /H20850 Comparing Eqs. /H2084912/H20850and /H2084913/H20850, we find that R=/H9254 /H9254H/H20885qdv, /H2084914/H20850 and analogously RL=dqLL dHe. /H2084914a /H20850 Since qmust be invariant with respect to the symmetry group of the crystal, the tensors /H9261ˆ1and/H9261ˆ2are determined byLow T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 769 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51that symmetry. For crystals of cubic symmetry /H9261ˆ1= diag /H20849/H92611,/H92611,/H92611/H20850,/H9261ˆ2= diag /H20849/H92612,/H92612,/H92612/H20850. /H2084915/H20850 For crystals of hexagonal symmetry /H9261ˆ1= diag /H20849/H926111,/H926111,/H926113/H20850,/H9261ˆ2= diag /H20849/H926121,/H926121,/H926123/H20850. /H2084916/H20850 And, for crystals of orthorhombic symmetry /H9261ˆ1= diag /H20849/H926111,/H926112,/H926113/H20850,/H9261ˆ2= diag /H20849/H926121,/H926122,/H926123/H20850. /H2084917/H20850 Formulas /H2084913/H20850and /H2084915/H20850–/H2084917/H20850give the relationship between the relaxation constants and the lattice symmetry. Comparatively recently a generalization of the Landau- Lifshitz relaxation law to the anisotropic case was given inthe paper by Safonov: 10 RS=1 M02/H20851M,/H9261ˆS/H20851M,H/H20852/H20852, /H2084918/H20850 where /H9261ˆSis a second-rank tensor which is determined by the symmetry of the crystal and is of a relativistic nature. III. MAGNETIZATION CONSERVATION LAWS AND THE DISSIPATIVE FUNCTION OF A FERROMAGNET The next step in the determination of the relaxation ten- sors/H9261ˆ1and/H9261ˆ2can be made using the conservation law for the magnetization: M=/H20885M/H20849r,t/H20850dV= const. /H2084919/H20850 For this conservation law there is a corresponding dynamical equation11 /H11509Mi /H11509t+/H11509/H9016ik /H11509xk=0 . /H2084920/H20850 The tensor /H9016ikis the flux density of the ith magnetization component through a unit area perpendicular to the kth axis. For this reason one should set /H9261ˆ1in formula /H2084914/H20850equal to zero in the exchange approximation. Then the dissipativefunction takes the form q=q ex=1 2/H11509H /H11509xk/H92612,ik/H11509H /H11509xi. /H2084921/H20850 We note that the Kamberskii relaxation term /H20851formula /H208495/H20850/H20852cannot describe the exchange relaxation, since RK,ex does not reduce to the divergence of a dissipative flow. Only in the linear approximation, when RK,ex=/H9261K/H20851/H9004M˙,M/H20852⇒/H9261K/H20851/H9004M˙,M0/H20852 =/H11509 /H11509xi/H9261K/H20875/H11509 /H11509xiM˙,M0/H20876, /H2084922/H20850 can one speak of its exchange nature. Let us consider a uniaxial ferromagnet whose symmetry axis lies along the zdirection. We shall start from the follow- ing expression for the nonequilibrium free energy density W of a uniaxial ferromagnet: W=Wex+Wa, /H2084923/H20850 where Wexis the exchange part of the energy density:Wex=/H9251ik 2/H11509M /H11509xi/H11509M /H11509xk+1 8/H9273/H20648M02/H20849M2−M02/H208502, /H2084924/H20850 andWais the anisotropy energy density of the uniaxial fer- romagnet: Wa=1 2K1Mz2+1 4K2Mz4. /H2084925/H20850 In formulas /H2084924/H20850and /H2084925/H20850the following notation is used: K1 andK2are the magnetic anisotropy constants, /H9273/H20648is the lon- gitudinal magnetic susceptibility, Mis the magnetization vector of the ferromagnet, M0is the saturation magnetiza- tion, and /H9251is the inhomogeneous exchange interaction. The anisotropy constants depend on the temperature T. The lon- gitudinal magnetic susceptibility /H9273/H20648describes the homoge- neous exchange interaction, and it is equal in order of mag-nitude to /H9262M0/J, where /H9262is the Bohr magneton, and Jis the exchange integral. From arguments of lattice symmetry for a uniaxial fer- romagnet /H20849Z/H11009/H20850, the dissipative-function density can be writ- ten in the form q=1 2/H926111/H20849Hx2+Hy2/H20850+1 2/H926113Hz2+1 2/H92612,ik/H20873/H11509H /H11509xi/H11509H /H11509xk/H20874. /H2084926/H20850 In this case there is a conservation law only for the magne- tization vector component M z=/H20848MzdV, and the correspond- ing equation of motion for the magnetic moment has theform /H11509Mz /H11509z+/H11509/H9016zk /H11509xk=0 . /H2084927/H20850 The effective magnetic field Hconsists of two parts: a field due to the exchange interaction: Hex=/H9251/H9004M−M2−M02 2/H9273/H20648M02M, /H2084928/H20850 and a field due to the magnetic anisotropy energy: Ha=−ez/H20849K1+K2Mz2/H20850Mz. /H2084929/H20850 The effective total magnetic field is determined by the ex- pression H=Hex+Ha=/H9251/H9004M−M2−M02 2/H9273/H20648M02M−ez/H20849K1+K2Mz2/H20850Mz. /H2084930/H20850 Corresponding to this field is a relaxation term R: R=/H20849/H926111+/H92612/H9004/H20850/H20873/H9251/H9004−M2−M02 2/H9273/H20648M02/H20874M/H11036 −/H20849/H926113+/H92612/H9004/H20850/H20873/H9251/H9004−M2−M02 2/H9273/H20648M02−/H20849K1+K2Mz2/H20850/H20874M /H20648. /H2084931/H20850 In this formula M/H11036=Mxex+Myey;M /H20648=Mzez. To refine the form of the dissipative function further, we invoke the con-servation law for M z. Examining formula /H2084931/H20850, one is con- vinced that the quantity Rzcan be reduced to a divergence only in the case when /H926113=0. This means that when the M z770 Low T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51conservation law is taken into account, the dissipative- function density of a uniaxial ferromagnet takes the form q=1 2/H926111/H20849Hx2+Hy2/H20850+1 2/H92612,ik/H20873/H11509H /H11509xi/H11509H /H11509xk/H20874. /H2084932/H20850 A microscopic calculation in the framework of spin-wave theory leads to a dissipative function with this structure. Us-ing the spin-wave damping decrements given in Ref. 12, onecan easily find the temperature dependence of the constants/H9261 11and/H92612at low temperatures T/H11270TC, where TCis the Curie temperature. We note that the Landau-Lifshitz relaxationterm is inconsistent with the M zconservation law for a uniaxial ferromagnet. For a ferromagnet of tetragonal symmetry /H20849Z/H11009 →/H20853Z4;X2;Y2/H20854/H20850the expression for the free energy density W remains the same except for the part corresponding to the anisotropy energy /H2084925/H20850: Wa=1 2K1Mz2+1 4K2Mz4+1 2K3Mx2My2. /H2084933/H20850 In the case of a tetragonal crystal there is no conservation law for the magnetization vector components and, conse-quently, the dissipative-function density of a tetragonal fer-romagnet will have the form in /H2084926/H20850. IV. SPIN-WAVE DAMPING It is well known13,14that in the case of anisotropy of the form /H2084926/H20850there exist the following three ground states /H20849phases /H20850of a uniaxial ferromagnet: a phase /H9021/H20849/H20648/H20850in which the magnetization M0is parallel to the symmetry axis, a phase /H9021/H20849/H11028/H20850in which the magnetization M0is oriented at an angle /H9258to the symmetry axis, and a phase /H9021/H20849/H11036/H20850in which the magnetization M0lies in the basal plane. For a description of these three phases it is sufficient to choose the magnetic an-isotropy energy in the form /H2084925/H20850. Knowing the free energy /H2084924/H20850and /H2084925/H20850of a uniaxial ferromagnet and the effective field /H2084930/H20850corresponding to it, the dissipative function /H2084932/H20850and the components of the re- laxation term /H2084931/H20850corresponding to it, and also the equation of motion /H208497/H20850for the magnetization, one can find the spin- wave dispersion relation and damping and the relaxation ofthe magnitude of the magnetization M 2in the three phases /H9021/H20849/H20648/H20850,/H9021/H20849/H11028/H20850, and/H9021/H20849/H11036/H20850. The relaxation of M2will be determined from the equa- tion dM2 dt=2MR, /H2084934/H20850 which is obtained by multiplying the Landau-Lifshitz equa- tion by M. The results of the calculations are presented below:1The/H9021/H20849/H11036/H20850phase: /H9258=/H9266/2,M2=M02. Stability condi- tions: K1/H110220. The magnitude of the magnetization changes with time in a manner determined by the following relaxation time andequation: /H9003 M=/H208491//H9270/H11036/H20849k/H20850/H20850=/H20849/H926111+/H92612k2/H20850//H9273/H20648,M2/H20849t/H20850=M02+2M0m/H20849k,0/H20850exp /H20851−t//H9270/H11036/H20849k/H20850/H20852. /H2084935/H20850 Here mis a small deviation from the ground state. We note that the relaxation time is diminished on account of the smalllongitudinal susceptibility /H20849the factor /H9273/H20648/H20850in comparison with the characteristic spin-wave relaxation times /H9270sdetermined by the relaxation constants: 1/ /H9270s/H11011/H926111;/H92612k2. We note that in the phase /H9021/H20849/H11036/H20850relaxation of both inhomogeneous and ho- mogeneous deviations of the magnitude of the magnetization occur. This is because in the ground state there is a nonzerocomponent of the magnetization in the basal plane. The frequency and damping of the spin waves are deter- mined by the formulas /H9275=/H9275s/H20849k/H20850−i/H9003s/H20849k/H20850; /H9275s/H20849k/H20850=/H208491/2 /H20850/H208814/H92532M02/H9251k2/H20849/H9251k2+K1/H20850−k4/H20851/H92612K1−/H9251/H926111/H208522; /H9003s/H20849k/H20850=/H208491/2 /H20850/H20851/H20849/H926111+/H92612k2/H20850/H9251k2+/H92612k2/H20849/H9251k2+K1/H20850/H20852. /H2084936/H20850 The term under the radical, which is proportional to the re- laxation constants /H926111,/H92612in the expression for the spin-wave frequency, describe the standard frequency decrease due todissipation. Interestingly, when the condition /H9251/H926111−/H92612K1=0 holds, the spin-wave frequency is unaffected by dissipation. The condition for the existence of spin waves in the /H9021/H20849/H11036/H20850phase holds because in the long-wavelength limit we have /H9275Im /H9275Re——→ k→0k/H20849/H926111/H9251+/H92612K1/H20850 2/H9253M0/H20881/H9251K1——→ k→00. /H2084937/H20850 A comparison of the relaxation time of the magnitude of the magnetization and the relaxation time of spin waves /H20851see formula /H2084935/H20850and /H2084936/H20850/H20852shows that /H9003M=/H208491//H9270/H20849k/H20850/H20850is much greater than /H9003s/H20849k/H20850: /H9003M//H9003s/H20849k/H20850/H11015/H20849 1//H9273/H20648/H20850/H112711. /H2084938/H20850 This inequality means that relaxation in a ferromagnet has a two-step character. In the first, fast step an equilibriumdistribution of the magnetization over magnitude is estab-lished on account of the exchange interaction. This process isdescribed by formula /H2084935/H20850. In the second, slow step a preces- sion of the magnetization around its equilibrium value occursat the spin-wave frequency, accompanied by damping of thespin-wave amplitude with a relaxation time /H9270s/H20849k/H20850=/H208491//H9003s/H20849k/H20850/H20850. These arguments as to the two-step character of the relax- ation process in a ferromagnet are valid not only for the/H9021/H20849 /H20648/H20850,/H9021/H20849/H11036/H20850, and /H9021/H20849/H11028/H20850phases discussed here but also for any other phases of a ferromagnet. This circumstance is due to the fact that expression /H2084931/H20850forRdemonstrates the quan- tity /H20849M2−M02/H20850/2/H9273/H20648M02with factors proportional to the relax- ation constants /H9261. For the equation of relaxation of the mag- nitude of the magnetization /H2084934/H20850this term gives a factor of 2Meq2m/H20648/2/H9273/H20648M02=m/H20648/2/H9273/H20648on the right-hand side of the equa- tion. Here m/H20648is the component of the deviation of the mag- netization along the equilibrium direction Meq. As a result, the structure of the relaxation equation for the magnitude ofthe magnetization takes the formLow T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 771 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51dM2/dt=−2 /H20849/H9261//H9273/H20648/H20850m/H20648Meq=− /H20849/H9261//H9273/H20648/H20850M2 independently of which of the phases is being considered. 2. The canted phase /H9021/H20849/H11028/H20850: cos2/H9258=−K1/K2M02,M2 =M02. The conditions of stability for this phase have the form: K2/H110220; −K2M02/H11021K1/H110210. For simplicity, in this and the next case /H20849/H9021/H20849/H20648/H20850/H20850we shall not write out the formulas for the spin-wave frequency with allowance for the renormalization due to damping, sincethose formulas do not describe any new effects. In the absence of damping the dispersion relation for this phase takes the form /H9275/H20849/H92752−/H9275S2/H20850=0 . /H2084939/H20850 The zero frequency corresponds to relaxation of the magnetic moment, and the formula /H9275S2=/H92532M02/H9251k2/H20849/H9251k2−2K1sin2/H9258/H20850/H20849 40/H20850 describes the spin-wave frequencies in neglect of the renor- malization due to the dissipative terms. Relaxation of the magnitude of the magnetization in the phase /H9021/H20849/H11028/H20850is described by the formula M2/H20849t/H20850=M02+2M0m/H20849k,0/H20850exp /H20851−t//H9270/H11028/H20849k/H20850/H20852, /H2084941/H20850 where 1/ /H9270/H11028/H20849k/H20850=/H20849/H926111sin2/H9258+/H92612k2/H20850//H9273/H20648. We note that for /H9258=/H9266/2 the expression for /H9270/H11028/H20849k/H20850goes over to formula /H2084935/H20850, and for /H9258=0, to /H2084947/H20850. In the approximation linear in /H926111,/H92612the spin-wave damping /H9003s/H20849k/H20850is simpler to find not from the dispersion relation but by using the following formula: /H9003s/H20849k/H20850=q W, /H2084942/H20850 where qandWare considered in the absence of the relax- ation terms in the equation of motion of the magnetic mo-ment. From formulas /H2084924/H20850,/H2084925/H20850, and /H2084932/H20850we obtain /H9003 s/H20849k/H20850=1 2/H20875/H20849/H926111+/H92612k2/H20850/H9251k2/H20849/H9251k2/H208491 + cos2/H9258/H20850−2K1sin2/H9258/H20850 /H9251k2−2K1sin2/H9258 +/H92612k2/H20849/H9251k2−2K1/H208502sin2/H9258 /H9251k2−2K1sin2/H9258/H20876. /H2084943/H20850 It is easy to check that at the points of the phase transi- tions /H9021/H20849/H11028/H20850⇔/H9021/H20849/H11036/H20850and/H9021/H20849/H11028/H20850⇔/H9021/H20849/H20648/H20850this formula goes over to the formulas for spin-wave damping in the phases /H9021/H20849/H11036/H20850and/H9021/H20849/H20648/H20850, respectively. The existence condition for spin waves in the /H9021/H20849/H11028/H20850 phase also holds, since in the long-wavelength limit we have /H9003s/H20849k/H20850 /H9275S——→ k→0k/H20849/H926111/H9251+/H926122K1/H20850 2/H9253M0/H208812/H9251/H20841K1sin2/H9258/H20841——→ k→00. /H2084944/H20850 It is seen from formulas /H2084936/H20850,/H2084940/H20850, and /H2084943/H20850that in the /H9021/H20849/H11036/H20850and/H9021/H20849/H11028/H20850phases, in which the ground state is degen- erate with a continuous degeneracy parameter /H92720/H20849/H92720is the angle between the magnetization and the xaxis in the basal plane /H20850, the spin-wave spectrum is activationless, and the damping is much less than the frequency and goes to zero asthe wave vector approaches zero. This result is a demonstra-tion of the general theorems of Goldstone and Adler for ourparticular case. 153. The/H9021/H20849/H20648/H20850phase: /H9258=0,M2=M021−2/H9273/H20648K1/1+2/H9273/H20648M02K2. Stability condition: K1+K2M02/H110210. The spin-wave dispersion relation and the spin-wave damping are determined by the formulas /H9275=/H9275s/H20849k/H20850−i/H9003s/H20849k/H20850; /H9275s/H20849k/H20850=/H9253M0/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852; /H9003s/H20849k/H20850=/H20849/H926111+/H92612k2/H20850/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852. /H2084945/H20850 The relaxation of the magnetic moment is determined by the formulas M2/H20849t/H20850=M021−2/H9273/H20648K1 1−2/H9273/H20648K2M02+2M0m/H20849k,0/H20850exp /H20851−t//H9270/H20849k/H20850/H20852, /H2084946/H20850 where 1/t/H20849k/H20850=i/H9003M=/H92612k2//H9273/H20648. /H2084947/H20850 We note that, first, the relaxation time of the magnitude of the magnetic moment, /H9270/H20849k/H20850/H11015/H9273/H20648//H92612k2is shortened on ac- count of the longitudinal susceptibility /H9273/H20648/H112701. Second, the relaxation time /H9270/H20849k/H20850/H110151/k2is inversely proportional to the wave vector. The absence of relaxation of the homogeneous deviations of the magnetizations is due to the conservationlaw for the magnetization component M z. In other words, in the relaxation process the small-scale inhomogeneities of themagnetic moment vanish first, and a magnetization of uni-form magnitude is established in the medium. To describethe slower relaxation of the homogeneous deviations of themagnetization from its equilibrium values it is necessary totake into account the interactions that violate conservation of M z. In order for spin waves to exist in the ground state, it is necessary that the imaginary part of the spin-wave frequency/H9003 s/H20849k/H20850=/H9275Im, which is responsible for energy dissipation, be much smaller than the real part /H9275Re/H20849activation frequency of spin waves /H20850. As is seen from Eq. /H2084945/H20850, this condition, /H9275Im /H9275Re=/H926111+/H92612k2 /H9253M0/H112701, /H2084948/H20850 is fulfilled if the relaxation constants /H9261of the medium are small compared to the characteristic frequency /H9253M0of the ferromagnet. V. SPIN-WAVE SPECTRA AND DAMPING CALCULATED WITH THE LANDAU-LIFSHITZ RELAXATION TERM In this case, of course, there is no relaxation of the mag- nitude of the magnetization, since M2=const. More pre- cisely, for /H9273→0 the relaxation time of the magnitude of the magnetization tends toward zero /H20851see formula /H2084935/H20850,/H2084941/H20850, and /H2084947/H20850/H20852. The stability conditions for the phases are written above, and we will not give them here. The spin-wave dispersionsand damping are determined by the following formulas: 1. The/H9021/H20849/H11036/H20850phase:772 Low T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51/H9275=i 2/H9261/H208492/H9251k2+K1/H20850±1 2/H208814/H92532M02/H9251k2/H20849/H9251k2+K1/H20850−/H92612K12. /H2084949/H20850 From this formula one can quickly conclude that there are no homogeneous oscillations in a uniaxial ferromagnet withmagnetic anisotropy of the easy-plane type. Indeed, at k=0 the frequency of the spin waves become imaginary: /H9275 =i/H9261K1=−i/2/H9270s. A calculation shows that the damping of the magnetization components mzandmywith time is described by the formulas mz/H20849t/H20850=mz/H208490/H20850exp /H20851−t//H9270s/H20852; my/H20849t/H20850=− /H208492/H9253M0/H9270s/H20850mz/H208490/H20850exp /H20851−t//H9270s/H20852. /H2084950/H20850 If we compare the spin-wave frequencies in the phases /H9021/H20849/H11036/H20850and/H9021/H20849/H20648/H20850/H20851see formula /H2084954/H20850/H20852, we are confronted with the following paradox. In the /H9021/H20849/H20648/H20850phase the spin waves exist as weakly damped magnetization waves, while in the /H9021/H20849/H11036/H20850 phase these waves become strongly damped. This contradic- tion can be presented in the following form. As is seen fromEq. /H2084949/H20850, the existence condition for weakly damped spin waves in the ground state /H9021/H20849/H11036/H20850is violated, not only for k →0 but also at wave vectors ksatisfying the condition /H20849 /H9253M0/H208502/H9251k2/H11021/H92612K1. /H2084951/H20850 This condition on the wave vectors presupposes that the spin-wave frequencies are imaginary. Thus we are faced withthe paradox that damped magnetization waves arise in a fer-romagnet with weak dissipation /H20849 /H9253M0/H20850/H11022/H9261 at the transition from one phase to another. 2. The/H9021/H20849/H11028/H20850phase: /H9275=i/H9261/H20849/H9251k2−K1sin2/H9258/H20850 ±/H20881/H92532M02/H9251k2/H20849/H9251k2−2K1sin2/H9258/H20850−/H92612K12sin4/H9258. /H2084952/H20850 For the /H9021/H20849/H11028/H20850phase there is a whole region of small wave vectors in which spin waves are waves with imaginary fre- quencies: /H20849/H9253M0/H208502/H9251k2/H11021/H92612/H20841K1/H20841/H20849sin/H9258/H208502. /H2084953/H20850 The spin-wave damping results /H2084949/H20850and /H2084952/H20850obtained with the use of a relaxation term in the Landau-Lifshitz formare in qualitative disagreement with the results /H2084936/H20850and /H2084943/H20850 obtained with the use of a relaxation term corresponding tothe dissipative function /H2084932/H20850. This disagreement means only one thing: the relaxation term in the Landau-Lifshitz formdoes not describe the relaxation for states with a continuousdegeneracy parameter. The /H9021/H20849/H11036/H20850and/H9021/H20849/H11028/H20850phases are just such states.3. The/H9021/H20849 /H20648/H20850phase: /H9275=i/H9261/H20849/H9251k2−/H20849K1+K2M02/H20850/H20850±/H9253M0/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852. /H2084954/H20850 In this phase the existence condition for spin waves is satis- fied if the relaxation constant /H9261is small. In that case we have /H9275Im /H9275Re=/H9261 /H9253M0/H112701. /H2084955/H20850 The situation is different if one is considering spin-wave damping at phase transition points. Under such conditionsthe damping obtained with the use of a relaxation term of theLandau-Lifshitz form is in only quantitative disagreementwith the results /H2084936/H20850and /H2084943/H20850obtained with the use of a relaxation term corresponding to the dissipative function/H2084931/H20850. These results are presented in Table I. VI. FERROMAGNET WITH TETRAGONAL SYMMETRY The study of spin waves in crystals with tetragonal sym- metry permits one to consider the real situation in whichactivation exists in the spin-wave spectrum and to identifythe particular anisotropy to which the spin-wave damping isdue. Since the results on the relaxation of the magnetizationagree with the results on the relaxation of M 2for the /H9021/H20849/H11028/H20850 phase of a uniaxial crystal, we shall not present them here. In the case of tetragonal symmetry the direction of the magnetization vector Min the xOy plane /H20849the azimuthal angle/H9272/H20850becomes important. The angle /H9272in the ground state can take the values /H9272=0 and /H9272=/H9266/2. For simplicity and brevity we shall present the results only for ground stateswith /H9272=0, since the results are fundamentally the same as for /H9272=/H9266/2. Using the formulas for the dissipative-function density /H2084926/H20850and the free-energy density /H2084924/H20850and /H2084933/H20850of a tetragonal ferromagnet and the equation of motion for the magnetiza-tion /H208497/H20850, we find the dispersion relation with the the spin- wave damping taken into account for all the different phasesat /H9272=0: 1. The /H9021/H20849/H20648/H20850phase: /H9272=0,/H9258=0, M2=M021−2/H9273/H20648K1/1 +2/H9273/H20648M02K2. Stability conditions: Ka=−/H20849K1+K2M02/H20850/H110220. The spin-wave dispersion relation and damping are de- termined by the formulas /H9275=−i/H20849/H926111+/H92612k2/H20850/H20849/H9251k2−/H20849K1+K2M02/H20850/H20850 ±/H9253M0/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852. /H2084956/H20850TABLE I. Ratio of the dissipative part to the activation part of the spin-wave frequency at the boundaries of the phase transitions in a uniaxial ferromagnet in the long-wavelength approximation /H20849k→0/H20850. Low T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 773 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51It is seen that the existence condition for spin waves holds in the ground state /H9021/H20849/H20648/H20850: /H9275Im /H9275Re=/H926111+/H92612k2 /H9253M0/H112701. /H2084957/H208502. The /H9021/H20849/H11036/H20850phase: /H9253=0,/H9258=/H9266/2,M2=M02. Stability condition: K3/H110220;K1/H110220. The spin-wave dispersion relation and damping are given by the formulas /H9275=−i 2/H20849/H20849/H926111+/H92612k2/H20850/H20849/H9251k2+K3M02/H20850+/H20849/H926113+/H92612k2/H20850/H20849/H9251k2+K1/H20850/H20850 ±1 2/H208814/H92532M02/H20849/H9251k2+K1/H20850/H20849/H9251k2+K3M02/H20850−/H20851/H20849/H926111/H20849/H9251k2+K3M02/H20850−/H92612k2/H20849K1−K3M02/H20850−/H926113/H20849/H9251k2+K1/H20850/H20850/H208522. /H2084958/H20850 The formulas for the homogeneous ferromagnetic reso- nance in a sample of spherical shape, when the demagnetiz-ing factors do not have to be taken into account, have theform /H9003 s/H208490/H20850=1 2/H20849/H926111K3M02+/H926113K1/H20850, /H2084959/H20850 /H9275s/H208490/H20850=1 2/H208814/H92532M02K1K3M02−/H20849/H926111K3M02−/H926113K1/H208502. /H2084960/H20850 Let us discuss these formulas. We start with the damping /H9003s/H208490/H20850.I fw es e t K3=0 in the anisotropy energy of a tetrago- nal crystal, we recover the model of a uniaxial crystal. As we have seen, for that model /H926113=0. This means that /H926113 =/H926113/H20849K32/H20850=bK32. By what arguments can we assert that /H926113 depends on the square of K3? There are two: the relaxation constant /H926113should not depend on the sign of the anisotropy constant K3; the relaxation constant /H926113is determined by the probability of processes in which spin waves scatter on eachother, which are conditional upon the anisotropy energy W a =K3Mx2My2/2, when one goes from the magnetic moment op- erators to the spin-wave creation and annihilation oeprators/H20849see Ref. 12 /H20850. Since the possibility of spin-wave scattering processes is proportional to the square of the matrix elementof the interaction energy, the relaxation constant /H9261 13/H11015K32. Analogous arguments can be adduced in an analysis of theconstant /H9261 11. This latter constant is equal to zero in the ex- change approximation, when the anisotropy constants K1,K2, andK3are equal to zero. For this reason /H926111=cK12. The esti- mates given permit us to write the spin-wave damping for atetragonal ferromagnet as /H9003 s/H208490/H20850=1 2/H20849cK12K3M02+bK32K1/H20850/H110151 2cK12K3M02 =1 2/H926111K3M02, /H2084961/H20850 where we have taken into account that K12/H11271K32. Let us now turn to a discussion of the formula for the spin-wave frequency /H9275s/H208490/H20850. As is well known, the negative term under the radical in formula /H2084958/H20850represents a decreaseof the oscillation frequency on account of relaxation. The condition that /H9275s/H208490/H20850be real is, of course, the same as the existence condition for spin waves. If /H92532M02K1K3M02/H11271/H20849/H926111K3M02−/H926113K1/H208502, /H2084962/H20850 then spin waves will be well-defined quasiparticles. 3. The /H9021/H20849/H11028/H20850phase: /H9272=0, cos2/H9258=−K1/K2M02,M2=M02. Stability conditions: − K2M02/H11021K1/H110210;K3/H110220;K2/H110220. Let us give an assessment of the spin-wave existence condition as we did for the case of a uniaxial ferromagnet.The spin-wave frequencies in the absence of damping areexpressed by the formula /H9275S2=/H92532M02/H20849/H9251k2+K3M02sin2/H9258/H20850/H20849/H9251k2−2K1sin2/H9258/H20850. /H2084963/H20850 Starting from formula /H2084942/H20850, we obtain the spin-wave damp- ing/H9003s/H20849k/H20850in the approximation linear in /H926111,/H926113, and/H92612: /H9003s/H20849k/H20850=1 2/H20849/H926111+/H92612k2/H20850/H20849 64/H20850 /H11003/H9251k2/H9251k2cos2/H9258+/H20849/H9251k2+K3M02sin2/H9258/H20850/H20849/H9251k2−2K1sin2/H9258/H20850 /H9251k2−2K1sin2/H9258 +1 2/H20849/H926113+/H92612k2/H20850/H20849/H9251k2−2K1/H208502sin2/H9258 /H9251k2−2K1sin2/H9258. The existence condition for spin waves in the /H9021/H20849/H11028/H20850phase has the following form: /H9003s/H20849k/H20850 /H9275S——→ k→0/H926111K3M02sin2/H9258−/H9261132K1 2/H9253M0/H208812/H20841K1/H20841K3M02sin4/H9258/H112701. /H2084965/H20850 Let us also give the dispersion relation for spin waves in the ground states of a tetragonal ferromagnet when a relax-ation term of the Landau-Lifshitz form /H208492/H20850is used: 1. The/H9021/H20849 /H20648/H20850phase: /H9275=i/H9261/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852±/H9253M0/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852. /H2084966/H20850 The spin-wave existence condition is /H9275Im /H9275Re=/H9261 /H9253M0/H112701. 2. The/H9021/H20849/H11036/H20850phase:774 Low T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51/H9275=i 2/H9261/H208492/H9251k2+K1+K3M02/H20850±1 2/H208814/H92532M02/H20849/H9251k2+K3M02/H20850/H20849/H9251k2+K1/H20850−/H92612/H20849K1−K3M02/H208502. /H2084967/H20850 The spin-wave existence condition is /H9275Im /H9275Re——→ k→0/H9261/H20849K1+K3M02/H20850 /H208814/H92532M02K1K3M02−/H92612/H20849K1−K3M02/H208502/H112701. 3. The/H9021/H20849/H11028/H20850phase: /H9275=i 2/H9261/H208492/H9251k2−2K1sin2/H9258+K3M02sin2/H9258/H20850±1 2/H208814/H92532M02/H20849/H9251k2+K3M02sin2/H9258/H20850/H20849/H9251k2−2K1sin2/H9258/H20850−/H92612/H208492K1+K3M02/H20850sin4/H9258./H2084968/H20850 The spin-wave existence condition is /H9275Im /H9275Re——→ k→0/H9261/H20849K3M02−2K1/H20850 /H208814/H92532M022/H20841K1/H20841K3M02−/H92612/H208492K1+K3M02/H208502 /H112701. As above, we present an analysis of the results obtained in Tables II and III. We see that the use of the isotropic Landau-Lifshitz re- laxation term and the anisotropic relaxation term in the form/H2084923/H20850in the case of a tetragonal ferromagnet leads to results that differ only quantitatively. In this case no contradictionsarise if /H9261/H11270 /H9253M0. This is due to the fact that a tetragonal ferromagnet has no ground states with a continuous degen-eracy parameter.Spin-orientation phase transitions in “viscous” ferromagnets Here we discuss the possibility of spin-orientation phase transitions in ferromagnets with large relaxation parameters/H9261/H11271 /H9253M0. In this case spin waves in the /H9021/H20849/H20648/H20850phase are strongly damped waves /H20851see Eq. /H2084948/H20850/H20852. The quantum indeter- minacy in the energy of these waves, /H9004/H9275/H11015/H9261, becomes larger than the spin-wave activation frequency /H92750/H20849T/H20850=/H9253M0Ka/H20849T/H20850. For this reason it makes no sense to speak about a condition of the form /H92750/H20849T/H20850=0 for the phase transition /H9021/H20849/H20648/H20850↔/H9021/H20849/H11028/H20850. Similar arguments apply to the phase transition /H9021/H20849/H11036/H20850↔/H9021/H20849/H11028/H20850in viscous media, for which /H9261/H11271/H9253M0. The case of phase transitions in viscous media will be considered separately below.TABLE II. Ratio of the dissipative part of the activation part of the spin-wave frequency in the ground states of a tetragonal ferromagnet in the long-wavelength approximation /H20849k→0/H20850. TABLE III. Ratio of the dissipative part of the activation part of the spin-wave frequency at the boundaries of phase transitions in a tetragonal ferromagnet in the long-wavelength approximation /H20849k→0/H20850. Low T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 775 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51VII. DISCUSSION OF THE RESULTS. CONCLUSIONS It is customary to treat spin waves under conditions of small damping. This case is that of small relaxation constantsin a phenomenological treatment of spin waves: it is assumedthat the condition /H9261/H11270 /H9253M0is fulfilled. In this approximation, as we show in this paper, one can give a systematic descrip- tion of the high-frequency and relaxation properties of a fer-romagnet and also describe the spin-orientation transitions inthe framework of Landau theory. The case of systems with degenerate ground states re- quires a special treatment, as is shown in Sec. IV . The stan-dard description of relaxation processes, which originated inthe paper by Landau and Lifshitz, 1leads to the conclusion that homogeneous oscillations of the magnetization vanishafter the transition from the easy-axis to the easy-plane phasein a uniaxial ferromagnet, even under the condition /H9261 /H11270 /H9253M0. More precisely, after such a transition the spin waves become strongly damped in the easy-plane phase. The sameresult is obtained for the canted-phase state. In other words,we are faced with the following paradox. In this crystal, spinwaves exist in the easy-axis phase and vanish /H20849become re- laxation modes /H20850in the easy-plane and canted-phase states. This means that in individual cases it can be incorrect to usea relaxation term of the Landau-Lifshitz or Gilbert form fordescribing spin-wave damping. Contradictions arise whenone is considering ferromagnets with high symmetry /H20849ex- change approximation, uniaxial ferromagnet /H20850, which have a degenerate ground state. This is due to the circumstance thatrelaxation terms of the Landau-Lifshitz or Gilbert form areconstructed with less than full regard for the symmetry of thecrystal. The direct use of symmetry arguments in construct-ing the dissipative function does not resolve the paradox.Only by invoking the conservation laws for the total magne-tization components has it been possible to formulate therequirements on the dissipative function that resolve theparadox /H20849see Sec. III /H20850. Since the conservation laws for the total magnetization components derive from the symmetry of the ferromagnet inspin space and coordinate space, the unusual, nonstandardstructure of the dissipative function is due, in our opinion, toa systematic and complete account of symmetry arguments. The dissipative function in the form proposed in this paper, constructed with allowance for Landau’s argumentsabout the importance of symmetry, gives correct results andleads us to the conclusion that spin-wave damping does notimpose any restrictions on Landau theory for the descriptionof spin-orientation transitions. The use of the dissipative function proposed by the au- thors allows one to describe damping of the magnitude of themagnetic moment, an impossibility when relaxation terms ofthe Landau-Lifshitz or Gilbert form are used. Analysis of a ferromagnet with tetragonal symmetry is of great methodological interest. It has been shown for thisexample that the number of constants describing the relax-ation of the uniform magnetization components is equal totwo rather than one, as in Landau-Lifshitz-Gilbert theory. Inthe general case of a crystal with orthorhombic symmetry,the number of relaxation constants corresponding to homo-geneous oscillations is equal to three.Using the standard relationship among the magnetic an- isotropy constants, K 2M02,K3M02/H11270K1, and the idea that the relaxation constants /H9261depend on the magnetic anisotropy constants /H20849see Sec. VI /H20850, one can particularize Bogolyubov’s general concept of quasi-averages to the description of spin-wave spectra and damping in systems with a continuous pa-rameter describing a degenerate vacuum. The prescriptionfor calculating these quantities is as follows. Choose themagnetic anisotropy so as to completely lift the ground-statedegeneracy that is characteristic of the exchange interaction.Construct the dissipative function corresponding to the sym-metry of the spin Hamiltonian and the lattice of the material.Calculate the spin-wave spectra and damping for the possiblephases of the ferromagnet. In the formulas obtained, pass tothe model of a uniaxial crystal /H20849K 3→0/H20850at a fixed wave vector k. Here the relaxation constant /H926113goes to zero along with K3. The transition to the exchange model is made by letting K1→0. Then /H926111also goes to zero. The spin-wave spectra and damping obtained by this procedure satisfy theexistence condition for weakly damped spin waves in all thepossible phases of the ferromagnet, including those with adegenerate ground state. The condition of weak spin-wavedamping for all the phases is /H9261/H11270 /H9253M0. Let us see how a rotation in the basal plane is related to spin waves. For sim-plicity we consider the /H9021/H20849/H11036/H20850phase. Suppose that in the equilibrium state the equilibrium value of the magnetic mo- ment is directed along the xaxis. Rotation by a small angle /H92721about the symmetry axis zmeans that the new ground state has a nonzero component /H20855My/H20856/H11015/H92721/H20855/H20849a0++a0/H20850/H20856. The averaging in this formula is done over coherent spin-wave states with wave vector k=0, for which the num- ber of excited spin waves obeys a Poisson distribution.16The average number of spin waves in the Poisson distribution isproportional to /H20855M y/H20856or/H92721. Thus a uniform rotation by an angle/H92721in a degenerate-vacuum system, as in any spin sys- tem, is equivalent to the excitation of spin waves. Systemswith a degenerate vacuum differ from ordinary systems inthat for them, spin waves with k=0 have zero frequency, /H92750=0, and for that reason the rotation is static. In nondegen- erate systems a deviation of the magnetization from theground state causes damped spin waves and relaxation to-ward a fixed ground state. This is what happens in any realsystem, since real systems are nondegenerate. Our detailing of Bogolyubov’s principle of quasi- averages for the case of spin systems with a degenerateground state was undertaken on the advice of S. M. Ryab-chenko, who also offered some valuable observations. Theauthors thank him for that and his interest in this study. Wealso thank B. A. Ivanov for valuable discussions. APPENDIX 1 Dissipative function and damping of spin waves in the approximation of zero longitudinal susceptibility The approximation /H9273/H20648=0 was used in the original paper by Landau and Lifshitz.1In this approximation they pro- posed both a dynamic and a dissipative part of the equation.The approximation /H9273/H20648=0 means that only two of the three components of the magnetic moment are independent. As the776 Low T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51two independent variables they chose the polar /H9258and azi- muthal /H9272tilt angles of the magnetization vector M: M=/H20853M0sin/H9258cos/H9272;M0sin/H9258sin/H9272;M0sin/H9258sin/H9272;/H20854. The energy of the ferromagnet in this approximation takes the form W=1 2/H9251M02/H20849/H20849/H11612/H9258/H208502+ sin2/H9258/H20849/H11612/H9272/H208502/H20850+1 2K1M02cos2/H9258 +1 4K2M04cos4/H9258. /H20849A.1.1 /H20850 The equation of motion of the magnetic moment is /H11509/H9272 /H11509t=/H9254W /H9254/H20849cos/H9258/H20850+R/H9272=−f/H9258+R/H9272, /H11509/H20849cos/H9258/H20850 /H11509t=−/H9254W /H9254/H9272+R/H9258=f/H9272+R/H9258, /H20849A.1.2 /H20850 where R/H9272andR/H9258are the relaxation terms corresponding to the variables /H9272and cos /H9258. To determine the form of R/H9272andR/H9258, we find the varia- tion of the free energy with time: /H11509W /H11509t=/H20885/H20877/H9254W /H9254/H20849cos/H9258/H20850/H11509/H20849cos/H9258/H20850 /H11509t+/H9254W /H9254/H9272/H11509/H9272 /H11509t/H20878dV. /H20849A.1.3 /H20850 Substituting the values of the time derivatives, we get /H11509W /H11509t=−/H20885/H20853R/H9258f/H9258+R/H9272f/H9272/H20854dV. /H20849A.1.4 /H20850 It follows from this formula that the relaxation terms can be chosen in the form R/H9258=/H9261/H9258/H9258f/H9258+/H9261/H9258/H9272f/H9272, R/H9272=/H9261/H9272/H9258f/H9258+/H9261/H9272/H9272f/H9272. /H20849A.1.5 /H20850 For simplicity we are neglecting terms with derivatives of the “generalized” forces f/H9258,f/H9272with respect to the coordi- nates. The dissipative function corresponding to these relax- ation terms is: Q=1 2/H20885/H20849/H9261/H9258/H9258f/H92582+/H9261/H9272/H9272f/H92722+2/H9261/H9258/H9272f/H9258f/H9272/H20850dV. /H20849A.1.6 /H20850 It is easy to see from formulas /H20849A.1.5 /H20850and /H20849A.1.6 /H20850that R/H9258=/H9254Q//H9254f/H9258,R/H9272=/H9254Q//H9254f/H9272. /H20849A.1.7 /H20850 Knowing the dissipative function, the free energy, and the equation of motion, we easily calculate the spin-wavespectrum and damping. The free energy /H20849A.1.1 /H20850corresponds to three ground states: /H9258=0 /H20849/H9021/H20849/H20648/H20850/H20850,/H9258=/H9266 2/H20849/H9021/H20849/H11036/H20850/H20850, cos2/H9258=−K1 M02K2/H20849/H9021/H20849/H11028/H20850/H20850. The value of the magnetic moment in the /H9273/H20648=0 approxima- tion remains unchanged: M2=M02.Let us consider the spectrum and damping of spin waves in the /H9021/H20849/H11028/H20850phase. In the linear approximation /H20849cos/H9258 =cos/H92580+cos/H92581,/H9272=/H92721/H20850the equation of motion /H20849A.1.2 /H20850takes the form /H11509/H9272 /H11509t=−/H9261/H9272/H9251K1+K2M02 K2M02/H116122/H9272−/H208752K1M02cos/H92581 −/H9251K2M04 K1+K2M02/H116122/H20849cos/H92581/H20850/H20876, /H11509/H20849cos/H92581/H20850 /H11509t=−/H9251K1+K2M02 K2M02/H116122/H20849cos/H92581/H20850. /H20849A.1.8 /H20850 Going over to the Fourier components with respect to time and the coordinates, we find the dispersion relation forthe system. The solution of this equation has the form /H9275=−i/H9261/H9272K1+K2M02 2K2/H9251k2±/H20881/H9251k2/H20875/H9251k2+2/H20849K1+K2M02/H20850K1 K2/H20876. /H20849A.1.9 /H20850 It is seen from formula /H20849A.1.9 /H20850that the imaginary part of the spin-wave frequency is much smaller than the real part. At the point of the phase transition /H9021/H20849/H11028/H20850→/H9021/H20849/H20648/H20850the condition K1+K2M02=0 holds and damping is absent, and one needs to take the next order of approximation into ac-count in constructing the dissipative function; e.g., the relax-ation terms depending on the derivatives of the forces f /H9258,f/H9272 with respect to the coordinates. At the phase transition /H9021/H20849/H11028/H20850→/H9021/H20849/H11036/H20850the condition K1 =0 holds, and in this case the imaginary part of the spin- wave frequency is much smaller than the real part. In writing the dissipative function we have not put any restrictions on the symmetry of the crystal. In principle, for-mula /H20849A.1.6 /H20850is applicable to crystals of low symmetry. For crystals of high symmetry /H20849isotropic, isotropic-exchange ap- proximation, uniaxial ferromagnet approximation /H20850restric- tions on the form of the dissipative function arise on accountof the magnetization conservation law. For example, in the case of a uniaxial crystal the com- ponent M z/H20849cos/H9258/H20850is conserved, and the equation for this component should have the form /H11509/H20849cos/H9258/H20850 /H11509t+ div P=0 , /H20849A.1.10 /H20850 where Piis the ith component of the flux of Mz. It is easy to check that the equation for cos /H9258with the relaxation terms taken into account assumes the form of a conservation lawonly in the case when /H9261 /H9258=/H9261/H9258/H9272=0. This is the case considered in our calculation of the spin-wave spectrum in the /H9021/H20849/H11028/H20850 phase. We have written this appendix to show how things relate to the mainstream papers on the calculation of the spectraand damping of spin waves. We note that the calculation ofthe spin-wave spectra and damping in the /H9273/H20648=0 approxima- tion is much simpler than when /H9273/H20648is taken into account. In the/H9273/H20648=0 approximation the spin-wave dissipative frequency, corresponding to relaxation of the magnetization magnitude,is lost.Low T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 777 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51APPENDIX 2 Dissipative function in the Gilbert form Here we give an expression for the Gilbert dissipative function QG. This dissipative function can be obtained /H20849as we have done /H20850by considering the variation in time of the energy of a ferromagnet /H11509W /H11509t=/H20885/H11509W /H11509M/H11509M /H11509tdV=−/H20885H/H11509M /H11509tdV. /H20849A.2.1 /H20850 Substituting for /H11509M//H11509tits value from the Landau-Lifshitz equation with a relaxation term of the Gilbert form, /H11509M /H11509t=−/H9253/H20851M,H/H20852+a M0/H20875M,/H11509M /H11509t/H20876, /H20849A.2.2 /H20850 we obtain /H11509W /H11509t=a M0/H20885/H11509M /H11509t/H20851M,H/H20852dV, and again using Eq. /H2084956/H20850, we obtain /H11509W /H11509t=a /H9253M0/H20885/H20873/H11509M /H11509t/H208742 dV=QG. /H20849A.2.3 /H20850 This dissipative function was found by Gilbert in his disser- tation. In the variables /H9258and/H9272the Gilbert dissipative function has the form QG=a /H9253M0/H20885/H20877/H20873/H11509/H9258 /H11509t/H208742 + sin2/H9258/H20873/H11509/H9272 /H11509t/H208742/H20878dV. /H20849A.2.4 /H20850 An advantage of this dissipative function is that it is expressed in terms of the time derivatives, unlike the dissi-pative function /H2084915/H20850. A shortcoming is that it does not take into account the restrictions related to the symmetry of thecrystal. a/H20850E-mail: bar@imag.kiev.ua 1L. D. Landau and E. M. Lifshits, Phys. Z. Sowjetunion 8,1 5 3 /H208491935 /H20850. 2T. L. Gilbert, Phys. Rev. 100, 1243 /H208491955 /H20850. 3V . G. Bar’yakhtar, Fiz. Tverd. Tela /H20849Leningrad /H2085029, 1317 /H208491987 /H20850/H20851Sov. Phys. Solid State 29, 754 /H208491987 /H20850/H20852. 4V . Kamberskii, Czech. J. Phys., Sect. B 22,5 7 2 /H208491972 /H20850. 5E. M. Lifshits, J. Phys. /H20849USSR /H208508, 337 /H208491944 /H20850. 6C. Kittel, Phys. Rev. 70, 965 /H208491946 /H20850. 7C. Kittel, Rev. Mod. Phys. 21, 541 /H208491949 /H20850. 8L. D. Landau and E. M. Lifshitz, Statistical Physics , Parts 1 and 2, 3rd ed., Pergamon Press, Oxford /H208491980 /H20850, Nauka, Moscow /H208491982 /H20850. 9L. T. Tsymbasl, Ya. B. Bazaliy, G. N. Kakazei, and P. E. Wigen, Ukr. Phys. J. 50, 883 /H208492005 /H20850. 10V . L. Safonov, J. Appl. Phys. 91, 8653 /H208492002 /H20850. 11L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media , 2nd ed., rev. and enl., by E. M. Lifshitz and L. P. Pitaevskii, PergamonPress, Oxford /H208491984 /H20850, Nauka, Moscow /H208491982 /H20850. 12A. I. Akhiezer, V . G. Bar’yakhtar, and S. V . Peletminskii, Spin Waves , North-Holland, Amsterdam /H208491968 /H20850, Nauka, Moscow /H208491967 /H20850. 13E. A. Turov, Physical Properties of Magnetically Ordered Crystals , Aca- demic Press, New York /H208491965 /H20850, Izd-vo AN SSSR, Moscow /H208491963 /H20850. 14A. G. Danilevich, Ukr. Fiz. Zh. /H20849Russ. Ed. /H2085051,6 6 8 /H208492006 /H20850. 15V . G. Bar’yakhtar, V . N. Krivoruchko, and D. A. Yablonski /c142,Green’ s Functions in the Theory of Magnetism /H20851in Russian /H20852, Naukova Dumka, Kiev /H208491984 /H20850. 16L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory , 3rd ed., Pergamon Press, Oxford /H208491977 /H20850, cited Russian edition: Fizmatlit, Moscow /H208492004 /H20850, p. 100, Problem 3. Translated by Steve Torstveit778 Low T emp. Phys. 32/H208498-9/H20850, August-September 2006 V. G. Baryakhtar and A. G. Danielevich 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: 134.99.128.41 On: Mon, 06 Jan 2014 18:25:51

1.4868250.pdf

Coupling of spinwave modes in wire structures H. G. Bauer, J.-Y. Chauleau, G. Woltersdorf, and C. H. Back Citation: Applied Physics Letters 104, 102404 (2014); doi: 10.1063/1.4868250 View online: http://dx.doi.org/10.1063/1.4868250 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/104/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Width dependent transition of quantized spin-wave modes in Ni80Fe20 square nanorings J. Appl. Phys. 116, 163912 (2014); 10.1063/1.4900616 Effect of the spin-twist structure on the spin-wave dynamics in Fe55Pt45/Ni80Fe20 exchange coupled bi-layers with varying Ni80Fe20 thickness J. Appl. Phys. 115, 17D105 (2014); 10.1063/1.4855015 Observation of propagating edge spin waves modes J. Appl. Phys. 114, 213905 (2013); 10.1063/1.4839315 Nonuniform apparent relaxation from dephasing of magnetostatic wave modes in a confined microdisk J. Appl. Phys. 106, 113904 (2009); 10.1063/1.3262613 Normal mode splitting in interacting arrays of cylindrical permalloy dots J. Appl. Phys. 99, 08C701 (2006); 10.1063/1.2150806 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: 129.49.170.188 On: Fri, 19 Dec 2014 13:00:19Coupling of spinwave modes in wire structures H. G. Bauer, J.-Y . Chauleau, G. Woltersdorf, and C. H. Back Department of Physics, University of Regensburg, 93040 Regensburg, Germany (Received 31 January 2014; accepted 27 February 2014; published online 11 March 2014) We investigate the magnon dispersion of spinwaves propagating in a Permalloy stripe and compare it to a plain film. We use a Scanning Kerr Microscope to directly image their spatial propagation- and decay properties as well as micromagnetic simulations to extract additional information about the static magnetization distribution. For low external fields, we find anti-crossings for the dispersionbranches corresponding to different transverse wave vectors that also occur as minima of the decay length for the spinwave. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4868250 ] Spinwave excitation in thin magnetic wires is used in var- ious experiments to study, e.g., spinwave modes1,2or spin- wave decay lengths.3In thin magnetic wires, quantized modes have been observed4and used to demonstrate focusing of spin waves by superposition of modes.5Similar to wires, spinwave propagation has been studied in great detail in periodically nanostructured ferromagnets often dubbed magnonic crystals.6–8Spinwaves can be bent around corners in metallic wires with the help of an electrical current9and thus might even serve as interconnects in wires of various shapes. Most recently, in magnetic wires, the interaction between propagat-ing spinwaves and spin polarized currents was found to lead to a spinwave Doppler shift 10–14detected mostly in propagat- ing spinwave spectroscopy experiments.10–13In these experi- ments, cw-excited spinwaves are measured non-locally,10,15 and the decay length of spinwaves is the crucial parameter for the signal amplitude. To understand and control this parameteris highly desirable. It is thus interesting to study the details of spinwave propagation and decay in nanostrutured magnetic wires. In contrast to the physics of waves in isotropic systemssuch as electromagnetic waves in vacuum or surface waves in water, spinwaves in magnetic films have highly anisotropic properties at low wave-vectors. This anisotropy is caused bythe long ranged dipolar interaction which leads to a situation where spinwave frequency and group velocity not only depend on the wave-number but also on the direction of wavepropagation. In this Letter, we will first discuss the spinwave proper- ties in a thin magnetic film. Based on this, the discussion isextended to spinwaves propagating in quasi one-dimensional structures, i.e., magnetic stripes and compared to our time resolved magneto-optic Kerr (TR-MOKE) experiments inwhich we determine the dispersion and decay characteristics for propagating spinwaves in Ni 80Fe20wire structures. Micromagnetic simulations are then used to clarify theappearance of anti-crossings and to identify the shortcomings of the effective stripe model. In general, the dynamic behavior of a ferromagnetic sys- tem in the micromagnetic limit can be well described by the Landau-Lifshitz-Gilbert equation. 16–18Due to the damping term in this equation, the spinwave excitations have a char-acteristic decay time typically in the nanosecond range for Ni 80Fe20. The magnetic linear response of a system to an external time-periodic stimulus is characterized by thecomplex magnetic susceptibility vðx;~kÞ. The properties of the excited spinwaves can be calculated very generally fromthe susceptibility of the magnetic medium. The dynamic magnetization can be expressed as an integral in k-space 19–21 mx¼ðþ1 /C01vðx;kÞhðkÞeikxdk; (1) where the harmonic time dependence has been dropped and h(k) is the Fourier spectrum of the excitation field. In gen- eral, the complex tensor v(x,k) has poles k0in the complex k-plane for a fixed frequency x. Due to these pole(s) which represent the modes of the linear dynamic system, thedynamic magnetization in Eq. (1)can be rewritten as m¼m poleþmimag/C25mpole/C25hðk0Þ2piResjk0ðvÞeik0x:(2) This means that the spatial decay of the spinwave is determined by Im( k0), and the spectral width of the excita- tion h(k) is not important as h(k0) enters only as a factor in the amplitude of the spinwave. Therefore, the wavelength k and the decay length are independent of the excitation spec-trum. In the special experimental case considered here, it is even advantageous to use a simple coplanar waveguide (CPW) structure instead of a more complicated periodicantenna structure 10as a broad excitation spectrum allows to measure the dispersion over a wider wave vector range (/C251/C08lm/C01). For the simple case of a thin film in Damon Eshbach (DE, ~k?~M), geometry v(k) is known and the dispersion can be calculated.20In particular, the damping length ldamp is given by ldamp¼1 Imðk0Þ¼dM2 sl0c 2axðMsþ2H0Þ¼vgs; (3) with the group velocity vg¼dðcl0MsÞ2 4xand the decay time s¼2 al0cðMsþ2H0Þ. For a plain film, vgvaries smoothly as 1/ x with frequency and therefore Im ðk0Þ/xfor DE spinwaves in the film. The dispersion for spinwaves propagating in stripes is usually discussed in terms of a simple model comprising the dispersion for a plain film and a quantized wavevector com-ponent in the direction perpendicular to the propagation 0003-6951/2014/104(10)/102404/4/$30.00 VC2014 AIP Publishing LLC 104, 102404-1APPLIED PHYSICS LETTERS 104, 102404 (2014) 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: 129.49.170.188 On: Fri, 19 Dec 2014 13:00:19direction [Fig. 1(a)]. The transverse wave vector thus becomes quantized as ky¼np w, where w is the width of the stripe. A spinwave with kx¼0 corresponds to a spinwave in a plain film in Backward-Volume geometry ( ~kjj~H) [Fig. 1(a)]. For small wavenumbers ky, the resonance frequency first decreases and then increases again for larger wave numbers. Thus, for the dispersion f(kx) of a spinwave in a stripe, it depends on the transverse wavenumber if it starts below orabove the dispersion of the n¼1 spinwave mode. Therefore, anti-crossings of the lowest transverse modes with the n¼1 mode can be observed for narrow and thin stripes, whereasfor wider and thicker stripes only co-propagation of different transverse spinwave modes is seen instead. 22 In the plain film, the spinwaves are eigenmodes and do not interact with each other. Therefore, the dispersion branches cross each other without repulsion [Fig. 1(b)]. In reality, however, the eigenmodes of the plain film are no longer eigenmodes for the stripe. This is a consequence of the mode profiles as the contribution of the dipolar inter- action is modified due to the finite width of the stripe.Therefore, spinwaves from different branches do interact with each other, leading to anti-crossings in the dispersion. Usually, the interaction is strongest when the frequencies oftwo branches (nearly) coincide and a repulsion of the two branches can be observed (anti-crossing). Due to the differ- ent symmetries of even and odd transverse modes, onlymodes with odd ncan be excited by the CPW structure (shown in Fig. 1) and they only interact with other odd modes. Therefore, only interactions between two nearlydegenerate modes of the same symmetry lead to repulsion of the two branches. From this, we expect that the group veloc- ity decreases for the main propagating mode (the one withthe highest group velocity) close to anti-crossings. The decrease in the group velocity is then directly measurable as a decreasing decay length (i.e., one expects a peak in Im( k)) of the propagating spinwave. In order to investigate spinwave dispersion, we measure the magnetic excitation in a magnetic wire using TR-MOKEmicroscopy. The sample is excited by rf-fields generated in the vincinity of a terminated CPW with a distance of 2 lm between the ground lines and 600 nm width of the signal lineas shown in Fig. 2. A fixed microwave frequency fis used to excite the magnetization in the wire. The magnetic excitation profile is recorded by spatially scanning the sample underthe objective lens as shown in Fig. 2. From these data, the spatial decay is extracted and fitted to an exponentiallydecaying wave mðxÞ¼Ae ikx;k2C. By performing this pro- cedure as a function of microwave frequency for a fixed magnetic field H applied perpendicular to the long axis ofthe wire, the complex spinwave dispersion can be measured directly as shown in Fig. 3(a). We observe DE-like dispersion relations with a group velocity of about 2.9 km/s and decay lengths of up to 4.5 lm which translates to amplitude decay times around 1.5 ns and a Gilbert damping parameter aof 0.008 which is typical for Ni 80Fe20thin films. Clearly visible are discontinuities in the FIG. 1. (a) In the effective stripe model, the susceptibility of a thin plain film is used and only quantized transverse wave numbers ky¼np w;n ¼1;2;… are allowed. (b) The branch with the highest group velocity (n¼1) crosses other branches. These crossings become anti-crossings in the full micromagnetic model. FIG. 2. Spatial map of the out-of-plane component of the dynamic magnet- ization recorded by TR-MOKE at a bias magnetic field of 31 mT applied perpendicular to the stripe (DE geometry). A rf-current oscillating at5.44 GHz in the shorted coplanar wave guide produces an Amperian rf- magnetic field that excites the spinwave. FIG. 3. (a) Experimental complex spinwave dispersion for propagating spin- waves in a 15 nm thick and 800 nm wide Ni 80Fe20stripe. The propagation direction of the spinwaves is perpendicular to the applied bias magnetic field of 21 mT (upper panel), 31 mT and 41 mT (lower panel) along the stripeaxis. The dissipative part of the dispersion relations is shown on the right side. (b) Corresponding simulated dispersion.102404-2 Bauer et al. Appl. Phys. Lett. 104, 102404 (2014) 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: 129.49.170.188 On: Fri, 19 Dec 2014 13:00:19spinwave dispersion in Fig. 3(a)at excitation frequencies of 4.72 GHz (for l0H¼21 mT) and 5.2, 6.0, and 6.2 GHz (l0H¼31 mT). For the measurements at l0H¼41 mT, only small deviations from a smooth dispersion with a monotonicbehavior of the group velocity v gare observed. At the fre- quencies of the discontinuities, we also observe maxima in the imaginary part of the wave-number. Even forl 0H¼41 mT, the corresponding maxima in Im( k) can be clearly identified. These maxima correspond to reduced val- ues of the damping length ldamp, i.e., spinwaves are more quickly attenuated. Due to the finite resolution of TR-MOKE microscopy (/C25250 nm), it is not possible to resolve transverse mode pro- files of higher order modes with sufficient quality. In the fol- lowing, we therefore use micromagnetic simulations23to obtain further insight into the spinwave propagation in the wire. We simulated a magnetic volume of 25.6 lm /C2840 nm /C213.5 nm, divided into 4096 /C2128 cells with peri- odic boundary conditions along the long dimension and veri- fied that the stripe is sufficiently long in order to avoidunphysical effects due to the finite length of the stripe. l 0Ms¼0.95 T and a¼0.008 were used to model typical Ni80Fe20material properties. The magnetization was relaxed into an equilibrium state [Fig. 4(a)] prior to the excitation through a CPW like structure that was applied for 100 excita- tion periods. Although there are two possible equilibriumstates (“C-shaped” and “S-shaped”), only the “S-shaped” magnetization configuration was realized in the experiment as even very small external field components parallel to thestripe axis favor the “S-shaped” configuration of the magnet- ization across the stripe width. This expectation was con- firmed by additional simulations. The simulated dispersion [Fig. 3(b)] is over all in good agreement with the measurements [Fig. 3(a)], i.e., group velocities and damping lengths match very well and the anti-crossings are also reproduced for all measured fields. Minor differences in the frequencies where the anti-crossings appear might be caused by imperfect knowledge of the sam-ple parameters or small misalignment of the external field. We verified the latter through simulations and found that especially the peak height of the anti-crossing in Im( k) depends strongly on the in-plane misalignment angle from the transverse direction and we can therefore conclude that it was better than 1 /C14.The spatial amplitude distribution of the out-of-plane magnetization component across the stripe width is shown in Fig. 4for two frequencies that are separated by an anti- crossing. For a given applied field, the spectral weight ofhigher transverse wave numbers increases with the excitation frequency and the mode profile changes significantly from one side of an anti-crossing to the other. This shows that theobserved phenomena are indeed associated with the anti- crossing of two spinwave dispersion branches characterized by different transverse wave numbers rather than a single mode dispersion. The anti-crossings are due to the finite width of the stripe and are not found for a plain film (as we are dealing with thin films higher perpendicular quantized modes have much higher eigen frequencies). Therefore, the questions arise whether thisis mostly due to the different static or due to dynamic fields. Thus, we separate dynamic and static effects of the finite stripe width with the aid of additional simulations. We achieve this by artificially compensating the static magnetic surface charges at the edges of the stripe thus leav- ing the equilibrium magnetization uniform in the direction ofthe applied external field. The static effective field is also uniform along y, i.e., the edge channels with reduced effec- tive fields are removed and the edge modes which normallypropagate in these regions of reduced effective fields disap- pear. Although the magneto statics are equal to a plain film, the dynamic fields induced by the magnetization precessionare different and thus spinwave modes in the stripe differ from those of the plain film especially at the edges. Simulations of the spinwave dispersion in these artificiallyaltered stripes reveal anti-crossings with slightly reduced coupling strength. Therefore, we conclude that the main con- tribution to the anti-crossings is due to the dynamic demag-netizing fields differing from those of a plain film and changes in the equilibrium magnetization configuration are of minor importance. As the dipolar fields scale with the thickness of the sample, we also measured a thinner sample (10 nm at 31 mT, w /C251lm) and found only very weak (almost no) signs of the anti-crossings, as expected. This confi rms that the anti-crossings are formed due to dipolar rather than exchange interactions. To show the importance of taking the spatial decay part of propagating spinwaves properly into account, we doubled the damping parameter ain additional simulations and found that Im( k) also doubles as expected except near the anti- FIG. 4. Cross sections of simulated ground states are shown, where bright color represents magnetization aligned with the external field, i.e., perpe ndicular to the stripe axis. The width of the edge channels (dark color) decreases with the external field leading to less pronounced anti-crossings. The transver se mode profiles of the dynamic Mzcomponent (upper row) show that the characteristic transverse mode number increases from one side of the anti-crossings to the other.102404-3 Bauer et al. Appl. Phys. Lett. 104, 102404 (2014) 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: 129.49.170.188 On: Fri, 19 Dec 2014 13:00:19crossings where mode coupling dominates the relaxation. Therefore, the interaction between two spinwave branches at the anti-crossing must have decreased due to the change in a. This is also seen in the real part of the dispersion relation,which displays less pronounced discontinuities (not shown here). In conclusion, we investigated propagating spinwaves in magnetic thin films and wires and found anti-crossings in the dispersion relation of spinwaves when a weak external mag- netic field was oriented perpendicular to the stripe axis. Furthermore, we demonstrated that the signature of anti- crossings in stripes crucially depends on the external field aswell as on its thickness. Micromagnetic simulations allowed us to study the influence of the magnetic ground state and the damping parameter on the anti-crossings. As the propa-gation properties of spinwaves change quite drastically in the vincinity of anti-crossings, small changes in field or fre- quency, e.g., by spin transfer torques significantly alter thespinwave propagation properties. Therefore, the accurate knowledge of real and imaginary parts of the spinwave dis- persion is inevitable for precise quantitative measurementson propagating spinwaves and possibly for their future use in applications. This work was supported by the German Research Foundation (DFG) through programs WO1144 and by the European Research Council through starting grant ECOMAGICS. J.-Y.C. gratefully acknowledges financialsupport from the Alexander von Humboldt Foundation. 1J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. Lett. 89, 277201 (2002).2J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K. Y. Guslienko, A. N. Slavin, D. V. Berkov, and N. L. Gorn, Phys. Rev. Lett. 88, 047204 (2002). 3D. R. Birt, K. An, M. Tsoi, S. Tamaru, D. Ricketts, K. L. Wong, P. Khalili Amiri, K. L. Wang, and X. Li, Appl. Phys. Lett. 101, 252409 (2012). 4C. Mathieu, J. Jorzick, A. Frank, S. O. Demokritov, A. N. Slavin, B. Hillebrands, B. Bartenlian, C. Chappert, D. Decanini, F. Rousseaux, and E. Cambril, Phys. Rev. Lett. 81, 3968 (1998). 5V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko, and G. Reiss, Phys. Rev. B 77, 064406 (2008). 6S. Nikitov, P. Tailhades, and C. Tsai, J. Magn. Magn. Mater. 236, 320 (2001). 7S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009). 8V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010). 9K. Vogt, H. Schultheiss, S. Jain, J. E. Pearson, A. Hoffmann, S. D. Bader, and B. Hillebrands, Appl. Phys. Lett. 101, 042410 (2012). 10V. Vlaminck and M. Bailleul, Science 322, 410 (2008). 11V. Vlaminck and M. Bailleul, Phys. Rev. B 81, 014425 (2010). 12M. Zhu, C. L. Dennis, and R. D. McMichael, Phys. Rev. B 81, 140407 (2010). 13K. Sekiguchi, K. Yamada, S.-M. Seo, K.-J. Lee, D. Chiba, K. Kobayashi, and T. Ono, Phys. Rev. Lett. 108, 017203 (2012). 14J.-Y. Chauleau, H. G. Bauer, H. S. K €orner, J. Stigloher, M. H €artinger, G. Woltersdorf, and C. H. Back, Phys. Rev. B 89, 020403 (2014). 15S. Neusser, G. Duerr, H. G. Bauer, S. Tacchi, M. Madami, G. Woltersdorf, G. Gubbiotti, C. H. Back, and D. Grundler, Phys. Rev. Lett. 105, 067208 (2010). 16L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935). 17T. L. Gilbert, Phys. Rev. 100, 1243 (1955). 18T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 19K. J. Harte, J. Appl. Phys. 39, 1503 (1968). 20B. A. Kalinikos and A. N. Slavin, J. Phys. C: Solid State Phys. 19, 7013 (1986). 21R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F. Egelhoff, J. Appl. Phys. 83, 7037 (1998). 22V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko, and G. Reiss, Appl. Phys. Lett. 92, 232503 (2008). 23A. Vansteenkiste and B. V. de Wiele, J. Magn. Magn. Mater. 323, 2585 (2011).102404-4 Bauer et al. Appl. Phys. Lett. 104, 102404 (2014) 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: 129.49.170.188 On: Fri, 19 Dec 2014 13:00:19

1.370218.pdf

Switching dependence on fabrication accuracy of tapered ends of a single giant magnetoresistance memory cell in word disturb condition M. Redjdal , P. W. Gross , A. Kazmi , and F. B. Humphrey Citation: Journal of Applied Physics 85, 6193 (1999); View online: https://doi.org/10.1063/1.370218 View Table of Contents: http://aip.scitation.org/toc/jap/85/8 Published by the American Institute of PhysicsSwitching dependence on fabrication accuracy of tapered ends of a single giant magnetoresistance memory cell in word disturb condition M. Redjdal, P. W. Gross, A. Kazmi, and F. B. Humphrey Electronic and Computer Engineering Department, Boston University, 8 Saint Mary’s Street, Boston, Massachusetts 02215 Direct integration of the Landau–Lifshitz–Gilbert equation in a three-dimensional rectangular lattice shows that fabrication asymmetry of tapered ends affects the magnetic switching behavior ofthe magnetic layers of a pseudospin valve ~PSV!memory cell in word disturb condition. When a 10 nmasymmetryisintroducedwiththelongersidesofthetapersonthesamesideofthePSVmemorycell, a ‘‘360°’’ wall in the storage layer is formed when a 40 Oe uniform word disturb field isapplied opposite to the magnetization direction in the storage layer. The initial magnetization stateis recovered when the field is turned off. When the longer sides of the tapers are on opposite sidesof the cell, complete magnetization reversal occurs in the storage layer. The sense layer and thestorage layer relax to an antiparallel configuration magnetization that is opposite to the initialconfiguration in the memory cell when the field is turned off. The information is lost. Successfuloperation of the memory depends upon a fabrication accuracy that is better than 10 nm. © 1999 American Institute of Physics. @S0021-8979 ~99!38908-8 # I. INTRODUCTION The pseudospin valve ~PSV!memory is being proposed as a potential giant magnetoresistance random accessmemory ~GMRAM !. 1–3The cell design is based on unpinned sandwich materials with magnetic layers of different thick-ness. The memory is configured as a coincident–currentmemory array. Each memory cell is addressed by the appli-cation of two currents in coincidence that generate two crossfields, word and sense fields, to switch the selected magneticbit. These same currents disturb all of the other cells on thesame row or column. 4Bit and word disturb stability are criti- cal in the coincident current mode operation. One design1 uses a large ‘‘plus–minus’’ word field for the writing of ‘‘0’’or a ‘‘1.’’ In this design, the word disturb field is particularlyimportant since it is rather large ~40 Oe !uniform field. In the present work, the dependence of the magnetic switching behavior of the magnetic layers of a PSV memorycell with tapered ends in word disturb condition, i.e., when auniform field is applied to the nonselected memory cell, isinvestigated as a function of fabrication asymmetry in thetapered ends. II. SIMULATION TECHNIQUE Simulations5were performed by solving the Landau– Lifshitz–Gilbert ~LLG!equation numerically in a Cartesian lattice with 50 3250324 nodes, or 300000 points. At each grid point, the simulation uses the phenomenological mag-netic parameters of exchange, anisotropy, demagnetization,saturation magnetization, and applied external field. Freeboundary conditions are used. A three-dimensional fast Fou-rier transform ~FFT!is combined with zero padding to evalu- ate the demagnetizing field in the isolated sample. The simu-lations were performed on a the Silicon Graphics/CrayResearch Origin 2000 Supercomputer System. A PSV memory cell with tapered ends is considered in this investigation. 1The thickness differential of the magneticthin films is implemented by making the storage layer ~or hard layer !10 nodes thick ~5n m!and the sense layer, or soft layer, 7 nodes thick ~3.5 nm !in the direction normal to the plane of the memory cell. The copper layer is represented bya 7 node thick ~3.5 nm !void between the two magnetic lay- ers. The total simulation volume is 0.5 mm wide 32.5mm long312 nm thick using a 10 nm 310 nm 30.5 nm unit cell. The asymmetry is introduced in our simulations by makingone side of the taper 10 nm longer than the other side. The material parameters are exchange constant, A53.0 10 26erg/cm; anisotropy field, Hk520Oe in the transverse direction; saturation magnetization, 4 pMs5800G; Gilbert damping constant, a50.5; gyromagnetic ratio, g520.0178 Oe21ns21. The damping constant, a, is purposely chosen large to speedup the calculation of dynamic equilibriumstates. III. RESULTS The effects of taper fabrication asymmetry on the switching behavior of a PSV memory cell is investigated fortwo cases, defined by the location of the asymmetry. The FIG. 1. Initial equilibrium magnetization distribution showing the magneti- zation direction in the sense and the storage layers in the memory cell. The10 nm asymmetry is in the lower left ~ll!and lower right ~lr!corners of the tapers in both magnetic layers. Contours of the transverse magnetization component, m t, are shown in the tapered ends.JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 8 15 APRIL 1999 6193 0021-8979/99/85(8)/6193/3/$15.00 © 1999 American Institute of Physicsideal case where end shapes are perfectly symmetrical is not considered since some fabrication asymmetry will always bepresent. A. Same side asymmetry Figure 1 shows the sense and storage layers of the PSV memory cell magnetized in an antiparallel equilibrium con-figuration. Since the magnetization change was found to besmall through the layer thickness, only the magnetization inthe center of each layer is shown. A 10 nm fabrication asym-metry is in the lower left ~ll!and lower right ~lr!corners of the tapers in both layers. Contours of the transverse magne-tization component, m t, are also shown in the tapered re- gions. The contour nearest to the edge is for 60.10Msand the inner one for 60.05Ms. It can be seen that the edge regions next to the longer sides of the tapers, indicated by ~ll! and~lr!, have a larger tranverse magnetization component. The magnetization change depends on the torques aris- ing from coupling the applied field with the magnetization.In the center region of the sense or storage layers, the mag-netization is either directly parallel or antiparallel to the ap-plied field and the corresponding torque is equal to zero.However, in the tapered regions, the magnetization tends tolie parallel to the lateral sides to reduce stray fields. Conse-quently, a transverse magnetization component ( m t) is gen- erated in these end regions. For this PSV memory cell, theword write field is 640 Oe. 1Figure 2 shows the transverse magnetization contour ( mt50.09Ms) for the first 15% of the reversal time when a 40 Oe external field, Hext, is applied opposite to the magnetization direction in the storage layer.This operation is referred to as a word disturb condition. Themagnetization reversal takes place via local magnetizationrotation, originating in the tapered regions and progressingtowards the body of the memory cell. Figure 3 shows the magnetization distribution in both thesense and the storage layers after the external field of 240 Oe has been applied long enough for the memory cell tocome to equilibrium. The magnetization has undergone alarge rotation in the storage layer. A stable ‘‘360°’’ wall isformed in the center region of the storage layer. 2In contrast, the body of the sense layer although magnetostaticallycoupled to the storage layer, remains, on average, magne-tized parallel to the applied field. Ferromagnetic exchangeinteraction and the length of the memory cell limit the widthof the wall along the longitudinal direction of the memorycell. No further magnetization change is observed with the240 Oe field. The configuration in Fig. 3 is considered a blocked state. Figure 4 shows the magnetization distribution in both the sense and storage layers after the external field is turned off.Both layers remain magnetostatically coupled as evidencedby the antiparallel configuration of corresponding dipoles inboth the sense and storage layers. The 360° wall is slowlyunwinding towards the initial magnetization distributionshown previously in Fig. 1. The information in the memorycell is recovered from the effect of a uniform write disturbfield applied opposite to the magnetization in the storagelayer. B. Opposite side asymmetry Figure 5 shows the zero-field antiparallel equilibrium magnetization distribution in the sense and storage layerswhen the asymmetry is in the lower left and upper rightcorners of the tapers in both layers. Contours of the trans-verse magnetization, m t560.05Msand60.10Ms, are also shown in the tapered regions as was done previously. Now FIG. 3. Equilibrium magnetization distribution in the sense and the storage layers when a 40 Oe external field has been applied opposite to the originalmagnetization in the storage layer. A 360° wall is formed in the storagelayer. FIG. 4. Magnetization distribution slowly relaxing towards an equilibriumantiparallel configuration in the sense and the storage layers after the exter-nal field is turned off. FIG. 2. Equal mtcontour of the transverse magnetization component ( mt 520.09Ms) in the storage layer are shown at zero applied field ~a!and with increasing equal intervals of time during the first 15% of the reversal~b!~c!~d!~e!when a 240 Oe field is applied. FIG. 5. Initial equilibrium magnetization distribution showing the magneti- zation direction in the sense and the storage layers in the memory cell. The10 nm asymmetry is located in the lower left ~ll!and upper right ~ur!corners of the tapers in both magnetic layer. Contours of equal transverse magneti- zation component, m t, are shown in the tapered ends.6194 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 Redjdalet al.the larger transverse magnetization regions are on opposite sides of the cell into the lower left ~ll!and upper right ~ur! corners. Figure 6 shows contours of equal transverse magnetiza- tion component, mt, when a uniform 40 Oe external field is applied opposite to the magnetization in the storage layer.Dashed contours ~curve ‘‘a’’ !represent the position of the contours in the initial state ( m t50.1Ms). Solid contours, ‘‘b’’ and ‘‘c,’’ show the progression of the reversal towardsthe body of the memory cell for the first 30% of the reversal.It can be seen that the reversal originates in the regionswhere the asymmetries are located, i.e., the lower left andupper right corners of the tapered ends. After the reversal is complete, the magnetization distri- bution in both the sense and the storage layers appears asshown in Fig. 7. The magnetization in the storage layer isfully reversed, with no wall structure. The sense and storagelayers are magnetized in parallel and in the direction of theexternal field. No further magnetization change is observed.The magnetization configuration in Fig. 7 has reversed with-out creating any blocking structure. Figure 8 shows the magnetization distribution in the sense and storage layers as it relaxes slowly towards equilib-rium after the word disturb field is removed. The switching speed of both layers is a function of the magnetostatic cou-pling between the two layers. Since the thickness of the stor-age layer is greater than that of the sense layer, the resultingmagnetostatic differential field forces the sense layer toswitch while the storage layer remains unchanged. The mag-netization in the sense layer is at an angle to the longitudinalaxis of the memory cell and opposes its initial direction. Thesense layer and the storage layer relax slowly to an antipar-allel configuration that is opposite to the initial configurationin the memory cell. The information in the memory cell islost because of the application of the 40 Oe uniform writedisturb field. IV. CONCLUSION Fabrication asymmetry in the tapered ends of a PSV memory cell has been shown to affect the magnetic switch-ing behavior of the storage layer under a uniform 40 Oewrite disturb field. When the asymmetry is on the same sideof the tapered ends of the magnetic layers, a 360° wall isformed and the stored bit is recovered. However, when theasymmetry is on opposite sides of the tapered ends, the mag-netization configuration in both layers was fully reversed andthe stored bit was lost. Fabrication accuracy better than 10nm is required for a viable memory cell of this design. 1J. Gadbois, J.-G. Zhu, W. Vavra, and A. Hurst, IEEE Trans. Magn. 34, 1066 ~1998!. 2Y. Zheng and J.-G. Zhu, IEEE Trans. Magn. 33, 3286 ~1997!. 3B. A. Everitt and A. V. Pohm, IEEE Trans. Magn. 33, 3289 ~1997!. 4D. D. Tang, P. K. Wang, V. S. Speriosu, S. Le, and K. K. Kung, IEEE Trans. Magn. 31, 3206 ~1995!. 5M. Redjdal, Ph.D. thesis, Boston University, Boston, 1997. FIG. 6. Equal mtcontour of the transverse magnetization component ( mt 50.1Ms) in the storage layer in ~a!the initial state and ~b!~c!in interme- diate states during the first 30% of the magnetization reversal. FIG. 7. Equilibrium magnetization distribution showing a parallel magneti-zation configuration in the sense and the storage layers when a 40 Oe ex-ternal field has been applied opposite to the original magnetization in thestorage layer. The storage layer is reversed. FIG. 8. Magnetization distribution slowly relaxing towards an equilibriumantiparallel configuration in the sense and the storage layers after the 240 Oe external field is turned off.6195 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 Redjdal et al.

1.5007744.pdf

The growth temperature and measurement temperature dependences of soft magnetic properties and effective damping parameter of (FeCo)-Al alloy thin films Yusuke Ariake , Shuang Wu , Isao Kanada , Tim Mewes , Yoshitomo Tanaka , Gary Mankey , Claudia Mewes , and Takao Suzuki Citation: AIP Advances 8, 056119 (2018); doi: 10.1063/1.5007744 View online: https://doi.org/10.1063/1.5007744 View Table of Contents: http://aip.scitation.org/toc/adv/8/5 Published by the American Institute of Physics Articles you may be interested in Soft magnetic properties and damping parameter of (FeCo)-Al alloy thin films AIP Advances 7, 056105 (2017); 10.1063/1.4975995 Enhanced room-temperature spin Seebeck effect in a YIG/C 60/Pt layered heterostructure AIP Advances 8, 055906 (2018); 10.1063/1.5007233 Low Gilbert damping in Co 2FeSi and Fe 2CoSi films Journal of Applied Physics 120, 083904 (2016); 10.1063/1.4960705 The temperature dependence of magnetic anisotropy of Nd-Fe-B thin films AIP Advances 8, 056213 (2018); 10.1063/1.5006517 Spin wave modes of width modulated Ni 80Fe20/Pt nanostrip detected by spin-orbit torque induced ferromagnetic resonance Applied Physics Letters 111, 172407 (2017); 10.1063/1.4999818 Antidamping spin-orbit torques in epitaxial-Py(100)/ β-Ta Applied Physics Letters 111, 232407 (2017); 10.1063/1.5007202AIP ADV ANCES 8, 056119 (2018) The growth temperature and measurement temperature dependences of soft magnetic properties and effective damping parameter of (FeCo)-Al alloy thin films Yusuke Ariake,1,2,aShuang Wu,1,3Isao Kanada,1,2Tim Mewes,1,3 Yoshitomo Tanaka,2Gary Mankey,1,3Claudia Mewes,1,3 and Takao Suzuki1,4,5 1The Center for Materials for Information Technology, The University of Alabama, Tuscaloosa, AL 35487, USA 2Materials Development Center, TDK Corporation, Narita, Chiba 286-0805, Japan 3Department of Physics and Astronomy, The University of Alabama, Tuscaloosa, AL 35487, USA 4Department of Metallurgical and Materials Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA 5Department of Electrical and Computer Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA (Presented 10 November 2017; received 2 October 2017; accepted 19 November 2017; published online 5 January 2018) The soft magnetic properties and effective damping parameters of Fe 73Co25Al2alloy thin films are discussed. The effective damping parameter effmeasured by fer- romagnetic resonance for the 10 nm-thick sample is nearly constant ( 0.004 0.0008) for a growth temperature T sfrom ambient to 200C, and then tends to decrease for higher temperatures and effis 0.0020.0004 at T s= 300C. For the 80 nm-thick sample, the effseems to increase with T sfrom eff= 0.0010.0002 at Ts= ambient to eff= 0.0020.0004. The effis found nearly constant ( eff= 0.004 0.0008) over a temperature range from 10 to 300 K for the 10 nm films with the different T s(ambient, 100 and 200C). Together with an increasing non-linearity of the frequency dependence of the linewidth at low T s, extrinsic contributions such as two-magnon scattering dominate the observed temperature dependence of effective damping and linewidth. © 2018 Author(s). All article content, except where oth- erwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5007744 I. INTRODUCTION For future high frequency device applications of soft magnetic thin films, further improve- ment in saturation magnetization (M s), permeability ( ), coercivity (H c), eddy current loss and damping parameter ( eff) is indispensable.1Among many candidates of materials to choose, FeCo-based alloy thin films are attractive because they exhibit high M sand low eff.2–5 A recent work on Fe 73Co25Al2thin films reported the thickness dependence of effective damp- ing parameter effand showed the values of eff= 0.0004 at about 85nm, indicating an attractive candidate as soft magnetic materials for future high frequency device applications.5 However, since the coercivity for those films was still high for any practical use, lowering coercivity is desirable. The present paper describes a systematic study of the growth- and measurement- temperature dependences of soft magnetic properties and damping parameter of Fe 73Co25Al2thin films. aElectronic mail: ariakey@jp.tdk.com 2158-3226/2018/8(5)/056119/6 8, 056119-1 ©Author(s) 2018 056119-2 Ariake et al. AIP Advances 8, 056119 (2018) II. EXPERIMENTAL Multilayers of [Fe (0.45 nm)/Fe 66Co34(1.3 nm)/Al (0.038 nm)] x N were sputter-deposited onto MgO (100) by DC magnetron sputtering in Ar atmosphere of 4 mTorr, where N = 5 and 37, corresponding to the total thickness of 10 and 80 nm, respectively. The substrate-deposition temperatures (T s) were varied from ambient to 350C. In order to induce a uniaxial magnetic anisotropy, an in-plane field of 50 Oe was applied during deposition. A 5 nm thick Ru layer was over-coated at ambient temperature for protection. The base pressure prior to deposition was better than 2 x 10-7Torr. The deposition rates for Fe, Fe 66Co34and Al were 0.15, 0.17 and 0.038 nm/s, respectively. The film thicknesses were estimated by X-ray reflectivity. Structural analyses were performed by X-ray diffraction (XRD) with Cu K radiation, high resolution transmission electron microscopy (TEM) and energy-dispersive X-ray spectroscopy (EDX). The composition of these samples was estimated as Fe 73Co25Al2by EDX.5Measurements of magnetic properties were carried out by a vibrating sample magnetometer (VSM) in fields up to 10 kOe. Two different types of measurements for magnetization dynamics were carried out by ferromagnetic resonance (FMR); namely over a frequency range from (i) 12 to 66 GHz at room temperature and (ii) 10 to 40 GHz for a temperature range from 10 to 300 K. III. RESULTS AND DISCUSSION A. Growth temperature dependence Figure 1 shows the XRD patterns for 80 nm-thick Fe 73Co25Al2alloy thin films deposited at various substrate temperatures onto MgO (100) substrate. The samples deposited at higher than 100C show the diffraction peaks of (200) bccat 2at around 64.8 degree. (The diffraction of (110) bcc can’t be seen since the peak from the MgO (100) substrate overlaps.) It was found that the low angle XRD pattern for the 10 nm-thick samples deposited above 100C showed the four peaks of FIG. 1. XRD patterns for Fe 73Co25Al2alloy thin films deposited onto MgO (100) substrate. Low angle XRD patterns for the films deposited at 100C are shown in the inserted figure where blue line shows the pattern of (200) FeCoAl and red line shows of (200) MgO.056119-3 Ariake et al. AIP Advances 8, 056119 (2018) FIG. 2. TEM cross-sectional image and the electron diffraction pattern for Fe 73Co25Al2alloy thin films deposited at ambient onto MgO (100) substrate.3 (200) bccseparated at every 90 degree. In addition, it was also found that these peaks were shifted by 45 degree with respect to the peaks of (200) MgO, indicating that the film is a single crystalline film with<100>FeCoAl //<110>MgO. Figure 2 shows the TEM cross sectional image, together with the diffraction patterns for 50 nm-thick Fe 73Co25Al2alloy thin films deposited at ambient temperature onto MgO (100) sub- strate.3The TEM image shows the columnar structure grown along the film normal with the average width of about 20-30 nm. The diffraction patterns indicate that the film is polycrystalline film and its growth direction is <110>, which is different of the films deposited above 100C. Figure 3(a) are the M-H curves for the 10 nm-thick samples deposited at various T s. These curves were measured along the direction of <100>FeCoAl and<110>FeCoAl by VSM. The shape of the curve measured along the direction of <100>FeCoAl is significantly changed from the sample with the T sof 100C to 200C, where the remanence M rbecomes much higher with higher T s. The values of M s, Hcand M r/Msfor the samples with the thicknesses of 10 and 80 nm deposited at various T sare summarized in Figure 3(b). It is seen that M sremains unchanged (1,600 emu/cm3) with T s. On the other hand, H cmeasured along the direction of <100>FeCoAl changes with T s, becoming minimum at around 100 and 200C for the 10 and 80 nm-thick samples, respectively. The observed decrease of H cis probably caused by reducing a residual stress in the films which induces magnetic anisotropy through magneto-elastic effect. The M r/Msmeasured along the <110>FeCoAl FIG. 3. (a) M-H curves and (b) growth temperature T sdependence of saturation magnetization M s, coercivity H cand M r/Ms ratio for Fe 73Co25Al2alloy thin films deposited onto MgO (100) substrate. The <110>FeCoAl and <100>FeCoAl indicate the direction of applied field during the measurement, corresponding to the direction of <100>MgO and<110>MgO, respectively.056119-4 Ariake et al. AIP Advances 8, 056119 (2018) decreases with T s, becoming minimum at T s= 200C. The M r/Msalong the<100>FeCoAl , on the other hand, increases with T sand it becomes higher than that along the <110>FeCoAl above T sof 150C. Figure 4(a) shows the FMR linewidth
H as a function of resonance frequency f resfor Fe73Co25Al2alloy thin films deposited at various T s. The linewidth
H tends to decrease with T s, and the linearity of these relationships is improved for higher T sover a wide range of frequency for the 10 nm-thick sample. However, for the 80 nm-thick film, the nonlinear frequency dependence of
H is found for T s= ambient and 100C. As T sgoes higher, the crystallinity becomes improved, as found by XRD and TEM. Therefore, one would expect the contributions of crystalline anisotropy field to the linewidth broadening which varies from grain to grain.6For the 10nm thick sample fabricated at Ts= ambient temperature, the nonlinear linewidth evolution can be attributed as the characteristic of two-magnon scattering due to the boundary of Ru/FeCoAl, as shown by Lenz et al.7Theeff was estimated based on the linear relationship, as shown by a dotted line, fitted to the following equation.
H=
H0+4p 3 efffres. The result of effas a function of T sis shown in Figure 4(b). It is found that the efffor the 10 nm-thick sample is nearly constant ( 0.0040.0008) for T sfrom ambient to 200C, and then tends to decrease for higher temperatures and effis 0.0020.0004 at T s= 300C. For the 80 nm-thick sample, although there is much scatter in the data points, the effseems to increase with Tsfromeff= 0.0010.0002 at T s= ambient to eff= 0.0020.0004. It should be pointed out that the sample fabricated at T s= 150C has eff= 0.00070.0002, in agreement within an error with the value reported.5As pointed out by Li et al.,8there can be a contribution from eddy current even in relatively thin films. An estimation of damping parameter contribution of eddy current loss in a 80 nm-thick cobalt film is about 0.001, and therefore in the present study its contribution to the mea- sured effective damping parameter may not be negligible for the 80 nm-thick samples. However, due to the quadratic dependence on the film thickness this contribution is negligible for the 10 nm-thick film. Angle dependent measurements of the H resand
H were also performed, showing a four fold symmetry for the resonance field, which is consistent with the in-plane XRD measurements, with the easy axis along the <100>FeCoAl . FIG. 4. (a) FMR linewidth
H as a function of resonance frequency f resand (b) growth temperature T sdependence of effective damping parameter efffor Fe 73Co25Al2alloy thin films deposited onto MgO (100) substrate. The reported result in Ref. 5 is shown as the red square.056119-5 Ariake et al. AIP Advances 8, 056119 (2018) FIG. 5. Measurement temperature dependence of effective damping parameter efffor Fe 73Co25Al2alloy thin films deposited onto MgO (100) substrate at ambient, 100 and 200C with 10 nm thickness. Also shown are the results of Permalloy thin films9() and of YIG thin films10(
). B. Measurement-temperature dependence Figure 5 shows the temperature dependence of efffor the 10 nm-thick films with different growth temperatures T s, together with the data of permalloy9and YIG10thin films reported. For all the samples under consideration, the effare nearly constant ( eff= 0.0040.0008) over a temperature range from 10 to 300 K for the three different T s(ambient, 100 and 200C). On the other hand, the permalloy and the YIG thin films exhibit the decrease with decreasing temperature, although the permalloy film shows a slight increase for a range from 100 to 50 K. The present result of effvs. T is at variance with those results. It is noted that in the present study an increas- ing non-linearity of the frequency dependence of the linewidth at low growth temperatures was observed, therefore it is likely that extrinsic contributions such as two-magnon scattering dominate the observed temperature dependence of effective damping and linewidth.6Although further stud- ies are necessary, the present result of the effwhich is insensitive to temperature suggests that a thin (around 10 nm-thick) Fe 73Co25Al2alloy thin film may be useful for high frequency device applications. IV. SUMMARY The growth- and measurement-temperature dependences of soft magnetic properties and effective damping parameters of Fe 73Co25Al2alloy thin films are discussed. The saturation magnetization Msis about 1,600 emu/cm3, independent of thickness and the substrate deposition temperature T s. Coercivity H cis found to decrease with T sup to around 100200C, which is probably caused by reducing the stress in the film. The effective damping parameter effmeasured by ferromagnetic resonance (FMR) over a fre- quency range from 12 to 66 GHz at room temperature and over a temperature range from 10 to 300 K. For the 10 nm-thick sample the effective damping parameter is nearly constant ( 0.004 0.0008) for T sfrom ambient to 200C, and then tends to decrease for higher temperatures and effis 0.0020.0004 at T s= 300C. For the 80 nm-thick sample, the effseems to increase with T sfrom eff= 0.0010.0002 at T s= ambient to eff= 0.0020.0004. As pointed out by Li et al.,8there can be a contribution from eddy currents even in relatively thin films. However, due to its quadratic thickness dependence eddy currents do not contribute significantly to the linewidth of the 10 nm film.056119-6 Ariake et al. AIP Advances 8, 056119 (2018) The temperature dependence of effobtained for a frequency range from 10 to 40 GHz for the 10 nm films with the different growth temperatures T s(ambient, 100 and 200C) shows that the eff is nearly constant ( eff= 0.0040.0008) over a temperature range from 10 to 300 K. Together with an increasing non-linearity of the frequency dependence of the linewidth at low growth temperatures, extrinsic contributions such as two-magnon scattering are likely responsible for the observed absence of a temperature dependence of the effective damping and linewidth. ACKNOWLEDGMENTS The authors would like to thank Dr. Michael Buettner for his technical assistance. The present work was supported in part by the MINT-TDK collaboration program. 1H. Masumoto and H. Saito, “On the effect of heat treatment on the magnetic properties of iron-aluminium alloys. II. The change of permeabilities, intensity of magnetization and magnetic hysteresis loss due to quenching and a new alloy ‘Alperm’,” Sci. Rep. RITU A4, 321–337 (1952). 2F. Schreiber, J. Pflaum, Z. Frait, Th. M ¨uhge, and J. Pelzl, “Gilbert damping and g-factor in Fe xCo1-xalloy films,” Solid State Commun. 93, 965–968 (1995). 3I. Kanada, A. Cruce, T. Mewes, S. Wu, C. Mewes, G. Mankey, and T. Suzuki, “Soft magnetic properties and damping parameter of (FeCo)-Al alloy thin films,” AIP Advances 7, 056105 (2017). 4M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva, H. T. Nembach, O. Eriksson, O. Karis, and J. M. Shaw, “Ultra-low magnetic damping of a metallic ferromagnet,” Nat. Phys. 12, 839–842 (2016). 5Y . Ariake, I. Kanada, T. Mewes, G. Mankey, Y . Tanaka, S. Wu, C. Mewes, and T. Suzuki, “The thickness dependence of soft magnetic properties of (FeCo)-Al alloy thin films,” IEEE Trans. Mag. 53, 2003404 (2017). 6C. Mewes and T. Mewes, in Handbook of Nanomagnetism (Pan Stanford, 2015), p. 71. 7K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy, and A. J ´anossy, “Two-magnon scattering and viscous Gilbert damping in ultrathin ferromagnets,” Phys. Rev. B 73, 144424 (2006). 8Y . Li and W. E. Bailey, “Wave-number-dependent Gilbert damping in metallic ferromagnets,” Phys. Rev. Lett. 116, 117602 (2016). 9Y . Zhao, Q. Song, S. H. Yang, T. Su, W. Yuan, S. S. P. Parkin, J. Shi, and W. Han, “Experimental investigation of temperature- dependent Gilbert damping in permalloy thin films,” Scientific Reports 6(2016). 10M. Haidar, M. Ranjbar, M. Balinsky, R. K. Dumas, S. Khartsev, and J. Åkerman, “Thickness- and temperature-dependent magnetodynamic properties of yttrium iron garnet thin films,” J. Appl. Phys. 117, 17D119 (2015).

1.2759166.pdf

Numerical simulations of fluid-structure interactions in single-reed mouthpieces Andrey Ricardo da Silvaa/H20850and Gary P . Scavone Computational Acoustic Modeling Laboratory, Schulich School of Music, McGill University, Montreal, Quebec, Canada Maarten van Walstijn Sonic Arts Research Centre, School of Electronics, Electrical Engineering, and Computer Science, Queen’ sUniversity Belfast, Belfast, United Kingdom /H20849Received 23 April 2007; revised 18 June 2007; accepted 21 June 2007 /H20850 Most single-reed woodwind instrument models rely on a quasistationary approximation to describe the relationship between the volume flow and the pressure difference across the reed channel.Semiempirical models based on the quasistationary approximation are very useful in explaining thefundamental characteristics of this family of instruments such as self-sustained oscillations andthreshold of blowing pressure. However, they fail at explaining more complex phenomenaassociated with the fluid-structure interaction during dynamic flow regimes, such as the transient andsteady-state behavior of the system as a function of the mouthpiece geometry. Previous studies havediscussed the accuracy of the quasistationary approximation but the amount of literature on thesubject is sparse, mainly due to the difficulties involved in the measurement of dynamic flows inchannels with an oscillating reed. In this paper, a numerical technique based on the latticeBoltzmann method and a finite difference scheme is proposed in order to investigate thecharacteristics of fully coupled fluid-structure interaction in single-reed mouthpieces with differentchannel configurations. Results obtained for a stationary simulation with a static reed agree verywell with those predicted by the literature based on the quasistationary approximation. However,simulations carried out for a dynamic regime with an oscillating reed show that the phenomenonassociated with flow detachment and reattachment diverges considerably from the theoreticalassumptions. Furthermore, in the case of long reed channels, the results obtained for the vena contracta factor are in significant disagreement with those predicted by theory. For short channels, the assumption of constant vena contracta was found to be valid for only 40% of the duty cycle. ©2007 Acoustical Society of America. /H20851DOI: 10.1121/1.2759166 /H20852 PACS number /H20849s/H20850: 43.75.Pq, 43.75.Ef /H20851NHF /H20852 Pages: 1798–1809 I. INTRODUCTION The study of the acoustical properties of single-reed in- struments has followed a paradigm first proposed byHelmholtz, 1with these systems divided into linear and non- linear components representing the instrument’s bore andmouthpiece-reed, respectively. Previous research on the resonator component has pro- vided an extensive list of experimental and theoretical stud-ies since the pioneering work of Bouasse. 2Much light has been shed on the behavior of this system and, consequently,many satisfactory models have been proposed. Conversely, the amount of available literature on the mouthpiece-reed component is considerably smaller and themajority of models rely on the quasistationary approximationto describe the flow behavior. That is, the flow in a mouth-piece with an oscillating reed is assumed to be equal, at anyinstant, to the flow in a mouthpiece with a static reed havingthe same configuration. 3Moreover, the flow is considered to be frictionless and incompressible. Consequently, the depen-dence of the volume flow Uon the pressure difference acrossthe reed /H9004pand on the reed opening his normally described by the Bernoulli obstruction theory based on the stationaryBernoulli equation, given by U B=hw/H208812/H20841/H9004p/H20841 /H9267sgn /H20849/H9004p/H20850, /H208491/H20850 where wis the channel’s width and /H9267is the density of the fluid. This approach was first presented by Backus,4whose semiempirical model was limited to low blowing pressureregimes. Years later, Worman 5presented a more complex model addressing, in further detail, phenomena such asthreshold of pressure and Bernoulli forces acting on the reed.Wilson and Beavers 6coupled the previous model to an ide- alized cylindrical resonator. More recent models involvingthe same approach were developed by Fletcher, 7,8Saneyoshi et al.9Kergomard,10and Olivier.11 The quasistationary approximation has also been used to derive a steady viscous flow representation by Hirschberget al. 12Their semiempirical model was based on the results obtained from the simulation of flow in a two-dimensional/H20849Borda /H20850tube based on the theory of potential flow. They noticed that, for Reynolds numbers Re /H1102210, two patterns of a/H20850Electronic mail: andrey.dasilva@mail.mcgill.ca 1798 J. Acoust. Soc. Am. 122 /H208493/H20850, September 2007 © 2007 Acoustical Society of America 0001-4966/2007/122 /H208493/H20850/1798/12/$23.00 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55flow may occur simultaneously, depending on the ratio l/h, where lis the length of the channel and his its height. The flow is fully detached along the channel, for short channels/H20849l/h/H110211/H20850, whereas for long channels /H20849l/h/H110223/H20850the flow is reattached at a roughly fixed point, l r, measured from the channel’s entrance. They also observed that, in the case ofshort channels, the vena contracta factor /H9251=Tj/hwas ap- proximately constant with a value /H112290.6, where Tjis the thickness of the jet formed at the detached portion of theflow. van Zon et al. 13provided an experimental validation of Hirschberg’s model using an idealized prototype of themouthpiece with a static reed and assuming the flow to betwo dimensional. They also derived a more sophisticatedflow model in which the transition between fully separated toattached flow is represented by a boundary layer solution.Other stationary measurements using realistic mouthpieceshave found the same flow behavior, such as those conductedby Valkering 14and by Dalmont et al. ,15in the case of clari- net, and by Maurin,16in the case of saxophones. However, previous attempts to characterize flow in dy- namic regimes,13,17i.e., flow in a mouthpiece with a moving reed, have suggested that the stationary behavior observed byvan Zon et al. is unrealistic. This is particularly evident in the case of the steadiness associated with the detachment/reattachment phenomenon, which is strongly affected bysubtle modifications of the reed channel geometry as the reedmoves. The unsteadiness of the flow modulates the aerody-namic forces acting on the reed and plays an important rolein the reed’s behavior. In fact, this unsteadiness is respon-sible for the self-sustained oscillations in systems whoseacoustic coupling between the resonator and the exciter isweak or even absent. This is the case in the harmonium, 18in the accordion,19and in the human phonatory system.20–24 Moreover, the unsteadiness of the flow can explain why small modifications in a mouthpiece geometry can corre-spond to enormous changes in the transient behavior and thesteady-state sound of single-reed instruments. 25,26 Unfortunately, the accurate quantification and visualiza- tion of a dynamic flow controlled by a moving boundary /H20849in this case, the reed /H20850is a rather complicated task. For this reason, previous attempts to do so are limited to qualitativeoutcomes. 13,17Similar difficulties are found when tackling the problem with unsteady numerical flow simulations usingtraditional computational fluid dynamic /H20849CFD /H20850techniques based on the continuum theory. 27,28 The objectives of this paper are the presentation of a numerical model of fully coupled fluid-structure interactionin a single-reed mouthpiece in order to address the majoraspects of dynamic flow and its dependency on the reedchannel geometry and to verify the validity of the quasista-tionary theory in dynamic regimes. To accomplish that, weimplement a two-dimensional dynamic model of a single-reed mouthpiece based on a hybrid numerical approach in-volving the lattice Boltzmann method /H20849LBM /H20850, to represent the fluid and acoustic domains, and on a finite differencescheme to resolve the distributed model of the reed withvarying cross section, as proposed by Avanzini and vanWalstijn. 29The main advantage of this approach consists inits simplicity in providing solutions of second-order accuracy to represent the fluid-structure interaction involving a mov-ing boundary. 30This simplicity is contrasted with the com- plexity and high computational demand associated with tra-ditional CFD techniques. Furthermore, the LBM can solvethe different scales associated with the flow and acousticfields in a single calculation, thus allowing the direct repre-sentation of the acoustic-flow interaction. 31 The influence of the player’s lip and the coupling of the proposed system with the instrument’s bore and player’s vo-cal tract is not considered in this paper. Furthermore, thecontribution of aeroacoustic sources on the instrument’ssound content due to undisturbed flow will be left to futurework. This paper is organized as follows: Section II describes the model by presenting the lattice Boltzmann techniqueused, as well as the finite difference scheme to resolve thedistributed reed model and the coupling between both tech-niques. Section III compares the results obtained from a sta-tionary simulation with those provided by the quasistationarytheory. Section IV evaluates the characteristics of a dynamicflow in a reed-mouthpiece system without acoustic couplingfor three different reed channel geometries and compares theresults with those expected by the quasistationary theory. Fi-nally, the conclusions and remarks for future investigationsare presented in Sec. V. II. THE REED-MOUTHPIECE MODEL The following describes the implementation of the two- dimensional model of the mouthpiece-reed system. Themouthpiece is represented by the LBM, which includes solidstatic boundaries associated with the mouthpiece walls /H20849face, rails, and cavity walls /H20850and the fluid domain, described in terms of acoustic and fluid fields. The moving boundary as-sociated with the reed is represented by a distributed modelof a clamped-free bar with varying cross section and re-solved with an implicit finite difference scheme, as proposedby Avanzini and van Walstijn. 29 A. The lattice Boltzmann method The LBM is classified as a particle ornonequilibrium technique. It simulates the space-temporal evolution of fluid-acoustic systems based on a time-space discretization of theBoltzmann equation, known as the lattice Boltzman equation/H20849LBE /H20850/H20851see Eq. /H208492/H20850/H20852. Xe and Luo 32have demonstrated that the Navier-Stokes and continuity equations can be fully recovered from theLBE for low Mach numbers, namely /H20849Ma/H110210.2/H20850, by applying the Chapman-Enskog expansion, thus providing a physical validity for the method. Detailed descriptions of the LBMare provided by Succi 33and Gladrow.34 The LBE controls two essential operations: advection and collision of fluid particles. These particles are describedin terms of velocity distribution functions and can propagatein a discrete set of directions within the lattice. In this paper we use an isothermal two-dimensional model known as D2Q9, after Qian et al. 35In this sense, the lattice grid is represented by squared two-dimensional lattice J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces 1799 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55cells containing nine sites each /H20849eight propagation directions and one rest site /H20850, as depicted in Fig. 1. Each cell connects to eight neighbor cells by the unity vectors ci, where i =1,2,...,8, indicates the propagation direction associated with each site. The null vector c0is associated with a non- propagating site and plays an important role in improving theaccuracy of the model by removing the unphysical velocitydependency of pressure. 36The two main operations, namely, advection and colli- sion, are controlled by the LBE, fi/H20849x+ci/H9004t,t+/H9004t/H20850−fi/H20849x,t/H20850=−1 /H9270/H20849fi−fiM/H20850, /H208492/H20850 where fiis the distribution function associated with the propagation direction iat the site x/H11032and time t/H11032./H9270is the relaxation time or collision period, which acts to control thekinematic viscosity of the fluid, and f iMis the equilibrium distribution function for direction i, which depends on the local fluid velocity u/H11032/H20849x/H11032,t/H11032/H20850and local fluid density /H9267/H11032/H20849x/H11032,t/H11032/H20850. Here and in the following, variables indicated with a prime are adimensional. The general expressions of the equilibrium function fiMassociated with the D2Q9 model are fiM=/H20902/H9267/H11032/H9280i/H208751+3 ci·u/H11032+9 2/H20849ci·u/H11032/H208502−3 2u/H110322/H20876 fori= 1,2, ... ,8 /H9267/H11032/H208754 9−2 3u/H110322/H20876 fori=0/H20901/H208493/H20850 with/H92801=/H92802=/H92803=/H92804=1/9 and /H92805=/H92806=/H92807=/H92808=1/36. The left-hand side of Eq. /H208492/H20850represents the advection operation and determines the diffusion of the distributionfunctions f iover the lattice grid. The right-hand term deter- mines the rate at which fichange due to intermolecular col- lisions between particles. This term is defined by a simplifiedcollision function, known as BGK, after Bhatnagar, Grass,and Krook, 37which is controlled by a single relaxation time /H9270for all the advection directions i. This process, known as relaxation, forces fitoward equilibrium and restitutes the vis- cosity of the fluid, recovering its nonlinear form whereby thecontinuity and Navier-Stokes equations are satisfied. The local macroscopic variables /H9267/H11032andu/H11032are obtained in terms of moments of the local distribution functions fiby /H9267/H11032/H20849x,t/H20850=/H20858 ifi/H20849x,t/H20850,/H9267/H11032/H20849x,t/H20850u/H11032/H20849x,t/H20850=/H20858 ifi/H20849x,t/H20850ci. /H208494/H20850 Other macroscopic parameters such as lattice pressure p/H11032, lattice viscosity /H9263/H11032, and lattice speed of sound c0/H11032are obtained by expanding the LBE into the Navier-Stokes equa-tion and are expressed by p /H11032=/H9267/H11032 3,/H9263/H11032=2/H9270−1 6,c0/H11032=1 /H208813. /H208495/H20850 The adimensional lattice variables /H9267/H11032,p/H11032,u/H11032,x/H11032,t/H11032, and /H9263/H11032can be easily related to their respective physical counter- parts/H9267,p,u,x,t, and /H9271by the following relations: /H9267=/H9267/H11032, p=/H9267c02,u=u/H11032c0/c0/H11032,x=x/H11032/H9004x,t=/H9004x/H20849c0/H11032/c0/H20850t/H11032, and /H9263 =/H20849c0/c0/H11032/H20850/H9004x/H9263/H11032, where c0is the physical speed of sound.B. The mouthpiece model The mouthpiece model was implemented in a lattice grid containing 1002 /H11003502 cells. The physical dimensions of the system are depicted in Fig. 2, as well as the dimensions of the grid. The lattice pitch was /H9004x=4/H1100310−5m and the time step/H9004t=6.792 /H1100310−8s. As a matter of convenience, we have opted to use an undisturbed fluid density /H92670/H11032=/H92670=1.0 kg/m3. The relaxation time /H9270was chosen to be 0.505, which implies a lattice viscosity /H9263/H11032=1.68 /H1100310−3and a physical kinematic viscosity /H9263=3.95 /H1100310−5m2/s, using c0=340 m/s as the ref- erence speed of sound. Although the choice of values for /H92670and/H9263differ con- siderably from those of air in normal playing conditions, thedynamic similarity with the real system is obtained by forc-ing Re /H112291200 for a maximum Ma=0.1. These parameters also allow the two essential criteria of the lattice BoltzmannBGK model to be met: the maximum compressibility /H20849Ma FIG. 2. Lattice grid representing the two-dimensional model of the mouthpiece-reed system. FIG. 1. The squared grid for the D2Q9 lattice Boltzmann model. 1800 J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55/H110210.1/H20850before numerical instabilities34and a minimum grid resolution /H208495.6 lattices per wavelength /H20850to avoid spurious dis- persion and dissipation effects associated with the numericalbulk viscosity, as described by Crouse et al. 38 C. The reed model The reed is represented as a clamped-free bar with non- uniform thickness b/H20849x/H20850, constant width w, and driven by a force F/H20849x,t/H20850. The partial differential equation describing the vertical displacement y/H20849x,t/H20850as a function of F/H20849x,t/H20850is given by /H9267rA/H20849x/H20850/H115092y /H11509t2/H20849x,t/H20850+/H115092 /H11509x2/H20875YI/H20849x/H20850/H208731+/H9257/H11509 /H11509t/H20874/H115092y /H11509x2/H20849x,t/H20850/H20876=F/H20849x,t/H20850, /H208496/H20850 where x/H33528/H208510,L/H20852is the horizontal position, A/H20849x/H20850=wb/H20849x/H20850is the cross section, /H9267ris the material density, Yis the Young’s modulus, I/H20849x/H20850is the moment of area about the longitudinal axis, and /H9257is the viscoelastic damping coefficient. Table I shows the values used in the simulation, obtained experimen-tally by Avanzini and van Walstijn. 29Equation /H208496/H20850considers only reed motion associated with flexural waves in the ver-tical direction and, thus, torsional and longitudinal modes areneglected. This is similar to the approach used by Chaigneand Doutaut 39to simulate xylophone bars. In our model, a term associated with the energy dissipation of the reed due tothe work exerted on the surrounding fluid is neglected in Eq./H208496/H20850. However, this is taken into account implicitly by the fully coupled fluid-structure interaction scheme presented inSec. II E. Equation /H208496/H20850is solved by performing a space-temporal discretization based on an implicit finite difference schemedescribed by Chaigne and Doutaut. 39This results in a matri- cial difference equation in which the spatial coordinate isvectorialized, y/H20849n+1/H20850=A 0·y/H20849n/H20850+A1·y/H20849n−1/H20850+AF·F/H20849n/H20850, /H208497/H20850 where y/H20849n+1/H20850,y/H20849n/H20850, and y/H20849n−1/H20850represent the displacement vector at successive time instants and A0,A1, and AFare coefficient matrices. F/H20849n/H20850is a vector representing the longi- tudinally distributed force on the reed. The interaction be- tween the reed and the mouthpiece lay is considered to beinelastic. This is achieved by nullifying the kinetic energy ofthose reed sections that collide with the mouthpiece side rail,which presents an upper boundary to the reed. The inelasticassumption for the reed/lay interaction is discussed and jus-tified in Ref. 29.D. Initial and boundary conditions The algorithm assumes a no-slip condition of flow at the walls by implementing a bounce-back scheme proposed byBouzidi et al. 40The bounce-back scheme works to invert the direction of propagation of a distribution function fijust be- fore it reaches a solid boundary. This procedure creates a nullfluid velocity at the walls and provides second-order accu-racy to represent viscous boundary layer phenomena. The problem of a moving boundary, the reed, within the lattice is tackled by using an interpolation scheme proposedby Lallemand and Luo. 30This technique preserves second- order accuracy in representing the no-slip condition and thetransfer of momentum from the boundary to the flow. Oneconstraint of this approach is the velocity limit defined byMa/H110210.5, Ma= u b/c0,ubbeing the velocity of the boundary. However, such a limitation does not represent a problem inour simulation because it corresponds to values of velocitymuch higher than those found for reeds at normal playingconditions. The mean flow is initiated by using a fairly well known technique in CFD called absorbing boundary conditions/H20849ABC /H20850. This technique has been adapted to LBM by Kam et al. 41and consists in using a buffer between the fluid region and the open boundary to create an asymptotic transitiontoward a target flow defined in terms of target distributionfunctions f iT. This is done by adding an extra term to Eq. /H208492/H20850 to represent the transition region, fi/H20849x+ci/H9004t,t+/H9004t/H20850−fi/H20849x,t/H20850=−1 /H9270/H20849fi−fiM/H20850−/H9268/H20849fM−fiT/H20850, /H208498/H20850 where /H9268=/H9268m/H20849/H9254/D/H208502is the absorption coefficient, /H9268mis a constant, normally equal to 0.3, is the distance measured from the beginning of the buffer zone, and Dis the width of the buffer. The fiTis constant and can be obtained in the same manner as fiMusing Eq. /H208493/H20850, where the local velocity u/H11032and local density /H9267are replaced by the desired target flow uTand target density /H9267T, respectively. Another desired feature of this technique is the anechoic characteristic that avoids any re-flection or generation of spurious waves at the open bound-aries. E. Numerical procedures Seven different operations are executed at every time step in order to couple the lattice Boltzmann model with thefinite difference scheme. The sequence of operations is de-picted as a flowchart in Fig. 3. Before the simulation begins, the initial conditions associated with the fluid and reed vari-ables are set and the definition of solid boundaries within thelattice are defined. The reed variables such as displacement y/H20849x,0/H20850, velocity y˙/H20849x,0/H20850, and force F/H20849x,0/H20850are set to zero, as well as the variables associated with the fluid domain, such as the local fluid velocities u. The initial fluid variables are used to define the initial distribution functions based on Eq./H208493/H20850, so that, in the first time step f i=fiM. The flow is started by prescribing a target pressure difference at the ABC layer,defined as /H9004p T=/H20849/H9267inT−/H9267outT/H20850c02, where the indexes “in” andTABLE I. Characteristics of a plastic reed /H20849Plasticover /H20850obtained by Avan- zini and van Walstijn /H20849Ref. 29/H20850. Length Lreed=34/H1100310−3m Width w=10/H1100310−3m Density /H9267reed=500 kg/m3 Young’s modulus Y=5.6/H11003109N/m2 Viscoelastic const. /H9257=6.0/H1100310−7s Fluid damping coef. /H9253air=100 s−1 J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces 1801 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55“out” indicate inlet and outlet, respectively. The values of /H9004pTdepend on the type of simulation being conducted and are described in the next sections of this paper. With respect to the flowchart in Fig. 3, the following operations take place after the initial conditions are set: /H20849a/H20850 calculate the relaxation functions fiM’s using Eq. /H208493/H20850;/H20849b/H20850 propagate fito all directions, ignoring the presence of pre- defined solid boundaries and perform their relaxation basedon Eq. /H208492/H20850;/H20849c/H20850find the lattice positions of f ithat have crossed solid boundaries during the propagation step in /H20849b/H20850;/H20849d/H20850re- place fifound by the previous operator with new values based on two different interpolation strategies: fiat crossed static boundaries are replaced by values calculated using thesimple bounce-back scheme, proposed by Bouzidi et al.40 Otherwise, fiare replaced by values calculated using the moving boundary scheme proposed by Lallemand and Luo.30 In this case, the calculation of the new firequires the actual values of y˙/H20849x,t/H20850in order to take into account the transfer of momentum from the reed to the flow; /H20849e/H20850determine new values of u/H11032and/H9267/H11032using Eq. /H208495/H20850;/H20849f/H20850evaluate the new dis- tributed force F/H20849x,t/H20850on the reed model based on local lattice pressures across the reed boundary; and /H20849g/H20850calculate the reed’s new position y/H20849x,t/H20850and velocity y˙/H20849x,t/H20850. III. QUASISTATIC MODEL The following compares the results obtained for a sta- tionary simulation /H20849static reed /H20850using the model described in Sec. II with the results provided by the quasistationary ap-proximation for two-dimensional flows in single-reed mouth-pieces with constant channel cross section, as proposed byvan Zon et al. 13The model was based on the results from a stationary measurement involving an idealized two-dimensional prototype of the mouthpiece-reed system. Simi-lar measurements using real mouthpieces have found identi-cal results and were conducted by Valkering 14and Dalmont,15in the case of a clarinet, and by Maurin,16for the saxophone. A. Overview of the analytical model Hirschberg et al.12derived a semiempirical analytical model for the viscous steady flow in a two-dimensionalsingle-reed mouthpiece channel with constant height. Themodel was based on the numerical study of a two-dimensional channel /H20849Borda tube /H20850using a potential flow scheme. They observed two types of flow for Reynolds numbers Re/H1102210/H20849Re=U/w /H9271/H20850, depending on the ratio between the channel height hand its length l. In both cases, a jet is formed at the sharp edges of the channel’s entrance. Forsmall ratios /H20849l/h/H333551/H20850, the jet does not reattach along the channel walls, whereas, for high ratios /H20849l/h/H113503/H20850, the jet re- attaches at a fixed point l r/H11229hmeasured from the channel’s entrance. Thus, in the case of short channels, the flow is described by the Bernoulli equation /H20851Eg. /H208491/H20850/H20852scaled with a constant vena contracta factor /H9251, whereas, in the case of long chan- nels, the detached segment is represented by the Bernoulliflow and the reattached part is represented by the Poisseuilleflow. These results were confirmed experimentally by van Zon et al. 13who derived a more accurate steady flow model in which the transition between fully separated to Poisseuilleflow is described by a boundary layer flow. In this case, thevelocity profile u/H20849x,y/H20850within the boundary layer of thickness /H9254/H20849x/H20850is assumed to increase linearly with the distance yfrom the wall. Similar to the model proposed by Hirschberg et al. , the flow in short channels /H20849l/h/H333551/H20850is given by FIG. 3. Flowchart of the integrated algorithm. 1802 J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55U=/H9251UB, /H208499/H20850 where UBis the Bernoulli flow given by Eq. /H208491/H20850and/H9251, 0.5 /H33355/H9251/H333550.61 is the constant vena contracta factor whose value depends on the external geometry of the mouthpiece. For long channels /H20849l/h/H113504/H20850and/H9254/H20849l/H20850/H11022/H9254c, where /H9254cis the critical boundary layer thickness, van Zon et al. describe the volume flow by U=/H9263w ch/H20849lc−lr/H20850. /H2084910/H20850 The term /H20849lc−lr/H20850is the length of the transition between fully separated flow to Poisseuille flow, given by lc−lr l−lr=12c/H208491−/H9254*/H208502 24c−1/H208751−/H208811−h4/H2084924c−1/H20850/H9004p 72/H9267/H92632/H20849l−lr/H208502/H208491−/H9254*/H208502/H20876, /H2084911/H20850 where /H9254*is the generalization of the critical boundary layer thickness /H9254cfor a channel of arbitrary height h, expressed by /H9254*=/H9254c h=4 9/H208731−/H208815 32/H20874= 0.2688 /H2084912/H20850 and c=1 6/H208754/H9254*+9l n /H208491−/H9254*/H20850+5/H9254* 1−/H9254*/H20876= 0.01594. /H2084913/H20850 B. Stationary results The stationary simulations were conducted for different cases involving geometries with the same characteristics asthat shown in Fig. 2, but with different channel profiles as depicted in Fig. 4. For each geometry, different steady-state values of Uare achieved by prescribing different target pres- sure values /H9004p Tfrom 0 to 9 kPa. The simulations used the same characteristics described in Sec. II in terms of initialand boundary conditions, lattice discretization, and fluidproperties. However, in this case the reed is maintained fixed/H20849or static /H20850throughout the simulations. Figure 5/H20849a/H20850presents the numerical results obtained for the three cases in terms of vena contracta factor /H9251=U/UBas a function of the modified Reynolds number proposed byvan Zon et al. 13These results are compared with those pre- dicted by the quasistationary model presented in the previoussection. For short channels, the values of /H9251were chosen to represent two geometry cases, namely, a slit in an infinitewall and a tube with sharp edges /H20849Borda tube /H20850. According to potential flow theory, 42/H9251is determined by the turning angle of the upstream flow into the channel, which depends on thecharacteristics of the external geometry. For the slit in aninfinite wall, one finds /H9251=0.61, whereas for the Borda tube /H9251=0.5. Therefore, in the case of a short single-reed mouth- piece channel /H20849l/h/H333551/H20850, one should expect an intermediate value between the two extreme cases, i.e., 0.5 /H33355/H9251/H333550.61. Figure 5/H20849b/H20850plots the same simulation results in terms of volume flow Uas a function of the pressure difference /H9004p and compares that with the theory provided for short andlong channels as presented in Sec. III A.In general, Figs. 5/H20849a/H20850and5/H20849b/H20850show that the results ob- tained for geometries 1, 2, and 3 agree very well with thetheory presented in Sec. III A. However, Fig. 5/H20849a/H20850shows that the result for geometry 1 is in considerable disagreement forhRe/ /H20849l−s/H20850/H1102160 when compared with the limits provided by the theory for short channels and fully detached flow /H208490.5 /H33355 /H9251/H333550.61 /H20850. This type of disagreement is commonly re- ported in the literature and is attributed to the influence of viscous effects at low Reynolds numbers, as described byDurrieu et al. 43and Blevins.44Curiously, the results obtained for geometry 2 are very similar to those found for geometry3 and agree very well with those predicted by the theory forlong channels /H20851Eq. /H2084910/H20850/H20852, despite the fact that geometry 3 has a rather diverging channel profile due to the presence of thechamfer. The flow profiles were found to be roughly constant for all geometries. In the first geometry, the flow remained fullyseparated for Re /H1102230, whereas in geometries 2 and 3 the flow separated at the beginning of the channel and reattachedatl r/H112292hfor Re /H1102260. IV. DYNAMIC RESULTS The goal of the following is to investigate the main as- pects of the fluid-structure interaction in dynamic regimes byusing the same geometries investigated in Sec. III /H20849Fig. 4/H20850. We also intend to substantiate the validity of quasistationarytheory by evaluating the main assumptions associated with FIG. 4. Different reed channel profiles used in the simulation: /H20849a/H20850L/h=1, /H20849b/H20850L/h=4, and /H20849c/H20850L/h=4 with a chamfer. J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces 1803 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55the steadiness of the flow reattachment point and steadiness of the vena contracta factor when the oscillation of the reedis taken into account.For all three cases, we use the same initial and boundary conditions described in Sec. II. The flow is initiated by pre-scribing /H9004p T=5 kPa. This value corresponds to a middle point between the threshold of oscillation and the maximumpressure found for a clarinet mouthpiece. 15It must be stressed that the ABC scheme used at the inlet and outlet ofthe system /H20849Fig. 2/H20850provides a complete anechoic behavior, which avoids any sort of acoustic coupling between the reedand the upstream and downstream chambers. Therefore, thereed can only move if an aerodynamic force F Bexists due to flow detachment with an ensuing reattachment.25This aero- dynamic force can explain the movement of the reed duringthe transient state of the flow but its existence alone is, how-ever, insufficient to explain a self-sustained oscillatory re-gime. This can only happen when the net energy exchangedbetween the flow and the reed during one duty cycle is posi-tive: E=/H20848 0TFB·y˙tip/H110220, where FBis the space averaged aero- dynamic force on the reed and y˙tipis the velocity of the reed measured at its tip. In other words, the amount of energyabsorbed by the reed from the flow during one duty cycle hasto be greater than the energy imparted to the flow by thereed. As explained by Hirschberg, 25in the absence of acous- tic coupling, a positive net energy after one duty cycle ispossible due to several reasons: /H20849a/H20850the difference in the reed channel geometry between opening and closing phase; /H20849b/H20850 the inertia of the flow in the channel; 18and /H20849c/H20850variability of the separation/reattachment point behavior.21 A. General results Figure 6depicts the time histories associated with dis- placement of the reeds measured at their tips for all geom-etries. The self-sustained oscillation regime is achieved forall geometries in Fig. 4. The long channel geometries de- picted in Figs. 4/H20849b/H20850and4/H20849c/H20850present very similar behavior with high oscillation amplitudes, which forces the tip of thereed to close the channel completely. For the geometry withthe short channel /H20851Fig.4/H20849a/H20850/H20852, the reed oscillation is roughly FIG. 5. Comparison between theory and numerical results for a stationary reed: /H20849a/H20850Vena contracta factor as a function of the modified Reynolds num- ber and /H20849b/H20850pressure difference across the reed channel as a function of the volume flow. FIG. 6. Time histories of the reed dis-placement measured at the tip for dif-ferent channel geometries. 1804 J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55sinusoidal and the average value of the tip displacement ytip/H112298.0/H1100310−4m. Long-channel geometries present similar oscillation periods that are /H112296.5% shorter than those found for the short-channel geometry. This vibratory behavior at afrequency close to the reed’s first natural frequency f 0was expected, given the absence of acoustic coupling between thereed and the downstream and upstream cavities. Further analysis was carried out by investigating the dy- namic characteristics of one single oscillation period. Theselected duty cycles are related to the sixth oscillation periodof each case and are indicated between dashed lines in Fig. 6. Figure 7/H20849b/H20850shows the normalized energy flows E˙ = FBy˙tipas a function of time in terms of fraction of one duty cycle. The negative areas indicate transfer of energy to theflow due to the work of the reed. They take place during thephases associated with the opening of the reed, as shown inFig. 7/H20849a/H20850. Conversely, the positive areas in Fig. 7/H20849b/H20850take place when the reed is closing and represent the energy ab-sorption by the reed due to flow work. In the regions ofnegative energy flow, y˙ tipandFBare out of phase but become in phase as the reed starts to close again. In all cases, theshift from negative to positive energy flow also coincideswith the maximum volume flow U, shown in Fig. 7/H20849c/H20850. These results present the same behavior found in the experimentsconducted by Thomson 23for an idealized model of the hu- man larynx. The high amplitudes of oscillation found in the case of long-channel geometries are explained by the higher ratiosbetween absorbed E +and lost energy E−during one cycle, as shown in Table II. The excess of energy given to the reed is dissipated internally by the viscous damping predicted by thethird term on the right-hand side of Eq. /H208496/H20850and by the in- elastic collision of the reed against the side lays of themouthpiece. Furthermore, Fig. 7/H20849b/H20850shows that the reeds in the long-channel geometries start to receive energy from theflow at 0.6 T of the duty cycle, which represents a delay of 0.13 T compared to the short channel geometry. This is dueto a higher flow inertia caused by larger fluid volume withinlong channels and due to the effect of flow driven by themoving reed, as will be discussed later in this paper. Table II presents some aspects related to the oscillation frequenciesachieved by each geometry, as well as aspects related to theenergy exchange between the flow and the reed. Figure 8provides a better understanding of the results presented in Figs. 7/H20849a/H20850–7/H20849c/H20850by depicting snapshots of the normalized velocity field u norm=/H20849ux2+uy2/H208501/2/max /H20849ux/H20850in the mouthpiece models, taken at four different instants within the same duty cycle. In all cases, a jet is formed at the channel’s entrance as the reed starts to open. At this point, the jet rapidly adheresto the rail tip but remains detached elsewhere. This situationcontinues until the gradient of pressures between the jet andthe reed is enough to force the jet to attach to the reed’ssurface. The gradient is originated by the entrainment of flowbetween the jet and the reed wall due to viscous momentumtransfer and it is proportional to the downstream volumeflow. This phenomenon, known as the Coanda effect, playsan important role in the self-sustained oscillations in vocal folds 20–24and in reed instruments such as the accordion19and the harmonium.18 During the opening stage the volume flow Uin the short channel accelerates earlier into the mouthpiece chamber. Infact, for the same channel aperture y tip, the volume flow into the short channel is much higher than that into the long- FIG. 7. Oscillation characteristics as function of time in terms of fraction of one duty cycle: /H20849a/H20850channel aperture, /H20849b/H20850normalized energy flow, and /H20849c/H20850 volume flow. J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces 1805 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55channel geometries, as shown in Figs. 7/H20849a/H20850and7/H20849c/H20850. The early acceleration provides the necessary pressure gradientfor the jet to detach from the rail tip and adhere on the reedat/H112290.5 T, in contrast with the long channel geometries in which the same phenomenon happens at /H112290.7 T, as depicted in Fig. 8. The separation/adhesion phenomenon is confirmed by the determination of the skin friction based on the shearstress on the reed surface. As already mentioned, there are two explanations for the early volume acceleration in the case of the short channel.First, the fluid volume within the channel has a reduced in-ertia. The second reason is attributed to the effect of the flowdriven by the reed U wall. This is because, in the case of a dynamic regime, the effective volume flow can be expressedbyU=U /H9004p+Uwall, where U/H9004pis the flow driven by the pres- sure difference /H9004pacross the reed channel. Thus, during theopening stage the reed exerts work on the flow by pulling it out of the mouthpiece chamber in the upstream direction,which means that U /H9004pandUwallare out of phase. In short channels, the influence of Uwallon the effective flow Uis much smaller than in the case of long channels, which ex-plains the early acceleration. The effect of U wallalso becomes significant at instants near the complete closure of the channel /H208490.9T/H33355t/H333551/H20850. Dur- ing this period, U/H9004pandUwallare in phase and Uwallmay become higher than U/H9004p, which could explain the consider- able unsteadiness of the flow at this fraction of the dutycycle. This phenomenon has been reported by Devergeet al. 45in the case of experiments involving prototypes of the human glottis. In their observations, however, the effect ofU wallseems to be more evident in channels with constantTABLE II. Aspects of dynamic flow in the different channel profiles. L/hf /H20849Hz/H20850 f/f0 E /H20841E+/E−/H20841 Geometry 1 1 1760.3 1.00 120.28 1.10 Geometry 2 4 1877.2 1.07 188.59 1.24Geometry 3 4 1855.7 1.06 179.30 1.22 FIG. 8. /H20849Color online /H20850Snapshots of the velocity field for different instants within the same duty cycle: /H20849a/H20850L/h=1, /H20849b/H20850L/h=4, and /H20849c/H20850L/h=4 chamfered. 1806 J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55height. This fact contrasts with our case in which the reed channel becomes divergent near the closure stage. Furthermore, the aerodynamic force FBcaused by the pressure gradient increases when the jet attaches to the reedand becomes proportional to the attachment length. This ex-plains the higher oscillation amplitudes in geometries 2 and3. The increase in F Bacts to decelerate the reed until it stops. At this point, y˙tipandFBbecome in phase and the reed starts to receive energy from the flow. The stronger FBin long channels also explains the positive pitch shift in these geom-etries, because a stronger F Bforces the reed to close more rapidly. B. Discrepancy from the quasistationary predictions The snapshots of flow during one duty cycle depicted in Fig.8show some fundamental deviations between the qua- sistationary assumptions and the numerical results regardingthe detachment/adhesion phenomenon. In the case of theshort channel geometry, Fig. 8/H20849a/H20850, the constant fully sepa- rated flow assumed in the quasistationary theory has not beenobserved. In fact, for the first half of the duty cycle the flowis detached from the reed but remains attached to the rail tip.For the second half of the duty cycle the flow attaches toboth reed and rail tip. The results for geometries 2 and 3, Figs. 8/H20849b/H20850and8/H20849c/H20850, are very self-similar. The presence of a chamfer in geometry3 did not play a significant role on the stability of the attach-ment phenomenon. In those cases, the flow remains detachedfrom the reed for nearly /H1122970% of the duty cycle. At /H112290.7 T, the flow adheres to the reed and gradually detaches from therail tip until the complete channel closure. This pattern con-trasts with the theory, which assumes a constant separationregion between the channel’s entrance and l r=2hand full attachment of the flow afterwards. As expected, the numerical results for the vena contracta factor also diverge considerably from the theoretical predic-tions. Figure 9depicts the comparison between numerical and theoretical values for /H9251along one duty as a function of the modified Reynolds number proposed by van Zon et al. The hysteresis observed for all cases in Fig. 9agrees quali- tatively with that found in a dynamic flow measurement con-ducted by van Zon et al. 13The hysteresis observed in the short channel geometry /H20851Fig.9/H20849a/H20850/H20852is much smaller than that observed in the remaining cases /H20851Figs. 9/H20849b/H20850and9/H20849b/H20850/H20852. This is probably due to a less significant influence of the flow drivenby the reed U wall, as previously discussed. Furthermore, the flow adhesion segment is much shorter in the case of geom-etry 1, which minimizes the contribution of shear dissipationon the hysteresis. Figure 10depicts the numerical values of /H9251 as a function of time in terms of fraction of a duty cycle. For the short channel, the values of /H9251remain constant for only 35% of the duty cycle, namely 0.50 T /H33355t/H333550.85 T. The val- ues of /H9251become very unstable as the reed approaches the closed position. As already discussed, this characteristic isattributed to the effect of U wall, which becomes higher than the flow driven by the pressure difference across the reedchannel.V. CONCLUSIONS We propose a numerical technique based on the lattice Boltzmann and finite difference methods to represent theproblem of fully coupled fluid-structure interaction in singlereed mouthpieces. The model provides second-order accu-racy at representing boundary layer phenomena and was usedto evaluate the behavior of three different reed channel ge-ometries in two types of regimes, namely, stationary and dy-namic. The stationary results agree very well with those pre-dicted by the quasistationary theory, in terms of volume flow FIG. 9. Numerical and theoretical results for the vena contracta factor as function of the modified Reynolds number: /H20849a/H20850geometry 1, /H20849b/H20850geometry 2, and /H20849c/H20850geometry 3. J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces 1807 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55and vena contracta factor. Furthermore, we observed the same behavior found experimentally by van Zon et al. ,13 associated with the steadiness of the vena contracta factor for different Reynolds numbers, in the case of short channels,and with the steadiness of the detachment / reattachmentphenomenon in long channels. However, the results obtained during the dynamic simu- lations are very different from those predicted by the quasis-tationary theory. For the short channel geometry, /H9251was found to be constant for only /H1122940% of the duty cycle, and for long channels, the values of /H9251were in stark disagreement with the quasistationary predictions. Moreover, the patternsobserved in stationary measurements such as fully detachedflow, in the case of short reed channels, and the twofoldpattern, in the case of long channels, were not observed inthe dynamic simulations. The main difference in the flowbehavior between short and long channels was found to bethe time taken by the flow to adhere on the reed wall withinone duty cycle. This characteristic was attributed to the effectof inertia associated with different fluid volumes within thereed channel and to the flow driven by the reed. The resultsalso show that different levels of self-sustained oscillationscan be achieved in the absence of acoustic feedback due tothe complexity of hydrodynamic forces acting on the reed,which supports the hypothesis proposed by Hirschberget al. 12,25in the case of single reed mouthpieces. The two-dimensional nature of our numerical approach restricts the results to a qualitative analysis. Another limita-tion is associated with the lack of acoustic feedback, whichneglects eventual influences of the acoustic field on the flowwithin the reed channel. Nevertheless, we feel it is worth-while to focus on the aerodynamicaly oscillating situationpresented in this paper. The widespread assumptions made inmodeling wind instrument reeds that have been reportedmany times in this journal and others are based on a quasis-tationary assumption that itself does not take into account theinfluence of the acoustic field on the flow behavior. Thesimulations reported in this paper show that there are signifi-cant deviations from these long held assumptions that callinto question the validity of the currently accepted model.These deviations might easily be obscured in the presence ofacoustic feedback. Furthermore, the approach presented inthis paper contributes to our understanding of the behavior ofdynamic flow in single-reed mouthpieces and its dependency on the characteristics of the reed channel geometry. More investigations are needed in order to understand the behavior of the dynamic flow when the acoustic couplingbetween mouthpiece-reed system and resonator is taken intoaccount. Another interesting step could be taken in order toinvestigate the mechanisms of energy transfer between flowand the acoustic field, as well as the characterization ofaeroacoustic sources in the mouthpiece and its contributionto the instrument’s sound content. ACKNOWLEDGMENTS The authors would like to thank Professor Luc Mogeau and the anonymous reviewers of this paper for their helpfulsuggestions. A.R.d.S wishes to thank CAPES /H20849Funding Council of the Brazilian Ministry of Education /H20850for support- ing his doctoral research. 1H. L. F. Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music /H20849Dover, New York, 1877 /H20850. 2H. Bouasse, Instruments à Vent (Wind Instruments) /H20849Blanchard, Paris, 1929 /H20850, Vols. I and II. 3S. S. Chen, “A general theory for dynamic instability of tube arrays in crossflow,” J. Fluids Struct. 1, 35–53 /H208491987 /H20850. 4J. Backus, “Small-vibration theory of the clarinet,” J. Acoust. Soc. Am. 35, 305–313 /H208491963 /H20850. 5W. E. Worman, “Self-sustained nonlinear oscillations of medium ampli- tude in clarinet-like systems,” Ph.D. thesis, Case Western Reserve Univer-sity, Cleveland, OH, 1971. 6T. A. Wilson and G. S. Beavers, “Operating modes of the clarinet,” J.Acoust. Soc. Am. 56, 653–658 /H208491974 /H20850. 7N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments /H20849Springer, New York, 1998 /H20850. 8N. H. Fletcher, “Air flow and sound generation in musical instruments,” Annu. Rev. Fluid Mech. 11, 123–146 /H208491979 /H20850. 9J. Saneyoshi, H. Teramura, and S. Yoshikawa, “Feedback oscillations in reed woodwind and brasswind instruments,” Acustica 62, 194–210 /H208491987 /H20850. 10J. Kergomard, “Elementary considerations on reed-instruments oscilla- tions,” in Mechanics of Musical Instruments , edited by A. Hirschberg, J. Kergomard, and G. Weinreich /H20849Springer, Wien, 1995 /H20850. 11S. Ollivier, “Contribution á l’étûde des oscillations des instruments à vent à anche simple: Validation d’un modèle élémentaire /H20849Contribution to the study of oscillations in single-reed instruments: Validation of an elemen-tary model /H20850,” Ph.D. thesis, University of Maine, Maine, 2002. 12A. Hirschberg, R. W. A. van der Laar, J. P. Marrou-Mauriere, A. P. J. Wijnands, H. J. Dane, S. G. Kruijswijk, and A. J. M. Houtsma, “A quasi-stationary model of air flow in the reed channel of single-reed wind in-struments,” Acustica 70, 146-154 /H208491990 /H20850. 13J. van Zon, A. Hirschberg, J. Gilbert, and A. P. J. Wijnands, “Flow through the reed channel of a single reed instrument,” in Congrs Franais d’Acoustique Sup. J. Phys. Colloque de Physique, Paris, Vol. 54, pp. 821- 824 /H208491990 /H20850. 14A. M. C. Valkering, “Characterization of a clarinet mouthpiece,” Technical Report No. R-1219-S, Vakgroep Transportfysica, TUE, Eindhoven, 1993. 15J. P. Dalmont, J. Gilbert, and S. Ollivier, “Nonlinear characteristics ofsingle-reed instruments: Quasistatic volume flow and reed opening mea-surements,” J. Acoust. Soc. Am. 114, 2253–2262 /H208492003 /H20850. 16L. Maurin, “Confrontation théorie-expérience des grandeurs d’entrée d’un excitateur à anche simple /H20849Theoretical and experimental comparisons of input variables in a single-reed oscillator /H20850,” Ph.D. thesis, University of Maine, Main, 1992. 17J. Gilbert, “Étude des instruments de musique à anche simple /H20849A study of single-reed musical instruments /H20850,” Ph.D. thesis, University of Maine, Maine, 1991. 18A. O. St-Hilaire, T. A. Wilson, and G. S. Beavers, “Aerodynamic excita-tion of the harmonium reed,” J. Fluid Mech. 49, 803–816 /H208491971 /H20850. 19D. Ricot, R. Causse, and N. Misdariis, “Aerodynamic excitation and sound production of a blown-closed free reeds without acoustic coupling: The FIG. 10. Numerical values for the vena contracta factor as function of timefor one duty cycle. 1808 J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55example of the acordion reed,” J. Acoust. Soc. Am. 117, 2279–2290 /H208492005 /H20850. 20X. Pelorson, A. Hirschberg, R. R. van Hassel, A. P. J. Wijnands, and Y. Auregan, “Theoretical and experimental study of quasisteady-flow separa-tion within the glottis during fonation. Application to a modified two-massmodel,” J. Acoust. Soc. Am. 96, 3416–3431 /H208491994 /H20850. 21B. D. Erath and M. W. Plesniak, “The occurrence of the coanda effect in pulsatile flow through static models of the human vocal folds,” J. Acoust.Soc. Am. 120, 1000–1011 /H208492006 /H20850. 22S. L. Thomson, L. Mongeau, and S. H. Frankel, “Aerodynamic transfer of energy to the vocal folds,” J. Acoust. Soc. Am. 118, 1689–1700 /H208492005 /H20850. 23S. L. Thomson, “Fluid-structure interactions within the human larynx,” Ph.D. thesis, Purdue University, West Lafayette, IN, 2004. 24I. R. Titze, “The physics of small-amplitude oscillations of the vocalfolds,” J. Acoust. Soc. Am. 83, 1536–1552 /H208491988 /H20850. 25A. Hirschberg, J. Gilbert, A. P. Wijnands, and A. M. C. Valkering, “Mu- sical aero-acoustics of the clarinet,” J. Phys. IV 4, 559–568 /H208491994 /H20850. 26A. H. Benade, Fundamentals of Musical Acoustics /H20849Oxford University Press, New York, 1976 /H20850. 27W. Shyy, Computational Fluid Dynamics with Moving Boundaries , Series in Computational Methods and Physical Processes in Mechanics and Ther-mal Sciences /H20849Taylor and Francis, New York, 1995 /H20850. 28M. Liefvendahl and C. Troeng, “Deformation and regeneration of the computational grid for cfd with moving boundaries,” in Proceedings of the 45th AIAA Aerospace Science Meeting , Nevada, 2007. 29F. Avanzini and M. van Walstijn, “Modelling the mechanical response of the reed-mouthpiece-lip system of a clarinet. 1. A one-dimensional distrib-uted model,” Acta Acust. 90, 537–547 /H208492004 /H20850. 30P. Lallemand and L. S. Luo, “Lattice Boltzmann method for moving boundaries,” J. Comput. Phys. 184, 406–421 /H208492003 /H20850. 31X. M. Li, R. C. K. Leung, and R. So, “One-step aeroacoustics simulation using lattice Boltzmann method,” AIAA J. 44, 78–89 /H208492006 /H20850. 32X. He and L. S. Luo, “Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation,” Phys. Rev. E 56, 6811–6817 /H208491997 /H20850. 33S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond/H20849Oxford University Press, Oxford, 2001 /H20850. 34D. A. Wolf-Gladrow, Lattice Gas Cellular Automata and Lattice Boltz- mann Models: An Introduction , Lecture Notes in Mathematics, Vol. 1725 /H20849Springer, Berlin, 2004 /H20850. 35Y. Qian, D. d’Humires, and P. Lallemand, “Lattice BGK models for the Navier-Stokes equation,” Europhys. Lett. 17, 479–484 /H208491992 /H20850. 36Y. Qian, S. Succi, and S. A. Orszag, “Recent advances in lattice Boltz- mann computing,” Annu. Rev. Comput. Phys. 3, 195–242 /H208491995 /H20850. 37P. L. Bhatnagar, E. P. Gross, and M. Krook, “A model for collision pro- cesses in gases. I. Small amplitude processes in charged and neutral one-component systems,” Phys. Rev. 94, 511–525 /H208491954 /H20850. 38B. Crouse, D. Freed, G. Balasubramanian, S. Senthooran, P. Lew, and L. Mongeau, “Fundamental capabilities of the lattice Boltzmann method,” inProceedings of the 27th AIAA Aeroacoustics Conference , Cambridge, 2006. 39A. Chaigne and V. Doutaut, “Numerical simulations of xylophones. 1.Time-domain modeling of the vibrating bars,” J. Acoust. Soc. Am. 101, 539–557 /H208491997 /H20850. 40M. Bouzidi, M. Firdaouss, and P. Lallemand, “Momentum transfer of a Boltzmann-lattice fluid with boundaries,” Phys. Fluids 13, 3452–3459 /H208492001 /H20850. 41E. W. S. Kam, R. M. C. So, and R. C. K. Leung, “Non-reflecting boundary for one-step lbm simulation of aeroacoustics,” in 27th AIAA Aeroacous-tics Conference, Cambridge, MA, 2006, pp. 1–9. 42R. H. Kirchhoff, Potential Flows ,Mechanical Engineering /H20849CRC, New York, 1985 /H20850. 43P. Durrieu, G. Hofmans, G. Ajello, R. Boot, Y. Aurégan, A. Hirchberg, and M. C. A. M. Peters, “Quasisteady aero-acoustic response of orifices,” J. Acoust. Soc. Am. 110, 1859–1872 /H208492001 /H20850. 44R. D. Blevins, Applied Fluid Dynamics Handbook /H20849Van Nostrand Hein- hold, New York, 1984 /H20850. 45M. Deverge, X. Pelorson, C. Vilain, P.-Y. Lagreé, F. Chentouf, J. Willems, and A. Hirschberg, “Influence of collision on the flow through in-vitrorigid models of the vocal folds,” J. Acoust. Soc. Am. 114, 3354–3362 /H208492003 /H20850. J. Acoust. Soc. Am., Vol. 122, No. 3, September 2007 da Silva et al. : Dynamic flow in single-reed mouthpieces 1809 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 216.165.126.139 On: Sun, 05 Oct 2014 02:43:55

1.2165599.pdf

Effect of thermal fluctuation field on the noise performance of a perpendicular recording system Sharat Batra, Werner Scholz, and Thomas Roscamp Citation: Journal of Applied Physics 99, 08E706 (2006); doi: 10.1063/1.2165599 View online: http://dx.doi.org/10.1063/1.2165599 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/99/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin-transfer torque magnetization reversal in uniaxial nanomagnets with thermal noise J. Appl. Phys. 114, 033901 (2013); 10.1063/1.4813488 Exchange spring media for perpendicular recording Appl. Phys. Lett. 87, 012504 (2005); 10.1063/1.1951053 The effect of thermal fluctuation on tilted perpendicular media J. Appl. Phys. 97, 10E314 (2005); 10.1063/1.1852853 Write field gradient effect on perpendicular recording J. Appl. Phys. 93, 7837 (2003); 10.1063/1.1557755 Effect of pole tip anisotropy on the recording performance of a high density perpendicular head J. Appl. Phys. 93, 6543 (2003); 10.1063/1.1555377 [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: 129.100.253.50 On: Mon, 24 Nov 2014 19:59:42Effect of thermal fluctuation field on the noise performance of a perpendicular recording system Sharat Batra,a/H20850Werner Scholz, and Thomas Roscamp Seagate Research, 1251 Waterfront Place, Pittsburgh, Pennsylvania 15212 /H20849Presented on 1 November 2005; published online 20 April 2006 /H20850 We have studied the effect of thermal fluctuations on the transition jitter of a perpendicular recording system using a micromagnetic recording model. A stochastic thermal field term is addedto the total effective field in the Landau-Lifshitz-Gilbert equation of motion to simulate thermalfluctuations. The recording model uses a head field generated by a finite element model in thepresence of a soft underlayer. A Monte Carlo simulation of 300 isolated transitions is done tocalculate the variation in transition jitter. Transition jitter is significantly increased in the presenceof a stochastic thermal field for different media parameters. Our results highlight that the thermalfluctuation fields must be included in a realistic assessment of the system performance. © 2006 American Institute of Physics ./H20851DOI: 10.1063/1.2165599 /H20852 I. INTRODUCTION It is believed by many in the storage industry that con- ventional perpendicular magnetic recording1,2will be able to extend the areal density up to 1000 Gb/in2. The recording industry has relied on scaling down the grain size to reducemedia noise while simultaneously increasing the anisotropyfield H Kand the coercivity of the media for thermal stability. As grain volume decreases, the stochastic thermal fluctuationfield increases the probability for magnetization to switch itsorientation. 3The thermally assisted switching gives rise to the time dependence of magnetization decay.4,5Given that hard disk drives are not operated at 0 K, a question is raisedon the effect of thermal fluctuations on the quality of writtentransition at drive operating temperatures. The effect of ther-mal fluctuations can degrade the transition as it is written andwill lead to a low signal-to-noise ratio /H20849SNR /H20850for the record- ing system. Media damping plays a critical role in the writeprocess as it defines how quickly magnetization settles intoits final state. In this paper, we have explored the effects ofthermal fluctuations and media damping on the quality ofwritten transitions. II. MODEL A finite element model6/H20849FEM /H20850is used to generate head fields in the presence of a soft underlayer /H20849SUL /H20850. In this model, the SUL is treated as a nonlinear, magnetic permeablematerial with an initial relative permeability value of 500.The write field profiles obtained by the FEM are used in amicromagnetic media model 7to simulate the recording pro- cess. We have also used a large-scale micromagneticmodel 8,9to model the write head and the SUL to compute the write field. The dynamic write process uses the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation of motion to calculate the temporal evolution of the magnetization vector of each grainwithin the medium. The effect of the thermal fluctuation fieldis included in micromagnetic simulations by adding a sto-chastic thermal field to the local magnetic field. 3After add- ing the thermal field, we get the stochastic LLG equationwith H locshown below, dm dt=−/H9253 1+/H92512m/H11003/H20851Hloc+/H9251m/H11003Hloc/H20852. where Hloc=HK+Hdemagnetization +Hexchange +Happlied +Hthermal . The thermal field is modeled as a Gaussian stochastic pro-cess with the following statistical properties: /H20855H th,i/H20849t/H20850/H20856=0 , /H20855Hth,i/H20849t/H20850Hth,j/H20849t/H11032/H20850/H20856=2/H9251kBT /H9253VM s/H9254ij/H9254/H20849t−t/H11032/H20850. The medium has a pseudo-Voronoi grain structure, which is based on a simulated grain growth process using a regularlyspaced /H208491.5 nm /H20850lattice of square subgrains. This naturally leads to a log-normal grain-size distribution. The mediumalso includes a log-normal distribution of the crystalline an-isotropy field magnitude and a Gaussian distribution for itsorientation. Demagnetizing fields are calculated by Fouriertransforms over the underlying uniform grid. Written patternsare calculated by integrating the Landau-Lifshitz-Gilbertequation as the head passes over the media. The written tran-sition pattern is read back using a read sensitivity function inconjunction with a reciprocity-based read process. 7The read process does not include any time-dependent degradation ofwritten transition. A Monte Carlo simulation of 300 isolatedtransitions is done to calculate the transition jitter as thevariation in the zero-crossing position of the read-back volt-age of each transition. The transition parameter is calculatedby an hyperbolic tangent fit to the isolated transition. We usea bootstrap method 10to calculate the lower and upper bounds of the transition jitter and transition parameter at 95% confi-dence level. III. RESULTS AND DISCUSSION Several cases of heads and media parameters were mod- eled with and without the thermal fluctuation field. These areshown as cases 1–12 in Table I. First we discuss the results a/H20850Electronic mail: sharat.batra@seagate.comJOURNAL OF APPLIED PHYSICS 99, 08E706 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08E706/3/$23.00 © 2006 American Institute of Physics 99, 08E706-1 [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: 129.100.253.50 On: Mon, 24 Nov 2014 19:59:42for a head design presented previously.11The head 1 design includes a down-track shield /H20849DS /H20850. The modeled head has a physical track width of 40 nm, a relatively thick pole /H20849320 nm /H20850, a short throat length /H2084920 nm /H20850, and a head-to-media spacing /H20849HMS /H20850of 5 nm to maximize the write field. The maximum effective field /H20849Heff/H20850for this head at a distance of 10 nm from the pole is calculated to be 1.87 T. The relevant media parameters are 4 /H9266Msof 579 emu/cm3, thickness of 12 nm, and grain size of 5.7 nm. Anisotropy field HK, damp- ing constant /H20849/H9251/H20850, and intergranular exchange hefor two dif- ferent media are listed in Table I. In addition, a log-normal grain size with volume distribution of 30%, a log-normaldistribution in the crystalline anisotropy field of 5% in mag-nitude, and a Gaussian distribution for its orientation of 3.5°are assumed. Figure 1 shows the results of a Monte Carlo simulation for an isolated transition. A significant transitioncurvature is observed, and a jitter value of 1.63 nm is ob-tained for case 1. When a thermal field corresponding to atemperature of 293 K is turned on, we see an increase in thewidth of the written transition. This effect is understood byrealizing that the coercivity of the media is lowered at highertemperatures, leading to a larger write bubble. The largerwrite bubble leads not only to a wider write width but also toan increase in the transition parameter /H20849a parameter /H20850from 4.67 to 4.91 nm and a shift in the transition location from5.78 to 9.43 nm. In addition, the transition gets noisier, asconfirmed by the transition jitter value of 2.16 nm /H20849case 2 /H20850. This effect is shown in Fig. 2. Next, we have used a differenthead configuration that gives a larger head field /H20849H effof 2.2 T at a distance of 10 nm from the pole /H20850and a higher field gradient and is able to write on the higher HKmedia. Again, we see an increase in aparameter from 3.29 to 3.70 nm, with a corresponding increase in jitter from 1.06 to 1.37 nm /H20849cases 3 and 4 /H20850and a shift in the transition location from 3.86 to 5.42 nm as stochastic thermal fluctuation fields are turned on. We use the system criterion that jitter is less than 10% of the bit cell as an estimate of linear density. Based on thiscriterion, the linear density will degrade from 1558 kbpi/H20849case 1 /H20850to 1176 kbpi /H20849case 2 /H20850for head 1, i.e., a decrease of 24% in the presence of thermal field. The results for the head2 that is capable of producing larger field and a larger fieldgradient at the media show that linear density will degrade from 2405 kbpi /H20849case 3 /H20850to 1854 kbpi /H20849case 4 /H20850, i.e., a degra- dation of 23% in the presence of thermal fluctuation field.The increase in jitter value includes all aspects of writing andis a complex function of write field, write field gradient, andmedia parameters such as H K, grain size, 4 /H9266Ms, and ex- change parameter he. We attribute the increase in the jitter values to the fact that the thermal fluctuation field is smallyet significant in comparison to the total effective field with-out the thermal fluctuation field term. Next, we have modeled media with different damping values “ /H9251” ranging from 0.05 to 0.4 corresponding to cases 5, 6, 7, and 8 using head 1. As the damping value is in-TABLE I. Simulation parameters and results. CaseHK /H20849kOe /H20850Stochastic thermal field /H9251 he Jitter+/−95% confidencea parameter+/−95% confidence 1 15 off 0.10 0.09 1.63 0.13 4.67 0.14 2 15 on 0.10 0.09 2.16 0.19 4.91 0.193 17 off 0.10 0.07 1.06 0.08 3.29 0.094 17 on 0.10 0.07 1.37 0.12 3.70 0.115 15 off 0.05 0.09 2.02 0.15 4.96 0.166 15 off 0.10 0.09 1.63 0.13 4.67 0.147 15 off 0.20 0.09 1.31 0.10 4.21 0.128 15 off 0.40 0.09 1.14 0.09 4.02 0.119 15 on 0.05 0.09 2.44 0.19 5.19 0.21 10 15 on 0.10 0.09 2.16 0.19 4.91 0.19 11 15 on 0.20 0.09 1.78 0.12 4.62 0.16 12 15 on 0.40 0.09 1.76 0.13 4.43 0.15 FIG. 1. Monte Carlo simulation results using micromagnetic write model for an isolated transition showing an increase in write width. The upperpanel is without thermal fluctuation field and the lower panel includes ther-mal fluctuation field.08E706-2 Batra, Scholz, and Roscamp J. Appl. Phys. 99, 08E706 /H208492006 /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: 129.100.253.50 On: Mon, 24 Nov 2014 19:59:42creased from 0.05 to 0.4, we observe a decrease in the jitter value and a decrease in the a parameter. These values withassociated 95% confidence level /H20849average of +/−95% /H20850are listed in Table I. This is expected as an increasing dampingvalue reduces the time it takes for magnetization to settleinto its final state. However, there is an optimal value ofdamping for the recording system when flux rise-time con-siderations are included. 12As the thermal fluctuation field term is included in the model, we see a similar decrease inthe absolute value of jitter and aparameter /H20849cases 9–12 /H20850. The results for the two cases for the transition jitter and the a parameter as functions of damping /H9251are plotted in Fig. 3. Our results show that the percentage increase in the jittervalue or jitter ratio /H20849thermal field on/thermal field off /H20850in- creases as a function of damping value /H9251. IV. CONCLUSIONS We have considered the effect of the thermal fluctuation field corresponding to a temperature of 293 K in micromag-netic simulations of a perpendicular recording system. Athigher temperatures, the size of the write bubble increasesdue to a reduction of the media coercivity. This leads to anincrease in the transition width and a shift in the transitionlocation. The inclusion of the thermal fluctuation field in-creases the probability of magnetization switching, thatcoupled with a larger write bubble, leads to a larger transitionjitter. Using a system criterion that transition jitter is lessthan 10% of the bit cell, we find that the inclusion of the thermal fluctuation field degrades the linear density by ap-proximately 24% for the two cases studied. Therefore, weconclude that including thermal fluctuation field degrades thesystem performance. We have studied this effect as a func-tion of damping value of the media. With increasing damp-ing value, the absolute values of the jitter decrease, confirm-ing that a sharper transition is written as media settle quicklyinto its final state. The inclusion of thermal fluctuation in-creases the probability of magnetization reversal of indi-vidual grains. Consequently, when thermal fluctuation fieldsare included, there is an increase in jitter value for all damp-ing values /H9251compared to the case when thermal fluctuation field is not included. Our results show that the percentageincrease in the jitter values in the presence of thermal fluc-tuation field increases for increasing damping values of themedia. The fact that operating temperature requirements forthe hard disk drives are between 300 and 350 K, the effect ofthermal fluctuation fields and media damping needs to beincluded to optimize a recording system for high areal den-sity. ACKNOWLEDGMENTS The authors thank Pierre Asselin and Jason Goldberg for their collaboration with the micromagnetic recording modeland Hong Zhou for valuable discussion for thermal stabilityof media. The authors are members of IEEE. 1M. Mallary, A. Torabi, and M. Benakli, IEEE Trans. Magn. 38,1 7 1 9 /H208492002 /H20850. 2K. Gao and H. N. Bertram, IEEE Trans. Magn. 38,3 6 7 5 /H208492002 /H20850. 3W. F. Brown, Phys. Rev. 130, 1677 /H208491963 /H20850. 4S. H. Charap, P.-L. Lu, and Y. He, IEEE Trans. Magn. 33,9 7 8 /H208491997 /H20850. 5D. Weller and A. Moser, IEEE Trans. Magn. 35, 4423 /H208491999 /H20850. 6FLUX3D , Magsoft Corp., 1223 Peoples Ave., Troy, New York. 7T. Roscamp, E. Boerner, and G. Parker, J. Appl. Phys. 91,8 3 6 6 /H208492002 /H20850. 8MAGPAR , parallel finite element micromagnetics package; http:// magnet.atp.tuwien.ac.at/scholz/magpar/ 9W. Scholz, J. Fidler, T. Schrefl, D. Suess, R. Dittrich, H. Forster, and V.Tsiantos, Comput. Mater. Sci. 28, 366 /H208492003 /H20850. 10B. Effron and B. R. J. Tibshirani, An introduction to the Bootstrap /H20849Chap- man and Hall, New York, 1993 /H20850. 11S. Batra, J. D. Hannay, H. Zhou, and J. S. Goldberg, IEEE Trans. Magn. 40,3 1 9 /H208492004 /H20850. 12W. Scholz and S. Batra, IEEE Trans. Magn. 41,7 0 2 /H208492005 /H20850. FIG. 3. Transition jitter and aparameter with and without thermal fluctua- tion field as functions of damping value /H9251. FIG. 2. Wave forms and associated noise for cases 1 and 2 based on aMonte Carlo simulation for 300 isolated transitions as thermal fluctuationfields /H20849lower panel /H20850are included.08E706-3 Batra, Scholz, and Roscamp J. Appl. Phys. 99, 08E706 /H208492006 /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: 129.100.253.50 On: Mon, 24 Nov 2014 19:59:42

1.4803050.pdf

Radio-frequency amplification property of the MgO-based magnetic tunnel junction using field-induced ferromagnetic resonance K. Konishi, D. K. Dixit, A. A. Tulapurkar, S. Miwa, T. Nozaki et al. Citation: Appl. Phys. Lett. 102, 162409 (2013); doi: 10.1063/1.4803050 View online: http://dx.doi.org/10.1063/1.4803050 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v102/i16 Published by the American Institute of Physics. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 02 May 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsRadio-frequency amplification property of the MgO-based magnetic tunnel junction using field-induced ferromagnetic resonance K. Konishi,1D. K. Dixit,2A. A. Tulapurkar,2S. Miwa,1T. Nozaki,3H. Kubota,3A. Fukushima,3 S. Yuasa,3and Y . Suzuki1 1Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan 2Department of Physics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India 3National Institute of Advanced Industrial Science and Technology (AIST), Spintronics Research Center, Ibaraki 305-8568, Japan (Received 9 January 2013; accepted 15 April 2013; published online 25 April 2013) The radio-frequency (RF) voltage amplification property of a tunnel magnetoresistance device driven by an RF external-magnetic-field-induced ferromagnetic resonance was studied. Theproposed device consists of a magnetic tunnel junction and an electrically isolated coplanar waveguide. The input RF voltage applied to the waveguide can excite the resonant dynamics in the free layer magnetization, leading to the generation of an output RF voltage under a DC biascurrent. The dependences of the RF voltage gain on the static external magnetic field strength and angle were systematically investigated. The design principles for the enhancement of the gain factor are also discussed. VC2013 AIP Publishing LLC [http://dx.doi.org/10.1063/1.4803050 ] The discovery of giant tunneling magneto-resistance (TMR) in MgO-based magnetic tunnel junctions (MTJs)1,2 accelerates the development of spin devices using MTJs, such as reading heads for hard disk drives (HDDs) and spin-transfer torque magnetic random access memories(STT-MRAMs). 3Further, spin-torque oscillators4and spin- torque diode effect5using MgO-based MTJs have attractive characteristics because of their large magneto-resistanceeffect. In addition to these devices, a MgO-based MTJ can be used to amplify radio-frequency (RF) signals with frequency tunability. Thus far, several RF amplification properties of anMTJ structure have been proposed and demonstrated experi- mentally. The study by Slonczewski 6was the first to propose the above mentioned properties. The other concepts of RFamplification have been proposed using negative differential resistance, 7vortex-core resonance,8,9and spin-torque-induced ferromagnetic resonance in an MTJ.10These devices for RF amplification were combined with the spin-transfer torque effect.11,12On the other hand, we proposed a spin transistor using an MTJ, which is driven by a current-induced magneticfield pulse, and demonstrated a direct current (DC) power gain of more than 1 at room temperature. 13In this paper, we present the RF amplification properties in the current-inducedmagnetic field driven spin transistor. A schematic structure of the proposed device with the measurement circuit is shown in Fig. 1(a). When a coplanar waveguide (CPW) is combined with an MTJ, an RF magnetic field is generated around the CPW under an application of RF current to the CPW. Further, ferromagnetic resonance (FMR) isinduced when the frequency of the RF magnetic field is tuned to the resonant frequency of magnetization of the free layer in the MTJ. Generally, magnetization precession can be excitedefficiently under the resonance c ondition, leading to the genera- tion of a large RF output voltage from the MTJ under a large DC bias current. If the output voltage exceeds the input voltage,the proposed device can be used for the amplification of RFs. MTJ films with a structure of buffer layers/PtMn/CoFe/ Ru/CoFeB(3)/MgO(1.1)/CoFeB(3)/Ru(1.5)/NiFe(2)/cappinglayers (nm in thickness) were deposited on Si/SiO 2substrates using a magnetron sputtering method (Canon ANELVAC7100). From the current-in-plane tunneling (CIPT) meas- urements, the resistance-area product and the magneto- resistance ratio were evaluated to be 3.8 Xlm 2and 110%, respectively. The multilayer film was patterned into junc- tions (0.3 /C20.6lm2) with an electrically isolated coplanar waveguide. The RF power, which generates the RF magneticfield, was applied at port-1 ( V in) of the vector network ana- lyzer (VNA). Further, a bias current was applied to the MTJ through a DC port of the bias Tee. The output signal fromthe MTJ ( V out) was detected by port-2 of the VNA. The RF amplification property was evaluated by monitoring the S 21 (¼Vout/Vin) parameter. The magnetization configurations of the pinned and free layers and the magnetic field are shown in Fig. 1(b). The x-axis (direction perpendicular to the CPW) is defined to be parallel to the easy axis of the free layer. Theexternal magnetic field, H ext, is applied along the direction with angle hfieldfrom the x-axis. The S 21parameter was measured under various Hextandhfieldconditions. The input RF power and the DC bias current were fixed at 20 lW and /C08 mA in all the measurements. Here, the positive current is defined as one in which the electrons flow from the free layerto the pinned layer. In order to extract the FMR signal clearly, the back- ground signal, which originates from the transmission prop-erty of the CPW, was subtracted from the raw S 21data. The background signal was obtained under the sufficiently large Hext¼1 kOe applied along the direction of the x-axis, because the FMR could not be induced in this configuration. In this paper, DS21indicates the measured S 21parameter without the background. Fig. 2(a) shows the typical magneto-resistance (MR) curve when hfieldis 0/C14. The obtained MR ratio became three times smaller than that obtained by the CIPT measurement,owing to the influence of the parasitic series resistance. A shift field of 50 Oe was obtained, which is caused by orange- peel coupling (a parallel state is preferred under zero 0003-6951/2013/102(16)/162409/4/$30.00 VC2013 AIP Publishing LLC 102, 162409-1APPLIED PHYSICS LETTERS 102, 162409 (2013) Downloaded 02 May 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsmagnetic field). The measured gain factor (i.e., the absolute value [ DS21], which is hereafter simply called DS21)a sa function of frequency is shown in Fig. 2(b), where hfieldwas fixed at 55/C14. Depending on the strength of the external mag- netic field, the amplitude of DS21and the resonant frequency were changed, indicating that the FMR excitation wasinduced by the application of the RF signal to the CPW. The value of DS 21becomes small when Hext¼50 Oe, possibly because of the inhomogeneous precession motion attributedto multi-domain formation. The observed resonant frequency was plotted as a function of the external magnetic field withh field¼55/C14in Fig. 2(c). The observed shift was well repro- duced by Kittel’s formula (red curve in the same figure). From the fitting, the in-plane and out-of-plane anisotropyfields were evaluated to be 15 Oe and 7500 Oe, respectively. Fig. 3(a) shows DS 21_max as functions of the strength and angle of Hext. Here, we define DS21_max as the peak height of DS21at each angle and external magnetic field. The red area indicates the high amplification gain, which reaches the maximum value of approximately 0.07. The output voltage from the MTJ ( Vout(x)) for small precession angles is described by Eq. (1)[see also supple- mentary material in Ref. 4] VoutðxÞ¼g0ðxÞRAP/C0RP 4ffiffi ffi 2p i0ðhprecÞsinðhtiltÞ; (1) where g0(x) is the efficiency of the RF circuit, RPandRAP are the resistances in parallel and anti-parallel states, respec- tively, i0is the bias current, hprecis the precession angle, and htiltdenotes the relative angle between the pinned and free layer magnetization. When hfield¼55/C14andHext¼100 Oe, a relative angle of htilt¼83/C14was obtained by resistance moni- toring. The largest RF gain factor was obtained at this field and angle. This behavior is consistent with Eq. (1). Furthermore, the resonant frequency decreases with a decrease in the external magnetic field, which in turn results in an increase in the precession angle. From the above FIG. 1. (a) Concept of the proposed device with the measurement circuit. The FMR in the free layer of the MTJ is induced by an RF magnetic field due to the application of an RF signal from port-1 of the VNA. Under DC bias, the MTJ produces an RF output, which is applied to port-2 of the VNA. The VNA measures the S 21(¼Vout/Vin) parameter, which represents the gain factor. (b) Schematic image of the configuration of magnetization and magnetic fields. M freeand M pindenote the magnetization of the free and pinned layer; htiltandhfieldindicate the relative angle and the angle of the external magnetic field; and hRF,Hext, and Hshiftindicate the RF magnetic field, the external magnetic field, and shifted field, respectively. FIG. 2. (a) Typical MR curve ( hfield¼0/C14). (b) Gain factor (absolute value [DS21]) as a function of frequency ( hfield¼55/C14). (c) Resonant frequency as a function of the external magnetic field ( hfield¼55/C14). The black dots and red curve show the experimental results and the fitting curve, respectively.162409-2 Konishi et al. Appl. Phys. Lett. 102, 162409 (2013) Downloaded 02 May 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsdiscussion, it can be concluded that it is important to reduce the resonant frequency and obtain a relative angle of 90/C14to obtain a large output. Here, we compare the experimental results with the sim- ulation results. Using the parameters obtained from our experiments and from Kittel’s fitting, we simulated the RFgain by using a macro-spin model based on the Landau- Lifshitz-Gilbert equation. Fig. 3(b) shows the simulated result as functions of H extandhfield. We assume that the damping factor a¼0.01 and temperature T¼300 K. Here, the influence of the thermal effect is treated as a random magnetic field.14The experimental results were well repro- duced qualitatively by this simple simulation. However, in the simulation, the maximum gain was obtained for both a small magnetic field and a small tilt angle at aroundH ext¼50 Oe and hfield¼15/C14. At this field and angle, the shift field and in-plane magnetic anisotropy are cancelled simulta- neously by the external magnetic field. Subsequently, theresonant frequency becomes very low and the RF gain increases. However, this condition is difficult to achieve in the experiment. Because the relevant size to obtain single do-main structure in a magnetic cell with in-plane magnetiza- tion is around 100 nm, 15a multi-domain structure can be easily formed in the low magnetic field. Further, the simu-lated gain is approximately 5 times greater than that obtained by the experimental result. In the experiment, the linewidth of the FMR peak is broadened because of non-uniform pre-cession attributed to a relatively large sample size. This leads to a small Q-value ( /C242) in the experiment, and consequently the gain factor becomes smaller than expected.The RF gain does not exceed 1 even in the simulation. One possible approach to realize high RF amplification is to introduce perpendicular magnetic anisotropy. When the out- of-plane anisotropy is cancelled by the external magneticfield, a high Q-value of 50 can be obtained. 16This value is 25 times greater than that obtained by our results. Thus, the RF gain factor can exceed 1 if we apply their condition (i.e.,using Fe-rich CoFeB free layer for perpendicular magnetic anisotropy 17,18and applying external magnetic field to can- cel demagnetization field) to our experiment. Another possible approach to enhance the RF gain factor is to utilize the spin transfer effect. In the present measure-ment conditions, the spin transfer torque has only a small influence on the RF gain, because h tiltis nearly 90/C14when the maximum DS21was obtained and the electric breakdown volt- age of the sample was relatively low. However, if the effect of the spin torque is sufficiently large, the effective damping fac- tor will be reduced,19inducing a large precession angle; hence, the output voltage from the MTJ can be increased, leading to the realization of a higher RF gain. This technique can be adapted to a magnetic field feedback oscillator.20 In summary, the amplification property of MTJs afforded by the magnetic-field-induced FMR was proposed and demonstrated. A maximum voltage gain of 0.07 wasachieved under the optimized external magnetic field condi- tion. In addition, the static external magnetic field strength and angle dependences of the voltage gain were systemati-cally investigated and reproduced qualitatively by a simple macro-spin model simulation. The improvement of the uni- form precession in the element, as well as the introduction ofperpendicular magnetic anisotropy or the spin transfer effect, were shown to be effective in achieving RF voltage gains greater than 1. This work was mainly supported by the New Energy and Industrial Technology Development Organization (NEDO) Spintronics Nonvolatile Device Project. K. Konishiis supported by a Research Fellowship of the JSPS for young scientists. 1S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando, Nat. Mater. 3, 868 (2004). 2S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, and S. H. Yang, Nat. Mater. 3, 862 (2004). 3T. Kishi, H. Yoda, T. Kai, T. Nagase, E. Kitagawa, M. Yoshikawa, K. Nishiyama, T. Daibou, M. Nagamine, M. Amano, S. Takahashi, M. Nakayama, N. Shimomura, H. Aikawa, S. Ikegawa, S. Yuasa, K. Yakushiji, H. Kubota, A. Fukushima, M. Oogane, T. Miyazaki, and K. Ando, Tech. Dig. – IEDM 2008 , 309. 4A. M. Deac, A. Fukushima, H. Kubota, H. Maehara, Y. Suzuki, S. Yuasa, Y. Nagamine, K. Tsunekawa, D. D. Djayaprawira, and N. Watanabe, Nat. Phys. 4, 803 (2008). 5A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S. Yuasa, Nature (London) 438, 339 (2005). 6J. C. Slonczewski, U.S. patent 5,695,864 (9 December 1997). 7H. Tomita, H. Maehara, T. Nozaki, and Y. Suzuki, J. Magn. 16, 140 (2011). 8S. Kasai, K. Nakano, K. Kondou, N. Ohshima, K. Kobayashi, and T. Ono,Appl. Phys. Express 1, 091302 (2008). 9T. Nozaki, H. Kubota, S. Yuasa, M. Shiraishi, T. Shinjo, and Y. Suzuki, Appl. Phys. Lett. 95, 022513 (2009). 10L. Xue, C. Wang, Y. T. Cui, J. A. Katine, R. A. Buhrman, and D. C. Ralph, Appl. Phys. Lett. 99, 022505 (2011). 11J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). FIG. 3. S 21_max as functions of hfieldandHext. (a) Experimental result and (b) simulated result.162409-3 Konishi et al. Appl. Phys. Lett. 102, 162409 (2013) Downloaded 02 May 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions12L. Berger, Phys. Rev. B 54, 9353 (1996). 13K. Konishi, T. Nozaki, H. Kubota, A. Fukushima, S. Yuasa, M. Shiraishi, and Y. Suzuki, Appl. Phys. Express 2, 063004 (2009). 14R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett. 92, 088302 (2004). 15F. J. Casta ~no, Y. Hao, S. Haratani, C. A. Ross, B. V €ogeli, M. Walsh, and H. I. Smith, IEEE Trans. Mag. 37, 2073 (2001). 16S. Ishibashi, T. Seki, T. Nozaki, H. Kubota, S. Yakata, A. Fukushima, S. Yuasa, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, and Y. Suzuki, Appl. Phys. Express 3, 073001 (2010).17S. Yakata, H. Kubota, K. Yakushiji, A. Fukushima, S. Yuasa, and K. Ando, J. Appl. Phys. 105, 07D131 (2009). 18H. Kubota, S. Ishibashi, T. Saruya, T. Nozaki, A. Fukushima, K. Yakushiji, K. Ando, Y. Suzuki, and S. Yuasa, J. Appl. Phys. 111, 07C723 (2012). 19S. Petit, C. Baraduc, C. Thirion, U. Ebels, Y. Liu, M. Li, P. Wang, and B. Dieny, Phys. Rev. Lett. 98, 077203 (2007). 20D. Dixit, K. Konishi, C. V. Tomy, Y. Suzuki, and A. A. Tulapurkar, Appl. Phys. Lett. 101, 122410 (2012).162409-4 Konishi et al. Appl. Phys. Lett. 102, 162409 (2013) Downloaded 02 May 2013 to 150.108.161.71. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions

1.3562214.pdf

Current-induced magnetization switching of synthetic antiferromagnetic free layer in magnetic tunnel junctions Seo-Won Lee and Kyung-Jin Lee Citation: J. Appl. Phys. 109, 07C904 (2011); doi: 10.1063/1.3562214 View online: http://dx.doi.org/10.1063/1.3562214 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v109/i7 Published by the American Institute of Physics. 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 07 Jun 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsCurrent-induced magnetization switching of synthetic antiferromagnetic free layer in magnetic tunnel junctions Seo-Won Lee and Kyung-Jin Leea) Department of Materials Science and Engineering, Korea University, Seoul 136-713, Korea (Presented 16 November 2010; received 24 September 2010; accepted 14 December 2010; published online 21 March 2011) Spin transport in a magnetic tunnel junction with a synthetic antiferromagnetic (SAF) free layer is investigated using the drift-diffusion model. Although the diffusive transport is inappropriate for the MgO tunnel barrier, the drift-diffusion model is found to capture the core features of in-plane spin-transfer torque (STT) through the tunnel barrier, and more importantly, it can describenon-negligible STT exerting on two ferromagnets in a SAF-free layer. STT in a SAF-free layer substantially changes the magnetization dynamics and induces a shift of the critical switching current. STT in a SAF-free layer suppresses current-induced parallel-to-antiparallel switching,whereas it encourages antiparallel-to-parallel switching. VC2011 American Institute of Physics . [doi: 10.1063/1.3562214 ] I. INTRODUCTION An electric current spin-polarized by a ferromagnet (FM) transfers its spin-angular momentum to a local magnet- ization of another FM, i.e., spin-transfer torque (STT).1In multilayer structures, it enables the full magnetization rever-sal 2and the steady precession motion of magnetization.3The current-induced magnetization switching and precession mode are applicable to magnetic memories and tunablemicrowave oscillators, respectively. Owing to the potential for these applications and interesting physics, STT has been extensively studied for over a decade. To analyze STT in multilayer structures, a number of transport theories have been developed with various approaches, ranging from a drift-diffusion theory in the diffu-sive regime 4–6to a quantum mechanical approach in the ballis- tic regime.7The drift-diffusion theory describes the spin transport in fully metallic multilayers, whereas the quantummechanical approach is appropriate for magnetic tunnel junc- tions (MTJ). Recently, a synthetic antiferromagnet (SAF), con- sisting of two FMs antiferromagnetically coupled via aRuderman–Kittel–Kasuya–Yosida (RKKY) interaction across a thin Ru layer, has been adopted in STT-active devices. Espe- cially, several experimental reports for MgO-based MTJs witha SAF-free layer (i.e., FM/MgO/FM/Ru/FM) are in debate about the usefulness of a SAF-free layer for a lower critical current density and a higher thermal stability. 8,9At h e o r e t i c a l study would be essential to resolve this debate. However, as it includes not only a tunnel barrier but also a metallic SAF-free layer, the spin transport of this system could not be classifiedinto either one of the transport theories. There have been sev- eral attempts to consider two regimes within one transport re- gime, 10but it is still an unconcluded issue. In this work, we investigated the spin transport in MTJ with a SAF-free layer in the frame of the drift-diffusion modelusing proper parameters for MgO and interfaces of MgO/FM. This model allows us to consider STT in a metallic SAF-freelayer as the spin transport in a SAF-free layer is diffusive. This model calculation suggests that when STT in a SAF-free layer is nonzero, it considerably affects the magnetization dynamics of the SAF-free layer. II. MODEL For the drift-diffusion model, we followed the formulas proposed by Barnas ´et al.5and extended it appropriately for the structure, including three ferromagnets. The structure used in the calculation was Cu(infinite)/Pinned layer(10)/ MgO(1)/FM1(2)/Ru( tRu)/FM2(2)/Cu(infinite) (all in nano- meters). Various Ru thicknesses of 0.6, 0.8, 1.2, and 2.2 nm were tested, corresponding to the antiferromagnetic cou- pling.8In this structure, STT is exerted on three interfaces; MgO/FM1, FM1/Ru, and Ru/FM2. STT exerted on FM1(STT FM1)i st h es u mo fS T T MgO/FM1 and STT FM1/Ru , and that exerted on FM2(STT FM2)i sS T T Ru/FM2 . In a drift- diffusion model, the magnitude of in-plane STT at each interface, sD in/C12/C12/C12/C12,i sg½m/C2ðm/C2pÞ/C138 jj ð ¼gsinhÞ,w h e r e gis he polarization factor and his angle between magnetizations of the free layer and the reference layer and it is described in the following: sD in/C12/C12/C12/C12¼/C0/C22h e2½RefG"#gðgycoshþgxsinhÞþImfG"#ggz/C138z¼d; (1) where in-plane magnetization is in the x–yplane, z-axis is the out-of-plane direction of current flow, giis the spin accu- mulation for electron spins oriented along the i-axis ( i¼x,y, andz),G"#is the spin-mixing conductance at each interface, anddis the thickness of nonferromagnetic layer adjacent to the interface. The polarization factor gdepending on the angle is obtained by sD in/C12/C12/C12/C12=sinh. With an obtained angle-de- pendent polarization factor, we simulated the magnetization dynamics of SAF-free layer within the macrospina)Author to whom correspondence could be addressed. Electronic mail: kj_lee@korea.ac.kr. 0021-8979/2011/109(7)/07C904/3/$30.00 VC2011 American Institute of Physics 109, 07C904-1JOURNAL OF APPLIED PHYSICS 109, 07C904 (2011) Downloaded 07 Jun 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsapproximation, using the Landau–Lifshitz–Gilbert (LLG) equation including the STT term written in dm dt¼/C0cm/C2Heffþam/C2dm dtþs; (2) where mis normalized magnetization vector, Heffis effective magnetic field, cis gyromagnetic ratio, ais damping parameter (¼0.02), and sstands for STT ( s¼sinþsout). In the STT term, in-plane STT is sin¼cð/C22h=2eÞðJ=MStÞg½m/C2ðm/C2pÞ/C138 and out-of-plane STT is sout¼bJm/C2p,w h e r e pis the mag- netization vector of the reference layer, Jis the current density, Msis the saturation magnetization, and tis the thickness of FM. It is known that the out-of-plane STT is non-negligible in MTJs,7,11however, in this work we ignored the out-of-plane STT in order to focus on the effect of STT in a SAF-free layer.The effective magnetic field in cludes the demagnetization, crystalline anisotropy field, dipolar, and RKKY-exchange cou- pling field. When the thicknesses of the Ru layer are 0.6, 0.8,1.2, and 2.2 nm, RKKY-exchange constants are assumed to be /C00.17, /C00.09, /C00.01, and /C00.02 erg/cm 2, respectively.8Satu- ration magnetizations of two FMs are 1000 emu/cm3and the crystalline anisotropy field along the x-axis is set as 10 Oe. The positive current corresponds to electrons flowing from PL to aSAF-free layer. The SAF-free lay er structure (FM1/Ru/FM2) is assumed to be patterned into 180 nm /C280 nm. In our con- vention, the parallel (P) [antiparallel (AP)] configuration is thatthe magnetizations of FM1 and FM2 are, respectively, aligned inþx-axis ( /C0x-axis) and /C0x-axis ( þx-axis) and the magnetiza- tion of PL is fixed along the þx-axis. The real part of G "#at the interface of MgO/FM (Re fG"#gMgO =FM) was assumed to be equal to that at Co/Cu [ ¼0.542 /C21015(Xm2)/C01]5whereas RefG"#gFM=Ruw a sa s s u m e dt ob e0 . 0 1 0 /C21015(Xm2)/C01.N o t e that Re fG"#gFM=Ruis unknown and an ab-initio calculation12 is required to find its true value. Owing to its ambiguity, we have varied the value of Re fG"#gFM=Ru, but found that it does not alter the main conclusion of this work. All of the parame- ters used in the calculation are listed in Table I. III. RESULTS AND DISCUSSION STT and the polarization factor at the interface of MgO/ FM1 are shown in Fig. 1(a). Here only the angle of FM1 magnetization changes, whereas the FM2 magnetization is fixed along the þx-axis. In a constant current (dotted lines), the angular dependence of STT at MgO/FM1 is asymmetric and the polarization factor has different values with theangle. In a constant voltage (solid lines), however, STT is symmetric and the polarization factor is constant. This dif-ferent angular dependence of STT is due to the strong varia- tion of the resistance with the angle shown in Fig. 1(b), calculated from the drift-diffusion model. The simple sin( h) dependence of STT in a constant voltage is one of the main features of STT through the MgO tunnel barrier. 7Thus, it implies that the drift-diffusion theory could describe the in-plane STT through MgO in a phenomenological way. The following simulations were performed based on the calculated angle-dependent polarization factors in a constantcurrent. The polarization factors at MgO/FM1, FM1/Ru, and Ru/FM2 are shown in Fig. 2when taking the STT across the Ru into account. In Fig. 2, one notable thing is that the polar- ization factors at FM1/Ru and Ru/FM2 are non-negligible compared with that at MgO/FM1. Thus, it is expected that this STT in a SAF-free layer could considerably affect themagnetization dynamics. Another interesting feature is that the polarization factors at interfaces of FM/Ru strongly depend on the angle of FM1, as well as the angle of FM2. Itis because the change in the angles causes a strong variation in the spin accumulation distribution in the whole layer structure, directly correlated to STT in the framework of thedrift-diffusion model. We calculated the switching probability P SWat 300 K for two cases where STT through Ru is considered (case I, solid line) and is ignored (case II, dotted line) [Fig. 3(a)]. Figure 3(b) summarizes the switching currents ISWat PSW¼0.5 for various cases. In case I, | ISW| for P-to-AP switching is larger than that for AP-to-P switching. This asymmetry is caused by the fact that the polarization factorsat the interface of MgO/FM1 for the initial P and AP states are 0.170 and 0.534, respectively [Figs. 3(c) and 3(d)].TABLE I. Spin transport parameters used in calculations.a Material or interfaceMeasured resistivity (mXcm)Bulk scattering asymmetry bMeasured RA (mXmm2)Interfacial scattering asymmetry cSpin–flip length (nm) PL,FM1,FM2 (CoFe) 7.00 0.60 50.00 Ru 9.50 0.00 14.00 Cu 0.50 0.00 500.00 MgO 1.00 /C21050.00 1.00 CoFe/Cu 0.48 0.77 CoFe/Ru 0.50 /C00.20 CoFe/MgO 3.00 /C21031.00 aReference 13. FIG. 1. (Color online) (a) Normalized spin-transfer torque and polarization factor in a constant voltage (solid line) and in a constant current (dashed line) (b) Calculated resistance from the drift-diffusion model.07C904-2 S.-W. Lee and K.-J. Lee J. Appl. Phys. 109, 07C904 (2011) Downloaded 07 Jun 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions|ISW|’s for both P-to-AP and AP-to-P switchings mostly decrease with increasing tRu[Fig. 3(b)]. In case II, | ISW|’s are smaller for P-to-AP switching, whereas they are larger for AP-to-P switching, as compared to case I. It means that STT through Ru suppresses the P-to- AP switching, whereas it enhances AP-to-P switching. This is because STT through Ru acts differently on a SAF-free layer for P-to-AP and AP-to-P switchings. The effect of STTon P-to-AP and AP-to-P switchings could be explained by the sign of STT. If the sign of STT exerted on an interface of FM is opposite (equal) to the magnetization of the STT-exerted FM, the magnetization is excited (stabilized). For P-to-AP switching, STT through Ru assists the excitation of FM1 but suppresses the excitation of FM2, and as a result, itcauses an increase of the switching current. On the other hand, for AP-to-P switching, the magnetization of FM1 is stabilized, but that of FM2 is excited by STT through Ru. Inthis case, the excitation of the SAF-free layer is mainly induced by STT at MgO/FM1 because of its relatively large value. Thus, the switching current slightly decreases.IV. CONCLUSION We investigated the spin transport in MgO-based MTJ with a metallic SAF-free layer in the frame of the drift-diffusion model. It was found that STT through Ru affects the magnetization dynamics o f a SAF-free layer and results in a shift of critical switching current. For P-to-AP switch-ing, STT through Ru stabilizes the magnetization of a SAF-free layer and induces the increases of switching cur- rents. On the other hand, for AP-to-P switching, theexcitation of a SAF-free layer is enhanced, resulting in a decrease of switching curre nt. Overall, the averaged switching current increases when taking into account STTthrough the Ru layer. ACKNOWLEDGMENTS This work was supported from the KOSEF through the NRL program funded by the Korean Ministry of Education, Science and Technology (Project No. M10600000198-06J0000-19810) and the DRC Program funded by the KRCF. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 54, 9353 (1996). 2E. B. Myers et al. ,Science. 285, 867 (1999). 3M. Tsoi et al .,Phys. Rev. Lett. 80, 4281 (1998); S. I. Kiselev et al. , Nature. 425, 380 (2003). 4A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. 84, 2481 (2000); A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Eur. Phys. J. B 22, 99 (2001). 5J. Barna ´set al.,Phys. Rev. B. 72, 024426 (2005). 6S. W. Lee and K. J. Lee, J. Kor. Phys. Soc. 55, 1501 (2009). 7I. Theodonis et al. ,Phys. Rev. Lett. 97, 237205 (2006); A. Manchon et al., J. Phys.: Condens. Matter. 19, 165212 (2007); C. Heiliger and M. D. Stiles, Phys. Rev. Lett. 100, 186805 (2008). 8J. Hayakawa et al.,Jap. J. Appl. Phys. 45, L1057 (2006). 9S. Ikeda et al.,IEEE Trans. Electron Devices. 54, 991(2007). 10A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep. 427, 157 (2006); V. S. Rychkov et al. ,Phys. Rev. Lett. 103, 066602 (2009). 11J. C. Sankey et al.,Nat. Phys. 4, 67 (2008); H. Kubota et al.,ibid. 4,3 7 (2007); S. C. Oh et al.,ibid.5, 898 (2009). 12M. Zwierzycki et al. ,Phys. Rev. B 71, 064420 (2005). 13H. Yuasa et al .,J. Appl. Phys. 92, 2646 (2002); K. Eid et al .,ibid. 91, 8102 (2002); K. Eid, W. P. Pratt, Jr., and J. Bass, ibid. 93, 3445 (2003); N. Strelkov, A. Vedyaev, and B. Dieny, ibid.94, 3278 (2003). FIG. 2. (Color online) Polarization factors with magnetization angles of FM1 and FM2 at interfaces of (a) MgO/FM1, (b) FM1/Ru, and (c) Ru/FM2 where the pinned layer is fixed along þx-axis. FIG. 3. (Color online) (a) Switching probability PSWfor P-to-AP and AP- to-P switching. Dotted lines and solid lines correspond to the case I and case II, respectively. (b) Critical switching currents at PSW¼0.5 as a function of the thickness of Ru layer. Open and solid symbols correspond to the case Iand the case II, respectively. Polarization factors at various t Ru’s at the three interfaces for (c) initial P state and (d) initial AP state.07C904-3 S.-W. Lee and K.-J. Lee J. Appl. Phys. 109, 07C904 (2011) Downloaded 07 Jun 2013 to 130.15.241.167. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.2177141.pdf

Intrinsic and nonlocal Gilbert damping parameter in all optical pump-probe experiments Marija Djordjevic, Gerrit Eilers, Anne Parge, Markus Münzenberg, and J. S. Moodera Citation: Journal of Applied Physics 99, 08F308 (2006); doi: 10.1063/1.2177141 View online: http://dx.doi.org/10.1063/1.2177141 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/99/8?ver=pdfcov Published by the AIP Publishing [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: 93.180.53.211 On: Wed, 12 Feb 2014 06:29:10Intrinsic and nonlocal Gilbert damping parameter in all optical pump-probe experiments Marija Djordjevic,a/H20850Gerrit Eilers, Anne Parge, and Markus Münzenberg IV . Institute of Physics, University of Göttingen, 37077 Göttingen, Germany J. S. Moodera Francis Bitter Magnet Laboratory, MIT, Cambridge, Massachusetts 02139 /H20849Presented on 1 November 2005; published online 5 May 2006 /H20850 The study of magnetization dynamics on the femtosecond time scale is an important task for the implementation of future ultrafast spintronics. With the time resolution inherent using femtosecondlaser pulses in all optical pump-probe experiments, the basic time constants of magneticprecessional modes as well as the energy dissipation processes, which determine the Gilbertdamping, can be studied. The dominant magnetic relaxation modes for the thin Ni films havefrequencies in the range of 1.5–13 GHz. The corresponding Gilbert damping parameter is found tobe dependent on the precession mode. The /H9251values range from 0.05 to 0.8 for highly damped modes. The nonlocal Gilbert damping due to evanescent spin currents and two-magnon scattering isstudied for double layers Ni/Cr/Si /H20849100 /H20850with varied Ni thicknesses. A large increase of the damping parameter for films with a thinner Ni layer is observed. © 2006 American Institute of Physics . /H20851DOI: 10.1063/1.2177141 /H20852 I. INTRODUCTION Fast developing spintronics demand new insight in mag- netization relaxation processes in the subnanosecond regime.The absorption of intensive laser pulses by a ferromagnetcauses a rearrangement of electrons and magnetic momentsby means of fundamental microscopical physical processes,such as electron-electron scattering, electron-phonon scatter-ing, and magnon generation. All these processes are acces-sible with femtosecond laser pulses. On the nanosecond timescale the time evolution of the magnetization is well de-scribed by the macroscopic Landau-Lifshitz-Gilbert /H20849LLG /H20850 Eq. /H208491/H20850. The magnetic resonant Eigen modes have been in- tensively explored with Brillouin light scattering 1and ferro- magnetic resonance2,3/H20849FMR /H20850experiments in the last years. dm dt=−/H9253m/H11003Heff+/H9251m/H11003dm dt. /H208491/H20850 The transition between those two regimes of magnetization relaxation, the basic ultrafast relaxation, and the coherentprecessional motion on the other side are very actual andintriguing questions. A lot of effort is made to understand theleading mechanisms as well as to explore the characteristictime scales. With a laser induced change in the anisotropyfield it is possible to trigger a precession of the magnetiza-tion in the thin ferromagnetic films. 4,5Within this article, the coherent precessional modes of a thin polycrystalline Ni filmare studied. In the first part, it will be discussed how thecoherent precessional modes can be controlled by the exter-nal magnetic field. In the second part basic studies ofintrinsic 6and nonlocal Gilbert damping will be presented.II. EXPERIMENTAL METHOD In all optical pump-probe experiments the Ti:sapphire laser pulses with a central wavelength of 800 nm amplifiedby a regenerative amplifier /H20849RegA 6400, 1 /H9262J pulse energy, 80 fs pulse width, 250 kHz repetition rate /H20850are used to excite the ferromagnetic film.7The genuine time-resolved magnetic signal is extracted with a double modulation technique, forwhich the intensity of the pump beam is modulated by achopper, and the polarization of the probe beam is modulatedby a photoelastic modulator /H20849PEM /H20850. The external magnetic fields, both in plane and out of plane, are controlled by anelectromagnet. It is possible to vary the magnetic field direc-tion by almost 360° without changing the pump-probe geom-etry: for all measurements, the pump beam is impinging or-thogonal to the film surface, while the probe beam is tiltedby an angle of 25°. They are focused to spot sizes of 60 , and30 /H9262m, respectively. The laser pump fluence determines the strength of the excitation in the ferromagnetic sample, whichcan be directly followed by the demagnetization rate ob-served in a reduction of the saturation Kerr signal of thehysteresis loop. The measurements have been performed onaN i /H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850film, as well as on double layers of Ni /H20849xnm/H20850/Cr/H208495n m /H20850/Si with Ni thicknesses ranging from 2 nm up to 100 nm. All films are capped with a 3 nm Cu layer for protection. The samples are grown by e-beamevaporation in a UHV chamber with a base pressure of ap-proximately 5 /H1100310 −10mbar. The roughness of the surface and interfaces of the double layers, which are determined bystandard low angle x-ray scattering measurements, is ap-proximately 3–4 Å. III. RESULTS AND DISCUSSION The time-resolved spectra of the Kerr signal /H20851time- resolved magneto-optical Kerr effect /H20849TRMOKE /H20850/H20852of aa/H20850Electronic mail: mdjordj@gwdg.deJOURNAL OF APPLIED PHYSICS 99, 08F308 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08F308/3/$23.00 © 2006 American Institute of Physics 99, 08F308-1 [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: 93.180.53.211 On: Wed, 12 Feb 2014 06:29:10Ni/H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850film for a laser pump fluence of 50 mJ/cm2are presented in Fig. 1. The external magnetic field is varied from 0 up to 100 mT, while keeping the fieldorientation 25° out of plane constant. The temperature rise onthe surface of the sample after absorption of the pump pulseis up to /H110151000 K. 8The spectrum is characterized by a strong negative peak shortly after the pump pulse reaches thesample, which represents an ultrafast demagnetization, fol-lowed by a damped sine oscillation plus a background thatdecays on the nanosecond time scale. The former reflectscoherence, while the latter reflects incoherent relaxation pro-cesses, respectively. A strong field dependence can be no-ticed: the coherent oscillations vary in amplitude and fre-quency. With smaller external magnetic fields, the period ofthe oscillations increases from 160 to 500 ps, and the exci-tation amplitude of the oscillations is decreased. Also theincoherent background contribution is reduced. In higherfields the contribution of a second shorter oscillation periodof about 75 to 90 ps becomes clearly visible. The spectra areanalyzed using a sum of two damped sine functions, and theresults are shown in Fig. 2. The dominant oscillation modefrequency /H92630is in the range of 1.5 up to 6.5 GHz. The sec- ond oscillation mode reaches frequencies /H92631from 11 up to 13 GHz. For the lower frequency mode, the field dependenceis fitted using Eq. /H208492/H20850, which is derived from the magnetic free energy density including an external magnetic field H and an out-of-plane anisotropy K z.9 /H9275=/H9253/H20881/H92620Hx/H20873/H92620Hx+2Kz Ms+/H92620Ms/H20874. /H208492/H20850 /H9253is the gyromagnetic ratio, Msis the saturation magnetiza- tion and Hxis the in-plane component of the external field.An anisotropy constant of Kz=−4.1 /H208492/H20850105J/m3is deter- mined. This value is considerably smaller than 1/2 /H92620Ms2 and therefore agrees with our experimental observation that the easy axis of the system is in plane. The higher frequencymode is attributed to the first order of the standing spinwave, which is expected for films thicker than the penetra-tion depth of the laser beam. This mode obeys the dispersionrelation /H92630=/H92631+Dk2, where k=/H9266/d, d is the thickness of the layer, and Dis the spin wave exchange constant. The value ofD=680 /H2084910/H20850meV Å2is determined. The frequency range is comparable with the data reported in all optical pump- probe experiments by van Kampen et al. and FMR experiments.4 So far, the only information extracted is about the pre- cession frequencies. Even more interesting, the spectra carrythe information about the energy dissipation processes andthus how fast the magnetization will align itself with theequilibrium orientation due to intrinsic and extrinsic damp-ing processes. To explore the nature of the magnetic damp-ing, the damping parameter /H9251is extracted from the fitting parameters of the TRMOKE spectrum using the relation /H9270 =1//H9275/H9251./H9270represents the characteristic exponential decay time, and /H9275is the frequency of the magnetic precession. In addition to the Ni /H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850layer, the double layer system with added Cr layer has been studied for various Ni layer thicknesses. Here we present only the resultsfor the double layer system Ni /H20849x/H20850/Cr/H208495n m /H20850/Si with x=12 and 40 nm. The laser pump fluence was 50 mJ/cm 2. The damaging effects of heating leading to alloying or even melt-ing of the sample are excluded by comparison of the fre-quencies and the damping parameters at lower pump fluen-cies. The tilting angle of the external magnetic field was 35°out of plane for Ni/Cr /H208495n m /H20850/Si/H20849100 /H20850films, and 25° out of plane for the Ni /H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850film. In order to vary the frequency of the precession, the magnetic field has been varied from 10 up to 100 mT for theNi/H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850film and from 30 up to 150 mT for the Ni/Cr /H208495n m /H20850/Si/H20849100 /H20850films. In Fig. 3 the damping parameter is plotted versus the precession fre- quency. In the case of the Ni /H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850 FIG. 1. Time-resolved MOKE /H20849TRMOKE /H20850for a Ni /H2084950 nm /H20850/ Cu/H208495n m /H20850/Si/H20849100 /H20850film. The external magnetic field, applied at an angle of 25° out of plane, is varied in amplitude. FIG. 2. Frequency spectrum for the Ni /H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850film. The lower frequency /H92630is fitted with Eq. /H208492/H20850and yields Kz=−4.1 /H208492/H20850105J/m3. The higher frequency /H92631belongs to the first standing spin wave and obeys the plotted magnon dispersion relation with D=680 /H2084910/H20850meV Å2.08F308-2 Djordjevic et al. J. Appl. Phys. 99, 08F308 /H208492006 /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: 93.180.53.211 On: Wed, 12 Feb 2014 06:29:10film, the phenomenological Gilbert damping parameter /H9251 shows a clear tendency to increase at lower frequencies. For higher fields the damping constant corresponds to the intrin-sic damping for nickel. Nickel is the itinerant ferromagnetwith the highest intrinsic damping. /H9251values range from 0.05 for high fields up to 0.17 for lower fields. For theNi/Cr /H208495n m /H20850/Si/H20849100 /H20850films, the damping parameter is strongly enhanced as compared to the Ni/H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850film and it has a very strong frequency dependence. The additional damping observed with insertion of a Cr layer shows further enhancement by areduction of the Ni layer thickness. For theNi/H2084912 nm /H20850/Cr/H208495n m /H20850/Si film /H9251values starts from 0.2 for high field and increases up to 0.8 for lower fields. This is a strong increase of the damping compared to theNi/H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850film. Remarkably, this value corresponds to a damping within /H9270/H1101560–110 ps only. The additional nonlocal energy dissipation processes are respon-sible for the observed enhanced damping which can be stud-ied by the time-resolved method in real time, compared toFMR where the damping is determined by a resonant mea-surement in frequency space. The thickness dependence ofthe damping parameter implicates that the nonlocal Gilbertdamping parameter is a direct consequence of the emissionof spin waves and dynamic spin currents. The effect of non-local damping by spin currents has been calculated in amodel by Tserkovnyak et al. 10and experimentally verified in Refs. 11 and 12. For thicker Ni films the precessional motiontakes place in a larger reservoir and its energy dissipation byprocesses at the interfaces affects the precessional modesless. A frequency dependence of the damping parameter iswell known from FMR measurements in various systems andcommonly attributed to scattering processes involving thecoherent precession mode and intrinsic spin waves. 2,3,6Time- resolved pulsed field measurements have been reported re-cently for Cr/Fe/GaAs films. They have been compared toearlier FMR experiments. 13The authors point out that two-magnon scattering, as a consequence of spatially inhomoge- neous exchange bias due to the antiferromagnet-ferromagnetinterface, induces a frequency dependence of /H9251. In our case, though, the effect is much more pronounced, and the damp-ing is measured for a broader field range. The quite complexTRMOKE spectra reveal two dominating modes and a largerincoherent background containing various different magneticand nonmagnetic excitations. The damping exceeds the ex-trinsic damping observed in Ref. 13. The strong difference inthe damping constant for both modes, as it is seen forNi/H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850film, can also point to mode mixing effects. 14The coherent mode decays into different resonant and nonresonant magnetic excitations. Interfacesand defects are needed for symmetry breaking to enhance themode conversion. The inserted Cr layer acts as a spin sinkdue to strong spin-orbit scattering and a complex magneticorder, and it is responsible for the strong additional dampingobserved. CONCLUSION We have demonstrated that all optical pump-probe ex- periments are a powerful method to explore the magneticanisotropies and energy dissipation processes in thin Nifilms. Two modes that dominate the spectrum are found forthe Ni /H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850film. One is a homoge- neous mode and the other is a standing spin wave. The time- resolved all optical method which determines the magneticcoherent precession modes also gives an access to the Gilbertdamping. The nonlocal Gilbert damping and magnon scatter-ing result in a strong increase of the damping. This can bestudied systematically for Ni/Cr /H208495n m /H20850/Si/H20849100 /H20850films with varied Ni thicknesses. Ultrastrong damping with damping constants as high as /H9251=0.8 in the low frequency range is observed. ACKNOWLEDGMENT The authors would like to acknowledge the DFG Schwerpunktprogramm SPP 1133 “Ultrafast magnetizationprocesses” for supporting this research. 1J. Jorzick et al. , Appl. Phys. Lett. 75, 3859 /H208491999 /H20850. 2B. Heinrich et al. ,Phys. Rev. Lett. 59, 1756 /H208491987 /H20850. 3J. Lindner et al. , Phys. Rev. B 68, 060102 /H208492003 /H20850. 4M. van Kampen et al. , Phys. Rev. Lett. 88, 227201 /H208492002 /H20850. 5G. Yu et al. , Phys. Rev. Lett. 82, 3705 /H208491999 /H20850. 6S. Bhagat and P. Lubitz, Phys. Rev. B 10,1 7 9 /H208491974 /H20850. 7T. Kampfrath et al. , Phys. Rev. B 65, 104429 /H208492002 /H20850. 8The maximum temperature is only reached at the direct surface for an ultrashort time scale. The average rise in temperature is a few 10 K for thehighest pump fluence. The spectra have been repeatedly checked for aninfluence of the heating processes on the sample in the spot area. 9M. Farle, Rep. Prog. Phys. 61, 755 /H208491998 /H20850. 10Y. Tserkovnyak et al. , Phys. Rev. Lett. 88, 117601 /H208492002 /H20850. 11S. Mizukami et al. , Y. Ando, and T. Miyazaki, J. Magn. Magn. Mater. 226–230 , 1640 /H208492001 /H20850. 12R. Urban et al. , Phys. Rev. Lett. 87, 217204 /H208492001 /H20850. 13G. Woltersdorf et al. , Phys. Rev. Lett. 95, 037401 /H208492005 /H20850. 14M. Buess et al. , Phys. Rev. Lett. 94, 127205 /H208492005 /H20850. FIG. 3. Gilbert damping parameter for the Ni /H2084950 nm /H20850/Cu /H208495n m /H20850/Si/H20849100 /H20850 film and for the Ni /H20849x/H20850/Cr/H208495n m /H20850/Si/H20849100 /H20850films with x=5 and 40 nm. The frequency is varied by a variation of the applied field. The lines are guidesto the eyes.08F308-3 Djordjevic et al. J. Appl. Phys. 99, 08F308 /H208492006 /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: 93.180.53.211 On: Wed, 12 Feb 2014 06:29:10

5.0038804.pdf

J. Appl. Phys. 129, 141101 (2021); https://doi.org/10.1063/5.0038804 129, 141101 © 2021 Author(s).Axion electrodynamics in topological materials Cite as: J. Appl. Phys. 129, 141101 (2021); https://doi.org/10.1063/5.0038804 Submitted: 26 November 2020 . Accepted: 23 February 2021 . Published Online: 09 April 2021 Akihiko Sekine , and Kentaro Nomura COLLECTIONS Paper published as part of the special topic on Topological Materials and Devices This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Materials for quantum technologies: Computing, information, and sensing Journal of Applied Physics 129, 140401 (2021); https://doi.org/10.1063/5.0050140 Topological magnonics Journal of Applied Physics 129, 151101 (2021); https://doi.org/10.1063/5.0041781 The physical properties of the Hall current Journal of Applied Physics 129, 144501 (2021); https://doi.org/10.1063/5.0044912Axion electrodynamics in topological materials Cite as: J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 View Online Export Citation CrossMar k Submitted: 26 November 2020 · Accepted: 23 February 2021 · Published Online: 9 April 2021 Akihiko Sekine1,a) and Kentaro Nomura2 AFFILIATIONS 1RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan 2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan Note: This paper is part of the Special Topic on Topological Materials and Devices. a)Author to whom correspondence should be addressed: akihiko.sekine@riken.jp ABSTRACT One of the intriguing properties characteristic to three-dimensional topological materials is the topological magnetoelectric phenomena arising from a topological term called the θterm. Such magnetoelectric phenomena are often termed the axion electrodynamics since the θ term has exactly the same form as the action describing the coupling between a hypothetical elementary particle, axion, and a photon. Theaxion was proposed about 40 years ago to solve the so-called strong CPproblem in quantum chromodynamics and is now considered a can- didate for dark matter. In this Tutorial, we overview theoretical and experimental studies on the axion electrodynamics in three-dimensional topological materials. Starting from the topological magnetoelectric effect in three-dimensional time-reversal invariant topological insula- tors, we describe the basic properties of static and dynamical axion insulators whose realizations require magnetic orderings. We alsodiscuss the electromagnetic responses of Weyl semimetals with a focus on the chiral anomaly. We extend the concept of the axion electrody-namics in condensed matter to topological superconductors, whose responses to external fields can be described by a gravitational topologi-cal term analogous to the θterm. © 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/ ).https://doi.org/10.1063/5.0038804 I. INTRODUCTION Conventionally, metals and insulators have been distinguished by the existence of bandgaps. In 2005, a novel phase of matter that doesnot belong to either conventional metals or insulators, called the topo-logical insulator, was discovered. 1–5It is notable that topological insu- lators have bulk bandgaps but also have gapless boundary (edge or surface) states. Furthermore, a topological insulator phase and a trivialinsulator phase cannot be connected adiabatically to each other. Inother words, bulk bandgap closin g is required for the transitions between topologically nontrivial an d trivial phases. In addition, before the establishment of the concept of topological insulators, different phases of matter had usually been distinguished from each other bythe order parameters that indicate spontaneous symmetry breaking.For example, magnetism can be understood as a consequence of spon- taneous spin rotational symmetry breaking. However, from the view- point of symmetry analysis, time-reversal invariant topologicalinsulators and time-reversal invari ant band insulators cannot be dis- tinguished. The ways to distinguish s uch topologically nontrivial and trivial insulator phases can be d ivided into two types (which, of c o u r s e ,g i v er i s et oe q u i v a l e n tr esults). One way is introducing a“topological invariant ”such as Z 2invariant,1,6–8which are calculated from the Bloch-state wave function of the system. The other way is the “topological field theory, ”9which describes the responses of topologi- cal phases to external fields and is the focus of this Tutorial. In the topological field theory, the responses of a topological phase to external fields are described by a topological term. In two spatial dimensions, it is well known that the quantum Hall effect of a time-reversal symmetry broken phase can be described by a Chern – Simons action with the quantized coefficient given by the first Chernnumber. 10,11In three spatial dimensions, time-reversal symmetry plays an important role. The topological magnetoelectric effect described bythe so-called θterm 9is a hallmark response of three-dimensional (3D) time-reversal invariant topological insulators to external electric a n dm a g n e t i cf i e l d s .I nt h ep r e s e n c eo ft i m e - r e v e r s a ls y m m e t r y , the coefficient of the magnetoelectric effect θtakes a quantized value θ¼π(mod 2 π) for topological insulators, while θ¼0i nt r i v i a li n s u - lators. However, in systems with broken time-reversal symmetry, e.g.,in magnetically ordered phases, the value of θcan be arbitrary, i.e., can deviate from the quantized value πor 0, which means that the value of θcan even depend on space and time as θ(r,t). It should beJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-1 ©A u t h o r ( s )2 0 2 1noted that spatial-inversion symme try breaking can also lead to the deviation of θfrom the quantized value πor 0. In the field theory literature, the phenomena described by the θterm is termed the axion electrodynamics12because the θterm has exactly the same form as the action describing the couplingbetween a hypothetical elementary particle, axion, and a photon. The axion was proposed about 40 years ago to solve the so-called strong CPproblem in quantum chromodynamics. 13–15By subse- quent studies in particle physics and astrophysics, the axion is nowconsidered as a candidate for dark matter. 16–19However, regardless of intensive experimental searches, the axion has not yet been found. Since the coefficient of the θterm, θ(r,t), is a field describ- ing the axion, observing the magnetoelectric responses in materialswhose effective action is described by a θterm is equivalent to real- izing the (dynamical) axion field in condensed matter. 20So far, it has been shown theoretically that in a class of magnetic insulators such as magnetically doped topological insulators, the value of θ(r,t) is proportional to the antiferromagnetic order parameter (i.e., the Néel field), i.e., the antiferromagnetic spin fluctuation isidentical to a dynamical axion field. 20InFig. 1 , a classification of 3D insulators in terms of the value of θis schematically shown. The effective action of the form of the θterm appears not only in insulator phases but also in semimetal phases. The key inthe case of topological semimetals is the breaking of time-reversalor spatial-inversion symmetry, which can lead to nonzero and nonquantized expressions for θ. For example, in a time-reversal broken Weyl semimetal with two Weyl nodes, its response toexternal electric and magnetic fields is described by a θterm with θ(r,t)¼2(b/C1r/C0b 0t),21–25where bis the distance between the two Weyl nodes in momentum space and b0is the energydifference between the two nodes. In contrast, in the case of topo- logical superconductors, their topological nature is captured only by thermal responses,26–28since charge and spin are not con- served. It has been heuristically suggested that the effective actionof 3D time-reversal invariant topological superconductors may bedescribed by an action which is analogous to the θterm but is written in terms of gravitational fields corresponding to a temper- ature gradient and a mechanical rotation. 29,30 In this Tutorial, we overview theoretical and experimental studies on the axion electrodynamics in topological materials. InSec. II, we start by deriving the topological magnetoelectric effect described by a θterm in phenomenological and microscopic ways in 3D time-reversal invariant topological insulators. We also reviewrecent experimental studies toward observations of the quantizedmagnetoelectric effect. In Sec. III, we review the basics and recent experimental realizations of the so-called axion insulators in which the value of θis quantized due to a combined symmetry (effective time-reversal symmetry), regardless of the breaking of time-reversalsymmetry, focusing on MnBi 2Te4family of materials. In Sec. IV, we consider generic expressions for θin insulators and extend the derivation of the θterm in a class of insulators with broken time- reversal and inversion symmetries whose realization requires anti- ferromagnetic orderings. In Sec. V, we describe emergent dynami- cal phenomena from the realization of the dynamical axion field intopological antiferromagnetic insulators. In Secs. VIand VII,w e extend the study of the axion electrodynamics in condensed matter to Weyl semimetals and topological superconductors, respectively,whose effective action can be described by topological terms analo-gous to the θterm. In Sec. VIII, we summarize this Tutorial and outlook future directions of this fascinating research field. FIG. 1. Schematic of a classification of 3D insulators in terms of time-reversal symmetry and the orbital magnetoelectric coupling coefficient θ. In the first classification process, 3D insulators are divided into two types: insulators with or without time-reversal symmetry. In the second classification process, 3D insu lators with time-reversal symmetry are divided into types: topological insulators and normal (trivial) insulators. T opological insulators are characterized by the topolog ical magnetoelectric effect with the quantized coefficient θ¼π(mod 2 π). In the second classification process, 3D insulators with broken time-reversal symmetry are divided into two types: axion insula- tors and magnetic insulators. In axion insulators, time-reversal symmetry is broken but an “effective ”time-reversal symmetry represented by a combination of time-reversal and a lattice translation is present, leading to the topological magnetoelectric effect with the quantized coefficient θ¼π(mod 2 π). In magnetic insulators, the value of θis arbitrary, including θ¼0. In a class of magnetic insulators termed topological magnetic insulators, θis proportional to their magnetic order parameters Msuch as the Néel vector (i.e., antiferromagnetic order parameter), and the fluctuation of the order parameter realizes a dynamical axion field δθ(r,t)/δM(r,t) in condensed matter. Here, note that spatial-inversion symmetry must be broken in order for the value of θto be arbitrary, i.e., in the magnetic insulators we have mentioned above, whereas its breaking is not required in the other three phases. See also T able I for the role of inversion symmetry.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-2 ©A u t h o r ( s )2 0 2 1II. QUANTIZED MAGNETOELECTRIC EFFECT IN 3D TOPOLOGICAL INSULATORS In this section, we describe the basics of the topological mag- netoelectric effect, one of the intriguing properties characteristic to 3D topological insulators. We derive phenomenologically and microscopically the θterm in 3D topological insulators, which is the low-energy effective action describing their responses to exter-nal electric and magnetic fields, i.e., the topological magnetoelectriceffect. We also review recent theoretical and experimental studies toward observations of the topological magnetoelectric effect. A. Overview As has been briefly mentioned in Sec. I, topological phases can be characterized by their response to external fields. One of thenoteworthy characters peculiar to 3D topological insulators is the topological magnetoelectric effect, which is described by the so-called θterm. 9Theθterm is written as Sθ¼ð dtd3rθe2 4π2/C22hcE/C1B, (1) where h¼2π/C22his the Planck ’s constant, e.0 is the magnitude of the electron charge, cis the speed of light, and EandBare external electric and magnetic fields, respectively. From the variation of thisaction with respect to Eand B, we obtain the cross-correlated responses expressed by P¼θe2 4π2/C22hcB,M¼θe2 4π2/C22hcE, (2) with Pbeing electric polarization and Mbeing magnetization. We see that Eq. (2)clearly exhibits a linear magnetoelectric effect, as schematically illustrated in Fig. 2 . Since E/C1Bis odd under time reversal (i.e., E/C1B!/C0 E/C1Bunder t!/C0 t), time-reversal symme- try requires that the action (1)is invariant under the transforma- tion θ!/C0 θ. Then, it follows that in the presence of time-reversal symmetry θtakes a quantized value θ¼π(mod 2 π) for topologicalinsulators, while θ¼0 in trivial insulators. A simple and intuitive proof of this quantization has been given.31However, in systems with broken time-reversal symmetry, e.g., in magnetically ordered phases, the value of θcan be arbitrary, i.e., can deviate from the quantized value πor 0,32which means that the value of θcan even depend on space and time as θ(r,t). A similar argument can be applied to spatial-inversion symmetry. Namely, θtakes a quantized value θ¼πorθ¼0 (mod 2 π) in the presence of inversion sym- metry,33,34and inversion symmetry breaking can also lead to the deviation of θfrom the quantized value, because E/C1Bis also odd under spatial inversion. Table I shows the constraints on the value ofθby time-reversal and spatial-inversion symmetries. B. Symmetry analysis of the magnetoelectric coupling The magnetoelectric effect is the generation of bulk electric polarization (magnetization) by an external magnetic (electric)field. The linear magnetoelectric coupling coefficient is generically described by α ij¼@Mj @Ei/C12/C12/C12/C12 B¼0¼@Pi @Bj/C12/C12/C12/C12 E¼0, (3) where i,j¼x,y,zindicates the spatial direction, EandBare exter- nal electric and magnetic fields, and PandMare the electric polar- ization and the magnetization. In general, both time-reversal andspatial-inversion symmetries of the system must be broken, since the occurrence of nonzero P(M) breaks spatial-inversion (time- reversal) symmetry. This requirement is consistent with the con-straints on the value of θby time-reversal and spatial-inversion symmetries (see Table I ). Among several origins of the magneto- electric effect, we are particularly interested in the orbital (i.e., elec- tronic band) contribution to the linear magnetoelectric coupling of the form α ij¼e2θ 4π2/C22hcδij, (4) where δijis the Kronecker delta. Here, note that θis a dimension- less constant. Equation (4) implies the Lagrangian density L¼(e2θ=4π2/C22hc)E/C1B, since the magnetization and polarization can be derived from the free energy of the system FasM¼/C0 @F=@B and P¼/C0 @F=@E. Notably, the susceptibility of the topological FIG. 2. Schematic picture of the topological magnetoelectric effect in a 3D topo- logical insulator. (a) Magnetization Minduced by an external electric field E.jH is the anomalous Hall current on the side surface induced by the electric field. (b) Electric polarization Pinduced by an external magnetic field B. Surface states are gapped by magnetic impurities (or a proximitized ferromagnet) whose magnetization direction is perpendicular to the surface, as indicated by greenarrows.TABLE I. Constraints on the value of θby time-reversal and spatial-inversion sym- metries. The mark ✓(×)indicates the presence (absence) of the symmetry. Here, the notation of time-reversal symmetry in this table includes an “effective ”time- reversal symmetry represented by a combination of time-reversal and a lattice trans- lation, as well as “true ”time-reversal symmetry. Time reversal Inversion Value of θ(mod 2 π) ✓✓ 0o r π ✓ ×0 o r π × ✓ 0o r π × × ArbitraryJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-3 ©A u t h o r ( s )2 0 2 1magnetoelectric effect in Eq. (4)with θ¼πreads (in SI units) e2 4π/C22hc1 μ2 0c≃24:3p s =m, (5) which is rather large compared to those of prototypical magneto- electric materials, e.g., the total linear magnetoelectric susceptibilityα xx¼αyy¼0:7p s=m of the well-known antiferromagnetic Cr 2O3 at low temperatures.35,36 It should be noted here that we need to take into account the presence of boundaries (i.e., surfaces) of a 3D topological insulator,when we consider the realization of the quantized magnetoelectriceffect in a 3D topological insulator. This is because, as is mentionedjust above, finite PandMrequire the breaking of both time-reversal and spatial-inversion symmetries of the whole system, whereas the bulk of the topological insulator has to respect both time-reversal andinversion symmetries. As we will see in the following, the occurrenceof the quantized magnetoelectric effect is closely related to the (half- quantized) anomalous Hall effect on the surface, which requires a somewhat special setup that breaks both time-reversal and inversionsymmetries as shown in Fig. 2 . In this setup, time-reversal symmetry is broken due to the surface magnetization. Inversion symmetry isalso broken because the magnetization directions on a side surface and the other side surface are opposite to each other (spatial inver- sion does not change the direction of spin). C. Surface half-quantized anomalous Hall effect Before deriving the quantized magnetoelectric effect in 3D topo- logical insulators, we briefly consider the anomalous Hall effect onthe surfaces in which the Hall conductivity takes a half-quantized value e 2=2h. Let us start with the effective Hamiltonian for the surface states of 3D topological insulators such as Bi 2Se3,w h i c hi s described by 2D two-component massless Dirac fermions,37 Hsurface (k)¼/C22hvF(kyσx/C0kxσy)¼/C22hvF(k/C2ez)/C1σ, (6) where vFis the Fermi velocity of the surface state (i.e., the slope of the Dirac cone) and σx,σyare the Pauli matrices for the spin degree of freedom. The energy eigenvalues of the Hamiltonian (6)are readily obtained as Esurface (k)¼+/C22hvFffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k2 xþk2 yq from a simple algebra H2 surface ¼/C22h2v2 F(k2 xþk2 y)12/C22. The Fermi velocity of the surface states in Bi 2Se3is experimentally observed as vF/C255/C2105m/s.38 Due to the spin-momentum locking, the surface states are robust against disorder, as long as time-reversal symmetry is pre- served. Namely, the backscattering of surface electrons from(k,")t o(/C0k,") are absent. 39Theoretically, it has been shown that 2D two-component massless Dirac fermions cannot be local-ized in the presence of nonmagnetic disorder. 40,41However, surface states are not robust against magnetic disorder that breaks time-reversal symmetry. This is because the surface Dirac fermionsdescribed by Eq. (6)can be massive by adding a term proportional toσ z, i.e., mσz, which opens a gap of 2 min the energy spectrum. More precisely, such a mass term can be generated by considering the exchange interaction between the surface electrons and magneticimpurities42–44such that Hexch :¼JP iSi/C1σδ(r/C0Ri), where Siis the impurity spin at position Ri. Then, the homogeneous part of the impurity spins gives rise to the position-independent Hamiltonian, Hexch :¼Jnimp/C22Simp/C1σ;m/C1σ, (7) where nimpis the density of magnetic impurities and /C22Simpis the averaged spin of magnetic impurities. Adding Eq. (7)to the Hamiltonian (6)leads to a gapped spectrum Esurface (k)¼+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (/C22hvFkxþmy)2þ(/C22hvFky/C0mx)2þm2 zq :(8) We see that mxandmydo not open the gap but only shift the posi- tion of the Dirac cone in the momentum space.Let us consider a general 2 /C22 Hamiltonian given by H(k)¼R(k)/C1σ.I nt h ec a s eo f massive Dirac fermions, R(k)i sg i v e nb y R(k)¼(vFky,/C0vFkx,mz). The Hall conductivity of the system with the Fermi level being in the gap can be calculated by45 σxy¼/C0e2 h1 4πð dkxdky^R/C1@^R @kx/C2@^R @ky/C18/C19 ¼/C0sgn(mz)e2 2h, (9) where ^R¼R(k)=jR(k)jis a unit vector. The integral is equivalent to the area where the unit vector ^Rmoves on the unit sphere, which, namely, gives the winding number of ^R.A tk¼0, the unit vector ^R points to the north or south pole, that is, ^R¼(0, 0, sgn( mz)). At large kwithjkj/C29j mzj,^Ralmost points to the horizontal directions. Hence, varying k,^Rcovers the half of the unit sphere, which gives 2 π. Equation (9)indicates that the anomalous Hall effect occurs on the surfaces of 3D topological insulators, when magnetic impurities are doped or a magn etic film is put on the surfa- ces.44,46The direction of the Hall current depends on the sign of mz, i.e., the direction of the magnetization of magnetic impurities or proximitized magnetization . Actually, the surface quantum anomalous Hall effect has been observed experimentally.47,48The observed surface quantum anomalous Hall effect in a thin film of Cr-doped (Bi,Sb) 2Te3is shown in Fig. 3 . Note that in those systems, the magnetization directions of top and bottom surfacesare the same, and thus the observed Hall conductivity is 2/C2e 2=(2h)¼e2=h. It can be seen from Fig. 3(b) that the Hall conductivity takes the quantize d value when the chemical poten- tial lies in the surface bandgap. D. Phenomenological derivation of the θterm We have seen in Sec. II C that the surface states of 3D topo- logical insulators can be gapped (i.e., the surface Dirac fermions can be massive) via the exchange interaction with magnetic impuri-ties or proximitized magnetization which breaks time-reversal sym-metry, giving rise to the surface half-quantized anomalous Hall effect. We show phenomenologically in the following that, as a con- sequence of the surface half-quantized anomalous Hall effect, theJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-4 ©A u t h o r ( s )2 0 2 1topological magnetoelectric effect [Eq. (2)] emerges in 3D topologi- cal insulators. Let us consider a case where the side surface of a cylindrical 3D topological insulator is ferromagnetically ordered due to mag-netic doping or the proximity effect, 9as shown in Fig. 2 . The resulting surface Dirac fermions are massive. When an external electric field Eis applied parallel to the cylinder, the surface anom- alous Hall current jHis induced as jH¼/C0sgn(m)e2 2h^n/C2E, (10) where ^nis a unit vector normal to the side surface. From the Ampère ’s law, the magnetization MwithjMj¼jjHj=c(cis the speed of light) is obtained as [see Fig. 2(a) ] M¼sgn(m)e2 2hcE: (11) Similarly, when an external magnetic field Bis applied parallel to the cylinder, the circulating electric field Eindnormal to the mag- netic field is induced as ∇/C2Eind¼/C0 @B=@t. Then, the induced electric field Eindgenerates the surface anomalous Hall currentparallel to the magnetic field as jH¼sgn(m)e2 2h@B @t: (12) On the other hand, a polarization current is equivalent to the time derivative of the electric polarization. Finally, the induced electricpolarization Pis given by [see Fig. 2(b) ] P¼sgn(m)e2 2hcB: (13) Equations (11) and (13) clearly show the magnetoelectric effect. Here, recall that the magnetization and polarization can be derived from the free energy of the system FasM¼/C0 @F=@Band P¼/C0 @F=@E. To satisfy the relations (11) and(13), the free energy must have the following form:9 F¼/C0ð d3re2 2hcE/C1B¼/C0ð d3rθe2 4π2/C22hcE/C1B, (14) where we have omitted sgn( m) for simplicity, and θ¼π. The inte- grand can be regarded as the Hamiltonian density. The equivalentaction is written as S θ¼ð d4xθe2 4π2/C22hcE/C1B¼ð d4xθe2 32π2/C22hcεμνρλFμνFρλ, (15) where d4x¼dtd3r,Fμν¼@μAν/C0@νAμwith Aμ¼(A0,/C0A) being the electromagnetic four potential, and εμνρλis the Levi –Civitá symbol with the convention ε0123¼1. Here, the electric field and the magnetic field are given, respectively, by E¼/C0∇A0/C0@A=@t andB¼∇/C2A. Note that e2=/C22hc(≃1=137) is the fine-structure constant. Equation (15) is indeed the θterm [Eq. (1)]. Under time- reversal ( t!/C0 t), electric and magnetic fields are transformed as E!EandB!/C0 B, respectively. Similarly, under spatial inversion (r!/C0 r), electric and magnetic fields are transformed as E!/C0 E andB!B, respectively. Hence, the term E/C1Bis odd under time- reversal or spatial inversion. On the other hand, 3D topologicalinsulators have time-reversal symmetry, which indicates that S θ remains unchanged under time-reversal. In other words, the value ofθmust be invariant under the transformation θ!/C0 θ. It follows that θ¼π(mod 2 π) in time-reversal invariant topological insula- tors and θ¼0 in normal (topologically trivial) insulators. Note that Sθis a surface term when the value of θis constant, i.e., independent of spatial coordinate and time, since we can rewrite the integrand of Sθin a total derivative form, εμνρλFμνFρλ¼4εμνρλ@μ(Aν@ρAλ), (16) which indicates that the topological magnetoelectric effect in the bulk is a consequence of the surface response to the electric and magnetic fields. However, as we shall see later, the presence of theθterm that is dependent of spatial coordinate and/or time results in an electric current generation in the bulk. Here, let us consider the inverse process of the derivation of theθterm (15). Namely, we derive the surface anomalous Hall FIG. 3. (a) Schematic illustration of an experimental setup to detect the quantum anomalous Hall effect in a ferromagnetically ordered topological insula- tor thin film. (b) Gate-voltage Vgdependence of the Hall conductivity σxyand the longitudinal conductivity σxxin a thin film of Cr-doped (Bi,Sb) 2Te3. Reproduced with permission from Chang et al ., Science 340, 167 (2013). Copyright 2013 American Association for the Advancement of Science.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-5 ©A u t h o r ( s )2 0 2 1current from Eq. (15). We have seen in Eq. (16) that the integrand of the θterm is a total derivative when the value of θis constant. For definiteness, let us see what happens at a given surface in the z direction. Using Eq. (16) and integrating out with respect to z, the surface term can be obtained from Eq. (15) as Ssurface¼ð d3xθe2 8π2/C22hcεzνρλAν@ρAλ, (17) where d3x¼dtdxdy . Recall that, in general, an electric current density jνin the νdirection can be obtained from the variation of an action with respect to the electromagnetic vector potential Aν: jν¼δS=δAν. Without loss of generality, we may consider the current in the xdirection, jx¼δSsurface δAx¼θe2 4π2/C22hcεzxρλ@ρAλ¼θe2 4π2/C22hcEy, (18) where Ey¼/C0 @yA0/C0@tAyis the electric field in the ydirection. Since θ¼πin topological insulators, Eq. (18) clearly shows the surface half-quantized anomalous Hall effect. More precisely, we should consider an electric current derived directly from the θterm. Namely, we should consider the spatial dependence of θsuch that θ¼0 in vacuum and θ¼πinside the topological insulator. Notice that the θterm can be rewritten as Sθ¼/C0ð dtd3re2 8π2/C22hεμνρλ[@μθ(r,t)]Aν@ρAλ: (19) Then, the electric current density is obtained as jx¼δSθ δAx¼e2 4π2/C22h@tθ(r,t)Bx/C0@zθ(r,t)Ey/C2/C3 : (20) The magnetic-field induced term is the so-called chiral magnetic effect,49which will be mentioned later. For concreteness, we require that the region z/C200(z.0) be the topological insulator (vacuum). The zdependence of θ(r,t) can be written in terms of the Heaviside step function as θ(z)¼π[1/C0Θ(z)], since θ¼π (θ¼0) inside (outside) the topological insulator. Then, we obtain @zθ¼/C0πδ(z), which gives rise to the half-quantized Hall conduc- tivity at the topological insulator surface z¼0. E. Microscopic derivation of the θterm So far, we have derived the topological magnetoelectric effect [Eq. (2)] from a surface property of 3D topological insulators. In this section, we derive the θterm microscopically from a low- energy effective model of 3D topological insulators. There are several ways to derive the θterm microscopically. One way is to use the so-called Fujikawa ’s method.50,51Another way is the dimen- sional reduction from (4+1)-dimensions to (3+1)-dimensions,9 which will be briefly mentioned in Sec. IV A . Here, we show the derivation of the θterm based on Fujikawa ’s method.1. Effective Hamiltonian for 3D topological insulators Let us start from the low-energy continuum model for proto- typical 3D topological insulators such as Bi 2Se3. The bulk electronic structure of Bi 2Se3near the Fermi level is described by two p-orbitals P1þ zandP2/C0 zwith+denoting parity. Defining the basis [jP1þ z,"i,jP1þ z,#i,jP2/C0 z,"i,jP2/C0 z,"i] and retaining the wave vector kup to quadratic order, the low-energy effective Hamiltonian around the Γpoint is given by37,52 Heff(k)¼M(k)0 A1kz A2k/C0 0 M(k) A2kþ/C0A1kz A1kzA2k/C0/C0M(k)0 A2kþ/C0A1kz 0 /C0M(k)2 66643 7775 ¼A 2kxα1þA2kyα2þA1kzα3þM(k)α4, (21) where k+¼kx+ikyand M(k)¼m0/C0B1k2 z/C0B2k2 ?. The coeffi- cients for Bi 2Se3estimated by a first-principles calculation read m0¼0:28 eV, A1¼2:2e V/C1Å,A2¼4:1e VÅ , B1¼10 eV Å2,a n d B2¼56:6e VÅ2.37,52Here, note that we have introduced a basis in Eq. (21) that is slightly different from that Refs. 37and52. The 4/C24 matrices αμare given by the so-called Dirac representation, αj¼0σj σj0/C20/C21 ,α4¼10 0/C01/C20/C21 , (22) where the Clifford algebra { αμ,αν}¼2δμν1is satisfied. The above Hamiltonian is nothing but an anisotropic 3D Dirac Hamiltonianwith a momentum-dependent mass. Before proceeding to the derivation of the θterm, it is informative to consider the lattice version of Eq. (21).H e r e , recall that the Z 2invariant,1,6–8which identifies whether a phase is topologically nontrivial or tr ivial, is calculated in lattice models. This means that we cannot directly show that the phase described by the effective Hamiltonian (21) represents a 3D topological insulator. From this viewpoint, we need to constructa lattice Hamiltonian from the continuum Hamiltonian (21). The simplest 3D lattice is the cubic lattice. We replace k iand k2 i terms by ki!sinkiand k2 i!2(1/C0coski). Although this replacement is valid only when ki/C281, as is shown below, it turns out that this replacement describes the topological insula-tor phase. We also simplify the co efficients to obtain the isotro- pic lattice Hamiltonian H eff(k)¼/C22hvF(α1sinkxþα2sinkyþα3sinkz) þm0þrX i¼x,y,z(1/C0coski)"# α4,( 2 3 ) where we have defined /C22hvF¼A1¼A2andr¼/C02B1¼/C02B2.A s is mentioned below, the Hamiltonian (23) is also called the Wilson –Dirac Hamiltonian,53–55which was originally intro- duced in lattice quantum chromodynamics. In cubic lattices, the eight time-reversal invariant momenta Λα, which are invariant under ki!/C0 ki, are given by (0, 0, 0), (π=a, 0, 0), (0, π=a, 0), (0, 0, π=a), (π=a,π=a, 0), ( π=a,0 ,π=a),Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-6 ©A u t h o r ( s )2 0 2 1(0,π=a,π=a), and ( π=a,π=a,π=a), where ais the lattice constant. We can calculate the Z2invariant of the system as6,8 (/C01)ν¼Y8 α¼1sgn m0þrX i¼x,y,z(1/C0cosΛi α)"# ¼/C01( 0.m0=r./C02,/C04.m0=r./C06) þ1(m0=r.0,/C02.m0=r./C04,/C06.m0=r):/C26 (24) Indeed, the topological insulator phase with 0 .m0=r./C02 satis- fies the above realistic value for Bi 2Se3;m0=r/difference/C00:1, where we have assumed the value of the lattice constant as a¼3Å . It should be noted here that the lattice Dirac Hamiltonian (23) is exactly the same as the Hamiltonian of the Wilson fermions , which was originally introduced in the lattice gauge theory to avoidthe fermion doubling problem. 53Namely, we can see that Eq. (23) around the Γpoint (0, 0, 0) represents the usual (continuum) massive Dirac fermions with mass m0, while Eq. (23) around other momentum points, e.g., ( π=a, 0, 0), represent massive Dirac fermi- ons with the mass m0þ2r. 2. Fujikawa ’s method Now, let us return to the continuum Hamiltonian (21) to obtain the θterm. As we have seen in Eq. (24), the lattice Hamiltonian (23) describes a topological insulator when 0.m0=r./C02. Without loss of generality, we can set m0,0 and r.0. Then, the Hamiltonian (21) with m0,0 and r.0, which describes a topological insulator, around the Γpoint can be simpli- fied by ignoring the terms second-order in kias HTI(k)¼/C22hvFk/C1αþm0α4, (25) where m0,0. Except for the negative mass m0, this is the usual Dirac Hamiltonian. In the presence of an external electromagneticvector potential A, minimal coupling results in k!kþeA,w i t h e.0 being the magnitude of the electron charge. In the presence of an external electromagnetic scalar potential A 0,t h ee n e r g yd e n s i t yi s modified as ψyH0ψ!ψy(H0/C0eA0)ψ.U s i n gt h e s ef a c t s ,t h ea c t i o n of the system in the presence of an external electromagnetic fourpotential A μ¼(A0,/C0A) is written in the usual relativistic form,56 STI¼ð dtd3rψyi(@t/C0ieA0)/C0[HTI(kþeA)] fg ψ ¼ð dtd3r/C22ψ[iγμ(@μ/C0ieAμ)/C0m0]ψ,( 2 6 ) where ψy(r,t) is a fermionic field representing the basis of the Hamiltonian (21) and /C22ψ¼ψyγ0. Here, the gamma matrices γμare given by the so-called Dirac representation as γ0¼α4¼10 0/C01/C20/C21 ,γj¼α4αj¼0 σj /C0σj0/C20/C21 , γ5¼iγ0γ1γ2γ3¼01 10/C20/C21 ,( 2 7 )which satisfy the relation { γμ,γν}¼2gμνwith gμν¼diag(þ1, /C01,/C01,/C01) being the metric tensor. It is convenient to study the system in the imaginary time notation, i.e., in Euclidean spacetime.Namely, we rewrite t,A 0,a n d γjast!/C0 iτ,A0!iA0,a n d γj! iγj(j¼1, 2, 3). The Euclidean action of the system is then written as SE TI¼/C0iSTI¼ð dτd3r/C22ψ[γμ(@μ/C0ieAμ)/C0m0eiπγ5]ψ,( 2 8 ) w h e r ew eh a v eu s e dt h ef a c tt h a t m0¼/C0m0(c o s πþiγ5sinπ) ¼/C0m0eiπγ5.N o t et h a t γ0and γ5are unchanged ( γ0¼γ0and γ5¼γ5), so that the anticommutation relation { γμ,γν}¼2δμνis sat- isfied. Note also that, in Euclidean spacetime, we do not distinguishbetween superscripts and subscripts. Now, we are in a position to apply Fujikawa ’s method 50,51to the action (28). First, let us consider an infinitesimal chiral trans- formation defined by ψ!ψ0¼e/C0iπdfγ5=2ψ,/C22ψ!/C22ψ0¼/C22ψe/C0iπdfγ5=2, (29) where f[[0, 1]. Then, the partition function Zis transformed as Z¼ð D[ψ,/C22ψ]e/C0SE TI[ψ,/C22ψ]! Z0¼ð D[ψ0,/C22ψ0]e/C0S0E TI[ψ0,/C22ψ0]:(30) The θterm comes from the Jacobian defined by D[ψ0,/C22ψ0]¼ JD[ψ,/C22ψ]. The action (28) is transformed as S0E TI¼ð dτd3r/C22ψ[γμ(@μ/C0ieAμ)/C0m0eiπ(1/C0df)γ5]ψ þi 2πð dτd3rdf@μ(/C22ψγμγ5ψ): (31) The Jacobian is written as50,51 J¼exp/C0ið dτd3rdfπe2 32π2/C22hcεμνρλFμνFρλ/C20/C21 : (32) Here, Fμν¼@μAν/C0@νAμ, and we have written /C22hand cexplicitly. We repeat this procedure infinite times, i.e., integrate with respectto the variable ffrom 0 to 1. Due to the invariance of the partition function, finally, we arrive at the following expression of SE TI: SE TI¼ð dτd3r/C22ψ[γμ(@μ/C0ieAμ)/C0m0]ψ þið dτd3rπe2 32π2/C22hcεμνρλFμνFρλ, (33) where we have dropped the irrelevant surface term. The first term is the action of a topologically trivial insulator, since the mass /C0m0 is positive. The second term is the θterm in the imaginary time, and we obtain Eq. (15) by substituting τ¼it.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-7 ©A u t h o r ( s )2 0 2 1F. Toward observations of the topological magnetoelectric effect 1. Utilizing topological insulator thin films As we have seen in Sec. II D, the experimental realization of the topological magnetoelectric effect in topological insulators requires that all the surface Dirac states are gapped by the magneticproximity effect or magnetic doping, resulting in the zero anoma-lous Hall conductivity of the system. However, such an experimen-tal setup is rather difficult to be realized. As an alternate route to realize the topological magnetoelectric effect, it has been proposed theoretically that the ν¼0 quantum Hall state, which attributes to the difference between the Landau levels of the top and bottomsurface Dirac states, can be utilized. 57,58The ν¼0 quantum Hall state has been experimentally observed in topological insulator (Bi1/C0xSbx)2Te3films,59as shown in Fig. 4(a) . The two-component Dirac fermions in a magnetic field are known to show the quantumHall effect with the Hall conductivity, σ xy¼nþ1 2/C18/C19e2 h, (34) where nis an integer. Note that, as we have seen in Eq. (9), the1 2 contribution arises as a Berry phase effect. The total Hall conduc- tivity contributed from the top and bottom surfaces of a topological insulator film in a magnetic field is then written as σxy¼nTþnBþ1 ðÞe2 h;νe2 h: (35) The ν¼0 quantum Hall state is realized when the Landau levels of the top and bottom surface states are NT¼/C0N/C01a n d NB¼N(and vice versa), where Nis an integer.57This state cor- responds to nT¼/C0N/C01a n d nB¼Nin Eq. (35),w h i c hc a nb eachieved in the presence of an energy difference between the two surface states, as shown in Fig. 4(b) . Here, recall that the electron density is given by ne¼σxyB=e,w i t h Bbeing the magnetic field strength and ebeing the elementary charge. Using this fact, the charge densities ( ρ¼/C0ene) at the top and bottom surfaces are obtained as ρT¼(Nþ1 2)Be2=handρB¼/C0(Nþ1 2)Be2=h,r e s p e c - tively. We consider the case of N¼0, which is experimentally relevant.59The induced electric polarization in a topological insulator film of thickness dreads P¼1 2ddρTþ(/C0d)ρB ½/C138 ¼e2 2hB,( 3 6 ) which is indeed the topological magnetoelectric effect with the quan- tized coefficient θ¼π. Note that the case of N=0, which gives rise to θ¼(2Nþ1)π, still describes the topological magnetoelectric effect, since θ¼πmodulo 2 π. Another route to realize the topologi- cal magnetoelectric effect is a magnetic heterostructure in which themagnetization directions of the top and bottom magnetic insulatorsare antiparallel. 57,58Several experiments have succeeded in fabricat- ing magnetic heterostructures that exhibits a zero Hall plateau.60–62 In Ref. 60, a magnetic heterostructure consisting of a magnetically doped topological insulator Cr-doped (Bi,Sb) 2Te3and a topological insulator (Bi,Sb) 2Te3was grown by molecular beam epitaxy. A zero Hall conductivity plateau was observed in this study as shown in Fig. 5 , implying an axion insulator state. In Ref. 62, a magnetic heter- ostructure of a topological insulator (Bi,Sb) 2Te3sandwiched by two kinds of magnetically doped topological insulators V-doped (Bi,Sb) 2 Te3and Cr-doped (Bi,Sb) 2Te3was grown by molecular beam epitaxy. Importantly, as shown in Fig. 6 , the antiparallel magnetiza- tion alignment of the top and bottom magnetic layers was directly observed by magnetic force microscopy when the system exhibited azero Hall resistivity plateau. Note, however, that the above experi-ments did not make a direct observation of the magnetoelectric FIG. 4. (a) Quantum Hall effect in a topological insulator (Bi 1/C0xSbx)2Te3thin film. (b) Schematic illustration of the Landau levels of the top and bottom surface states in the presence of an energy difference between the two surfa- ces. Reproduced with permission from Yoshimi et al ., Nat. Commun. 6, 6627 (2015). Copyright 2015 Springer Nature. FIG. 5. (a) Schematic illustration of the magnetic heterostructure. Red arrows indicate the magnetization directions. (b) The observed Hall conductivity as afunction of an external magnetic field. Reproduced with permission from Mogiet al., Nat. Mater. 16, 516 (2017). Copyright 2017 Springer Nature.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-8 ©A u t h o r ( s )2 0 2 1effect, i.e., the electric polarization induced by a magnetic field or the magnetization induced by an electric field. 2. Faraday and Kerr rotations As has been known in particle physics12,63before the discovery of 3D topological insulators, the θterm modifies the Maxwell ’s equations. Since the Maxwell ’s equations describe electromagnetic wave propagation in materials, the presence of the θterm leads to unusual optical properties such as the quantized Faraday and Kerrrotations in topological insulators, 9,64,65which can be viewed as a consequence of the topological magnetoelectric effect. To see this, let us start from the total action of an electromagnetic field Aμ¼(A0,/C0A) in the presence of a θterm is given by S¼ð dtd3rα 4π2θE/C1B/C01 16πð dtd3rFμνFμν, (37) where α¼e2=/C22hc≃1=137 is the fine-structure constant and Fμν¼@μAν/C0@νAμis the electromagnetic field tensor. The electric and magnetic fields are, respectively, given by E¼/C0∇A0/C0 (1=c)@A=@tand B¼∇/C2A.N o t et h a t E/C1B¼(1=8)εμνρλFμνFρλ andFμνFμν¼2(B2=μ0/C0ε0E2). Here, recall that the classical equa- tion of motion for the field Aμis obtained from the Euler –Lagrange equation, δS δAμ¼@L @Aμ/C0@ν@L @(@νAμ)/C18/C19 ¼0, (38)where Lis the Lagrangian density of the system. From Eqs. (37) and (38), one finds that the Maxwell ’s equations are modified in the presence of a θterm9,12,63 ∇/C1E¼4πρ/C02α∇θ 2π/C18/C19 /C1B, ∇/C2E¼/C01 c@B @t, ∇/C1B¼0, ∇/C2B¼4π cJþ1 c@E @tþ2α c@ @tθ 2π/C18/C19 Bþc∇θ 2π/C18/C19 /C2E/C20/C21 :(39) The∇θterms in Eq. (39) play roles when there is a boundary, e.g., gives rise to the surface Hall current as we have seen in Eq. (20). The modified Maxwell ’s Eq. (39) can be solved under the boundary conditions (see Fig. 7 ). It is found that the Faraday and Kerr rotation angles are independent of the material (i.e., topologi-cal insulator thin film) parameters such as the dielectric constant and thickness. 64,65Specifically, in the quantized limit, the Faraday and Kerr rotation angles are given, respectively, by64,65 θF¼tan/C01(α)≃α,θK¼tan/C01(1=α)≃π 2: (40) These quantized angles have been experimentally observed in the anomalous Hall state66and the quantum Hall state [ Fig. 8(a) ].67,68 Also, as predicted in Ref. 64, a universal relationship in units of the fine-structure constant αbetween the Faraday and Kerr rotation angles has been observed [ Fig. 8(b) ].66,67 III. AXION INSULATORS In Sec. II, we have seen that the topological magnetoelectric effect with the quantized coefficient θ¼π(mod 2 π)o c c u r si n3 D time-reversal invariant topological insulators. In general, the value of FIG. 6. Magnetic field dependence of (a) Hall resistivity and (b) magnetic domain contrasts. (c) –( j) Magnetic force microscopy images of the magnetic domains. Red and blue represent, respectively, upward and downward parallel magnetization alignment regions, while green represents antiparallel magnetiza-tion alignment regions. Reproduced with permission from Xiao et al., Phys. Rev. Lett. 120, 056801 (2018). Copyright 2018 American Physical Society. FIG. 7. Schematic figure of a measurement of the quantized Faraday and Kerr rotations in a topological insulator thin film. Reproduced with permission from Maciejko et al., Phys. Rev. Lett. 105, 166803 (2010). Copyright 2010 American Physical Society.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-9 ©A u t h o r ( s )2 0 2 1θis no longer quantized and becomes arbitrary in systems with broken time-reversal symmetry. However, in a class of 3D antiferro-magnetic insulators, an “effective ”time-reversal symmetry repre- sented by a combination of time-reversal and a lattice translation is present, leading to the topological magnetoelectric effect with thequantized coefficient θ¼π(mod 2 π). In this section, we review the- oretical and experimental studies on such antiferromagnetic topologi-cal insulators, which are also called the axion insulators . Starting from the basics of the antiferromagnetic topological insulators, we focus on the MnBi 2Te4family of materials that are layered van der Waals compounds and have recently been experimentally realized. A. Quantized magnetoelectric effect in antiferromagnetic topological insulators Following Ref. 69, we consider a class of insulators in which time-reversal symmetry is broken but the combined symmetry of time-reversal and a lattice translation is preserved. We note here that the presence or absence of inversion symmetry does notaffect their topological classifi cation, although the presence of inversion symmetry greatly simplifies the evaluation of their topological invariants as in the case of time-reversal invariant topological insulators. 8Let us start from some general arguments on symmetry operations. The time-reversal operator Θfor spin-1/2 systems is generically given by Θ¼iσyKwithΘ2¼/C01, where σiare Pauli matrices and Kis complex conjugation opera- tor. In the presence of time-reversal symmetry, the BlochHamiltonian of a system H(k)s a t i s f i e s ΘH(k)Θ/C01¼H(/C0k): (41) Recall that momentum is the generator of lattice translation. An operator that denotes a translation by a vector xis given by T(x)¼e/C0ik/C1x. Then, the translation operator that moves a lattice by half a unit cell in the a3direction is written as T1=2¼e/C0(i=2)k/C1a301 10/C20/C21 ,( 4 2 ) where a3is a primitive translation vector and 1is an identity operator that acts on the half of the unit cell.69One can see that T2 1=2gives a translation by a3because T2 1=2¼e/C0ik/C1a3. Now, we con- sider the combination of Θand T1=2defined by S¼ΘT1=2.I t follows that S2¼/C0e/C0ik/C1a3, which means that the operator Sis antiunitary like Θ. Here, we have used the fact that ΘandT1=2are commute. Note, however, that S2¼/C01 only on the Brillouin zone plane satisfying k/C1a3¼0, while Θ2¼/C01. When a system is invari- ant under the operation S, the Bloch Hamiltonian H(k)s a t i s f i e s SH(k)S/C01¼H(/C0k), (43) which has the same property as time-reversal symmetry in Eq. (41). Therefore, the Z2topological classification can also be applied in systems with the Ssymmetry.69,70Figure 9 shows a schematic illustra- tion of an antiferromagnetic topological insulator protected by theS¼ΘT 1=2symmetry. In this simple model, the unit cell consists of nonmagnetic equivalent A1andA2atomic layers and antiferromag- netically ordered B1andB2atomic layers. The half-uni-cell transla- tion T1=2moves the B1layer to the B2layer, and time-reversal Θ changes a spin-up state into a spin-down state. Therefore, the systemis obviously invariant under the S¼ΘT 1=2transformation. Next, let us consider the resulting surface states. Since S2¼/C01 on the Brillouin zone plane satisfying k/C1a3¼0, the 2D subsystem on the ( k1,k2) plane is regarded as a quantum spin-Hall system with FIG. 8. (a) Magnetic field dependence of the Faraday rotation angle. From Dziom et al ., Nat. Commun. 8, 15197 (2017). Copyright 2017 Author(s), licensed under a Creative Commons Attribution (CC BY) License. (b) Evolution of the scaling function f(θF,θK)¼cotθF/C0cotθK cot2θF/C02 cotθFcotθK/C01as a function of dc Hall conductance towards the universal relationship f(θF,θK)¼α. From Okada et al., Nat. Commun. 7, 12245 (2016). Copyright 2016 Author(s), licensed under a Creative Commons Attribution (CC BY) license. FIG. 9. Schematic illustration of an antiferromagnetic topological insulator protected by the S¼ΘT1=2symmetry.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-10 ©A u t h o r ( s )2 0 2 1time-reversal symmetry. This means that the k1ork2dependence of the surface spectra must be gapless because the k/C1a3¼0l i n eo ft h e surface states is the boundary of the 2D subsystem (the k/C1a3¼0 plane) in the bulk Brillouin zone. In other words, at the surfaces thatare parallel to a 3, which preserve the Ssymmetry, there exist an odd number of gapless surface states (as in the case of a strong time- reversal invariant topological insulator). On the other hand, at the surfaces that are perpendicular to a3, which break the Ssymmetry, such a topological protection of the surface states no longer exists,and the surface states can have gapped spectra. As we have seen above, the presence of Ssymmetry results in a realization of a new 3D topological insulator. This implies that such topological insulators exhibit a quantized magnetoelectric effectdescribed by a θterm, as in the case of time-reversal invariant 3D topological insulators. To see this, recall that the magnetoelectric effect resulting from a θterm is expressed as P¼θe 2=(4π2/C22hc)B, andM¼θe2=(4π2/C22hc)E, where PandMare the electric polarization and the magnetization, respectively. Under time-reversal Θ,t h e coefficient θchanges sign θ!/C0 θ, because P!PandE!E, while M!/C0 MandB!/C0 B. On the other hand, the lattice trans- lation T1=2does not affect θ.69Combining these, the Soperation implies the transformation such that θ!/C0 θþ2πnwith nbeing an integer. Then, it follows that θ¼0o r θ¼πmodulo 2 π. B. MnBi 2Te4 1. Electronic structure of MnBi 2Te4bulk crystals With the knowledge of antiferromagnetic topological insula- tors with the Ssymmetry, here we review recent experimental reali- zations of the antiferromagnetic topological insulator state inMnBi 2Te4.71–81The crystal structure of MnBi 2Te4is shown in Fig. 10 . The septuple layer consisting of Te –Bi–Te–Mn –Te–Bi–Te is stacked along the [0001] direction by van der Waals forces. A theoretical calculation of the exchange coupling constants betweenMn atoms shows that the intralayer coupling in each Mn layer is ferromagnetic, while the interlayer coupling between neighboring Mn layers is antiferromagnetic.71The magnetic ground state is thus considered to be antiferromagnetic with the Néel vector pointingthe out-of-plane direction (i.e., the zdirection), which is called A-type AFM- z. The Néel temperature is reported to be about 25 K. 71,74,79,81The unit cell of the antiferromagnetic insulator state consists of two septuple layers ( Fig. 10 ), where τc 1=2is the half-cell translation vector along the caxis that connects nearest spin-up and spin-down Mn atomic layers. It can be easily seen that this inter-layer antiferromagnetism between the Mn atonic layers preserves theS¼Θτ c 1=2symmetry, indicating that the system is a topological antiferromagnetic insulator, which we have discussed in Sec. III A . Interestingly, the bulk bandgap is estimated to be about 0.2 eV,71,72 which is comparable to that of the time-reversal invariant topologi- cal insulator Bi 2Se3. The A-type AFM- zstate is invariant under spatial inversion P1with the inversion center located at the Mn atomic layer in each septuple layers. Importantly, P2Θsymmetry, the combination of spatial inversion P2with the inversion center located between two septuple layers and time-reversal Θ, is also preserved. The presence ofP2Θsymmetry leads to doubly degenerate bands even in the absence of time-reversal symmetry.73,82,83Here, following Refs. 72 and 97, we derive the low-energy effective Hamiltonian of the A-type AFM- zstate. P2Θsymmetry requires that (P2Θ)H(k)(P2Θ)/C01¼H(k), (44) since momentum kchanges sign under both P2andΘ. As in the case of Bi 2Se3[Eq. (21)], the low-energy effective Hamiltonian of the nonmagnetic state of MnBi 2Te4around the Γpoint is written in the basis of [ jP1þ z,"i,jP1þ z,#i,jP2/C0 z,"i,jP2/C0 z,"i], where the states jP1þ z,"#iandjP2/C0 z,"#icome from the pzorbitals of Bi and Te, respectively.72In this basis, P2¼τz/C101and Θ¼1/C10iσyK, where τiandσiact on the orbital and spin spaces, respectively, and Kis complex conjugation operator. P2Θsymmetry constrains the possible form of the 4 /C24 Bloch Hamiltonian H(k)¼P i,jdij(k)τi/C10σj. It follows that the following five matrices and the identity matrix are allowed by P2Θsymmetry: τx/C10σx,τx/C10σy,τx/C10σz,τy/C101,τz/C101, (45) due to the property ( P2Θ)(τi/C10σj)(P2Θ)/C01¼τi/C10σj. Note that these five matrices anticommute with each other, leading to doubly degenerate energy eigenvalues. Using these five matrices, the low- energy effective Hamiltonian around the Γpoint is written as72,97 H(k)¼τx(A2kyσx/C0A2kxσyþm5σz)þA1kzτyþM(k)τz, (46) where M(k)¼MþB1k2 zþB2(k2 xþk2 y). The mass m5is induced by the antiferromagnetic order. One can see that the Hamiltonian (46) is invariant under both P2andΘwhen m5¼0. Indeed, the surface states of the lattice model constructed from Eq. (46) i nas l a bg e o m e - try in the zdirection exhibit the half-quantized anomalous Hall con- ductivity σxy¼+sgn(m5)e2=2h, implying the axion insulator state.97 FIG. 10. Crystal and magnetic structure of the antiferromagnetic topological insulator state in MnBi 2Te4. The unit cell consists of two septuple layers. τc 1=2is the half-cell translation vector along the caxis that connects nearest spin-up and spin-down Mn atomic layers. From Hao et al ., Phys. Rev. X 9, 041038 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-11 ©A u t h o r ( s )2 0 2 1The surface states of antiferromagnetic MnBi 2Te4are some- what complicated. Theoretical studies have predicted that the (0001) surface state (i.e., at the surface perpendicular to the zaxis) which breaks the Ssymmetry of the A-type AFM- zstate is gapped,71,72as indicated by the property of antiferromagnetic topo- logical insulators (see Sec. III A ). The first experimental study reported that the (0001) surface state is gapped.71However, subse- quent studies reported that it is gapless.77–79,84,85Figure 11(a) shows an ARPES measurement of the bulk and surface states, inwhich the surface state is clearly gapless Dirac cone at the (0001)surface. Among possible spin configurations that are allowed by symmetry, Ref. 77proposed that the gapless surface state is pro- tected by the mirror symmetry M x, while the Ssymmetry is broken at the surface. (Note that the mirror symmetry Mxis broken in the A-type AFM- zstate.) In other words, A-type AFM with the mag- netic moments along the xaxis (i.e., the in-plane direction), whose bulk and surface spectra obtained by a first-principles calculation is shown in Fig. 11(b) , might be realized in MnBi 2Te4instead of theA-type AFM- zshown in Fig. 11(c) . These observations of the gapless surface states imply the occurrence of a surface-mediated spin reconstruction. As pointed in Ref. 72, it should be noted here that the antiferromagnetic order in MnBi 2Te4is essentially different from such an antiferromagnetic order in Fe-doped Bi 2Se3 which has been proposed to realize a dynamical axion field.20 In the latter case, time-reversal Θand inversion symmetries are both broken, allowing the deviation of the value of θfrom π. The antiferromagnetic fluctuation contributes to the dynam- ical axion field at linear order in the Néel field. In contrast, in MnBi 2Te4, an effective time-reversal Ssymmetry and inversion symmetry are both preserved, keeping the quantization θ¼π and making no contribution to the dynamical axion field atlinear order in the Néel field. 2. Transport properties of MnBi 2Te4thin films Due to the intralayer ferromagnetism and interlayer antifer- romagnetism of the Mn layers, the layered van der Waals crystal MnBi 2Te4exhibit interesting properties in its few-layer thin films. In even-septuple-layer films, P2andΘsymmetries are both broken, but P2Θsymmetry is preserved.73As we have seen above, the presence of P2Θsymmetry leads to doubly degenerate bands. On the other hand, in odd-septuple-layer films, P1symmetry is preserved, but ΘandP1Θsymmetries are both broken, leading to spin-split bands.73Consequently, the Chern number is zero in even-septuple-layer films as required by the P2Θsymmetry, while the Chern number in odd-septuple-layer films can be nonzero. Indeed, first-principles calculations show that there exist gapless chiral edge states in odd-septuple-layer films, whereas there donot in even-septuple-layer films. 73,86It should be noted that the zero-Chern-number state with σxy¼0 is realized by the combina- tion of half-quantized anomalous Hall states with opposite con- ductivities σxy¼+e2=2hat the top and bottom surfaces, as shown in Fig. 12(a) . In other words, this state is an axion insula- tor exhibiting a topological magnetoelectric effect with the quan- tized coefficient θ¼π(see Sec. II D for a phenomenological derivation of the topological magnetoelectric effect). In contrast,even-septuple-layer films have the quantized anomalous Hall con- ductivity σ xy¼+e2=hthat results from the half-quantized anom- alous Hall conductivity σxy¼+e2=2hof the same sign at the top and bottom surfaces, giving rise to the Chern number C¼+1a s shown in Fig. 12(b) . Experimental observations that are consistent with theoreti- cal predictions have been made. Figure 13 shows the resistivity measurement in a six-septuple-layer MnBi 2Te4film,87in which an axion insulator behavior with a zero Hall plateau at the zero magnetic field and a Chern insulator behavior with the quantized Hall resistivity h=e2in a strong magnetic field were clearly observed. Also, the change in the Chern number betweenC¼+1 was observed in response to the change in the magnetic field direction. Figure 14 shows the resistivity measurement in a five-septuple-layer MnBi 2Te4film,88in which a quantum anoma- lous Hall effect with the quantized Hall resistivity h=e2was clearly observed. FIG. 11. (a) Bulk and surface spectra of MnBi 2Te4obtained by an ARPES mea- surement. Bulk and surface spectra of MnBi 2Te4obtained by a first-principles calculation, which assumes (b) A-type AFM with the magnetic moments along thexaxis and (c) A-type AFM with the magnetic moments along the zaxis. From Hao et al ., Phys. Rev. X 9, 041038 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-12 ©A u t h o r ( s )2 0 2 1C. MnBi 2Te4family of materials Taking advantage of the nature of van der Waals materials, the layered van der Waals heterostructures of (MnBi 2Te4)m (Bi2Te3)ncan be synthesized. Here, it is well known that Bi 2Te3is a time-reversal invariant topological insulator.37So far, MnBi 4Te7 (m¼n¼1)89–93and MnBi 6Te10(m¼1a n d n¼2)89,93,94have been experimentally realized. Figure 15 shows schematic illustra- tions of MnBi 4Te7and MnBi 6Te10and their STEM images. In MnBi 4Te7, a quintuple layer of Bi 2Te3and a septuple layer of MnBi 2Te4stack alternately. In MnBi 6Te10, two quintuple layers of Bi 2Te3are sandwiched by septuple layers of MnBi 2Te4.A si n t h ec a s eo fM n B i 2Te4, interlayer antiferromagnetism (between Mn layers) develops with a Néel temperature TN¼13 K in MnBi 4Te789,90,93and TN¼11 K in MnBi 6Te10,93and this anti- ferromagnetic insulator state is protected by the S¼ΘT1=2sym- metry, which indicates that MnBi 4Te7and MnBi 6Te10are also antiferromagnetic topological insulators. It was reported that,due to the gradual weakening of the antiferromagnetic exchange coupling associated with the increasing separation distance between Mn layers, a competition between antiferromagnetismand ferromagnetism occurs at low temperature /C255K . 89,90Am a g - netic phase diagram of MnBi 4Te7is shown in Fig. 16 .A l s o ,t w o distinct types of topological surface states are realized depending on the Bi 2Te3quintuple-layer termination or the MnBi 2Te4 septuple-layer termination.91,92ARPES studies showed that the Bi2Te3quintuple-layer termination gives rise to gapped surface states, while the MnBi 2Te4septuple-layer termination gives rise to gapless surface states.91,92Note that these terminations break the Ssymmetry, which implies in principle gapped surface states (see Sec. III A ). It is suggested that the gap opening in the Bi2Te3quintuple-layer termination can be explained by the mag- netic proximity effect from the MnBi 2Te4septuple layer beneath and that the gaplessness in MnBi 2Te4septuple-layer termination can be explained by the restoration of time-reversal symmetry at FIG. 12. Schematic illustration of (a) an axion insulator state realized in an even-septuple-layer MnBi 2Te4film and (b) a quantum anomalous Hall insulator state realized in an odd-septuple-layer MnBi 2Te4film. In even-septuple-layer (odd-septuple-layer) films, the anomalous Hall conductivities of the top and bottom surfaces are opposite (the same) to each other, resulting in the total anomalous Hall conductivity σxy¼0(σxy¼+e2=h), or equivalently, the Chern number C¼0(C¼+1). From Li et al., Sci. Adv. 5, eaaw5685 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license. FIG. 13. Resistivity measurement in a six-septuple-layer MnBi 2Te4film, showing (a) an axion insulator behavior with a zero Hall plateau at zero mag- netic field and (b) a Chern insulator behavior with the quantized Hall resistivity h=e2in a magnetic field of 9 T. Reproduced with permission from Liu et al ., Nat. Mater. 19, 522 (2020). Copyright 2020 Springer Nature. FIG. 14. Resistivity measurement in a five-septuple-layer MnBi 2Te4film, showing a quantum anomalous Hall effect with the quantized transverse resistiv- ityh=e2at the zero magnetic field. Reproduced with permission from Deng et al ., Science 367, 895 (2020). Copyright 2020 American Association for the Advancement of Science.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-13 ©A u t h o r ( s )2 0 2 1the septuple-layer surface due to disordered spin.92On the other hand, an ARPES study of MnBi 6Te10observed a gapped Dirac surface state in the MnBi 2Te4septuple-layer termination.94 Since the bulk crystals of MnBi 4Te7and MnBi 6Te10are real- ized by van der Waals forces, various heterostructures in the 2D limit, which are made from the building blocks of the MnBi 2Te4 septuple layer and the Bi 2Te3quintuple layer, can be obtained byexfoliation. A theoretical calculation shows that such 2D hetero- structures exhibit the quantum spin-Hall effect without time- reversal symmetry and the quantum anomalous Hall effect.95 Theoretically, it is suggested that (MnBi 2Te4)(Bi 2Te3)nis a higher- order topological insulator hosting surface states with a Möbiustwist. 96In contrast to MnBi 2Te4in which the value of θis quan- tized to be π, it is suggested that the antiferromagnetic insulator phases of Mn 2Bi6Te11(with m¼2 and n¼1)97and Mn 2Bi2Te598 in which the Ssymmetry is absent, break both time-reversal and inversion symmetries, realizing a dynamical axion field. D. EuIn 2As2and EuSn 2As2 EuIn 2As2and EuSn 2As2have also been considered a candi- date class of materials for antiferromagnetic topological insulators with inversion symmetry.99Different from MnBi 2Te4which is a layered van der Waals material, EuIn 2As2has a three-dimensional crystal structure as shown in Fig. 17 . EuSn 2As2has a very similar crystal and magnetic structure to EuIn 2As2. Two metastable mag- netic structures with the magnetic moments parallel to the baxis (AFM kb) and the caxis (AFM kc) have been known in EuIn 2As2 and EuSn 2As2.100,101As in the case of MnBi 2Te4, the antiferromag- netic insulator phases of EuIn 2As2and EuSn 2As2are protected by theS¼ΘT1=2symmetry, with the half-unit-cell translation vector connecting four Eu atoms along the caxis. Indeed, ARPES measure- ments in EuIn 2As2102and EuSn 2As278suggests that they are antifer- romagnetic topological insulators. Theoretically, it is suggested thatantiferromagnetic EuIn 2As2(both AFM kband AFM kc) is at the same time a higher-order topological insulator with gapless chiral hinge states lying within the gapped surface states.99 IV. EXPRESSIONS FOR θIN INSULATORS We have seen in Sec. IIthat time-reversal symmetry and inver- sion symmetry impose the constraint on the coefficient θof the topological magnetoelectric effect such that θ¼πin 3D topologi- cal insulators and θ¼0 in 3D normal insulators. In this section, first, we derive a generic expression for θwhich is given in terms of FIG. 15. Schematic illustrations of (a) MnBi 4Te7and (b) MnBi 6Te10. STEM images of (c) MnBi 4Te7and (d) MnBi 6Te10, showing layered heterostrucrutures. Here, QL and SL indicate a quintuple layer of Bi 2Te3and a septuple layer of MnBi 2Te4, respectively. From Wu et al., Sci. Adv. 5, eaax9989 (2019). Copyright 2019 Author (s), licensed under a Creative Commons Attribution (CC BY) license. FIG. 16. Magnetic phase diagram of MnBi 4Te7as functions of temperature and out-of-plane magnetic field, showing a complex competition between antiferro-magnetism (AFM) and ferromagnetism (FM). From Wu et al ., Sci. Adv. 5, eaax9989 (2019). Copyright 2019 Author(s), licensed under a Creative Commons Attribution (CC BY) license. FIG. 17. Crystal and magnetic structures of EuIn 2As2. There are two metasta- ble magnetic structures where the magnetic moments align parallel to (a) the b axis and (b) the caxis. Reproduced with permission from Xu et al., Phys. Rev. Lett. 122, 256402 (2019). Copyright 2019 American Physical Society.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-14 ©A u t h o r ( s )2 0 2 1the Bloch-state wave function. Then, we show explicitly that the value of θcan be arbitrary in a class of antiferromagnetic insulators with broken time-reversal and inversion symmetries, taking amicroscopic tight-binding model called the Fu –Kane –Mele – Hubbard (FKMH) model as an example. A. General expression for θfrom the dimensional reduction It is known that the chiral anomaly in (1+1) dimensions can be derived from the dimensional reduction from the (2+1)DChern –Simons action. A similar way of deriving the effective action of (3+1)D time-reversal invariant topological insulators from thedimensional reduction from the (4+1)D Chern –Simons action was considered in Ref. 9. To see this, let k wbe the momentum in the fourth dimension and ( kx,ky,kz) be the momentum in 3D spatial dimensions. The second Chern number in 4D momentum space(k x,ky,kz,kw) is given by9,103,104 ν(2)¼1 32π2ð d4kεijkltrfijfkl/C2/C3 , (47) where fij¼@iAj/C0@jAi/C0i[Ai,Aj], Aαβ j¼ihuαj@kjjuβi:(48) Here, juαiis the periodic part of the Bloch wave function of the occupied band α. By substituting the explicit expression for fij(48) into Eq. (47), we obtain ν(2)¼1 8π2ð d4k@ @kwε4jkltrAj@kAl/C02 3iAjAkAl/C20/C21 /C26/C27 ;ð dkw@P3(kw) @kw, (49) where j,k,l¼1, 2, 3 indicate the 3D spatial direction. Here, note thatε4jkl¼/C0εjkl4;/C0εjkldue to the convention ε1234¼1. On the other hand, the corresponding topological action in (4+1) dimen-sion ( x,y,z,w) is given by S¼ν(2) 24π2ð dtd4xεμνρστAμ@νAρ@σAτ, (50) which can be rewritten as S¼ν(2) 8π2ð dtd3xdwε4νρστA4@νAρ@σAτ ¼1 32π2ð dtd3xθ(r,t)ενρστFνρFστ, (51) where we have used the identity ε4νρστ¼ενρστand defined θ(r,t);ν(2)f. Here, f¼Þdw A 4(r,w,t) can be regarded as the flux due to the extra dimension. In analogy with the (1+1)D case in which the first Chern number is given by ν(1)¼Ð df@P=@f with Pthe electric polarization, Eq. (49) indicates a relationbetween the generalized polarization P3and the Chern number ν(2). Then, it follows that P3¼ν(2)f=2π. Finally, we arrive at a general expression for θ,9,32 θ¼/C01 4πð BZd3kεijktrAi@jAk/C02 3iAiAjAk/C20/C21 , (52) where i,j,k¼1, 2, 3, d3k¼dkxdkydkz, and the integration is done over the Brillouin zone of the system. Equation (52) can be derived more rigorously and microscopically, starting from ageneric Bloch Hamiltonian and its wave function. 105,106Figure 18 shows a numerically calculated value of θusing Eq. (52) and other equivalent expressions for θin the Fu –Kane –Mele model on a diamond lattice with a staggered Zeeman field that breaks both time-reversal and inversion symmetries.32One can see that the value of θis no longer quantized once time-reversal symmetry is broken and varies continuously between θ¼0 corresponding to the case of a normal insulator and θ¼πcorresponding to the case of a topological insulator. B. Expression for θin topological magnetic insulators A generic expression for θ[Eq. (52)] is applicable to the arbi- trary band structure. However, some techniques (such as choosing a gauge for the Berry connection A) are required to calculate numerically. On the other hand, it has been shown that there existsan explicit expression for θthat can be calculated easily from the Bloch Hamiltonian of a certain class of insulators with brokentime-reversal and inversion symmetries, 20which calculation does not rely on a specific choice of gauge. Here, we consider a generic 4/C24 Bloch Hamiltonian of the form H(k)¼X5 i¼1Ri(k)αi, (53) FIG. 18. Numerically obtained value of θin the Fu –Kane –Mele model on a diamond lattice. Here, β¼tan/C01(jhj=δt1) with h(¼Un) being a staggered Zeeman field in the [111] direction of the diamond lattice, and δt1being the hopping strength anisotropy due to the lattice distortion in the [111] direction. When β¼π(β¼0), the system is a topological (normal) insulator. Reproduced with permission from Essin et al ., Phys. Rev. Lett. 102, 146805 (2009). Copyright 2009 American Physical Society.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-15 ©A u t h o r ( s )2 0 2 1with matrices αisatisfying the Clifford algebra { αi,αj}¼2δij1. Here, the matrix α4is invariant under both time-reversal and spatial inversion. Specifically, it has been known that the antiferro-magnetic insulator phases of 3D correlated systems with spin –orbit coupling, such as Bi 2Se3doped with magnetic impurities such as Fe20and 5 dtransition-metal oxides with the corundum struc- ture,109can be described by Eq. (53). More recently, it has been suggested that van der Waals layered antiferromagnets such asMn 2Bi6Te1197and Mn 2Bi2Te598can also be described by Eq. (53). In such systems, we can calculate the value of θusing the following expression:20,109 θ¼1 4πð BZd3k2jRjþR4 (jRjþR4)2jRj3εijklRi@Rj @kx@Rk @ky@Rl @kz, (54) where i,j,k,l¼1, 2, 3, 5, jRj¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP5 i¼1R2 iq , and the integration is done over the Brillouin zone. 1. Four-band Dirac model Let us derive a simpler expression for θin systems whose effective continuum Hamiltonian is given by a massive Dirac Hamiltonian. We particularly consider a generic Dirac Hamiltonian with a symmetry-breaking mass term of the form H(q)¼qxα1þqyα2þqzα3þm0α4þm5α5,( 5 5 ) which can be derived by expanding Eq. (53) around some momen- tum points Xand retaining only the terms linear in q¼k/C0X. Here, the matrix α4is invariant under both time-reversal and spatial inversion and the matrix α5¼α1α2α3α4breaks both time-reversal and inversion symmetries. In other words, the system has both time-reversal and inversion symmetries when m 5¼0. For concreteness, we require that the system be a time-reversal invariant topological insulator when m0,0, as we have considered in Eq. (25).T h e action of the system in the presence of an external electromagneticpotential A μis given by [see also Eq. (26)] S¼ð dtd3r/C22ψ(r,t)iγμ(@μ/C0ieAμ)/C0m0eiθγ5hi ψ(r,t), (56) where tis real time, ψ(r,t) is a four-component spinor, /C22ψ¼ψyγ0, m0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (m0)2þ(m5)2p ,c o s θ¼m0=m0,s i n θ¼/C0m5=m0,a n dw e have used the fact that α4¼γ0,α5¼/C0iγ0γ5and αj¼γ0γj (j¼1, 2, 3). Here, the gamma matrices satisfy the identities {γμ,γ5}¼0a n d{ γμ,γν}¼2gμνwith gμν¼diag(1, /C01,/C01,/C01) (μ,ν¼0, 1, 2, 3). One can see that the action (56) is identical to Eq.(28), except for the generic value of θin the exponent. By applying Fujikawa ’s method to the action (56),t h e θterm is obtained as110,111 Sθ¼ð dtd3re2 2πhθE/C1B,( 5 7 )where θ¼π 2[1/C0sgn(m0)]/C0tan/C01m5 m0/C18/C19 : (58) Here, the first term in Eq. (58) is 0 or π, which descri bes whether the system is topologically trivial or nontrivial. The second termin Eq. (58) describes the deviation from the quantized value due to the m 5mass. Note that tan/C01(m5=m0)/C25m5=m0, i.e., the devi- ation is proportional to m5when m5/C28m0. 2. Fu –Kane –Mele –Hubbard model on a diamond lattice In Eq. (58), we have seen that the m5mass term that breaks both time-reversal and inversion symmetries generates a deviation of the value of θfrom the quantized value πor 0. Here, following Ref.110, we discuss a microscopic origin of this m5mass term and derive an expression for θof the form of Eq. (58) in a 3D corre- lated system with spin –orbit coupling. To this end, we start with the Fu –Kane –Mele –Hubbard (FKMH) model on a diamond lattice, whose tight-binding Hamiltonian is given by6,8,110,111 H¼X hi,ji,σtijcy iσcjσþi4λ a2X hhi,jiicy iσ/C1(d1 ij/C2d2 ij)cjþUX ini"ni#,(59) where cy iσis an electron creation operator at a site iwith spin σ(¼",#),niσ¼cy iσciσ,a n d ais the lattice constant of the fcc lattice. d1 ijandd2 ijare the two vectors that connect two sites iandjon the same sublattice. σ¼(σx,σy,σz) are the Pauli matrices for the spin degree of freedom. The first through third terms in Eq. (59) represent the nearest-neighbor hopping, the next-nearest-neighbor spin –orbit coupling, and the on-site repulsive electron –electron inter- actions, respectively. In the mean-field approximation, the interaction term is decomposed as UP ini"ni#/C25UP ih hni#ini"þhni"ini#/C0hni"i/C2 hni#i/C0h cy i"ci#icy i#ci"/C0hcy i#ci"i/C2cy i"ci#þhcy i"ci#ihcy i#ci"ii . The spin – orbit coupling breaks spin SU(2) symmetry and, therefore, the directions of the spins are coupled to the lattice structure. Hence, we should parameterize the antiferromagnetic ordering betweenthe two sublattices AandB[seeFig. 19(a) ] in terms of the spheri- cal coordinate ( n,θ, w), hSi0Ai¼/C0 h Si0Bi¼(nsinθcosw,nsinθsinw,ncosθ) ;n1exþn2eyþn3ez(;n), (60) where hSi0μi¼1 2hcy i0μασαβci0μβi(μ¼A,B)w i t h i0denoting the i0th unit cell. It is convenient to express the mean-field Hamiltonian in terms of the 4 /C24αmatrices that anticommute with each other. We can define the basis ck;[ckA",ckA#,ckB",ckB#]Twith the wave vector ki nt h ef i r s tB r i l l o u i nz o n eo ft h ef c cl a t t i c e[ s e e Fig. 19(b) ]. Then, the single-particle Hamiltonian HMF(k)[HMF;P kcy kHMF(k)ck]i s written in the form of Eq. (53),6,8where the alpha matrices αiareJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-16 ©A u t h o r ( s )2 0 2 1given by the so-called chiral representation, αj¼σj0 0/C0σj/C20/C21 ,α4¼01 10/C20/C21 ,α5¼0/C0i i0/C20/C21 ,( 6 1 ) which satisfies { αi,αj}¼2δij1with α5¼α1α2α3α4.I nt h ep r e s e n t basis, the time-reversal operator and spatial-inversion (parity) operator are given by T¼1/C10(/C0iσ2)K(Kis the complex conjugation opera- tor) and P¼τ1/C101, respectively. We have introduced the hopping strength anisotropy δt1due to the lattice distortion along the [111] direction. Namely, we have set such that tij¼tþδt1for the [111] direction, and tij¼tfor the other three directions. When δt1¼0, the system is a semimetal, i.e., the energy bands touch at the three points Xr¼2π(δrx,δry,δrz)(r¼x,y,z)w i t h δxx¼δyy¼δzz¼1 (and otherwise zero) indicating a Kronecker delta. Finite δt1opens a gap of 2 jδt1jat the Xrpoints. It is notable that, in the ground state characterized by the anti- ferromagnetic order parameter (60), the Dirac Hamiltonians around the Xrpoints acquire another mass induced by α5that breaks both time-reversal and inversion symmetries. In the strongly spin –orbit coupled case when the condition Unf/C282λ(f¼1, 2, 3)is satisfied, we can derive the Dirac Hamiltonians around the ~Xr points, which are slightly deviated from the Xrpoints,110 HMF(~Xrþq)¼qxα1þqyα2þqzα3þδt1α4þUnfα5: (62) Here, the subscript fcan be regarded as the “flavor ”of Dirac fermi- ons. This Hamiltonian (62) has the same form as Eq. (55), which means that Fujikawa ’s method can be applied to derive the θterm in the FKMH model. It follows that110 θ¼π 2[1þsgn(δt1)]/C0X f¼1,2,3tan/C01Unf δt1/C18/C19 : (63) Here, note that this expression for θis valid only when the symmetry-breaking mass Unf(f¼1, 2, 3) is small so that the con- dition Unf/C282λis satisfied. In other words, the Dirac Hamiltonian of the form (62) must be derived as the effective Hamiltonian of the system. A comparison of the analytical result [Eq. (63)]w i t ha numerical result obtained from Eq. (52) in Ref. 32has been made.110In the numerical result ( Fig. 18 ), in which the Néel vector is set to be in the [111] direction as nx¼ny¼nz;h=U, the value of θhas a linear dependence on β/h=δt1when Unf=δt1/C281 (i.e., around β¼0o r β¼π). Thus, the analytical result [Eq. (63)] is in agreement with the numerical result when the deviation from the quantized value (0 or π) is small, since in Eq.(63),t a n/C01(Unf=δt1)/C25Unf=δt1when Unf=δt1/C281. C. Values of θin real materials from first principles In real materials, there are two contributions to the linear magnetoelectric coupling: electronic and ionic (i.e., lattice) contri- butions. These contributions can be further decomposed in to spin and orbital parts. Among the electronic contribution, Eq. (52) rep- resents on an electronic orbital contribution to the isotropic linear magnetoelectric coupling. Here, note that there exist two additional electronic orbital (but non-topological) contributions to the isotro- pic linear magnetoelectric coupling.105,106Cr2O3is an antiferro- magnetic insulator with broken time-reversal and inversionsymmetries and is well known as a material that exhibits a linearmagnetoelectric effect with α xx¼αyyandαzz.Figure 20 shows the value of θin Cr 2O3obtained from a first-principles calculation as a function of the nearest-neighbor distance on the momentum-spacemesh Δk. 36The value of θextrapolated in the Δk¼0 limit is θ¼1:3/C210/C03, which corresponds to αii¼0:01 ps =m (i¼x,y,z). This value is about two orders of magnitude smaller than the experimentally observed value (i.e., full response) of the linear magnetoelectric tensor in Cr 2O3. The values of θin other conventional magnetoelectrics have also been evaluated in Ref. 36 asθ¼0:9/C210/C04in BiFeO 3and θ¼1:1/C210/C04in GdAlO 3, which are both very small compared to the quantized value π.A sa different approach, it has been proposed that the value of θmay be extracted from experimental observed parameters.107,108 W h a ta r et h ec o n d i t i o n sf o rl a r g e rv a l u e so f θin real materials? It was also shown in Ref. 36,t h ev a l u eo f θin Cr 2O3is approximately proportional to the spin –orbit coupling strength, which implies that materials with strong spin –orbit coupling can have large values of θ. FIG. 19. (a) Schematic illustration of the antiferromagnetic order between the two sublattices (denoted by red and blue) in the FKMH model. (b) The firstBrillouin zone of an fcc lattice. Around the X rpoints with r¼x,y,z(repre- sented by green circles), massive Dirac Hamiltonians are derived.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-17 ©A u t h o r ( s )2 0 2 1In addition, as we have seen in Secs. IV A andIV B ,t h eb r e a k i n go f both time-reversal and inversion symmetries are necessary to inducethe deviation of θfrom the quantized values πor 0. The value of θ changes continuously from π[seeFig. 18 and Eq. (58)]. Therefore, a system that lies near a topological insulator phase such as magneti- cally doped topological insulators can be one of good candidate systems. It is notable that if a material has a large value of θ(/differenceπ), then it will exhibit a significantly large magnetoelectric effect of α ii¼e2θ=[(4π2/C22hc)(cμ2 0)]/difference24 ps =m. V. DYNAMICAL AXION FIELD IN TOPOLOGICAL MAGNETIC INSULATORS So far, we have seen the “static ”expressions for θin insulators. In other words, we have not considered what happens in a system with a θterm when the system is excited by external forces. In general, the total value of θcan be decomposed into the sum of the static part (the ground-state value) θ0and the dynamical part δθ(r,t)a s θ(r,t)¼θ0þδθ(r,t): (64) The dynamical part δθ(r,t) is often referred to as the dynamical axion field ,20since the θterm has exactly the same form as the action describing the coupling between a h ypothetical elementary particle, axion, and a photon. Namely, θ(r,t) in condensed matter can be regarded as a (pseudoscalar) field for axion quasiparticles. In this section, first, we derive the action of a xion quasiparticles in topological antiferromagnetic insulators. Th en, we consider the consequences of the realization of the dynamical axion field in the condensed matter. A. Derivation of the action of axion quasiparticles Here, following Refs. 20and 111, we derive the action of axion quasiparticles in topological antiferromagnetic insulators whose effective Hamiltonian is given by a massive Dirac Hamiltonian (55), which is applicable to magnetically dopedBi2Se3and the Fu –Kane –Mele –Hubbard model as we have seen. In this case, the presence of the mass term m5α5that breaks time- reversal and inversion symmetries results in nonquantized valuesofθ.H e r e ,l e tu sc o n s i d e rt h ef l u c t u a t i o no f m 5(which corre- sponds to the fluctuation of the Néel field) denoted by m5þδm5, and derive the action for δm5. For this purpose, it is convenient to adopt a perturbative method. The action of the antiferromag- netic insulator phase in the presence of an external electromag-netic potential A μis written as [see Eq. (56)] S¼ð dtd3r/C22ψ(r,t)iγμDμ/C0m0þiγ5(m5þδm5)/C2/C3 ψ(r,t), (65) where Dμ¼@μ/C0ieA μwith e.0 being the magnitude of the elec- tron charge. By integrating out the fermionic field ψ, we obtain the effective action Weffforδm5andAμas Z¼ð D[ψ,/C22ψ]eiS;eiWeff[δm5,Aμ] ¼exp Tr ln G/C01 0(1þG0V)/C2/C3 /C8/C9 ¼exp Tr ln G/C01 0/C0/C1 /C0X1 n¼11 nTr/C0G0VðÞn"# : (66) In order to obtain the action of the low-energy spin-wave excita- tion, i.e., the antiferromagnetic magnon, we set the Green ’sf u n c - tion of the unperturbed part as G0¼(iγμ@μ/C0m0þiγ5m5)/C01, and the perturbation term as V¼eγμAμþiγ5δm5.N o t et h a tw e have used that iγμDμ/C0m0þiγ5(m5þδm5)¼G/C01 0þV.I nt h e random phase approximation, the leading-order terms read iWeff[δm5,Aμ]¼/C01 2TrG0iγ5δm5/C0/C12 þTr G0eγμAμ/C0/C12G0iγ5δm5/C0/C1 hi ,( 6 7 ) where the first and second terms on the right-hand side corre- spond to a bubble-type diagram and a triangle-type diagram,respectively (see Fig. 21 ). To compute the traces of the gamma matrices we use the fol- lowing identities: tr( γ μ)¼tr(γ5)¼0, tr( γμγν)¼4gμν, tr(γμγνγ5)¼0, and tr( γμγνγργσγ5)¼/C04iεμνρσ. The first term in Eq.(67) is given explicitly by W1¼ðd4q (2π)4Π(q)δm5(q)δm5(/C0q) /C25iJð dtd3r(@tδm5)2/C0(vi@iδm5)2/C0m2(δm5)2/C2/C3 : (68) Here, J,vi, and mare the stiffness, velocity, and mass of the spin- wave excitation mode, which are given, respectively, by20 J¼@2Π(q) @q2 0/C12/C12/C12/C12 q!0¼ð BZd3k (2π)3P4 i¼1R2 i 16jRj5, (69) FIG. 20. Value of θin Cr 2O3obtained from a first-principles calculation as a function of the nearest-neighbor distance on the momentum-space mesh. Theline indicates the second-order polynomial extrapolation to an infinitely densemesh ( Δk!0). Reproduced with permission from Coh et al., Phys. Rev. B 83, 085108 (2011). Copyright 2011 American Physical Society.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-18 ©A u t h o r ( s )2 0 2 1Jm2¼Π(q)jq!0¼m2 5ð BZd3k (2π)31 4jRj3, (70) where jRj¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP5 a¼1R2 aq and q!0 indicates the limit of both q0!0 and q!0. The second term in Eq. (67) is the so-called tri- angle anomaly, which gives the θterm. The final result is112,113 W2¼ið dtd3re2 4π2/C22h/C0δm5(r,t) m0/C20/C21 E/C1B, (71) from which we find that the fluctuation of the m5α5mass term behaves just as a dynamical axion field. For concreteness, let us consider the antiferromagnetic insula- tor phase of Bi 2Se3family doped with magnetic impurities such as Fe.20In this case, the direction of the Néel field nin the ground state is along the zaxis: m5¼/C0(2=3)Unzandnx¼ny¼0, where Uis the on-site electron –electron interaction strength. Defining δθ(r,t)¼/C0δm5(r,t)=m0¼(2=3)Uδnz=m0and substituting this into Eqs. (68) and(71), we, finally, arrive at the action of the axion quasiparticle, Saxion¼g2Jð dtd3r(@tδθ)2/C0(vi@iδθ)2/C0m2δθ2/C2/C3 þð dtd3re2 4π2/C22hδθ(r,t)E/C1B, (72) where g2¼m2 0. Finally, we mention briefly the case of the FKMH model. We find from Eq. (62)that there exist three m5,fα5mass terms with m5,f¼Unf(f¼1, 2, 3). Namely, all the three spatial compo- nents of the Néel field nis contained in the kinetic part of the action of the axion field, which means that the kinetic part is described by the nonlinear sigma model for antiferromagnets.114This is interesting because an effective action of an antiferromagnet is naturally derivedalthough our original action (65) does not explicitly indicate that the mass m 5corresponds to a component of the Néel field. B. Emergent phenomena from axion electrodynamics In the following, we consider the consequences of the realization of a dynamical axion field in condensed matter. Among several theo- retical studies on the emergent phenomena from a dynamical axion field,20,111,115 –118we particularly focus on three studies on theresponses of topological antiferromagnetic insulators with a dynami- cal axion field δθ(r,t) to external electric and magnetic fields. 1. Axionic polariton It has been proposed that the presence of a dynamical axion field can lead to a new type of polariton, the axionic polariton.20 To see this, we start with the total action involving an axion field δθ[Eq. (72)] and an electromagnetic field Aμ¼(A0,/C0A), which is given by S¼g2Jð dtd3r(@μδθ)(@μδθ)/C0m2δθ2/C2/C3 þð dtd3rα 4π2δθE/C1B/C01 16πð dtd3rFμνFμν, (73) where α¼e2=/C22hc≃1=137 is the fine-structure constant and Fμν¼ @μAν/C0@νAμis the electromagnetic field tensor. Note that E/C1B¼ (1=8)εμνρλFμνFρλand FμνFμν¼2(B2=μ0/C0ε0E2). Here, recall that the classical equation of motion for a field fis generically obtained from the Euler –Lagrange equation, δS δf¼@L @f/C0@μ@L @(@μf)/C18/C19 ¼0, (74) where Lis the Lagrangian density of the system. We consider the case of a constant magnetic field B¼B0. Then, the equations of motion for the axion and electromagnetic fields are obtained from Eq. (74)as @2E @t2/C0c02∇2E/C0α πεB0@2δθ @t2¼0, @2δθ @t2/C0v2∇2δθþm2δθ/C0α 8π2g2JB0/C1E¼0,(75) where c0is the speed of light in the media and εis the dielectric cons- tant. Neglecting the dispersion of the axion field compared to theelectric field E, the dispersion of the electric field, i.e., the axionic polariton, ω +(k), is given by20 2ω+(k)¼c02k2þm2þb2 +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (c02k2þm2þb2)2/C04c02k2m2q , (76) with b2¼α2B2 0=8π3εg2J. The photon dispersion in the absence of the axion field is just ω(k)¼c0k. In the presence of the axion field, the photon dispersion ω+(k) has two branches separated by a gap between mandffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2þb2p .A ss h o w ni n Fig. 22 ,t h i sg a pg i v e sr i s et o a total reflection of incident light in the case when the incident lightfrequency is in the gap. The point is the tunability of the axionicpolariton gap by the external magnetic field B 0. 2. Dynamical chiral magnetic effect and anomalous Hall effect Next, we consider an electric current response in insulators with a dynamical axion field. To this end, we rewrite the θterm in the Chern –Simons form, which procedure becomes possible when FIG. 21. Schematic of (a) a bubble-type Feynman diagram and (b) a triangle- type Feynman diagram. The solid lines, wavy lines, and double lines indicatethe Green ’s function G 0, the electromagnetic field A, and the Néel field δm5, respectively.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-19 ©A u t h o r ( s )2 0 2 1a dynamical axion field is realized Sθ¼/C0ð dtd3re2 8π2/C22hεμνρλ[@μθ(r,t)]Aν@ρAλ: (77) Then, the induced four-current density jνcan be obtained from the variation of the above action with respect to the four potential Aν asjν¼δSθ=δAν¼/C0(e2=4π2/C22h)εμνρλ[@μθ(r,t)]@ρAλ:The induced electric current density and charge density are given by12 j(r,t)¼δSθ δA¼e2 4π2/C22h_θ(r,t)Bþ∇θ(r,t)/C2E/C2/C3 , ρ(r,t)¼δSθ δA0¼/C0e2 4π2/C22h∇θ(r,t)/C1B,(78) where _θ¼@θ(r,t)=@t. The magnetic-field induced current is the so-called chiral magnetic effect, which was first studied in nuclearphysics. 49The electric-field induced current is the anomalous Hall effect, since it is perpendicular to the electric field. Note that t h ee l e c t r i cc u r r e n t[ E q . (78)] is a bulk current that can flow in insulators:111the magnetic-field induced and electric-field induced currents are, respectively, understood as a polarizationcurrent @P=@t¼e 2=(4π2/C22h)_θBand a magnetization current ∇/C2M¼e2=(4π2/C22h)∇θ/C2E,w h e r e PandMare directly obtained from the θterm [see Eq. (2)]. The electric current given by Eq. (78) has been studied in the antiferromagnetic insulator phase of theFKMH model. 111As we have seen in Eq. (63), the dynamical axion field can be realized in the FKMH model by the fluctuation of theantiferromagnetic order parameter, i.e., by the antiferromagnetic spin excitation. The magnetic-field induced current in Eq. (78), i.e., the dynamical chiral magnetic effect, emerges due to the time depen-dence of the antiferromagnetic order parameter. The simplest situa- tion is the antiferromagnetic resonance. The dynamics of the sublattice magnetizations hS i0Ai¼mAandhSi0Bi¼mBcan bephenomenologically described by119 _mA¼mA/C2/C0 ωJmBþgμBBþωA(mA/C1en0) ½/C138 en0 fg , _mB¼mB/C2/C0 ωJmAþgμBBþωA(mB/C1en0) ½/C138 en0 fg ,(79) where ωJand ωAare the exchange field and anisotropy field, respectively. Here, we have considered the case where a microwave(i.e., ac magnetic field) of frequency ω rfis irradiated and a static magnetic field B¼Ben0is applied along the easy axis of the anti- ferromagnetic order. In the antiferromagnetic resonance state thatis realized when ω rf¼ω+, the antiferromagnetic order parameter is described as the precession around the easy axis,119 n+(t);[mA(t)/C0mB(t)]=2/C25n0en0þδn+eiω+t, (80) where ω+¼gμBB+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (2ωJþωA)ωAp are the resonance frequen- cies. Schematic illustration of the dynamics of mAandmBin the antiferromagnetic resonance state is shown in Fig. 23(a) . Substituting the solution (80) into the first term in Eq. (78), a sim- plified expression for the dynamical chiral magnetic effect isobtained around the phase boundary where Un f=M0/C281a s111 jCME(t)¼e2 4π2/C22hUD 1 M0BX a¼+ωaδnasinωatþα ðÞ , (81) where D1is a constant and δn+is a Lorentzian function of ωrf. Equation (81) means that an alternating current is induced by the antiferromagnetic resonance. The maximum value of the dynami- cal chiral magnetic effect (81)jjCMEjmax¼e2 4π2/C22hUjD1j jM0jBω+δn+is esti- mated as jjCMEjmax/difference1/C2104A=m2, which is experimentally observable. It should be noted that there is no energy dissipation FIG. 22. Axionic polariton phenomenon. (a) In the absence of a static magnetic field, the incident light can transmit through the media. (b) In the presence of astatic magnetic field parallel to the electric field of light, a total reflection of inci-dent light occurs when the incident light frequency is in the gap. Reproduced with permission from Li et al ., Nat. Phys. 6, 284 (2010). Copyright 2010 Springer Nature. FIG. 23. Schematic figures of (a) an antiferromagnetic resonance state and (b) a 1D antiferromagnetic domain wall.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-20 ©A u t h o r ( s )2 0 2 1due to Joule heat in the dynamical chiral magnetic effect, unlike the conventional transport regime under electric fields. The electric-field induced current in Eq. (78), i.e., the anoma- lous Hall effect, emerges due to the spatial dependence of the anti-ferromagnetic order parameter. As an example, we consider a 1Dantiferromagnetic spin texture of length Lalong the Zdirection, an orientational domain wall. 120,121As shown in Fig. 23(b) , the anti- ferromagnetic order parameter n(r)¼[mA(r)/C0mB(r)]=2 at the two edges has a relative angle δ, resulting in θ(Z¼0)¼θ0and θ(Z¼L)¼θ0þδin the original spherical coordinate. A simpli- fied expression for the anomalous Hall effect is obtained around the phase boundary where Unf=M0/C281a s111 JX AHE¼ðL 0dZ jX AHE(Z)¼e2 4π2/C22hUD 2 M0EY, (82) where D2(δ)¼P f[nf(θ0þδ)/C0nf(θ0)] is a constant and a static electric filed Eis applied perpendicular to the antiferromagnetic order as E¼EYeY. The Hall conductivity is estimated as σXY¼e2 4π2/C22hUD 2 M0/difference1/C210/C02e2=h, which is experimentally observ- able. Note that D2¼0 when δ¼0, which means that this anoma- lous Hall effect does not arise in uniform ground states. 3. Inverse process of the dynamical chiral magnetic effect In Eq. (81), we have seen that ac current is generated by the antiferromagnetic resonance. It is natural to consider the inverseprocess of the dynamical chiral magnetic effect, i.e., a realization of the antiferromagnetic resonance induced by the ac electric field. 116 To this end, we study a continuum model of an antiferromagnet whose free energy is given by122,123 F0¼ð d3ra 2m2þA 2X i¼x,y,z(@in)2/C0K 2n2 z/C0H/C1m"# , (83) where aandAare the homogeneous and inhomogeneous exchange constants, respectively, and Kis the easy-axis anisotropy along the z direction. nandmare the Néel vector and small net magnetization satisfying the constraint n/C1m¼0 with jnj¼1a n d jmj/C281. The fourth term is the Zeeman coupling with H¼gμBBbeing an exter- nal magnetic field. For concreteness, we consider the antiferromag- netic insulator phase of the FKMH model (see Sec. IV B 2 ). The θ term can be written in the free energy form [see also Eq. (14)] Fθ¼/C0e2 4π2/C22hffiffiffi 3p Un0 M0ð d3r(n/C1e[111])E/C1B, (84) where we have used the fact thatP f¼1,2,3nf¼ffiffiffi 3p n/C1e[111], with e[111] being the unit vector along the [111] direction of the original diamond lattice in the FKMH model. Phenomenologically, the antiferromagnetic spin dynamics can be described by the Landau –Lifshitz –Gilbert equation. From the total free energy of the system FAF¼F0þFθ, the effective fields fornandmare given by fn¼/C0δFAF=δnandfm¼/C0δFAF=δm.The Landau –Lifshitz –Gilbert equation is given by116,123 _n¼(γfm/C0G1_m)/C2n, _m¼(γfn/C0G2_n)/C2nþ(γfm/C0G1_m)/C2mþτSP,(85) where γ¼1=/C22h,G1and G2are dimensionless Gilbert-damping parameters, and τSP¼/C0GSP(_n/C2nþ_m/C2m) is the additional damping torque with a spin pumping parameter GSP.124,126Let us consider a case where an ac electric field Eac(t)¼Eaceiω0tezand a static magnetic field B¼Bezare both applied along the easy axis. Assuming the dynamics of the Néel field n(t)¼ezþδn(t) and the net magnetization m(t)¼δm(t) and solving the above Landau – Lifshitz –Gilbert equation, it is shown that the antiferromagnetic resonance can be realized by the ac electric field Eac(t). The reso- nance frequencies are116 ω+¼ωH+ffiffiffiffiffiffiffiffiffiffiffiωaωKp, (86) where ωH¼γgμBB,ωa¼γa, and ωK¼γK. The essential point is the coupling of the Néel field and the electric field through the θ term, as is readily seen in Eq. (84). Note that these resonance fre- quencies are not dependent on the parameters of the θterm. This is because the θterm acts only as the driving force to cause the resonance. As shown in Fig. 24 , in the resonance state, a pure dc spin current Jsgenerated by the spin pumping is injected into the attached heavy-metal layer through the interface.124The spin current is converted into an electric voltage across the transversedirection via the inverse spin-Hall effect: 125VSP(ω0)/αSHJs(ω0), where αSHis the spin-Hall angle. For example, in the case of B¼ 0:1 T and Eac¼1V=m with possible (typical) values of the param- eters, the magnitude of VSPin the resonance state is found to be VSP(ω+)/difference10μV,116which is experimentally observable. Furthermore, it should be noted that the above value of the ac elec-tric field, E ac¼1V=m, is small. Namely, from the viewpoint of lower energy consumption, the spin current generation using topo- logical antiferromagnets with the θterm has an advantage FIG. 24. Schematic figure of the electric-field induced antiferromagnetic reso- nance and its detection. An ac electric field Eac(t) induces the antiferromagnetic resonance. A dc pure spin current Jsgenerated by the spin pumping into the attached heavy metal (HM) such as Pt can be detected through the inversespin-Hall effect (ISHE) as a direct current J c(i.e., the voltage VSP).Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-21 ©A u t h o r ( s )2 0 2 1compared to conventional “current-induced ”methods that require such high-density currents as /difference1010A=m2.127 VI. TOPOLOGICAL RESPONSE OF WEYL SEMIMETALS So far, we have focused on the axion electrodynamics in 3D insulators. In this section, we overview topological responses ofWeyl semimetals to external electric and magnetic fields, which are described by the θterm. Although a number of novel phenomena have been proposed theoretically and observed experimentally inWeyl semimetals, 128 –130we here focus on the very fundamental two effects, the anomalous Hall effect and chiral magnetic effect,starting from the derivation of the θterm. We also discuss the neg- ative magnetoresistance effect that arises as a consequence of the condensed-matter realization of the chiral anomaly. A. Derivation of the θterm in Weyl semimetals The Weyl semimetals have nondegenerate gapless linear dis- persions around band-touching points (Weyl nodes). The low- energy effective Hamiltonian around a Weyl node is written as HWeyl(k)¼Q/C22hvFk/C1σ, (87) where Q¼+1 indicates the chirality, vFis the Fermi velocity, and σiare Pauli matrices. The two energy eigenvalues are +/C22hvFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k2 xþk2 yþk2 zq . In contrast to 2D Weyl fermions such as those on the topological insulator surfaces, the 3D Weyl fermions described by Eq. (87) cannot acquire the mass, i.e., cannot be gapped, since all the three Pauli matrices are already used. This indicates the stableness of a single Weyl node. Because the sumof the chiralities of the Weyl nodes (or equivalently the mono-poles in momentum space) in a system must be zero, the simplestrealization of a Weyl semimetal is one with two Weyl nodes of opposite chiralities. Note th at the minimal number of Weyl nodes in Weyl semimetals with broken inversion symmetry isfour, 131while it is two in Weyl semimetals with broken time- reversal symmetry. For concreteness, we consider a 4 /C24 continuum model Hamiltonian for two-node Weyl semimetals with broken time-reversal symmetry,129,132,134,135 H0(k)¼/C22hvF(τzk/C1σþΔτxþb/C1σ), (88) where τiand σiare the Pauli matrices for Weyl-node and spin degrees of freedom, respectively, and Δis the mass of 3D Dirac fermions. The term b/C1σrepresents a magnetic interaction such as the exchange interaction between conduction electrons andmagnetic impurities or the Zeem an coupling with an external magnetic field. Note that the Hamiltonian with b¼0 describes a topological or normal insulator depending on the sign of Δ[see Eq.(25)]. Therefore, the above Hamiltonian (88) can be regarded as a model Hamiltonian describing a magnetically doped (topo-logical or normal) insulator. Without loss of generality, we maysetb¼(0, 0, b). In this case the Weyl semimetal phase is real- ized when jb=Δj.1, and the Weyl nodes are located at (0, 0,+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b 2/C0Δ2p ).129Here, we outline the derivation of the θterm from the micro- scopic four-band model (88). In order to describe a more generic Weyl semimetal, we add the term μ5τzto the Hamiltonian, which generates a chemical potential difference 2 μ5between the two Weyl nodes. Note that this term breaks inversion symmetry. We also setΔ¼0 for simplicity, so that the momentum-space distance between the Weyl nodes are 2 b.Figure 25 shows a schematic illustration of the Weyl semimetal we consider. The action of the system in thepresence of external electric and magnetic fields with the fourpotential A μ¼(A0,/C0A) is given by [see also Eq. (26)] S¼ð dtd3rψyi(@t/C0ieA 0)/C0[H0(kþeA)/C0μ5τz] fg ψ ¼ð dtd3r/C22ψiγμ(@μ/C0ieAμ/C0ibμγ5)ψ, (89) where e.0,ψis a four-component spinor, /C22γ¼ψyγ0,γ0¼τx, γj¼τxτzσj¼/C0iτyσj,γ5¼iγ0γ1γ2γ3¼τz, and bμ¼(μ5,/C0b). Now, we apply Fujikawa ’s method50,51to the action. The procedure is the same as that in the case of topological insulators presented in Sec. II E 2 . Performing an infinitesimal gauge transformation for infinite times such that ψ!ψ0¼e/C0idfθ(r,t)γ5=2ψ,/C22ψ!/C22ψ0¼/C22ψe/C0idfθ(r,t)γ5=2, (90) with θ(r,t)¼/C02xμbμ¼2(b/C1r/C0μ5t) and f[[0, 1], the action of the system becomes21 S¼ð dtd3r/C22ψ[iγμ(@μ/C0ieAμ)]ψ þe2 2π2/C22hð dtd3r(b/C1r/C0μ5t)E/C1B, (91) where the first term represents the (trivial) action of massless Dirac FIG. 25. Schematic illustration of a Weyl semimetal with two Weyl nodes. 2 b and 2 μ5are the momentum-space distance and the chemical potential differ- ence between the Weyl nodes, respectively. Q+¼+1 are the chiralities of the Weyl nodes.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-22 ©A u t h o r ( s )2 0 2 1fermions and the second term is nothing else but a θterm [Eq. (1)] with θ(r,t)¼2(b/C1r/C0μ5t). It should be noted that nonzero, non- quantized expression for θis due to the time-reversal symmetry breaking by band the inversion symmetry breaking by μ5. B. Anomalous Hall effect and chiral magnetic effect Next, let us consider the consequences of the presence of a θ term in Weyl semimetals. As we have also seen in the case of insu- lators with a dynamical axion field, an electric current is induced in the presence of a θterm. The induced electric current density and charge density are given by12 j(r,t)¼δSθ δA¼e2 4π2/C22h_θ(r,t)Bþ∇θ(r,t)/C2E/C2/C3 , ρ(r,t)¼δSθ δA0¼/C0e2 4π2/C22h∇θ(r,t)/C1B:(92) In the present case of θ(r,t)¼2(b/C1r/C0μ5t), we readily obtain a static current of the form j¼e2 2π2/C22hb/C2E/C0μ5B ðÞ , (93) in the ground state. The electric-field induced and magnetic-field induced terms are the anomalous Hall effect and chiral magnetic effect, respectively.21–25,49,129,133,134 To understand the occurrence of the anomalous Hall effect in Weyl semimetals [the first term in Eq. (93)], let us consider a 2D plane in momentum space, which is perpendicular to the vector b. For clarity, we set b¼(0, 0, b) and Δ¼0. In this case, performing a canonical transformation, Eq. (88) can be rewritten in a block- diagonal form with two 2 /C22 Hamiltonians given by129 H+(k)¼/C22hvF(kxσxþkyσy)þm+(kz)σz, (94) with m+(kz)¼/C22hvF(b+jkzj). The two Weyl nodes are located at (0, 0,+b). It can be seen readily that mþ(kz) is always positive and that m/C0(kz) is positive when /C0b/C20kz/C20band otherwise nega- tive. As we have seen in Eq. (9), the Hall conductivity of 2D massive Dirac fermions of the form (94) is given by σ+ xy(kz)¼/C0sgn[m+(kz)]e2=2h. Therefore, we find that the total 2D Hall conductivity is nonzero in the region /C0b/C20kz/C20band other- wise zero, which gives the 3D Hall conductivity as σ3D xy¼ðb /C0bdkz 2πσþ xy(kz)þσ/C0 xy(kz)hi ¼be2 πh: (95) T h i sv a l u ei se x a c t l yt h es a m ea st h a to ft h ef i r s tt e r mi nE q . (93).T h e expression for the anomalous Hall conductivity can be generalizedstraightforwardly to the case of multi-node Weyl semimetals. 136The anomalous Hall conductivity in t wo-node Weyl semimetals [Eq. (95)] is robust against disorder in the sense that the vertex correction in theladder-diagram approximation is absent as long as the chemicalpotential lies sufficiently close to the Weyl nodes. 137,138 The chiral magnetic effect in Weyl semimetals [the second term in Eq. (93)] looks like a peculiar phenomenon. The chiral magneticeffect indicates that a direct current is generated along a static mag- netic field even in the absence of electric fields, when there exists a chemical potential difference δμ¼2μ5between the two Weyl nodes. If the static chiral magnetic effect ex ists in real materials, there will be substantial possible applications. The existence of the static chiral magnetic effect is, however, ruled out in crystalline solids as discussed in Ref. 134, which is also consistent with our understanding that static magnetic fields do not generate equilibrium currents. As shall be dis-cussed in detail below, the chiral magnetic effect can be realizedunder nonequilibrium circumstances, i.e., when the system is drivenfrom equilibrium, for example, by the combined effect of electric and magnetic fields, which has been experimentally observed as the negative magnetoresistance in Weyl semimetals. Another possiblesituation for realizing the chiral magnetic effect is applying theoscillating (low-frequency) magnetic field. 139 –142A related current generation by the oscillating magnetic field is the gyrotropic mag- netic effect (natural optical activity),141,142which is governed by the orbital magnetic moment of the Bloch electrons on the Fermisurface. This is in contrast to the chiral magnetic effect which isdriven by the chiral anomaly and governed by the Berry curvature. 22 Finally, we note that the dynamical chiral magnetic effect in topo- logical antiferromagnetic insulators shown in Sec. VB2 is also one of the dynamical realizations of the chiral magnetic effect. C. Chiral anomaly and the negative magnetoresistance As we have seen above, the chiral magnetic effect does not occur in equilibrium. This means that a chemical potential differ- ence between Weyl nodes δμ¼2μ5needs to be generated dynami- cally in order for the chiral magnetic effect to be realized in Weylsemimetals. In the case of Weyl semimetals, such a chemical poten-tial difference can be generated by the so-called chiral anomaly . The chiral anomaly in Weyl semimetals is referred to as the elec- tron number nonconservation in a given Weyl cone under parallelelectric and magnetic fields, in which the rate of pumping of elec-trons is given by 22,138,143 @Ni @t¼Qie2 4π2/C22h2cE/C1B, (96) where iis a valley (Weyl node) index and Qi¼ðd3k 2π/C22h@f0(εm k) @εm kvm k/C1Ωm k (97) is the chirality of the valley. Here, εm kis the energy of Bloch elec- trons with momentum kin band min a given valley i,f0(εm k)i s the Fermi distribution function, vm kis the group velocity, and Ωm k is the Berry curvature. The difference of the total electron number between the Weyl nodes leads to the difference of the chemical potential between the Weyl nodes δμ. As shown in Fig. 26 ,t h i s electron pumping can also be understood by the electron flowthrough the zeroth Landau level connecting Weyl nodes of oppo- site chiralities induced by a magnetic field. It should be noted here that electron pumping also occurs in parallel temperatureJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-23 ©A u t h o r ( s )2 0 2 1gradient and magnetic field,144,145 @Ni @t¼eB/C1∇T 4π2/C22h2cðd3k 2π/C22hεm k/C0μ T@f0(εm k) @εm kvm k/C1Ωm k,( 9 8 ) which can be termed the thermal chiral anomaly. Here, Tis the (unperturbed) temperature and μis the chemical potential. A phenomenon manifested by the chiral anomaly is a negative magnetoresistance (or equivalently positive magnetoconductance)quadratic in the magnetic field for parallel electric and magneticfields in Weyl and Dirac semimetals. 137,138,143,144Here, note that the usual magnetoresistance due to Lorentz force is positive. For concreteness, we consider the case of electric and magnetic fields along the zdirection. The positive quadratic magnetoconductivity arising from the chiral anomaly reads137,138,143,144 σzz(B2 z)¼e2 4π2/C22hc2(eBz)2v3 F μ2τinter, (99) where μis the equilibrium chemical potential and τinteris the inter- valley scattering time. This unusual magnetoconductivity holds in the low-field limit Bz!0, since it is derived from a semiclassical approach where the Landau quantization can be neglected.Expression (99) is understood as coming from j/(E/C1Bτ inter)B, which indicates that it is a consequence of the chiral magnetic effect [the second term in Eq. (93)]. It has been shown that the vertex correction in the ladder-diagram approximation is absentin the positive quadratic ma gnetoconductivity [Eq. (99)]. 138Such an unusual negative magnetoresistance has recently been experi-mentally observed in the Dirac semimetals Na 3Bi,146 Cd3As2,147,148and ZrTe 5,149and in the Weyl semimetals TaAs150 and TaP.151As shown in Fig. 27 , the observed conductance is pos- itive and proportional to B2in the low-field limit as expected from Eq. (99). Also, we can see that the enhancement of the con- ductance is largest when the angle between the applied current and magnetic field is zero (i.e., when they are parallel), which isin agreement with the theoretical prediction. However, it must be noted here that those experimental observations of the negativemagnetoresistance is now generally understood to be an artifact of“current jetting, ” 130which can be large in high-mobility semime- tals. The point is that disentangling precisely the intrinsic quantum effect of the chiral anomaly from the extrinsic classical effect of current jetting is not easy in experiments,152although its presence is manifested theoretically. VII. GRAVITATIONAL RESPONSE OF TOPOLOGICAL SUPERCONDUCTORS In this section, we discuss topological responses of 3D topo- logical superconductors and superfluids that can regarded as the thermodynamic analog of the axion electromagnetic responses oftopological insulators and Weyl semimetals. A well-known exampleof 3D topological superfluids is the superfluid 3HeBphase.153The topological nature of such topological superconductors and super- fluids will manifest itself in thermal transport properties, such as the quantization of the thermal Hall conductivity,26since charge and spin are not conserved while energy is still conserved. A. Derivation of a gravitational θterm The systematic classification of topologically nontrivial insula- tors and superconductors has been established in terms of symme- tries and dimensionality and has clarified that topologically nontrivial superconductors and superfluids with time-reversal sym-metry are also realized in three dimensions. 153 –155From the bulk- boundary correspondence, there exist topologically protected gapless surface states in topological superconductors. In particular, the superconductivity infers that the gapless surface states are their FIG. 26. Electron pumping due to the chiral anomaly in a Weyl semimetal under parallel electric and magnetic fields along the zdirection. FIG. 27. Magnetic field dependence of the longitudinal conductance in the Dirac semimetal Na 3Bi. The conductance shows a quadratic dependence on the magnetic field strength when the angle f0between the applied current and magnetic field is small, as expected from Eq. (99). Reproduced with permission from Xiong et al ., Science 350, 413 (2015). Copyright 2015 American Association for the Advancement of Science.Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-24 ©A u t h o r ( s )2 0 2 1own antiparticles and thus Majorana fermions.153Because of the fact that Majorana fermions are charge neutral objects, an electric- transport study such as quantum Hall measurement cannot charac-terize their topological nature of topological superconductors.Instead, since the energy is still conserved, thermal transport, espe-cially the thermal Hall conductivity, reflects the topological charac- ter of topological superconductors as κ H¼sgn(m)π2 6k2 B 2hT (100) for the massive Majorana fermion with mass m.26,29 A spatial gradient in energy is related to a temperature gradi- ent, as one can infer from the thermodynamic equality dU¼TdS as follows. Here, Uis the internal energy, Sis the entropy, and Tis the temperature. For simplicity, let us first divide the total system into two subsystems (subsystems 1 and 2). The equilibrium of the total system is achieved when the total entropy is maximized:dS¼dS 1þdS2¼0. Since the energy is conserved, dE2¼/C0dE1, and hence dS1=dE1/C0dS2=dE2¼0, i.e., T1¼T2. Let us now turn on a gradient in the “gravitational potential, ”so that the gravitational potential felt by subsystems 1 and 2 differs by δfg.I nt h i sc a s e ,w e have dE2¼/C0dE1(1þδfg). This suggests the generation of a tem- perature difference T2¼T1(1þδfg). In other words, we can view the “electric ”field Egassociated with the gradient of fg,w h i c hw e call a “gravitoelectric field, ”as a temperature gradient,156 Eg¼/C0∇fg¼/C0T/C01∇T: (101) In analogy with electromagnetism, let us next consider the fol- lowing quantity described in terms of a vector potential Ag, which we call a “gravitomagnetic field ”: Bg¼∇/C2Ag: (102) For example, in a system rotating with the angular velocity Ωz around the zaxis, Agcan be expressed as Ag¼(1=v)Ωzez/C2r,45,157 which gives Bg¼(2=v)Ωzez. Here, vis the Fermi velocity of the system. Therefore, the gravitomagnetic field Bgcan be understood as an angular velocity vector. A gravitomagnetic field Bgcan also be introduced as a quantity which is conjugate to the energy mag- netization (momentum of energy current) MEin the free energy of a Lorentz-invariant system.29It follows that ME¼(v=2)Lwith L the angular momentum in Lorentz-invariant systems, which alsoleads to B g¼(2=v)Ω.29,45 Now, we study the responses of 3D topological superconduc- tors to a temperature gradient Egand a mechanical rotation Bg.F o r simplicity, we consider a sample in a cylindrical geometry withheight ‘and radius ras illustrated in Fig. 28(b) . We assume that magnetic impurities are doped near the surface and themagnetization directions are all perpendicular to the surface so that a uniform mass gap is formed in the surface Majorana state. Let us first introduce a temperature gradient in the zdirection, which generates the energy current j E¼κH@zTon the surface. Since jE=v2corresponds to the momentum per unit area, total momentum due to the surface energy current is Pw¼(2πr‘)jE=v2and thus the induced orbital angular momentumper volume is given by LzjΩz¼rPw (πr2‘)¼2 v2κH@zT: (103) Similarly, upon rotating the cylinder with Ω¼Ωzez(without a temperature gradient), we obtain the induced thermal energy density (the induced entropy change) localized on the top andbottom surfaces, 29 ΔQ(z)jT¼2TΩz v2κt Hδ(z/C0‘=2)þκb Hδ(zþ‘=2)/C2/C3 , (104) where κt H(κb H) is the thermal Hall conductivity on the top (bottom) surface given by Eq. (100) . Here, κt H¼/C0κb Hbecause the magnetization directions on the top and bottom surfaces are oppo- site to each other, resulting in different signs of m[seeFig. 28(b) ]. In terms of the gravitoelectric field Eg¼/C0T/C01∇Tand the momentum of the energy current (i.e., energy magnetization) ME, Eq.(103) can be written as ME¼(TκH=v)Egfrom the relation ME¼(v=2)L. Furthermore, introducing the thermal polarization PEbyΔQ¼/C0∇/C1PE, Eq. (104) can be written similarly as PE¼(TκH=v)Bg. Combining these, we find the correspondence between topological insulators and topological superconductors, TI:@Ma @Eb¼@Pa @Bb, TSC:@Ma E @Eb g¼@Pa E @Bb g: (105) Since the orbital angular momentum is obtained from the internal energy functional as La¼/C0δUθ=δΩa, the coupling energy of the temperature gradient and angular velocity is written as29 Ug θ¼ð d3x2 v2κH∇T/C1Ω¼ð d3xk2 BT2 24/C22hvθgEg/C1Bg: (106) FIG. 28. Electromagnetic responses in (a) 3D topological insulators and thermal and mechanical (rotation) responses in (b) 3D topological superconduc-tors. In (a), an electric field Einduces the surface Hall current j. In (b), a tem- perature gradient E ginduces the surface thermal Hall current jE. A uniform mass gap is induced in the surface fermion spectra by doping magnetic impuri-ties near the surface of the 3D topological insulator (a) and topological super-conductor (b) such that the magnetization directions are all perpendicular to the surfaces (as indicated by red arrows).Journal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-25 ©A u t h o r ( s )2 0 2 1This is analogous to the axion electromagnetic response with e2=/C22hc$(πkBT)2=6/C22hvand θg¼πplaying the same role as θin theθterm. Here, note that we have considered the contribution from one Majorana fermion to the internal energy (106) .I n general, 3D time-reversal invariant (class DIII) topological super- conductors with topological invariant Npossesses Ngapless Majorana fermions localized at the surface.153When uniform mass gaps (of the same sign) are induced in these Majorana fer-mions, each Majorana fermion gives rise to the half-integert h e r m a lH a l le f f e c t[ E q . (100) ]. 27,30Therefore, it follows that θg¼Nπin Eq. (106) for this generalized case. In the case of 2D topological superconductors, the corre- sponding term is written as U¼ð d2x(2=v2)TκHfΩz: (107) This is the thermodynamical analog of the Chern –Simons term. A similar term has been derived in the context of 3D3HeAphase with point nodes,158where the current flows parallel to the Ω vector. A comparison between cross correlations in topological insulators and topological superconductors in two and three spatialdimensions is summarized in Table II . B. Gravitational instanton term Here, we overview a topological field theory approach to the gravitational (thermal) response of 3D topological superconductorsand superfluids. 27,28In Sec. VII A , we have introduced gravitoelec- tric and gravitomagnetic fields that are written in terms of (ficti- tious) scalar and vector potentials. Strictly speaking, the presence of a gravitational background should be described as a curved space-time. Let us consider the Bogoliubov –de Gennes Hamiltonian of the 3HeBphase, HBdG(k)¼(Δp=kF)k/C1αþξkα4, (108) where kFis the Fermi wave number, Δpis the p-wave pairing amplitude, ξk¼/C22h2k2=2m/C0μwith μthe chemical potential is thekinetic energy, and 4 /C24 matrices αμsatisfy the Clifford algebra {αμ,αν}¼2δμν. Clearly, Eq. (108) is a massive Dirac Hamiltonian. When μ.0(μ,0), the system is topologically nontrivial (trivial).153,159In the presence of such a gravitational background, the action of a 3D topological superconductor such as the3HeB phase is written as160 S¼ð d4xffiffiffiffiffiffi/C0gpL, L¼/C22ψeμ aiγa@μ/C0i 2ωab μΣab/C18/C19 ψ/C0m/C22ψψ,(109) where μ¼0, 1, 2, 3 is a spacetime index, a,b¼0, 1, 2, 3 is a flat index,ffiffiffiffiffiffi/C0gp¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C0det(g)p with gμνthe metric tensor, eμ ais the viel- bein, ωab μis the spin connection, and Σab¼[γa,γb]=(4i) is the gen- erator of Lorentz transformation. As in the case of topological insulators (Sec. II E 2 ) and Weyl semimetals (Sec. VI A ), we can apply Fujikawa ’s method to the action (109) , in which the topologi- cal term of a system comes from the Jacobian. After a calculation,we arrive at a gravitational effective action 27,28 Sg¼1 1536 π2ð d4xθεμνρσRα βμνRβαρσ, (110) where θ¼πandRα βμνis the Riemannian curvature tensor. It follows that the coefficient θin Eq. (110) isθ¼0o r π (mod 2 π) due to time-reversal symmetry. However, in 3D time- reversal invariant topological insulators with topological number N, topological actions should have θ¼Nπ. This is because the Hamiltonian of a noninteracting 3D time-reversal invariant topo-logical insulator with topological number Ncan be decomposed into Ncopies of the Hamiltonian of the form (108) . The gravita- tional effective action (110) provides only a Z 2classification of 3D time-reversal invariant topological insulators, which is weaker than theZclassification that they have. C. Emergent phenomena from a dynamical gravitational axion field As we have seen in Sec. VII A , the derivation of the internal energy term [Eq. (106) ] for 3D topological superconductors and superfluids is not a microscopic derivation but a heuristic onebased on the surface thermal Hall effect. It has been suggested thatthe fluctuation of θ gin apþis-wave superconductor can be written as a function of the relative phase between the two superconducting gaps.161,162Such a fluctuation of a relative phase is known as the Leggett mode and can depend on time. Also, in analogy with 3Dtopological insulators, it is expected that the internal energy term canbe extended to the form of an action (see Sec. II D for the derivation of the θterm in 3D topological insulators). Therefore, it would be appropriate to consider the action of the form 162,163 Sg θ¼k2 BT2 0 24/C22hvð dtd3rθg(r,t)Eg/C1Bg (111)TABLE II. Comparison between cross correlations in topological insulators (TIs) and topological superconductors (TSCs) in two and three spatial dimensions. In topologi-cal superconductors, the orbital angular momentum L(momentum of energy current M E) and the entropy S(thermal polarization PEin three dimensions) are generated by a temperature gradient Eg¼/C0 T/C01rTand by a mechanical rotation with angular velocity vector Ω=(v/2)Bg. In analogy with the orbital magnetoelectric polar- izability χab θ¼δabe2=(4π/C22hc) in 3D topological insulators, the gravitomagnetoelectric polarizability χab θ,g¼δabπk2 BT2=(24/C22hv) can be introduced in 3D topological super- conductors. Note that the relations for topological superconductors applies also to the thermal response of topological insulators. TI TSC 2DσH¼ec@Mz @μ¼ec@N @BzκH¼v2 2@Lz @T¼v2 2@S @Ωz 3Dχab θ¼@Ma @Eb¼@Pa @Bbχab θ,g¼@Ma E @Eb g¼@Pa E @Bb gJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-26 ©A u t h o r ( s )2 0 2 1for non-quantized and dynamical values of θg, instead of the internal energy Uθ g[Eq.(106) ]. In order to induce the deviation of θgfrom the quantized value Nπ(with Nthe topological number of the system), time- reversal symmetry of the bulk needs to be broken, as in the caseof insulators. It has been shown theoretically that the imaginary s-wave pairing in class DIII topological superconductors such as the 3HeBphase leads to the deviation of the value of θgfrom π such that θg¼πþtan/C01(ΔIm s=μ)w i t h ΔIm sthe imaginary s-wave pairing amplitude.161Such an imaginary s-wave pairing term in a Bogoliubov –de Gennes Hamiltonian corresponds to the chiral symme- try breaking term (which also breaks time-reversal symmetry) Γ¼ΘΞ,w h e r e ΘandΞare the time-reversal and particle-hole opera- tors, respectively.30,109,162Therefore, the resulting superconducting state belongs to the class D.153,155,164When we take into account the superconducting fluctuations ΔIm s¼jΔIm sjeiθs(r,t)andΔp¼jΔpjeiθp(r,t), t h er e l a t i v ep h a s ef l u c t u a t i o n θr(r,t);θs(r,t)/C0θt(r,t), i.e., the Leggett mode, gives rise to a dynami cal gravitational axion field, as δθg(r,t)/δθr(r,t).161,162 Let us briefly consider the consequences of the realization of a dynamical gravitational axion field in 3D topological superconduc- tors. In the presence of a dynamical gravitational axion field δθg(r,t), a bulk heat current is obtained from the action (111) as163 jT(r,t)¼k2 BT2 0 12/C22hv_θg(r,t)Bgþv∇θg(r,t):/C2Eg/C2/C3 : (112) This expression should be compared with an electric current (78) obtained from the θterm in insulators. The first term in Eq. (112) indicates that a heat current is induced in the bulk of a 3D supercon- ductor by a gravitomagnetic field, i.e., by a mechanical rotation. This phenomenon is called the chiral gravitomagnetic effect163and can be understood as the thermal analog of the chiral magnetic effect. Thesecond term in Eq. (112) indicates that a heat current is induced in the bulk by a gravitoelectric field, i.e., by a temperature gradient, which is the anomalous thermal Hall effect since this current is per- pendicular to the temperature gradient. VIII. SUMMARY AND OUTLOOK In this Tutorial, we have overviewed the responses of 3D condensed-matter systems to external fields, which are described by the topological terms in their low-energy effective actions. We haveseen microscopically that the so-called θterm, which originally appeared in particle theory, is derived in topological insulators andWeyl semimetals. In the case of insulators, the coefficient θin the θterm takes the quantized value πor 0 in the presence of either time-reversal or inversion symmetry, and it can be arbitrary in theabsence of both symmetries. The θterm with θ¼πleads to a hallmark response of topological insulators, the topological mag-netoelectric effect. We note that, in spite of intensive experimental efforts, the direct observation of the topological magnetoelectric effect, i.e., observing the electric polarization induced by a mag-netic field or the magnetization induced by an electric field, intopological insulators is yet to be realized. We have also seen that a dynamical axion field δθ(r,t), the deviation of θfrom the ground- state value θ 0, can be realized by the antiferromagnetic spinfluctuation in a class of antiferromagnetic insulators with a θterm. In general, it is possible that the fluctuation of order parameters other than the antiferromagnetic order parameter also realizes adynamical axion field. In the case of Weyl semimetals, the expres-sion for θh a sas i m p l e rf o r mg i v e ni nt e r m so ft h ed i s t a n c ei n momentum space and the energy difference between Weyl nodes. Theθterm leads to a realization of the chiral anomaly in condensed- matter systems, which has been experimentally observed in Weyl andDirac semimetals through the negative magnetoresistance effect dueto the chiral magnetic effect. In Sec. III, we have focused on recent experimental realiza- tions of the axion insulator state where θ¼πdue to an “effec- tive ”time-reversal symmetry in MnBi 2Te4family of materials. The MnBi 2Te4family of materials are layered van der Waals compounds and thus the synthesis of few-layer thin films thatcan realize exotic phases and phenomena is possible. Especially, because of the intrinsic ferromagnetism of the MnBi 2Te4septu- ple layer, the anomalous Hall conductivity of even-layer (odd-layer) thin films is zero (quantized). Such a magnetization con-figuration with zero anomalous Hall conductivity is indeed thesituation that has been pursued fo r the observation of the topo- logical magnetoelectric effect. T herefore, an experimental obser- vation of the topological magnetoelectric effect might beachieved in the near future. As we have seen in Sec. VB, the dynamical chiral magnetic effect and its inverse effect in insulators have an important feature that they are energy-saving. The dynamical chiral magnetic effectin insulators is an ac electric current generation by a magnetic fieldand, therefore, does not cause energy dissipation due to Joule heat,although the dynamical axion field needs to be excited by external forces (which may cause energy loss). Its inverse effect is an electri- cal excitation of a dynamical axion field and the applied ac electricfield does not cause energy dissipation due to Joule heat becausethe system is insulating. These effects might be utilized for low-energy consumption devices. Recently, it was proposed that topological antiferromagnetic insulators with a dynamical axion field can be utilized to detect(true) axion as dark matter, 165which will be certainly an interest- ing possible application of such topological antiferromagneticinsulators. The outline of the proposal is as follows. Inside the topological antiferromagnetic insulator, the (true) axion couples to the axionic polaritons (i.e., electric field), which are generatedin the presence of axion quasiparticles (see also Sec. VB1 ). At the topological antiferromagnetic insulator dielectric boundary, the axionic polaritons convert to propagating photons, which are finally detected in the THz regime. Such a conversion process isresonantly enhanced when the axion frequency is equal to theaxionic polariton frequency. A microscopic derivation of the gravitational θterm [Eqs. (106) and(111) ] in the bulk of 3D topological superconduc- tors remains an important open issue, since the derivation outlinedin Sec. VII A is based on the surface thermal Hall effect of Majorana fermions. Similarly, it has been suggested that Weylsuperconductors can exhibit the anomalous thermal Hall effect 166 and that a gravitational θterm should also be derived in Weyl superconductors,163considering the fact that topological insulators and Weyl semimetals are both described by the θterm. WhenJournal of Applied PhysicsTUTORIAL scitation.org/journal/jap J. Appl. Phys. 129, 141101 (2021); doi: 10.1063/5.0038804 129, 141101-27 ©A u t h o r ( s )2 0 2 1treating gravitoelectric and gravitomagnetic fields microscopically, we might need to introduce a torsion field.167 –169 ACKNOWLEDGMENTS A.S. acknowledges valuable discussions with Koji Ishiwata and Makoto Naka. A.S. is supported by the Special PostdoctoralResearcher Program of RIKEN. K.N. is supported by JST CRESTGrant No. JP-MJCR18T2 and JSPS KAKENHI Grant No. JP20H01830. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. L. Kane and E. J. 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1.5052706.pdf

Enhancement of damping in FeNiN film due to two-magnon scattering effect Zengtai Zhu , Hongmei Feng , Hongkang Xie , Qingfang Liu , and Jianbo Wang Citation: Appl. Phys. Lett. 113, 232402 (2018); doi: 10.1063/1.5052706 View online: https://doi.org/10.1063/1.5052706 View Table of Contents: http://aip.scitation.org/toc/apl/113/23 Published by the American Institute of Physics Articles you may be interested in Switching domain wall motion on and off using a gate voltage for domain wall transistor applications Applied Physics Letters 113, 232401 (2018); 10.1063/1.5053852 Linear and nonlinear spin-wave dynamics in ultralow-damping microstructured Co 2FeAl Heusler waveguide Applied Physics Letters 113, 232404 (2018); 10.1063/1.5038836 Investigation of Gilbert damping of a tetragonally distorted ultrathin Fe 0.5Co0.5 epitaxial film with high magnetic anisotropy Applied Physics Letters 113, 232406 (2018); 10.1063/1.5052721 Dzyaloshinskii-Moriya interaction at disordered interfaces from ab initio theory: Robustness against intermixing and tunability through dusting Applied Physics Letters 113, 232403 (2018); 10.1063/1.5049876 Measurements of interlayer exchange coupling of Pt in Py|Pt|Py system Applied Physics Letters 113, 142401 (2018); 10.1063/1.5050935 Temperature dependence of interlayer exchange coupling and Gilbert damping in synthetic antiferromagnetic trilayers investigated using broadband ferromagnetic resonance Applied Physics Letters 113, 042401 (2018); 10.1063/1.5040666Enhancement of damping in FeNiN film due to two-magnon scattering effect Zengtai Zhu,1Hongmei Feng,1Hongkang Xie,1Qingfang Liu,1and Jianbo Wang1,2,a) 1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, People’s Republic of China 2Key Laboratory for Special Function Materials and Structural Design of the Ministry of the Education, Lanzhou University, Lanzhou 730000, People’s Republic of China (Received 21 August 2018; accepted 10 November 2018; published online 4 December 2018) Ferromagnetic resonance is used to study the damping of FeNiN films prepared by magnetron sputtering technology. The experimental results show that nitrogen doping significantly increasesthe magnetic damping of the samples, while its effect on the coercivity is negligible. We attribute the enhanced damping to the two-magnon scattering effect, supporting this by the analysis of the angular dependence of the ferromagnetic resonance field. Our study provides a method to adjustthe magnetic damping and enriches the understanding of the relationship between the magnetic damping and the doping process. Published by AIP Publishing. https://doi.org/10.1063/1.5052706 The magnetic damping behavior of soft magnetic films has been widely investigated due to their applications in high frequency microwave devices, spintronic devices, and magnetic random access memory. 1–5Determination and adjustment of the damping constant are critical for the per-formance of devices. The process of magnetic moment precession can be described phenomenologically by the Landau-Lifshitz-Gilbert equation 6 dM dt¼/C0cM/C2Heff ðÞ þa MsM/C2dM dt; (1) whereMis the magnetization, Msis the saturation magneti- zation,Heffis the effective magnetic field, cis the gyromag- netic ratio, and ais the dimensionless Gilbert damping constant. Ferromagnetic resonance (FMR) is one of the most important methods to obtain the dynamic magnetic proper-ties of thin films. The linewidth of FMR spectra provides direct information of effective damping, which consists of intrinsic Gilbert damping and extrinsic relaxation mecha- nisms. Intrinsic Gilbert damping is considered to be related to spin-orbit coupling, 7,8and the extrinsic relaxation mecha- nism mainly consists of two-magnon scattering (TMS)9–11 and inhomogeneous broadening in the thin film system.12,13 TMS represents the process from the uniform precession mode degenerating to the spin wave mode and then dissipat- ing to lattice thermal vibration14and inhomogeneous broad- ening is from the inhomogeneous distribution of magnetic parameters.15Recently, some progress has been made for the research of intrinsic damping.8,16The breathing Fermi- surface model is used to calculate and predict Gilbert damp- ing, which agrees well with the experimental results.8With regard to extrinsic relaxation, many works about TMS have also been carried out since Arias and Mills discussed and cal- culated the FMR linewidth contribution from TMS.9Soon, TMS expression was extended to the case of the out-plane magnetization configuration.10TMS theory has explained the enhanced FMR linewidth in many magnetic materials, such as Fe 3Si, Co 2MnSi, FeCoV, and FeCoHfN.17–20Permalloy (Py) is one of the common soft magnetic materials for investigating magnetic damping due to its ultra- low Gilbert damping and excellent soft magnetic properties. It is known that low magnetic damping corresponds to a longrelaxation time, which will limit the reading and writingspeed for high speed magnetic recording devices by increas-ing the switching time. For further application in spin-torquenano-oscillators, the low damping constant is also not condu-cive to broadening the frequency range. 21Doping rare earth elements is always used to adjust the Gilbert damping of Pyby modifying the distance between magnetic atoms. 22,23 Unfortunately, rare earth dopants strengthen the spin orbit coupling, which will inevitably increase the anisotropy and degenerate the soft magnetic properties.24So, a larger mag- netic field or current is needed to reverse the magnetization.In order to maintain excellent soft magnetic properties, wetry to adjust the magnetic damping by controlling TMS inthe magnetic thin films. Inspired by this proposition, wechoose the N element as the dopant in Py films. As one ofthe light elements, the N element has a limited impact on thespin-orbit coupling, and thus, Py films can keep their softmagnetic properties as far as possible. Furthermore, N dop- ants can also increase the number of magnon scattering cen- ters in the films, which may strengthen TMS. FeNiN films were prepared by magnetron sputtering equipment. The mixture of Ar and N 2is used as sputtering gas. The target is 3-in. diameter Ni 80Fe20alloy. Five different sam- ples were prepared by varying the N 2partial pressure with 0, 5%, 10%, 15%, and 20%, which are represented as S1–S5,respectively. The thickness of samples was controlled as 30 nm by monitoring the deposition time. The crystalline struc- ture of the prepared samples is analyzed by X-ray diffraction(XRD PANalytical X’ pert Pro) with Cu K aradiation. X-ray photoelectron spectroscopy (XPS, Kratos AXIS Ultra DLD)w a s used to analyze the compositions of films. The static magneticproperties of samples were measured by using a vibrating sam-ple magnetometer (VSM, Microsence EV9), and its dynamicmagnetic properties were measured by electron spin resonance(ESR, JES-FA300) commercial equipment. Grazing incidence XRD scans in Fig. 1(a) show only Fe 20Ni80(111) peaks. All of the scans show an evident broada)Author to whom correspondence should be addressed: wangjb@lzu.edu.cn. Tel.: þ86-0931-8914171. 0003-6951/2018/113(23)/232402/5/$30.00 Published by AIP Publishing. 113, 232402-1APPLIED PHYSICS LETTERS 113, 232402 (2018) bump instead of the sharp peak, which means that the sam- ples possess a nano-crystalline microstructure. The diffrac- tion peaks switch to a lower angle with increasing N partial pressure, which probably implies that N atoms scatter intothe lattice interstitial and expand the lattices. Figure 1(b) shows the XPS spectra of the samples for the N element. Theresults confirm that the N element is present in all of the pre-pared films except for S1 (Py film). The hysteresis loops of all samples measured along the in-plane direction by VSM are summarized in Fig. 2. With increasing N partial pressure, 4 pM sdecreases from 7702 G to 3330 G, 43% reduction compared to the original value. Table Ishows the coercivity of different samples. It can be seen that the coercivity does not fluctuate significantly,remaining as several Oe. The remanence ratio is very highfor all the samples. It is obvious that the prepared FeNiNfilms possess good soft magnetic properties, especially atlow N partial pressure. The angular dependence of rema- nence on the in-plane polar angle (not shown here) shows a near-zero in-plane anisotropy field for all the samples. Field-swept FMR is one of the most powerful experi- mental techniques to study the dynamic magnetic propertiesof thin films. The effective anisotropy field, saturation magnetization, and damping constant of materials can be obtained by analyzing the FMR experimental data. Figure 3 shows the coordinate system used to analyze FMR formagnetic films in this paper. The total free energy per unit volume of magnetization is given as 13,25,26 F¼/C0 MsHsinhHsinhMcosuH/C0uM ðÞ þcoshH/C0hM ðÞ/C2/C3 þ2pM2 scos2hM/C0K?cos2hM; (2) where the first term is the Zeeman energy, the second term represents the demagnetization energy, and the third term means the perpendicular anisotropy energy. hH(M)is the polar angle between the dc magnetic field (magnetization) and thezaxis, and u H(M) is the azimuth angle between the dc mag- netic field (magnetization) and the xaxis. K?is the perpen- dicular anisotropy constant. The angular frequency ( x)o f the microwave magnetic field can be written as x c/C18/C192 ¼H1H2; (3) H1¼HrcoshH/C0hM ðÞ /C04pMeffcos2hM; (4) H2¼HrcoshH/C0hM ðÞ /C04pMeffcos2hM; (5) where Hris the resonance magnetic field. 4 pMeffis the effec- tive demagnetization field, which can be written as 4pMeff¼4pMsþH?; (6) where H?is the perpendicular anisotropy field; the sign con- vention means that the direction normal to the film surface isthe hard axis when H ?>0. Using Eqs. (3)–(5),4pMeffand g-factor ( c¼glB//C22h) can be obtained by the fitting results of Hras a function of hH. Figures 4(a)–4(e) present the experimental FMR differ- ential spectra measured at some typical polar angles for S1–S5, respectively. The resonance field HrvshHcurves are shown in Fig. 4(f), where the curves are the fitting data. The values of 4 pMeffand g-factor were deduced through the FIG. 1. (a) XRD scans of samples (S1–S5). (b) XPS spectra of the N element for the samples (S1–S5). FIG. 2. Hysteresis loops of all the samples measured along in-plane orienta- tion by VSM.TABLE I. Magnetic parameters obtained from FMR theoretical fitting and hysteresis loops. Sample S1 S2 S3 S4 S5 4pMeff(G) 8000 6300 4400 4500 2700 Hc(Oe) 1.51 1.22 4.82 2.96 1.08 g-factor 2.08 2.15 2.08 2.00 2.00a 0.009 0.010 0.013 0.015 0.024 C 0(mT) 3.7 7.5 60 72 75 FIG. 3. Coordinate system used for the measurement and analysis of FMR.232402-2 Zhu et al. Appl. Phys. Lett. 113, 232402 (2018)fitting data, as shown in Table I. One can notice that 4 pMeff and g-factor decrease with the rise of N partial pressure. The decrease in 4 pMeffis mainly due to the weakening of 4 pMs, which is consistent with the static results. The g-factor can be written as an equation related to the orbital and spin angu-lar momentum of an electron 27 g¼2me elSþlL ShiþLhi; (7) where me/emeans the mass-to-charge ratio of the electron, lSandlLrepresent spin and orbital magnetic moments, and hSiandhLirepresent spin and orbital angular momentum, respectively. Considering the microstructure of Py, the orbital motion of electrons is quenched due to the symmetriclattice, i.e., hLi¼ 0. While the N atom is doped into the films, lattice symmetry is destroyed and orbital angular momentum is no longer entirely quenched, i.e., hLi6¼0. It is reasonable that the g-factor decreases gradually. The measured angular dependences of the FMR line- width are shown in Fig. 5. It can be seen that DHstrongly depends on h H. For S1 and S2, DHexhibits one peak near the vertical direction of the film plane. While for S3–S5, the maximum value of DHbegins to shift towards the in-plane direction of films. DHmeasured in the film plane is larger than that in the vertical direction. Interestingly, the nature ofDHconforms to TMS theory, namely, the intensity of TMS decreases when the applied magnetic field rotates from the in-plane direction to the vertical direction. Generally, the peak-to-peak linewidth ( DHpp)o fF M Ri s presented as11,28 DHpp¼DHGil ppþDHinh ppþDHTMS pp; (8) where DHGil ppis the Gilbert type linewidth due to intrinsic damping; DHinh ppis the inhomogeneous broadening; DHTMS ppis the linewidth from TMS. The peak-peak linewidth due to intrinsic damping is given by29 DHGil pp¼2ffiffiffi 3pax c: (9) This equation shows that intrinsic broadening is proportional tox. In other words, the slope of DHGil ppchanging with xrep- resents the magnitude of intrinsic damping, which is the most commonly used method to fit the value of intrinsic damping. The inhomogeneous broadening can be written as13 DHinh pp¼1ffiffiffi 3pdHr d4pMeff ðÞ/C12/C12/C12/C12/C12/C12/C12/C12D4pMeff ðÞ þ1ffiffiffi 3pdHr dhH/C12/C12/C12/C12/C12/C12/C12/C12DhHðÞ ;(10) FIG. 4. (a)–(e) represent the experi- mental FMR differential spectra mea-sured at some typical h Hfor S1–S5, respectively; (f) angular dependence of the resonance magnetic field and the corresponding fitting curves for differ- ent samples. FIG. 5. The measured angular depend- ences of the FMR linewidth (dark blue dots) and the fitting data of different samples as a function of hH. (a)–(e) represent the results of S1–S5,respectively.232402-3 Zhu et al. Appl. Phys. Lett. 113, 232402 (2018)where D(hH) is the spread of crystallographic axes among various grains, and D4pMeffrepresents the inhomogeneity of the local demagnetizing field. The TMS linewidth from the FMR process was pro- posed by Arias and Mills. It can be written as9 DHTMS pp¼2ffiffiffi 3pC/C1sin/C01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½f2þf0=2ðÞ /C1381=2/C0f0=2 ½f2þf0=2ðÞ /C1381=2þf0=2vuut; (11) where f¼x/2p,f0¼cMeff/2p, andCrepresents the magni- tude of TMS. The equation points out a nonlinear variation ofDHTMS ppwith f. The angular dependence of the FMR linewidth is a com- plicated problem. Gilbert damping, TMS, and inhomoge-neous broadening all attribute to the total FMR linewidth.For the angular dependence of the FMR linewidth, the non- linear magnetic field dragging effect stemming from the strong magnetic anisotropy should be considered. The broad-ening of Gilbert damping and TMS must be multiplied by adragging function N(h H).17The modified expressions of the Gilbert linewidth and TMS linewidth varying with hHcan be rewritten as DHGil pphHðÞ ¼2ffiffiffi 3pax cNhHðÞ ; (12) DHTMS pphHðÞ ¼2ffiffiffi 3pChHðÞ/C1NhHðÞ /C1sin/C01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C0H2 H1þ4pMeffcos 2 hMðÞ cos2hMs : (13) It should be noticed that the expressions of DHGil pphHðÞ, DHTMS pphHðÞ, and ChHðÞ are all different from the previous discussions because hHis introduced to the equations as a variable. Usually, one can obtain the variation trend of TMSthrough the values of C(h H). At present, this is not suitable sinceC(hH) is not a fixed value but a function of hH. In order to solve this problem, an angle-independent parameter C0is extracted from the complex expression of C(hH)9 ChHðÞ /8H2 ?b2p pD¼C0; (14) where the defects in samples are supposed to be rectangular in nature, bis the height of defects, pmeans the defect cover fraction, and Drepresents the exchange stiffness coefficient. Obviously, the TMS linewidth is related to the magneticanisotropy and characteristics of defects in the samples as well as h H. To do this simply, C0was regarded as a parame- ter to judge the magnitude of TMS. In Fig. 5, we show the measured and fitting results of the FMR linewidth includingcontributions from Gilbert damping, TMS, and inhomoge-neous broadening. It can be seen that the fitting curves are ingood agreement with the experimental results. As N partialpressure increases, the TMS linewidth has an increasing weight among the whole FMR linewidth, which leads to a transformation of FMR linewidth curves from “field drag-ging type” to “TMS type.” The magnitude of TMS representsthe characteristics of defects in samples, which reflects theinfluence of N dopants on the microstructure of the samples. It is foreseeable that the increase in defects due to the dopingprocess is not only limited to the N element. For example, by analyzing the angular dependence of the FMR linewidth, Jiang et al. reported the enhanced in-plane linewidth because of TMS in FeGd films with Gd dopants. 30 Two of the most important fitting parameters, Gilbert damping constant and the values of C0, are shown in Table I. The Gilbert damping constant increases from 0.009 to 0.026, which is about 3 times larger than that of S1. Moreover, C0 rises significantly from 2 mT to 55 mT, which is almost 28 times larger. N dopants increase the number of magnon scat-tering centers in the films and thus strengthen TMS. On thepremise of a large increase in effective damping, the soft magnetic properties of samples, such as coercivity and rema- nence ratio, have not changed much. In summary, we have conducted a systematic investiga- tion on the damping of FeNiN films, prepared by the magne-tron sputtering technique. It is found that the N dopants canadjust the effective damping without affecting the soft mag- netic properties. Based on the analysis of the FMR linewidth onh H, TMS is supposed to be the factor that contributes to the enhancement of the FMR linewidth. Our results offer aroute for analyzing the enhanced damping stemming fromthe doping process. In particular, the prepared FeNiN filmsmaintain excellent soft magnetic properties with a large effective damping, which satisfies the requirements of many devices, such as spin-torque nano-oscillators. 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5.0039372.pdf

AIP Advances 11, 025325 (2021); https://doi.org/10.1063/5.0039372 11, 025325 © 2021 Author(s).Large amplitude slow ion-acoustic solitons, supersolitons, and double layers in a warm negative ion plasma with superthermal electrons Cite as: AIP Advances 11, 025325 (2021); https://doi.org/10.1063/5.0039372 Submitted: 02 December 2020 . Accepted: 27 January 2021 . Published Online: 17 February 2021 X. Mushinzimana , F. Nsengiyumva , and L. L. Yadav COLLECTIONS Paper published as part of the special topic on Chemical Physics , Energy , Fluids and Plasmas , Materials Science and Mathematical Physics ARTICLES YOU MAY BE INTERESTED IN Large amplitude ion-acoustic solitary waves in a warm negative ion plasma with superthermal electrons: The fast mode revisited AIP Advances 10, 065305 (2020); https://doi.org/10.1063/1.5127199 Large amplitude ion-acoustic solitons in warm negative ion plasmas with superthermal electrons AIP Advances 7, 115114 (2017); https://doi.org/10.1063/1.4996267 Electrostatic solitary structures in warm multi-ion dusty plasmas: The effect of an external magnetic field and nonthermal electrons Physics of Plasmas 27, 022112 (2020); https://doi.org/10.1063/1.5139195AIP Advances ARTICLE scitation.org/journal/adv Large amplitude slow ion-acoustic solitons, supersolitons, and double layers in a warm negative ion plasma with superthermal electrons Cite as: AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 Submitted: 2 December 2020 •Accepted: 27 January 2021 • Published Online: 17 February 2021 X. Mushinzimana,1,a) F. Nsengiyumva,2,b) and L. L. Yadav3,c) AFFILIATIONS 1Department of Physics, University of Rwanda-College of Science and Technology, P.O. Box 3900, Kigali, Rwanda 2Department of Civil Engineering, Institut D’Enseignement Superieur de Ruhengeri, P.O. Box 155, Musanze, Rwanda 3Department of Mathematics, Science and Physical Education, University of Rwanda-College of Education, P.O. Box 55, Rwamagana, Rwanda a)Author to whom correspondence should be addressed: muxavier2000@yahoo.fr b)Email: franco.nseng@gmail.com c)Email: yadavll@yahoo.com ABSTRACT The pseudopotential approach is used to investigate the ion thermal and electron superthermal effects on the slow mode solitary wave prop- agation characteristics in a negative ion plasma, comprising warm positive and negative ions and kappa-distributed electrons. The Sagdeev pseudopotential for the plasma model is derived and analyzed in a systematic way. While it is well known that a negative ion plasma supports the propagation of the fast mode normal solitons, it is found that it supports, in addition to the slow mode normal solitons, the propagation of the slow mode supersolitons and double layers for high values of the negative ion density. The double layers occur as the lower limit to the supersoliton existence range and as the limiting factor for the propagation of normal solitons. When the relative temperature of the two ion species decreases, it is found that the Mach number range supporting the propagation of the nonlinear structures reduces, while the ampli- tudes of solitons and supersolitons decrease, and these effects are enhanced by the superthermal behavior of the electrons. The amplitudes of the double layers increase with a decrease in the relative temperature of the two ion species but decrease with an increase in the electron superthermality. ©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0039372 .,s I. INTRODUCTION Theoretical and experimental investigations of the propagation of electrostatic waves in negative ion plasmas have been of inter- est in recent years. The interest in negative ion plasmas has been motivated, on one hand, by observations of negative ions in natu- rally occurring space plasmas. They have been observed in D and F regions of the Earth’s ionosphere,1in Halley’s cometary comae,2in the interstellar medium,3,4and in the solar wind.5Recently, negative ions have been found in the ionosphere of Saturn’s moons (Titan6–11 and Enceladus)8by the electron spectrometer on board Cassini spacecraft. On the other hand, these investigations were motivated by the importance of negative ions in human made technology. Theneutral beam injection based on negative ion sources is one of the most promising heating systems for plasma heating in fusion reac- tors,12and negative ion plasmas are used in the area of material processing such as plasma surface etching and ion implantation,13 to cite only a few examples. Experimental14–18and theoretical19–34studies have revealed that in any two-ion species plasma with finite temperature of at least one of the ion species, two wave modes with different phase speeds propagate: the fast mode and the slow mode. The differentiation between these two modes is made on the basis of the range of their phase speeds35vis à vis the ion thermal speeds, viz., vtc<vslow<vth<vfast<vte, (1) AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-1 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv where vtc,vth, andvteare the thermal speeds of colder ion species, hotter ion species, and electrons and vslowandvfastare the phase speeds of the slow and the fast wave modes, respectively. An important parameter in the study of negative ion plasmas is the ratio of negative-to-positive ion density that shows the per- centage of the negative ions in the plasma. Theoretical36and exper- imental16investigations have shown that as the negative-to-positive density ratio increases, the slow mode increasingly damps, while the fast mode is less and less damped, making it the dominant mode. However, in a numerical analysis of the dispersion relation, Ichiki et al.37have shown that the slow mode dominates the fast mode whenever the negative ion species is much lighter than the positive ion species. Subsequently, Ichiki et al.18observed, in the low tem- perature regime, the slow mode in the X+ e−F−−enegative ion plasma, having negative-to-positive ion mass ratio μ= 0.15. The propagation of the slow mode waves has also been observed experi- mentally by Handique et al.17in an experiment with the A+ r−F−−e plasma. In space plasmas, the propagation of the slow mode waves has been observed in the Earth’s plasma sheet boundary layer (PSBL) by the Cluster multispacecraft, as reported by Norgren et al.38and Kakad et al.39 Most of the theoretical studies on the slow mode have focused on the small amplitude solitons based on the KdV theory.21,25–28,40–43 However, it is well known that negative ion plasmas can support nonlinear structures, which cannot be described by the KdV the- ory.44,45An example of such a structure is the supersoliton.46A supersoliton is a nonlinear structure, characterized by its unusual Sagdeev potential and distorted electric field.47,48Such distorted elec- tric fields have already been observed in space plasmas.49Supersoli- tons have been predicted in multi-component plasmas.46,47,50–54A two-component plasma cannot, however, support supersolitons, as has been shown by Verheest et al.55We note here that Sagdeev pseu- dopotentials yielding supersolitons had been reported56,57before the name supersoliton was coined by Dubinov and Kolotkov,46but they were not recognized as new nonlinear structures. In most cases,21,41,43studies of the propagation of the slow mode electrostatic solitary waves supported by negative ion plasmas were based on a Maxwellian distribution of electrons. However, observa- tions and measurements by satellites in different space environments have shown the presence of particle distributions with a high energy tail, following a power law instead of the Maxwellian distribution.58 Among non-Maxwellian distribution functions that have been uti- lized to model such particles, the so-called generalized Lorentzian or kappa distribution58has been found to be appropriate in many cases. The three-dimensional isotropic kappa velocity distribution function fκ(v) for any particle species is given by59 fκ(v)=N (πκθ2)3/2Γ(κ+ 1) Γ(κ−1 2)(1 +v2 κθ2)−(κ+1) , (2) whereκis a parameter showing the departure from the Maxwellian distribution and is called the spectral index, vis the particle speed, Nis the species particle density, Γis the gamma function, and θis the effective thermal speed modified by the spectral index and is related to the normal thermal speed, vt, byθ= [(2κ−3)/κ]1/2vt, provided that the spectral index κ>3/2. For large values of κ(κ →∞), the function fκ(v) approaches the Maxwellian distribution,while low values of κindicate the presence of a large fraction of par- ticles with speeds greater than the thermal speed. Kappa-distributed particles with 2 <κ<6 have been observed in the solar wind,60 Saturn’s magnetosphere,61and Titan’s upper atmosphere.62A more complete review of regions in space environments where kappa- distributed particles are present can be found in the work of Pierrard and Lazar.63 Based on the kappa velocity distribution for the electrons and the fluid theory for both ion species, we, inter alia , show that a nega- tive ion plasma supports the propagation of the slow mode supersoli- tons and double layers. The paper is organized as follows: After this introductory section, we derive from the fluid equations an expres- sion for the Sagdeev pseudopotential in Sec. II. We then discuss in this same section the lower and upper limits of the soliton existence domain in terms of the Mach number range. Section III specializes to the ion thermal and electron superthermal effects on the soliton exis- tence domain, amplitude, and width for low-to-intermediate values of negative-to-positive ion density ratio. These same effects are also discussed in Sec. IV for higher values of the negative-to-positive ion density ratio, for which the plasma supports the propagation of slow mode supersolitons and double layers. A summary of the results is presented in Sec. V. II. BASIC FORMALISM A. Plasma densities and Sagdeev pseudopotential We consider a one-dimensional, collisionless, unmagnetized negative ion plasma comprising singly charged adiabatic positive (label p) and negative (label n) ion species, as well as nonthermal electrons (label e), distributed according to kappa distribution, and investigate the slow mode arbitrary amplitude ion-acoustic electro- static waves using the Sagdeev pseudopotential method. In light of Eq. (1), the slow mode exists provided the thermal speeds of the ion species are different. Accordingly, we treat the positive ion species to be the colder component and the negative ion species to be the hotter component. Both ion species are dynamic, and the nonlinear behav- ior of the ion acoustic waves obeys the following set of normalized partial differential equations in the fluid description: ∂np ∂t+∂(npup) ∂x=0, (3) ∂up ∂t+up∂up ∂x=−μ∂φ ∂x−f2μτpnp∂np ∂x, (4) ∂nn ∂t+∂(nnun) ∂x=0, (5) ∂un ∂t+un∂un ∂x=∂φ ∂x−nn∂nn ∂x, (6) where xis the space coordinate, tis the time coordinate, φis the electrostatic potential, and np,upandnn,unare the densities and fluid flow velocities of positive and negative ion species, respectively. The parameter μ=mn/mpis the negative to positive ion mass ratio, τp=Tp/Tnis the positive to negative ion temperature ratio, and τn =Tn/Teis the ratio of the negative ion temperature to the electron temperature. The variable f=nn0/np0is the ratio of negative ion AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-2 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv equilibrium density nn0to positive ion species equilibrium density np0and expresses the percentage of the negative ions in the plasma. Its values are in the range 0 <f<1, where f= 0 corresponds to the absence of negative ions and f= 1 corresponds to the absence of elec- trons. We will skip these extreme values in our analysis. Positive ion, negative ion, and electron densities are coupled through Poisson’s equation, ∂2φ ∂x2=ne+nn−np, (7) where the density of the inertialess electrons nein an electrostatic potentialφobtained by integrating expression (2) over velocity space is given by ne=1−f f(1−2τnφ 2κ−3)−κ+1/2 . (8) The system is closed by the equation of state. We have assumed the ion flow to be adiabatic with the relation between the jth (j=p,n) ion species pressure and its density to be pjn−γ j=constant , (9) whereγ= 3 is the polytropic index and the unperturbed pressure of the jth adiabatic species is defined as64pj0=nj0Tj/3. The vari- ables have been normalized as follows: the space coordinate xis normalized by λDn=(ε0Tn/nn0e2)1/2, the time is normalized by the inverse ofωpn=(nn0e2/ε0mn)1/2, the flow velocities are normal- ized by the negative ion species thermal velocity vtn=(Tn/mn)1/2, the electrostatic potential is normalized by the thermal potential Te/e, and the densities are normalized by the negative ion species equilibrium density. With this normalization, the thermal speeds of positive and negative ions are μτpand 1, respectively, and the bound- ary conditions far away from the perturbation, where the plasma is undisturbed, are np(∞)=1 f, nn(∞)=1, up(∞)=0, un(∞)=0, φ(∞)=dφ dξ(∞)=d2φ dξ2(∞)=0.(10) To find the ion densities npandnnin Eq. (7), the fluid equa- tions (3)–(6) are written in a frame where the nonlinear struc- ture is stationary, by assuming that all dependent variables depend on a single independent variable ξ=x−Mt, where the Mach number M=V/vtnis the speed Vof the nonlinear structure in the inertial frame, normalized by the negative ion thermal speed. Using this transformation and going through the necessary algebra, Eqs. (3)–(7) become −Mdnp dξ+d(npup) dξ=0, (11) −Mdup dξ+updup dξ=−μdφ dξ−f2μτpnpdnp dξ, (12) −Mdnn dξ+d(nnun) dξ=0, (13)−Mdun dξ+undun dξ=−dφ dξ−nndnn dξ, (14) d2φ dξ2=ne+nn−np. (15) After integrating the pairs of Eqs. (11) and (12) and (13) and (14) with the boundary conditions (10), we get a biquadratic equation for each of the ion species densities, μτpf4n4 p−(M2−2μφ+μτp)f2n2 p+M2=0, (16) for the positive ion density, and n4 n−(M2+ 2φ+ 1)n2 n+M2=0, (17) for the negative ion density. The solutions to these equations are, respectively, n2 p=1 2f2μτp[M2−2μφ+μτp±√ (M2−2μφ+μτp)2−4M2μτp] (18) and n2 n=1 2[M2+ 2φ+ 1±√ (M2+ 2φ+ 1)2−4M2]. (19) According to Eq. (1), the colder positive ion species is supersonic andV>vtporM>√μτpin normalized variables. Therefore, far from the perturbation, where φ= 0,√ (M2−μτp)2=M2−μτpand we choose the negative sign in front of the square root of Eq. (18) to get the correct limit np(∞) = 1/ f. For a similar reason, we have to choose the positive sign in front of the square root in Eq. (19) for the hotter subsonic negative ion species ( V<vtn). With this choice of signs, Eqs. (18) and (19) can be rewritten following the approach of Ghosh et al.65as nj=cj[√a±√ b], (20) where cj(j=p,n) are cp=1/2f√μτpand cn= 1/2, and aand b are unknowns, which are determined by substituting (20) in (18) and (19). This results in the simplified expressions for the ion densities as np=1 2f√μτp[√ (M+√μτp)2−2μφ−√ (M−√μτp)2−2μφ] (21) for the positive ion species, and nn=1 2[√ (1 +M)2+ 2φ+√ (1−M)2+ 2φ] (22) for the negative ion species. After introducing density expressions (8), (21), and (22) in Poisson’s equation (7), we multiply it by dφ/dξ and integrate with the boundary conditions (10), and obtain an energy-like equation 1 2(dφ dξ)2 +S(φ)=0, (23) AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-3 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv where−dφ/dξis the electric field associated with the electrostatic potentialφand S(φ,M)=1−f fτn[1−(1−2τnφ 2κ−3)−κ+3/2 ] +τp 6f(μτp)3/2{2(μτp)3/2+ 6M2(μτp)1/2 −[(M+√μτp)2−2μφ]3/2 +[(M−√μτp)2−2μφ]3/2 } +1 6{2 + 6 M2−[(1 +M)2+ 2φ]3/2−[(1−M)2+ 2φ]3/2} (24) is the Sagdeev pseudopotential. As is well known, Eq. (23) has solitary wave solutions if the Sagdeev pseudopotential satisfies the following conditions:66 (i) S(0,M) =S′(0,M) = 0; (ii) S′′(0,M)⩽0, the origin is unstable; (iii) S(φm,M) = 0 for some Min the soliton existence domain and φm≠0; (iv) S(φ,M)<0 for 0<|φ|<|φm|; and (v) for double layers, S(φm,M) =S′(φm,M) = 0 for some Min addition to ( i)−(iv). Here, primes denote the derivatives of the Sagdeev pseudopo- tential with respect to the electrostatic potential and φmis the soliton amplitude. B. Minimum and maximum Mach numbers One of the advantages of using the Sagdeev pseudopotential method is that the function S(φ,M) and its derivatives determine the conditions under which nonlinear structures exist in the plasma model it represents, going as far as determining the minimum and the maximum Mach numbers of these structures. Generally, the slow mode Mach number is constrained by Eq. (1), which fixes the small- est Mach number to the thermal speed of the colder ion species (positive ion species in our case), and the largest Mach number to the thermal speed of the hotter ion species (negative ion species in our case). Normalizing Eq. (1) by the negative ion thermal speed vtn and considering the interval containing only the slow mode Mach number Mslow, it is found that √μτp<Mslow<1. (25) These limits do not account for other physical constraints that may bring in some modifications, such as an ion species reaching its sonic point or the occurrence of a double layer. These supplemen- tary constraints reduce further the interval (25) of admissible Mach numbers, making it narrower. A closer look shows that the Sagdeev pseudopotential (24) already satisfies conditions ( i). The application of condition ( ii) often referred to as the soliton condition yields the constraint −(1−f)(2κ−1)τn 2κ−3+f M2−1+μ M2−μτp⩽0, (26) where the equality has been addressed by Refs. 64 and 44. Physi- cally, the constraint (26) means that the nonlinear structures corre- sponding to S(φ,M) are acoustic or super-acoustic,64by fixing theminimum Mach number Msfor their existence, viz., M2⩾M2 s=2κ−3 2(1−f)(2κ−1)τn⎧⎪⎪⎪⎨⎪⎪⎪⎩(1−f)(2κ−1)τn 2κ−3(1 +μτp)+f+μ ±⎡⎢⎢⎢⎢⎣((1−f)(2κ−1)τn 2κ−3(1−μτp)+f−μ)2 + 4fμ⎤⎥⎥⎥⎥⎦1/2⎫⎪⎪⎪⎬⎪⎪⎪⎭, (27) where Msis the ion acoustic speed, normalized by the negative ion thermal speed vtnand satisfies equation S′′(0,Ms) = 0. The + and −signs in Eq. (27) correspond to the fast and the slow wave modes, respectively, and for our study of the slow mode, we choose the nega- tive sign. For most plasma parameter values, the second term under the square root is much smaller than the first and we can expand (27) in the Taylor series, resulting in an approximate value of the slow mode acoustic speed as M2 s≈μτp+μ(2κ−3) (1−f)(2κ−1)τn. (28) This equation shows that Ms>√μτp, as required by (25). Further- more, Eq. (28) shows that the ion acoustic speed of the slow mode waves increases with increasing colder-to-hotter temperature ratio τp, or in other words, Msincreases when the relative temperature of the two ion species decreases. As of the dependence of Mson the electron superthermality, Eq. (28) shows that when the electron superthermality increases (the spectral index decreases) with other plasma parameters fixed, Msdecreases. Having found the soliton minimum Mach number from the physical constraint that the origin is unstable, the maximum Mach number is found from the constraint that the ion densities remain real valued functions. As is well known,21–23the polarity of the slow mode soliton in any two-ion species plasma is the same as the sign of the charge of the colder ion species. As a consequence, only pos- itive solitons are encountered in the model with colder positive ion species under study. This means that the negative ion species density [see Eq. (22)] remains real for all possible values of the electrostatic potentialφ. However, to remain a real valued function, the positive ion species density [see Eq. (21)] requires that φ⩽φlp=1 2μ(M−√μτp)2, (29) whereφlpis the limiting electrostatic potential. Putting this limiting value ofφlpinS(φlp,M) = 0, we find the solution of this equation, which gives M=Mlp. This allows us to determine a relation between Mlpand the negative-to-positive density ratio f in the form (α−β)f+β+δ=0, (30) where α=1 6⎧⎪⎪⎨⎪⎪⎩2 + 6 M2 lp−[(1 +Mlp)2+1 μ(Mlp−√μτp)2]3/2 −[(1−Mlp)2+1 μ(Mlp−√μτp)2]3/2⎫⎪⎪⎬⎪⎪⎭, β=1 τn⎡⎢⎢⎢⎢⎢⎣1−⎛ ⎝1−τn(Mlp−√μτp)2 μ(2κ−3)⎞ ⎠−κ+3/2⎤⎥⎥⎥⎥⎥⎦, AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-4 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv and δ=τp 6(μτp)3/2[2(μτp)3/2+ 6M2 lp(μτp)1/2−(4Mlp√μτp)3/2]. The solution to Eq. (30), Mlpas a function of fis the Mach num- ber of the slow mode if it satisfies simultaneously conditions (25) and (27). Because the positive ion density becomes complex valued for electrostatic potentials larger than φlp,φlpis the largest real root of Eq. (30). When this root is accessible from the undisturbed condi- tionφ= 0, the corresponding Mach number Mlpis the upper limit to the soliton Mach number range for corresponding plasma param- eter values. There are situations in which the root φlpis not accessible from the undisturbed condition φ= 0 because there is another root or more in between.67In this case, it is a double layer that occurs as a limiting factor to the soliton amplitude before the sonic point is reached, and the corresponding Mach number is the upper limit to the soliton Mach number range. We recall here that the double layer Mach number and corresponding amplitude are calculated from the system of equations23 {S(φ,M)=0, S′(φ,M)=0.(31) In addition to the occurrence of double layers as the limiting factor to the soliton Mach number range, double layers have been shown also to occur as the lower limit to a set of supersolitons in dif- ferent plasma models.45,47,48,57,68As is now known,47the supersoliton existence domain in the Mach number range and in compositional parameter space is limited, on the lower side, either by double lay- ers, if they exist, or by other boundaries, arising from the merging of consecutive extrema. This will be illustrated in Sec. IV B. Anticipating our results, we have found that standard slow mode solitons are found to propagate at low to intermediate values of negative-to-positive density ratio, while supersolitons and dou- ble layers are possible at higher values. For a complete analysis, we consider each region separately, starting by the region of low to intermediate values of f. III. SOLITARY WAVES SUPPORTED BY LOW TO INTERMEDIATE VALUES OF THE NEGATIVE-TO-POSITIVE ION DENSITY RATIO A. Soliton existence domains In negative ion plasmas, the replacement of light electrons by heavy negative ions reduces the electron shielding effect, an effect that increases with an increase in negative-to-positive ion density ratio f.69This implies that the negative-to-positive ion density ratio plays an important role in the study of negative ion plasmas. In the following analysis, we use Eq. (27) and solve numerically Eqs. (30) and (31) for different plasma parameter values to delineate the soli- ton existence domains in { f,M} space, where Mis the soliton Mach number. The soliton existence domains are depicted in Fig. 1 for dif- ferent plasma parameter values. In Fig. 1, we show the ion thermal effects on the soliton existence domains from the upper to lower panels and the electron superthermal effects are shown from left to right panels. To study the effects of the relative temperature of the two ion species, we fix the ratio of the negative ion speciestemperature to electron temperature to τn= 0.7 and vary τp. With fixedτn, an increase in τpimplies the increase in the positive ion temperature at fixed negative ion temperature, which means that the ions’ relative temperature decreases with increasing values of τp. The negative-to-positive mass ratio is also fixed to a value of μ= 0.15, corresponding to a X+ e−F−−eplasma used by Ichiki et al.18in their experiments. In the upper left panel, the positive-to-negative ion temper- atureτp= 10−6models effectively cold positive ions, whereas the spectral index value κ= 1000 shows that electrons are Maxwellian. For these parameter values, the slow mode supersonic limit is ≈0.4 ×10−3, while the subsonic limit is 1. The solid line (black color online) represents the minimum Mach number Msfor the existence of solitons. It lies well above the supersonic limit for the whole range of the negative-to-positive ion density ratio f. The dashed line (red color online) represents the upper limit Mlpdue to the occurrence of positive ion sonic point, where Mlpis determined numerically from Eq. (30). We see that this limit remains well below the subsonic limit 1. Solitons may propagate if their Mach numbers are between the minimum Msand the maximum Mlp. Therefore, for the plasma parameter values in the upper left panel, positive solitons limited by the occurrence of positive ion sonic point exist for all values of the negative-to-positive ion density ratio f. The lower left panel presents the soliton existence domain when τpis increased to 0.7, shifting the supersonic limit to ≈0.32. As a result of this increase, the minimum Mach number Msat a fixed density ratio falso increases to a higher value. This result is con- sistent with Eq. (28), which shows that Msincreases with increasing τp. The maximum Mach number Mlphas a different behavior when τpis increased. At low values of τp,Mlpdecreases very quickly with an increase in τpand the Mach number range supporting the prop- agation of solitary waves narrows. With further increase in τp, the values of Mlpgo through a minimum before they start increasing with increasing τp, but the soliton Mach number range continues to narrow due to a faster growing of the acoustic Mach number. This result is shown in detail in Fig. 2, where the Mach number difference δM=Mlp−Ms, calculated at a fixed value of negative-to-positive ion density ratio f= 0.5, is plotted as a function of τpfor two val- ues of the spectral index, κ= 1000 (Maxwellian electrons) and κ= 2 (strongly non-Maxwellian electrons). Figure 2 shows clearly that δM decreases with increasing τp, meaning that the values of MsandMlp get closer as τpincreases. We recall that with the adopted normaliza- tion by hotter negative ion characteristics, increasing τpmeans that the temperature of the colder positive ion species is increased and the supersonic limit moves toward the subsonic limit, therefore reduc- ing the possible slow mode Mach number range as given by (25). A continuous increase in τpmoves the supersonic limit to higher values toward the subsonic limit so that at τp≈6.7, both limits coin- cide. At values of τp≳6.7, the colder (positive) ion thermal speed√μτpis larger than the hotter (negative) ion thermal speed 1 and the inequality (25) is breached. The right panels of Fig. 1 shed light on the effects of increas- ing the electron superthermality by reducing the spectral index κ from 1000 to 2. In the upper right panel, positive ions are still cold with negative-to-positive ion temperature ratio τp= 10−6. Compar- ison with the upper left panel shows that an increase of the electron superthermality causes both the minimum Msand the maximum MlpMach numbers to shift to lower values, but Mlpundergoes a AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-5 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 1 . Existence domain of ion acoustic solitary waves and double layers in { f,M} space. The left panels show the effects of increasing the positive ion temperature when electrons are Maxwellian ( κ= 1000), and the right panels show the same effects when electrons are strongly non-Maxwellian ( κ= 2). The fixed plasma parameters are μ= 0.15 andτn= 0.7. more significant shift. As a result, the Mach number range support- ing the soliton propagation at a fixed density ratio fnarrows with an increase of the superthermal behavior of the electron. This result, obtained by considering extreme values of the parameter κ, is also FIG. 2 . Soliton existence domain in { τp,δM} space, where δM=Mlp−Msat density ratio f= 0.5 forκ= 1000 (dashed line) and κ= 2 (dashed-dotted line). In both cases, the soliton Mach number range δM, supporting the propagation of solitons, decreases with increasing τp.valid for intermediate values. The decrease in Msasκdecreases is also consistent with Eq. (28). The upper right panel of Fig. 1 also shows that there are two other effects on the soliton existence domain, which arise due to the increase of the electron superthermality. On one hand, it is observed that when the spectral index decreases from κ= 1000 toκ= 2, the upper limit to the soliton Mach number range is still the maximum Mach number Mlpat low-to-intermediate values of density ratio, but at higher values of f, double layers arise as the limiting factor. Double layers are discussed further in Secs. IV A and IV B. On the other hand, while the minimum Mach numbers decrease with increasing negative-to-positive density ratio fwhen electrons are Maxwellian ( κ= 1000) (left panels), they increase when electrons are strongly non-Maxwellian ( κ= 2) (right panels). This means that when κis varied with other parameters held fixed, there is a critical value κcof the spectral index at which the acoustic Mach number Msis constant for all values of the density ratio fandκcis a function of other plasma parameters. The Mscurve as a function of the density ratio fdecreases for values of κ>κcand increases when κ<κc. Whenμ,τn, and/orτpare changed, the value of κcfollows. As an example, for fixed values of μ= 0.15 and τn= 0.7, numerical simulations have shown that the values of κcare between κc≈3.1 whenτp= 10−6andκc≈2.57 whenτp= 1. This explains why, e.g., the minimum Mach number Msdecreases as a function of density ratio fin the upper left panel, where the spectral index κ= 1000 is AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-6 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv larger than the critical value κc= 3.1, while in the upper right panel, whereκ= 2 is smaller than the critical value κc= 3.1 and Msis an increasing function f. The lower right panel of Fig. 1 shows a combined effect of reducing the relative ion temperature by increasing τpand increas- ing the superthermal behavior of the electrons by decreasing the spectral index κ. In this panel, τp= 0.7 andκ= 2. As a result of increasingτpand decreasing κ, the soliton Mach number range is significantly reduced. This means that the superthermal behavior of the electrons and the ion thermal effect enhance each other. B. The ion thermal and electron superthermal effects on the soliton amplitude and width For low-to-intermediate values of negative-to-positive density ratio f, we take an indicative value of f= 0.5. At this value of f, we present in Fig. 3 the pseudopotentials for parameter values taken from the soliton existence domains presented in Fig. 1. From the upper to lower panel, we show the effects of reducing the relative ion temperature for Maxwellian electrons ( κ= 1000) (left panels) and strongly non-Maxwellian electrons ( κ= 2) (right panels). The effects of increasing the electron superthermality are presented from left to right panels.In the upper left panel, positive ions are cold with τp= 10−6 and electrons are Maxwell-distributed with κ= 1000. With these plasma parameter values, the plasma supports slow mode nor- mal solitons, propagating with Mach numbers in the range Ms<M⩽Mlp, where Ms= 0.398 47 and Mlp= 0.75812. The amplitude of the maximum speed soliton (dashed-dotted line, red color online) is φlp≈1.91. Asτpincreases, the amplitude of the positive solitons with maximum speed decreases and, at τi= 0.7 (lower left panel), φlp≈0.21. This decrease in amplitude is all the more important as the electron superthermality is higher. Hence, in the lower right panel with other plasma parameters set to the same values, the maximum speed soliton amplitude decreases from φlp≈0.21 forκ= 1000 (lower left panel) to φlp≈0.07 (lower right panel). Therefore, reducing the relative ion temperature has an effect of decreasing the maximum speed soliton amplitude, and this effect is enhanced by the electron superthermality. When the relative ion temperature increases to such a value that the ion thermal speeds of both ion species are equal, the slow mode vanishes. This result is presented more explicitly in Fig. 4, in which we have plotted the soliton profiles for plasma parameters, as con- sidered in Fig. 3. Moreover, Fig. 4 shows that when the soliton amplitude decreases, its width increases. FIG. 3 . Sagdeev pseudopotentials with parameter values taken in the soliton existence domain. Fixed plasma parameters are f= 0.5,μ= 0.15, andτn= 0.7. The upper to lower panels show the ion thermal effects, while the left to right panels show the electron superthermal effects on the soliton amplitude and width. AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-7 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 4 . Profiles of solitons with maximum speeds, showing the effects of positive ion thermal effects on the soliton amplitude and width for Maxwellian ( κ= 1000) (upper panel) and strongly non-Maxwellian ( κ= 2) (lower panel). In both panels, μ = 0.15,τn= 0.7, and f= 0.5. IV. SOLITARY WAVES SUPPORTED BY HIGH VALUES OF THE NEGATIVE-TO-POSITIVE ION DENSITY RATIO In the upper panels of Fig. 1, positive ions are effectively cold withτp= 10−6. While in the left upper panel, the soliton wave ampli- tude is limited by the occurrence of the positive ion sonic point for the whole density ratio range 0 <f<1, in the right panel, in which electrons are strongly non-Maxwellian, the occurrence of positive ion sonic point acts as the upper limit to soliton amplitude for low to intermediate values of f, but at higher values of f, the limiting factor is a double layer. The soliton existence domain at high val- ues of fhas been blown up and replotted in Fig. 5, omitting the curve for the acoustic Mach numbers for clarity. The dashed line (red color online) represents the occurrence of positive ion sonic point as found from Eq. (30), and the dashed-dotted line (blue color online) represents the double layer as a solution to the set of Eq. (31). Figure 5 shows that the double layer and positive ion sonic point curves cross over at a critical density ratio of fc≈0.918 with the same Mach number of Mdl=Mlp= 0.454 59. FIG. 5 . Double layer and supersoliton existence domains in { f,M} space: Blow up of the upper right panel of Fig. 1 at high values of f. Supersolitons with double lay- ers as the lower limit exist in the region shaded horizontally. In the region shaded vertically, normal solitons exist with a Mach number larger than Mlpand the double layer as the upper limit. Forf<fc, there is a range of ion density ratio values 0.88 ≲f ≲0.92 supporting the existence of double layers that do not repre- sent the soliton upper limit but occur as the lower limit to a set of supersolitons, with the positive ion sonic point as the upper lim- iting factor. For this range, the limiting electrostatic potential φlp is the largest solution to Eq. (30), and there is no other solution betweenφ= 0 andφ=φlp. The double layer curve for this case lies below the positive ion sonic point curve. The region of { f,M} space admitting supersolitons with double layer as the lower limit is shaded horizontally in Fig. 5. Starting from fc, double layers occur as the upper limiting factor to the soliton amplitude, and the cor- responding curve in { f,M} space lies above the positive ion sonic point curve. For this range, the limiting electrostatic potential φlp is also the largest solution to Eq. (30), but there are other solu- tions between φ= 0 andφ=φlpso thatφlpis not accessible from the origin. This region plotted between MlpandMdlcurves up to f= 0.94 for clarity is shaded vertically. In the following discussion, we analyze the nonlinear structures’ characteristics in each region separately. A. Existence domain of double layers as the soliton limiting factor We start our discussion by first considering the range of den- sity ratio f, supporting the occurrence of double layer as the upper limit to the soliton Mach number range. The pseudopotential at the critical negative-to-positive density ratio fc, at which a double layer occurs at the positive ion sonic point with Mach number Mdl =Mlp= 0.454 59 is shown in the upper panel of Fig. 6. At this point, the double layer with amplitude φdl≈0.63 signals the end of the soliton Mach number range before the sonic point with limiting electrostatic potential φlp≈0.68 is reached.23 In the lower panel of Fig. 6, the pseudopotential curves are plot- ted at a negative-to-positive density ratio of f= 0.919, slightly larger than fc, from the minimum Mach number Ms. At Mach number AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-8 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 6 . Upper panel: The pseudopotentials plotted at the critical negative-to- positive density ratio fc= 0.918. A double layer exists at the positive ion sonic point. Lower panel: The pseudopotentials are plotted at a slightly higher value f= 0.919. The positive ion sonic curve presents two roots before the limiting poten- tial is reached. In both panels, other plasma parameters are μ= 0.15,τn= 0.7, τp= 10−6, andκ= 2. Mlp= 0.454 77 corresponding to the sonic point, the pseudopoten- tial curve (dashed-dotted line, red color online) presents 2 roots, φ1 ≈0.60 andφ2≈0.65, before the maximum root φlp≈0.69 is reached. Therefore, φlpis not accessible from the undisturbed condition φ= 0, and only the first encountered root φ1yields a normal soli- ton. Because φ1is smaller than φlp, the positive ion density is a real valued function at φ1. It is thus possible to increase the Mach num- berMbeyond the value of Mlpwithout rendering the positive ion density a complex valued function. Therefore, the occurrence at Mlp of the two roots before the sonic point opens a window beyond Mlp, in which ordinary solitons exist with Mach numbers M>Mlp, but with amplitudes, smaller than the limiting amplitude φlp. A continu- ous increase in Mach number Mbeyond Mlpyields normal solitons with increasing amplitudes until, at Mdl= 0.455 07 >Mlp, a double layer occurs with amplitude φdl≈0.62, also smaller than the limiting amplitudeφlp. No solitons are found to exist for Mach numbers M larger than Mdl, which means that the soliton Mach number rangeis terminated by the occurrence of the double layer. This situation repeats itself for all values of f>0.918 but less than the limiting value f= 1, at which the model breaks down. B. Existence domain of supersolitons Figure 5 shows that double layers exist also in a narrow range of values of f, smaller than fc, but in this case they do not signal the end of the soliton Mach number range, rather they signal the lower boundary of a set of supersolitons that terminates when the sonic point is reached (horizontally shaded region). In the region below fc, the double layer curve lies below the positive ion sonic point curve in {f,M} space and the Mach number Mlpcorresponding to the max- imum amplitude φlpis still the limiting Mach number to the soliton Mach number range. Figure 7 is a plot of the pseudopotentials at f= 0.91, slightly smaller than fc, for different values of Mach number M, ranging from the acoustic speed Ms= 0.3498 up to the maximum Mach number Mlp= 0.453 39 due to the occurrence of positive ion sonic point. The dashed line (blue color online) is plotted at M= 0.435 and represents the normal soliton pseudopotential curve. The dashed- double-dotted line (blue color online) is plotted at M= 0.451 96 and represents a double layer. The pseudopotential curve plotted at a slightly larger Mach number M= 0.4523 (dotted line, blue color online) presents three subsidiary extrema between the undisturbed conditions φ= 0 and the nonlinear structure amplitude. As is well known, such a curve with two subwells represents a supersoliton.48 When the Mach number is increased further, we observe that the corresponding pseudopotentials represent supersolitons even at the maximum Mach number due to the occurrence of the positive ion sonic point (dashed-dotted line, red color online). At Mach num- bers greater than the maximum, no more supersolitons exist in the plasma as is the case for normal solitons. FIG. 7 . Pseudopotentials plotted at f= 0.91 from the minimum Msup to the max- imum Mlp. The dashed line (blue color online) represents a normal soliton, the dashed-double-dotted line (blue color online) represents a double layer, and both the dotted (blue color online) and dashed-dotted (red color online) lines repre- sent supersolitons. Other plasma parameters are μ= 0.15,κ= 2,τi= 10−6, and τe= 0.7. AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-9 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv Supersolitons found in this case are part of a set of solitons and the transition route from normal soliton to supersoliton is through the occurrence of a double layer. Normal solitons exist for Mach numbers between MsandMdl, while supersolitons are encountered for Mach numbers between MdlandMlp. The double layer is thus the lower boundary to this set of supersolitons. Double layers also occur as the lower boundary for the supersoliton sets for other values of f, ranging from ∼0.908 to∼0.918. At f= 0.918, the lower limit (double layer) and the upper limit (sonic point) occur with the same Mach number, signaling the end of the supersoliton domain on the high value of f. This range and corresponding upper limits form a region in {f,M} space that is shaded horizontally in Fig. 5. For values of flower than 0.908, supersolitons exist with dif- ferent limitations, arising from the merging of consecutive extrema. This case is illustrated in Fig. 8. In both panels, the three subsidiary extrema are labeled A,B, and C. The upper panel is plotted at f= 0.894. The pseudopotential curve representing a supersoliton (dashed line, blue color online) FIG. 8 . Upper panel: Pseudopotentials yielding supersolitons with the coalescence of rightmost extrema B and C as the lower limit to the supersoliton Mach number range and a supersoliton occurring at the sonic point. Lower panel: Pseudopoten- tials yielding supersolitons lying between two normal solitons, resulting from the coalescence of the two rightmost extrema B and C (solid line) and the two leftmost extrema A and B (dashed-dotted line).is found between two pseudopotential curves, the solid line (black color online), and the dashed-dotted line (red color online). The dashed-dotted line is plotted at the maximum Mach number Mlp and corresponds to the maximum speed supersoliton. The solid line (black color online) plotted at a Mach number of M= 0.449 05 results from the merging of rightmost extrema B and C so that it has a single well, appropriate for a normal soliton. Verheest et al.47used the terminology “coalescence” to designate the merging between two consecutive extrema, and the corresponding curve is designed as the curve of coalescence or the curve of inflection,51,52because it is defined such that it has a point of inflection between the undis- turbed conditions. To find the Mach number and the corresponding potential at which a coalescence occurs, one has to solve the system of equations S′(φ) =S′′(φ) = 0.47Supersolitons limited on the lower side by the coalescence of two subsidiary extrema and the sonic point on the upper side are found in the interval 0.889 ≲f≲0.908. The lower panel of Fig. 8 is plotted at f= 0.884. For this case, the supersoliton is sandwiched between two normal solitons result- ing from the coalescence of the two rightmost extrema B and C (solid line, black color online) and the two leftmost A and B (dashed- dotted line, red color online). Such supersolitons are found in the interval 0.88 ≲f≲0.889. At f≈0.88, the upper and lower limits coincide, signaling the end of the supersoliton existence domain on the lower side of f. The supersoliton domain in { f,M} space is plotted in Fig. 9. Based on the lower limit to the supersoliton set, Steffy and Ghosh51,52classified as Type I supersolitons with a double layer as the lower limit and as Type II those for which the lower limit is the coalescence of two subsidiary extrema. While in principle there is no difference between these two types, they can however be iden- tified based on the variation of the amplitude when there is tran- sition from a normal soliton to supersoliton. This identification is easily seen from Fig. 10, showing the phase diagrams for the two routes. While the Type I transition (upper panel) is accompanied by FIG. 9 . Supersoliton existence domain. Plasma parameter values are μ= 0.15,κ = 2,τp= 10−6, andτn= 0.7. At a fixed density ratio f, supersolitons exist for Mach numbers between the lower and the upper curves. From atob, the supersoliton lower limit is a coalescence, and from btoc, it is a double layer. On the upper side, the supersoliton Mach number ends by the coalescence between aanddand by the positive ion sonic point between dandc. AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-10 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 10 . Phase diagrams, plotting dφ/dξas a function of φ. The upper panel is plotted at f= 0.916 at which the lower boundary to the set of supersolitons is a double layer. The dashed gray curve delineates the potential range, which is not accessible from the undisturbed condition. The lower panel is plotted at f= 0.888 at which the lower boundary to the set of supersolitons is the coalescence of two consecutive extrema. Other parameter values are μ= 0.15,τn= 0.7,τp= 10−6, andκ= 2. the existence of a range of electrostatic potentials that are not accessi- ble from the undisturbed conditions (the range limited by the dashed curve, gray color online),48the amplitude varies continuously from the soliton to supersoliton for the Type II transition (lower panel). C. Ion thermal and electron superthermal effects on the range of the density ratio f, supporting supersolitons and double layers In Secs. IV A and IV B, it was observed that supersolitons and double layers exist when the negative ion density is high (low density of electrons). Thus, in the upper right panel of Fig. 1, when the pos- itive to negative ion temperature ratio is τp= 10−6and the spectral indexκ= 2, the negative-to-positive density ratio range supporting supersolitons extends from f≈0.88 up to f≈0.918, and the double layers as the limiting factor to the soliton propagation occur for f≳ 0.918. When the relative ion temperature is reduced or, equivalently,whenτpis increased, these two ranges shift to higher values of fand at the same time narrow. Hence, at τp= 2×10−3, e.g., supersoli- tons exist for density ratio ranging from f≈0.926 up to f≈0.931, while double layers occur for f≳0.931. When τpis continuously increased further, these ranges narrow more and shift to higher val- ues of funtil they vanish at τp≈0.2. For larger values of τp, only positive ion sonic point occurs as the limiting factor to the soliton amplitude for all values of negative-to-positive density ratio f. The lower right panel of Figure 1, which is plotted at τp= 0.7 larger than 0.2, shows that only normal solitons limited by the occurrence of positive ion sonic point exist for the whole range of f. Keepingτpfixed at 10−6and reducing the spectral index (increasing the superthermal behavior of electrons), it is found that the negative-to-positive ion density ranges supporting the super- solitons and double layers shift to lower values of fand become larger. For example, if the spectral index decreases from 2 to 1.8, these ranges shift from 0.88 ≲f≲0.918 and 0.918 ≲f≲1 to 0.77 ≲f≲0.813 and 0.813 ≲f≲1, respectively. When, on the con- trary, the spectral index is increased (decreasing the superthermal behavior of electrons), the density ratio ranges supporting super- solitons and double layers’ propagation shift to higher values of f and become narrower. For values of κ≳2.5, supersolitons and dou- ble layers disappear and only normal solitons limited by positive ion sonic points are encountered. This analysis shows that in the model under study, supersolitons and double layers exist when the gap between the ion temperatures is large and electrons are strongly non-Maxwellian. D. The ion thermal and electron superthermal effects on the double layer amplitude and width Figure 11 shows the ion thermal effects (left panels) and elec- tron superthermal effects (right panels) on the double layer ampli- tude and width. The plasma parameter values μ= 0.15,τn= 0.7, and f= 0.95 are the same for all panels in Fig. 11. The value of the den- sity ratio f= 0.95 has been chosen so that the soliton upper limiting factor is a double layer when τp= 10−6andκ= 2. The left panels show the effects of increasing the negative to positive ion temper- ature from τp= 10−6toτp= 5×10−3when the spectral index value isκ= 2. As a result of this increase, the double layer ampli- tude increases from φdl≈0.62 (solid line, blue color online) to φdl ≈0.64 (dashed-dotted line, red color online) and its width decreases as is seen in the lower left panel. At the same time, the range of the density ratio supporting the double layers as the upper limit to the soliton amplitude shifts to higher values of f, as discussed in Sec. IV C. This means that with sufficiently high value of τp, the double layer gives way to the occurrence of sonic point as the upper limiting factor. This is what happens when τpincreases to τp= 7×10−3. At this value, the soliton upper limit is a normal soli- ton with soliton amplitude φlp= 0.65, occurring at the positive ion sonic point. With this new limit, a further increase in τpleads to the decrease in the soliton amplitude as it was observed for a plasma with low-to-intermediate values of f. The electron superthermal effects on the double layer ampli- tude and width are shown in the right panels of Fig. 11, where the positive to negative ion temperature ratio is τp= 10−6. The dou- ble layer amplitude, which is φdl≈0.62 when the spectral index κ= 2 (solid line, blue color online), decreases to φdl≈0.33 when AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-11 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 11 . The ion thermal effects (left panels) and electron superthermal behavior effects (right panels) on the double layer amplitude and width. The fixed parameter values taken in the double layer existence domain are μ= 0.15,τn= 0.7, and f= 0.95. The left panels are plotted at τp= 10−6(solid line, blue color online) and 5 ×10−3 (dashed-dotted line, red color online) at fixed spectral index κ= 2. The right panels are plotted at κ= 2 (solid line, blue color online) and 1.8 (dashed-dotted line, red color online) at fixed τp= 10−6. κdecreases to 1.8 (dashed-dotted line, red color online). Therefore, when the electron superthermality increases, the double layer ampli- tude decreases and its width increases. We have shown in Sec. III A that for Maxwellian electrons, only normal solitons are found. This means that as the spectral index increases, the upper limit changes. Thus, while the upper limit is the double layer when κ= 2, atκ= 2.2, the upper limit is a supersoliton and it is a normal soliton when κ exceeds a value of 2.3. E. The ion thermal and electron superthermal effects on the supersoliton amplitude and width For some plasma parameter values, supersolitons occur at the positive ion sonic point with maximum speed. For a fixed density ratio f, the amplitude and width of a maximum speed supersoliton vary similar to the same characteristics of the maximum speed nor- mal soliton when the relative temperature of the two ion species is reduced by increasing τp, and when the electron superthermality is increased by decreasing the value of the spectral index κ. Namely, the supersoliton amplitude decreases and its width increases with an increase in τpand this effect is enhanced by the electron superther- mality. However, it is important to emphasize that a continuousincrease in τpand a continuous decrease in κlead, ultimately, to the change of the soliton amplitude upper limit. This transition is shown in Fig. 12 in which we have plotted the pseudopotentials (upper panels) of the maximum amplitude structures and the cor- responding profiles (lower panels). The values of the plasma param- etersμ= 0.15,τn= 0.7, and f= 0.91 are the same for all pan- els. The left panels, where the spectral index κ= 2, show the ion thermal effect on the soliton upper limit transition. As is shown in both upper and lower left panels, when τp= 10−6(solid blue line), the soliton upper limit to the soliton range is a supersoliton, occurring at the positive ion sonic point with amplitude φlp= 0.68. When the relative temperature of the two ion species decreases to τp= 0.1 (dashed red line), the negative-to-positive density ratio range supporting the propagation of supersolitons shifts to higher values, as discussed in Sec. IV C, and there is a transition from the supersoliton upper limit to normal soliton upper limit with lower amplitudeφlp= 0.35. In the upper right panels, we show the transition process when the electron superthermality is increased by decreasing the spec- tral indexκwhile keeping the positive to negative ion temperature fixed toτp= 10−6. In this case, the negative-to-positive ion den- sity ratio range supporting the propagation of supersolitons shifts AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-12 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv FIG. 12 . The transition from a supersoliton to a normal soliton as the positive ion temperature increases (left panels) and to a double layer as the superthermal behavior of electrons increases (right panels). The fixed parameter values taken from the supersoliton existence domain are μ= 0.15,τn= 0.7, and f= 0.91. The left panels are plotted atτp= 10−6(solid line, blue color online) and 0.1 (dashed-dotted line, red color online) at fixed spectral index κ= 2. The right panels are plotted at κ= 2 (solid line, blue color online), 1.8 (dashed-dotted line, red color online), and 2.1 (dashed line, black online) at fixed τp= 10−6. to lower values of fwith an extension of the double layer existence range to lower values of f. It is thus possible that the supersoliton upper limit gives way to a double layer upper limit if the decrease inκis sufficiently large. Thus, in the right panels of Fig. 12 where κ= 2 (solid line, blue color online), the soliton upper limit is the occurrence of a supersoliton with amplitude φlp= 0.68. When the spectral index decreases to κ= 1.8 (dashed-dotted line, red color online), there is a transition of the upper limit to a double layer with a lower amplitude φdl= 0.31. On the other side, for a suffi- ciently large increase in the parameter κ, there is a transition of the upper limit to a normal soliton in agreement with the upper panels of Fig. 1. This transition is clear in the upper right panel of Fig. 12, in which the dashed line (black line online) has been plotted at κ= 2.1. This curve has one potential well and represents a normal soliton, as is confirmed by the corresponding soliton profile in the lower right panel. For values of κ>2.1, only normal solitons limited by the occurrence of positive ion sonic points occur. We note here that, despite the transition, the amplitude of the limiting structure decreases continuously whether the relative ion temperature is con- tinuously decreased or the electron superthermality is continuously increased.V. SUMMARY AND CONCLUSIONS In this paper, we have used the Sagdeev pseudopotential method to investigate the ion thermal and electron superthermal effects on the slow mode ion-acoustic nonlinear waves in a nega- tive ion plasma, comprising adiabatic positive and negative ions and kappa-distributed electrons. It is well known that in a two ion com- ponent plasma, the slow mode exists provided the thermal speeds of both ion species are different, with the polarity of the slow mode nonlinear structures being determined by the sign of the colder ion species.23In the present study, we have considered positive ion species to be colder, and negative ion species to be hotter, and only positive potential nonlinear structures were found. The ion thermal effects are investigated through the parameter τp=Tp/Tn, the posi- tive to negative ion temperature ratio at fixed τn=Tn/Te, the nega- tive ion temperature to electron temperature ratio. Thus increasing τpsignifies increasing the temperature of positive ions at a fixed tem- perature of negative ions, and the relative temperature of the two ion species is reduced. Having normalized the flow and nonlinear struc- tures speeds by the negative ion thermal speed vtn, the values of τpare limited by the condition μτp<1, whereμis the negative to positive AIP Advances 11, 025325 (2021); doi: 10.1063/5.0039372 11, 025325-13 © Author(s) 2021AIP Advances ARTICLE scitation.org/journal/adv ion mass ratio. The electron superthermal effects, on the other hand, are studied by considering different values of the spectral index κof the kappa-distributed electrons. The main results from this study are summarized as follows: 1. For Maxwellian electrons, cold positive ions, and warm nega- tive ions, the plasma supports the propagation of normal pos- itive solitons, limited on the upper side by the occurrence of a positive ion sonic point. As τpincreases, i.e., as the relative tem- perature between the two ion species reduces, the range of the allowed Mach numbers also reduces but shifts to higher val- ues. This result differs significantly from that obtained for the fast mode,34viz. that at low temperature, a negative ion plasma supports both negative and positive potential fast mode soli- tary waves, and at high temperature, only negative fast mode solitons are found. The increase in τpis not however arbi- trary, being limited by the slow mode Mach number range√μτp<Mslow<1. 2. When the electron superthermality increases with other plasma parameters held fixed, the soliton Mach number shifts to a lower value and the Mach number range supporting the soliton propagation narrows. For strongly non-Maxwellian electrons, cold positive ions, and warm negative ions, the plasma support, in addition to normal solitons, the propaga- tion of slow mode positive supersolitons for a narrow range of negative-to-positive ion density ratio f. The soliton amplitude is limited, on the upper side, by the occurrence of positive ion sonic points for low to intermediate values of density ratio f, while the limiting factor is a double layer for higher values of f. As for the supersolitons, they are limited, on the lower side, either by the double layer or the coalescence of two consecutive extrema and on the upper side, by the coalescence of two con- secutive extrema or the occurrence of positive ion sonic point. When the double layer acts as the lower limit to the super- soliton existence range, there is a range of electrostatic poten- tials that are not accessible from the undisturbed conditions, as was reported for the first time by Baluku et al.57However, when the coalescence is the lower limit, the supersoliton ampli- tude varies continuously. We note here that these results dif- fer qualitatively from those obtained for Maxwellian electrons, while for the fast mode34they were qualitatively the same. This breaks down a common belief, according to which kappa dis- tribution does not bring any new qualitative differences from the results with Maxwellian distribution.64 3. As the relative temperature of the two ion species is reduced (τpis increased) with fixed spectral index kappa, the range of negative-to-positive ion density ratio fsupporting the propa- gation of double layers as the upper limit to the soliton ampli- tude shifts to higher values of f, increasing at the same time the density range supporting solitons limited by the occur- rence of positive ion sonic point. This means that at a high density ratio, the soliton upper limit is the double layer when positive ions are cold, but changes to the positive ion sonic point when positive ions are warm. Ultimately, with a further increase in τp, double layers disappear as the negative ion den- sity, necessary to support them, tends to be larger than the limiting value of f= 1, and from then, solitons are limited by the sonic point for the whole range of f. When it is the electronsuperthermality that increases (by virtue of decreasing the spectral index κ) under fixed τp, the density range support- ing double layers extends to lower values of f, decreasing at the same time the density range supporting solitons limited by the occurrence of positive ion sonic point. There is, at intermedi- ate values of f, the transition from the positive ion sonic point as the soliton upper limit to double layer as the soliton limiting factor. 4. The range of negative-to-positive ion density ratio fsupport- ing the propagation of double layers is preceded by a range sup- porting the propagation of supersolitons, the two being sepa- rated at a point where a positive potential double layer occurs at the positive ion sonic point. Similar to the range supporting the propagation of double layers, the density range support- ing the propagation of supersolitons shifts to higher values of fand narrows as the relative temperature of the two ions is reduced (τpis increased), and it shifts to lower values of fand becomes larger when it is the electron superthermality that increases (decreasing the spectral index κ). It is thus obvious that in the case under study, the existence of supersolitons and double layers is favored by a combination of cold positive ions, warm negative ions, and strongly non-Maxwellian electrons. 5. While the amplitudes of solitons and supersolitons decrease with a decrease in the relative temperature of the two ion species, an effect that is enhanced by the superthermal behav- ior of the electrons, and it is found that the amplitudes of the double layers increase with a decrease in the relative tempera- ture of the two ion species but decrease with an increase of the electron superthermality. ACKNOWLEDGMENTS Partial financial support from the Swedish International Devel- opment Cooperation Agency (SIDA) through the International Sci- ence Programme (ISP) to the University of Rwanda (UR) through the Rwanda Astrophysics, Space and Climate Science Research Group (Grant No. ISP/RWA : 01) is gratefully acknowledged. Further financial support from The World Academy of Sciences (TWAS) for the advancement of science in developing countries (Grant No. 19-123RG/ PHYS /AF/AC I-FR3240310162) through the Institut d’Enseignement Superieur de Ruhengeri (INES-Ruhengeri) is also acknowledged. 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1.1857652.pdf

Micromagnetic study of domain-wall pinning characteristics with grooves in thin films H. Asada, H. Ii, J. Yamasaki, M. Takezawa, and T. Koyanagi Citation: Journal of Applied Physics 97, 10E317 (2005); doi: 10.1063/1.1857652 View online: http://dx.doi.org/10.1063/1.1857652 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Underlying mechanism of domain-wall motions in soft magnetic thin-film nanostripes beyond the velocity- breakdown regime Appl. Phys. Lett. 93, 052503 (2008); 10.1063/1.2968138 Domain-wall pinning by steplike thickness change in magnetic thin film J. Appl. Phys. 99, 08B701 (2006); 10.1063/1.2162038 Field dependence of micromagnetic domain patterns in MnAs films J. Appl. Phys. 98, 063909 (2005); 10.1063/1.2060959 Influence of domain wall structure on pinning characteristics with self-induced anisotropy J. Appl. Phys. 93, 7447 (2003); 10.1063/1.1560703 Micromagnetic simulation of 360° domain walls in thin Co films J. Appl. Phys. 87, 5517 (2000); 10.1063/1.373390 [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: 155.33.16.124 On: Tue, 25 Nov 2014 23:15:11Micromagnetic study of domain-wall pinning characteristics with grooves in thin films H. Asadaa!and H. Ii Department of Symbiotic Environmental System Engineering, Graduate School of Science and Engineering, Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan J. Yamasaki and M. Takezawa Department of Applied Science for Integrated System Engineering, Graduate School of Engineering,Kyushu Institute of Technology, 1-1 Sensui-cho, Tobata-ku, Kitakyushu 804-8550, Japan T. Koyanagi Department of Symbiotic Environmental System Engineering, Graduate School of Science and Engineering,Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan sPresented on 10 November 2004; published online 9 May 2005 d The pinning characteristics of a 180° domain wall with grooves are investigated using the micromagnetic simulation. The depinning fields required to pull the wall out of the grooved regionwere strongly related to the pinning characteristics at each step edge.The depinning field differencebetween the wall movement directions was improved by the increase of the lower depinning fieldcompared to that with the steplike thickness change. It was also found that the depinning fields forvarious groove widths were almost constant and the wall displacement was further suppressed bythe narrower groove having the vertical edge.©2005 American Institute of Physics .fDOI: 10.1063/1.1857652 g I. INTRODUCTION Artificial wall pinning is effective in controlling the de- pinning field in sensor applications utilizing a largeBarkhausen jump and improving the properties of high-frequency material applications, such as the magnetic-fieldsensor and the core, due to the suppression of wall motion.Etched grooves in a garnet film having a perpendicular an-isotropy were used for stabilizing the stripe domain in aBloch line memory. 1,2In a narrow track single-pole head, grooves across the track can control the domain structure ofa main-pole film and suppress the 90° wall motion of closuredomains at the film edges when the magnetic field is appliedalong the longitudinal direction of the film. 3 In the previous work, we reported the micromagnetic simulation results of the pinning characteristics of the asym-metric Bloch wall caused by a steplike thickness change inthin films having an in-plane anisotropy. 4It has been clari- fied that the bidirectional pinning effect for magnetic fieldsapplied along the magnetic domain was obtained when thespin rotation from the Bloch wall in the film center to Néelcap at the grooved side surface was across the step. Thedepinning field for the negative applied field which drove thewall toward the nongrooved region was considerably largerthan that for the positive one which drove the wall towardthe grooved region and the pinning characteristics stronglydepended on the wall structure. In this article, the pinning characteristics of a domain wall with grooves in thin films are investigated using themicromagnetic simulation. The dependences of the depin-ning fields on film thickness and the groove width are dis-cussed.II. SIMULATION MODEL Numerical simulations were performed by integrating the Landau–Lifshitz–Gilbert equation numerically by an ex-plicit scheme of the modified Dufort–Frankel method. 5,6As illustrated in Fig. 1, the cross-section normal to the filmplane syzplane dcontaining a thickness change sDhdis taken to be the computation region, which is discretized into a two-dimensional array. Boundary conditions on the compu-tation region are such that the wall is in the xzplane and infinite in extent in the xdirection. Material parameters used in the simulation are as follows: saturation induction 4 pMs =8000 G, uniaxial anisotropy constant Ku=3200 ergs/cm3, exchange constant A=10−6ergs/cm, gyromagnetic ratio g =1.76 3107ssOed−1, and damping constant a=0.5.4The grid element spacings are 50 Å for the film thickness h ł0.3mm, and 100 Å for h.0.3mm, respectively. The easy axis is along the xdirection and magnetic fields sHpdare applied along the magnetic domain. The time transient of the orthogonal component of an effective field was used for de-termining the depinning field. 7 III. RESULTS AND DISCUSSION The depinning fields of the asymmetric Bloch wall with the steplike thickness changes, as shown in Figs. 2 sadand adElectronic mail: asada@yamaguchi-u.ac.jp FIG. 1. Schematic drawing of a groove in a thin film.JOURNAL OF APPLIED PHYSICS 97, 10E317 s2005 d 0021-8979/2005/97 ~10!/10E317/3/$22.50 © 2005 American Institute of Physics 97, 10E317-1 [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: 155.33.16.124 On: Tue, 25 Nov 2014 23:15:112sbd, for positive s+Hpdand negative s−Hpdapplied magnetic fields are investigated. The arrows in Fig. 2 represent the magnetization directions for every ninth s333dgrid ele- ments. For the positive applied fields s+xdirection d, the wall moves in the direction of the right-hand side in Figs. 2 sadand 2sbd. The depinning fields for the positive and negative ap- plied fields in Fig. 2 sadand for the positive applied field in Fig. 2 sbdas a function of the film thickness are plotted in Fig. 3. The step depth ratio to the film thickness sDh/hdis 0.2. The depinning field for − Hpin Fig. 2 sbdis not obtained sless than 0.2 Oe d. Similar to the film thickness dependence of the depinning fields in Fig. 2 sad, the depinning field for +Hpin Fig. 2 sbdalso decreases with increasing film thick- ness due to the decrease of the exchange energy per unitarea. 4The larger depinning fields are obtained when the wall moves toward the nongrooved region in both Figs. 2 sadand 2sbd. Next, the pinning characteristics of a domain wall with grooves were examined. Figure 4 shows the simulation re-sults of the pinned wall configurations with the groove hav-ing the depth Dh=300 Å and the width W=600 Å in a 0.15- mm-thick film for sadHp=0Oe sinitial state d,sbd −30 Oe, and scd+30 Oe, respectively. The arrows in the fig- ures represent the magnetization directions for every ninths333dgrid elements. When the magnetic field H pis appliedfor the wall of Fig. 4 sad, the wall moves in the direction of the left-hand side for − Hpand the right-hand side for + Hp, and is pinned at each grooved edge, as shown in Figs. 4 sbd and 4 scd, respectively. In both cases of Figs. 4 sbdand 4 scd, the wall returns to the initial position fFig. 4 sadgafter the magnetic field is off. Figure 5 shows the film thickness de-pendence of the depinning field caused by the groove. Thegroove width and the depth ratio to the film thickness are600 Å and 0.2. The depinning fields for + H pand −Hpare approximately equal to those for + Hpin Fig. 2 sadand for −Hpin Fig. 2 sbd, respectively. Therefore, the difference be- tween the depinning fields for + Hpand −Hpdue to the asym- metricwallstructureisimprovedbytheincreaseofthelowerdepinning field compared to that with the steplike thicknesschange, as shown in Fig. 3. The depinning fields for variousgroove widths in a 0.15- mm-thick film are indicated in Fig. FIG. 2. Magnetization configuration of the asymmetric Bloch wall with each steplike thickness change in a 0.15- mm-thick film without the magnetic field. The step depth is 300 Å. The arrows in the figures represent the mag-netization directions for every ninth s333dgrid elements. FIG. 3. Film thickness dependence of the depinning fields for the positive sPdand negative ssdmagnetic fields in Fig. 2 sadand for the positive mag- netic field sjdin Fig. 2 sbd. The step depth ratio to the film thickness is 0.2. FIG. 4. Magnetization configuration of the pinned wall with the groove for the magnetic fields of sad0O e sinitial d,sbd−30 Oe, and scd30 Oe in a 0.15- mm-thick film. The groove width and depth are 600 and 300 Å, respectively. FIG. 5. Dependence of the depinning field of the wall with grooves for thepositive sjdand negative ssdmagnetic fields on the film thickness. The groove width and the depth ratio to the film thickness are 600 Å and 0.2,respectively.10E317-2 Asada et al. J. Appl. Phys. 97, 10E317 ~2005 ! [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: 155.33.16.124 On: Tue, 25 Nov 2014 23:15:116. The groove depth is 300 Å. Both the depinning fields for the positive and negative magnetic fields are almost constant.These results suggest that the pinning effect is dominated bythe magnetization state in the vicinity of the step edge. Simu-lations showed, however, that the shape of step affected thepinning characteristics, for example, the depinning field for−H pin Fig. 2 saddrastically decreased, as the slop became gentler. The influence of groove shape is under investigation. Figure 7 shows the groove width dependence of wall displacement for the applied magnetic fields of 10, 20, and30 Oe. The wall displacement is defined as the distance be- tween the pinned wall positions for − Hpand for + Hp. The initial wall position is at the left-hand edge of the groove, asshown in Fig. 4 sad. As mentioned before, after the magnetic field was off, the wall which pinned at the right-hand edgefor the positive applied magnetic field returned to the initialposition for the grooves having the width up to 600 Å. Onthe other hand, for the grooves having the width more than900 Å, the wall still existed near the right-hand edge. Thewall displacement linearly increases with increasing groovewidth more than 300 Å for H p=20 and 30 Oe. For Hp =10 Oe, the wall displacement is quite small.This is because that the amplitude of the applied field is lower than the de-pinning field for + H pin Fig. 2 sadssee Fig. 3 dand the wall is pinned for + Hpat the left-hand edge of the groove. The wall displacement is further suppressed with the narrower groovehaving the vertical edge. IV. CONCLUSIONS We have simulated the pinning characteristics of a do- main wall with grooves in thin films. The amplitudes of thedepinning fields of the wall with grooves are approximatelyequal to those with each steplike thickness change for theapplied field which drives the wall to the nongrooved region,respectively. The depinning field difference between the wallmovement directions is improved by the increase of thelower depinning field compared to that with the steplikethickness change. Simulation results suggested that the pin-ning effect is dominated by the magnetization state in thevicinity of the step edge. This character is reflected in thedepinning fields and the wall displacement for variousgroove widths. As a result, the wall displacement is furthersuppressed by the narrower groove having the vertical edge. 1D. Klein and J. Engemann, J. Magn. Magn. Mater. 45,3 8 9 s1984 d. 2T. Suzuki et al., IEEE Trans. Magn. 22,7 8 4 s1986 d. 3K. Ise and Y. Nakamura, J. Magn. Soc. Jpn. 15, 167 s1991 d. 4H. Asada, Y. Hyodo, J. Yamasaki, M. Takezawa, and T. Koyanagi, IEEE Trans. Magn. 40,2 1 1 0 s2004 d. 5S. Konishi, K. Matsuyama, N. Yoshimatsu, and K. Sakai, IEEE Trans. Magn.24,3 0 3 6 s1988 d. 6G. Ronan, K. Matsuyama, E. Fujita, M. Ohbo, S. Kubota, and S. Konishi, IEEE Trans. Magn. 21, 2680 s1985 d. 7H. Asada, K. Matsuyama, M. Gamachi, and K. Taniguchi, J. Appl. Phys. 75,6 0 8 9 s1994 d. FIG. 6. Depinning fields for the positive sjdand negative ssdmagnetic fields as a function of the groove width in a 0.15- mm-thick film. The groove depth is 300 Å. FIG. 7. Groove width dependence of the wall displacement for each appliedmagnetic field.10E317-3 Asada et al. J. Appl. Phys. 97, 10E317 ~2005 ! [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: 155.33.16.124 On: Tue, 25 Nov 2014 23:15:11

1.440485.pdf

The stable states picture of chemical reactions. II. Rate constants for condensed and gas phase reaction models Richard F. Grote and James T. Hynes Citation: J. Chem. Phys. 73, 2715 (1980); doi: 10.1063/1.440485 View online: http://dx.doi.org/10.1063/1.440485 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v73/i6 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsThe stable states picture of chemical reactions. II. Rate constants for condensed and gas phase reaction modelsa) Richard F. Grote and James T. Hynesb) Department of Chemistry. University of Colorado. Boulder. Colorado 80309 (Received 2 May 1980; accepted 6 June 1980) The time correlation function (tet) formulas for rate constants K derived via the stable states picture (SSP) of chemical reactions are applied to a wide variety (a~) of gas and solution phase reaction models. (a) For gas phase bimolecular reactions, we show that the flux tcf governing K corresponds to standard numerical trajectory calculation methods. Alternate formulas for K are derived which focus on saddle point surfaces, thus increasing computational efficiency. Advantages of the SSP formulas for K are discussed. (b) For gas phase unimolecular reactions, simple results for K are found in both the strong and weak coupling collision limits; the often ignored role of product stabilization is exposed for reversible isomerizations. The SSP results correct some standard weak coupling rate constant results by as much as 50%. (c) For barrier crossing reactions in solution. we evaluate K for a generalized (nonMarkovian) Langevin description of the dynamics. For several realistic models of time dependent friction. K differs dramatically from the popular Kramers constant friction predictions; this has important implications for the validity of transition state theory. (d) For solution reactions heavily influenced by spatial diffusion, we show that the SSP isolates short range reaction dynamics of interest and includes important barrier region effects in structural isomerizations often missed in standard descriptions. I. INTRODUCTION In the preceding paper! (hereafter I), Northrup and Hynes showed how chemical reaction rate constants are related to molecular dynamics by time correlation func tion (tcf) formulas. This was accomplished via the stable states picture (SSP), in which attention is focused on the dynamical flux out of a stable reactant region and the ensuing flux into a stable product region. The rate constant is determined by the dynamiCS in an interme diate, or "barrier, " region located between these two stable states. In this paper, we demonstrate the power and range of this approach by determining rate constants for a wide variety of reaction models. These range from gas phase bimolecular and unimolecular reactions to condensed phase potential barrier crossing and diffusion-controlled reactions. This work continues and significantly extends prior work in this direction by Northrup and Hynes. 2-5 Our present efforts provide insight on major features of importance for various reaction models and lead to some significant new resl,llts. The major example pre sented here is a new rate constant result for condensed phase barrier crOSSing based on a fairly realistic de scription of dynamical interactions with the solvent. To keep this paper relatively self-contained, we brief ly list and discuss some of the tcf rate constant formulas derived in I. The stable states picture (SSP) focuses on three regions. The first two describe the stable reac tants (R) and stable products (P). These regions are separated by a high energy intermediate or barrier re gion (1). There are thus two dividing surfaces SR and S p associated with Rand P, respectively. The forward {f) reaction rate constant is the barrier rate constant a)Supported in part by the National Science Foundation and the Alfred P. Sloan Foundation. b)John Simon Guggenheim Memorial Fellow, 1979. University of Colorado CRCW Faculty Fellow, 1979. Kf= l~ dtUj(SR)j:(Sp, t)R • o (1.1) This tcf expression involves (a) the inward flux jj(SR) from R into I across S R and (b) the outward flux jo(S p) from I into P across S p. The asterisk superscript de notes the dynamical condition that no recrossing into re gion I is allowed for trajectories that have crossed SR or Sp from I into the stable reactant or product regions. Thus, Kf is determined by short-lived nonequilibrium dynamics occurring in region I and not by the slowly varying Rand P populations. The brackets «( ... )R in Eq. (1. 1) denote an equilib rium average over initial conditions of (a) any buffer gas or solvent molecules present and (b) the atoms and molecules participating in the reaction. The subscript R denotes a normalization of the latter by the reactants' partition function QR' The equilibrium average arises from the key SSP assumption of internal equilibrium in the stable reactants. There are several equivalent ways of writing Kf. 1 Two important alternate forms are Kf= Uj(SR)R + I~ dt(jj(SR)j:(SR' t)R , o (1. 2) which focuses solely on fluxes across the reactant sur face (cf. Sec. III of I) and Kf= I~ dt(ji(SR)j*(S, t)R' o (1. 3) which involves the total flux j at some arbitrary dividing surface S lying within I and away from SR and Sp (cf. Sec. III of I). Expressions analogous to Eqs. (1.1)-(1. 3) hold for reverse rate constants Kr• Since Kf and Kr are simply related by Kf=KeQK r, where KeQ is an equilibrium con stant, we consider only forward reactions. The identification of Kf as the true reaction rate con- J. Chem. Phys. 73(6). 15 Sept. 1980 0021·9606/80/182715·18$01.00 © 1980 American Institute of Physics 2715 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions2716 R. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II stant depends on the SSP assumption of rapid internal equilibration in the reactant and product regions. 1 This allows (a) the restriction of the flux into region I to be that flux arising from an equilibrium reactant distribu tion and (b) the neglect of events occurring after passage into the stable product is accomplished. The rapid equilibration assumption breaks down in a few important cases. Slow internal equilibration within the stable reactant and product states can then influence the rate. According to the SSP, 1.3-5 the rate constant in this case is modified to kf = [1 + (K f/KR)tIKf (1. 4) for an irreversible reaction and to (1.5) for a reversible reaction.6 Here KR and Kp are internal relaxation rate constants governing equilibration rates within R and P, respectively. They are related to tcf's of population fluctuations at the stable state surfaces in Sec. IV of I. When KR:5 Kf, the rate depends significantly on dynamics within the stable reactant. The rate of "flow" up to surface S R can compete with the passage rate from SR to Sp governed by Kf• When Kp:5 Kr, the rate can depend significantly on the equilibration rate in the stable product. This equilibration can compete with the passage back to reactants (Sp-SR) governed by Kr• When Kf« K Rand Kr «Kp, these two phenomena are negligible and we recover the barrier rate constant Kf• The outline of this paper is as follows: Sections II and III deal with gas phase reaction models. Bimolecu lar reactions are discussed in Sec. II, while Sec. III deals with unimolecular reactions in two extreme limits of energy transfer. In Sec. IV, we turn to reactions in solution and evaluate rate constants for fairly realistic models of dynamical interaction with the solvent. In Sec. V, we consider reactions for which slow spatial diffusion plays a key role. We attempt to illustrate the key features and important directions for future exten sions in each section. Section VI provides a summary. II. GAS PHASE BIMOLECULAR REACTIONS In this section, we apply our tcf formulas for rate constants to bimolecular gas phase reactions. We re late our SSP results to more standard expressions and indicate some advantages of our own formulation. A. Collision viewpoint Our tcf expression [Eq. (1.1)] for Kf, i. e. , Kf = f~ dt (jf (S R)j :(S P' t)) R , o (2.1) has a simple and direct correspondence with (classical) collisional trajectory calculations for gas phase bimo lecular reactions. To see this, we consider a standard representation in Fig. 1 for a collinear atom-transfer reaction7: A+BC-AB+C. The reactant and product surfaces are placed away from the interaction zone I. They are thus located in the Sp I c __ --+-_ R Be FIG. 1. Potential surface diagram for a collinear A + Be reac tion. The dividing surfaces discussed in the text are displayed. asymptotic reactant (R) and product (p) regions. Tra jectories crossing SR into I are sampled from an equilib rium distribution of reactants. Such trajectories are then followed throughout a collision until either (a) prod ucts are formed by crossing of S p (AB and C separate) or (b) reactants are reformed by crossing of S R (A and BC separate). This corresponds precisely to our SSP discussion of Eq. (2.1) in Sec. II of I and Sec. I. Since no forces here return reactants or products to the intermediate zone, the absorbing boundary conditions in the dynamiCS in Eq. (2.1) are moot. Then Eq. (2.1) takes the Simpler form (2.3) where now there are no restrictions on the dynamics. 8 This is the Simplest tcf formula for the rate constant from the collision point of view. With suitable quasi classical definitions of the internal quantum states of reactants and products, the state-to-state rate constant analogs of Eq. (2.3) can also be derived. 9 A more familiar expression for the rate constant ob tained from the "collision" viewpoint is the equilibrium averagelO (2.4) Here the characteristic function X(S R) equals unity if a trajectory originating from reactants is reactive; it equals zero if the trajectory is nonreactive. We can demonstrate the identity of Eqs. (2.3) and (2.4) as fol lows: We first rewrite Eq. (2.3) as Kf= f~ dt(jf(SR)j(Sp, t))R • o (2.5) This replacement of the outward flux jo(Sp, t) by the total flux j(S p, t) across the product surface is exact because (a) all trajectories arise from reactants and (b) no forces return separated products to the intermediate region. The time integral of the flux f~ dtj (Sp, t) in Eq. (2.5) is just X(SR); contributing trajectories must arise from the reactants and ultimately form products. This estab- J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsR. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II 2717 lishes the equivalence of Eqs. (2.3) and (2. 4). There are at least two advantages of the tcf (2.3) (and its state-to-state analogs) over Eq. (2.4). First, the flux tcf itself as a function of time gives a more detailed picture of reactive dynamics than does its integrated time area Kf. Since the tcf involves some averaging over initial conditions, its information content lies between that of the enormous detail of individual trajectories and that of the fully averaged and time independent rate con stant. Second, we will develop evaluation techniques in Sec. IV for the rate constant where the reactant motion is coupled to that of a liquid solvent. Similar techniques can account for the effect on motion in the reaction coor dinate of intramolecular degrees of freedom orthogonal to this coordinate. 11 This is under study. 12 B. Saddle point initial conditions The collision viewpoint is sometimes inconvenient. Many trajectories originating from reactants may re cross SR' i. e., are unreactive. From a computational viewpoint then, the collision picture can often be inef ficient; only a small fraction of these trajectories are "interesting, " i. e., reactive. An alternate perspective which circumvents this dif ficulty to some extent was developed by Keck13 and Anderson14 and has been discussed by a number of au thors.15 The basic idea is to relate the rate constant to the fate of trajectories originating from the saddle point. While the population in this region is low, many such trajectories lead to reaction, thereby increasing com putational efficiency. 13-15 Our tcf expressions for rate constants will now be cast in this "saddle point" or "transition state" language. We begin with Eq. (1. 3) for Kf rewritten as Kf= I~ dt(jj(SR)j(SS' t))R . o (2.6) Here Ss denotes a saddle point dividing surface (Fig. 1). We have again used the fact that the absorbing Be's are moot to write j*(Ss, t) =j(Ss, t) in Eq. (2.6) 0. e., unre stricted dynamics). We next bring the propagator in eILtj(Ss) in Eq. (2.6) around to the left in the average and reverse the momenta to find that Kf= J~ dt(j(SS)jI(SR' -t)R = J~ dt(j(SS)jO(SR' t)R· o 0 (2.7) Equation (2. 7) does not yet involve the saddle point flux and the flux into the product zone. Its form, however, suggests that the reverse rate constant Kr can be related to the form we wish. Indeed, similar manipulations of the formula (2.8) give the expression (2.9) NOW, since the ratio Kfl Kr is the equilibrium constant Ke\ Le., (2.10) and the reactant and product averages are related by « ... )R =Keq« ... )p, we can combine Eq. (2.9) with Eq. (2.10). This gives the desired result for the for ward rate constant or its equivalent Kf= J~ dt(j(Ss)j(Sp, t)R . o (2.11) (2. 12) Similarly, combination of Eqs. (2.7) and (2.10) gives the reverse rate constant as Kr= J~ dt(j(SS)j(SR, t)p . o (2.13) Equations (2.12) and (2.13) give the desired rate con stant expressions in terms of the tcf's of (a) an initial flux across the saddle point and (b) the subsequent flux across one of the stable state surfaces. They provide very effiCient computational routes to the calculation of rate constants. 16 The saddle point analog of the collision picture expres sion (2.4) has also been discussed. 10 It is given by (2. 14) The characteristic function X(Ss) equals unity if a tra jectory starting on the surface S s ultimately exits to re gion P. Otherwise, X(Ss) equals zero. Equations (2.14) follows directly from our Eq. (2.12) by the identity fa dtj(Sp, t)=X(Ss) in this equation. C. Connection to transition state theory yet another tcf representation of Kf is useful in the discussion of transition state theory (TST). 17 Since the rate constant is independent of any choice of dividing surface in the intermediate zone (cf. Sec. III of I), we can just as well write Eq. (2.12) as18 (2. 15) TST focuses on the saddle point surface; it is assumed that no trajectory crossing S s from the side containing R towards the side containing P recrosses S5' 3.10.16.17 Then K/= KjST is determined solely by the initial delta function Singularity in the flux tcf (j(Ss)j(Ss, t)R arising from those trajectories crossing S s initially from the side of R to the side of p.19 This relationship of KrT to a tcf singularity was evidently first pointed out by Zwanzig20 and has been discussed in a number of con texts. 3.21 It is often stated that TST is exact if there is no re crossing of a surface located in what we have called the intermediate zone. This statement must be carefully interpreted; its validity requires an equilibrium distri bution of those particles that actually cross the sur face.22 This condition is often violatedU in, e. g. , gas phase unimolecular reactions, to which we now turn. J. Chem. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions2718 R. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II Sp I p ------SR FIG. 2. Schematic model for unimolecular decomposition. Only a very few of the possible levels and transitions are shown. The first level in I has the threshold energy value Eo. III. GAS PHASE UNIMOLECULAR REACTION MODELS Many unimolecular reactions can be characterized as escape from a potential energy well. 23.24 Once the mo lecular energy exceeds some threshold, reaction can oc cur irreversibly unless deactivating collisions with sur rounding buffer gas molecules intervene and return the molecule to the well. The reaction is then conveniently discussed in terms of (a) intrinsic reaction steps of those molecules with sufficient energy to react and (b) colliSional activating and deactivating steps for their formation and depletion. We now show how our SSP for mulation describes such reactions. A. Strong coupling limit We focus first on the strong coupling limit where large amounts of energy can be exchanged upon collision. 1. Irreversible decomposition A well-known schematic model for unimolecular de composition is displayed in Fig. 2. Collisions of the molecule with surrounding buffer gas molecules M trans fer the molecule from and to energy levels of its stable chemical state R to and from its high energy reactive states in R*, here labeled [: R(i) + M w/j R*(j) + M . (3.1) wji The intermediate region [ states begin above a threshold energy Eo. A molecule in some state i within [ may uni molecularly decompose to produce product P with a uni molecular rate constant W/P: (3.2) Here the rate constants w/P are regarded as known or to be modeled. 25 As in Sec. II, we could consider tcf expressions for these quantities. Here, however, we are one level removed from such a microscopic descrip tion. We characterize the reaction via master, or ki netic, equations13•26: apl/at= -LWIJPI + LWJIPJ -WIPPi . (3.3) J J Our task is then to evaluate the rate constant at this level by regarding Rand P as the stable states and the excited states R* as comprising the intermediate re gion [. We consider this model first in the so-called strong collision limit. 27 Then the collisional transition proba bilities (rates) from any state i to another state j are24 (3.4) Here Z is the collision frequency; p;" is the j state equi librium probability pf = e-IlEJ/Q, where Q = QR + Qr is the partition function for the molecule. The total colli sional loss Li from level i, of population PI' then de pends only upon the collision frequency and the level population: Li = -LWiJPI = -ZPI (3.5) j A simple steady state solution28 of Eqs. (3.3) and (3.4) shows that the reactant R levels remain in internal equilibrium; the levels in [, however, generally have some nonequilibrium distribution due to the perturbation of reactive loss. Along with the irreversible passage into the P region, this precisely matches the SSP view point. We evaluate the rate constant equation (1.1): Kf= 1~ dt (ji (SR)j:(Sp, t)R o (3.6) for this model in Appendix A. For the usual case of re action threshold energy large compared to thermal en ergy ({3Eo» 1), we find that Kf=Lp7QzWiP(Z +WiPr1 iEI (3.7) This is, of course, the well-known RRKM form. 23.24.29 At high collision frequency, equilibrium prevails in I and Kf -LiEr P~QwIP' This TST-like result involves the production rates for the product as weighted by an equi librium distribution in [. At intermediate and low den sities, partial to major nonequilibrium obtains in [. This is reflected in Eq. (3.7) by the effective [region distribution Z p~"(Z + Wi pr1 • In the extreme low density limit, region [ is drastically depleted of population. This allows a major Simplification in the rate constant description to which we now turn. 2. Low density collision-dominated limit In the very low density limit, the rate constant is de termined solely by the rate limiting activation steps to region I. For example, the strong collision model re sult Eq. (3.7) gives (3.8) This can be rewritten as (3.9) J. Chem. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsR. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II 2719 with detailed balance, when (JEo» 1. Therefore, the low density rate constant is the equilibrium (in R) rate of production of the energetic intermediate states. We can establish this as a general feature from the alternate rate constant formulation (1.2): Kf= (jl(SR»R + J'" dt(h(SR)j:(SR' t»R • o (3.10) In the low density limit, all molecules promoted to the intermediate region decompose before they suffer a de activating collision. We can therefore neglect the sec- 0nd member of Eq. (3.10); there will be vanishing re turn flux j:(S R' t) to the stable reactant. The rate con stant is then given by the initial, short time contribution (3.11) i. e., the production rate of energized molecules from an equilibrium distribution in the stable reactant. In terms of quantum levels and general transition proba bilities WI}' this is (3.12) which generalizes Eq. (3.9). In terms of a classical description of energy transfer in terms of a transition rate kernel K(E -E '), this is Kf= JEo dE P:(E) J'" dE' K(E -E') , o EO (3.13) first obtained by Bak and co_workers3o (see also Ref. 31). In the strong coupling limit, internal equilibrium is maintained in the stable reactant state below the thresh old energy. We now turn to the weak coupling limit where this equilibrium assumption breaks down to some extent; a slightly different stable state definition is re quired. B. Dissociation via energy diffusion Our final low denSity model, due to Kramers, 32 is for dissociative escape from a potential well (Fig. 3). The reaction is viewed as a slow and rate determining dif fusion in energy of an oscillator up' to a dissociation en ergy Ed; the oscillator then rapidly dissociates. The SSP approach can be applied to this model by (a) recog nizing the irreversible flux across the energy surface E = Ed and (b) finding an energy surface in the reactant well below which internal equilibrium holds. Slow energy diffusion via weak collisions is clearly the opposite extreme of the strong coupling limit of Sec. III A. It is most appropriate for massive diatomics im mersed in gases of small, light molecules. 33 It also has relevance for gas phase ion recombination, 34(a) molecu lar sticking on surfaces, 34(b) and possibly intramolecu lar energy flow. 35 1. Model for dissociation The specific model assumptions are (a) the time be tween oscillator (O)-gas molecule (M) collisions is large compared to characteristic oscillator periods w-1; (0) v x FIG. 3. schematic definition diagrams for dissociation of a Morse oscillator. (a) Phase curves, including that for disso ciation. (b) Potential energy curve indicating the stable reac tant surface SR' the phase of 0 is then "randomized" between O-M col lisions; (b) the energy of 0 is only very slightly changed per collision; (c) on the phase curve of 0 at energy Ed' rapid irreversible passage to products occurs by an un stable oscillation; the true unimolecular dissociation rate then plays no role in the reaction rate constant. The reaction rate constant will then be proportional to a "friction constant" , governing energy transfer to and from O. The constant, is proportional to the mean square energy transfer per collision and the gas den sity.24(a) This is the low density bimolecular limit of a unimolecular reaction. With the above assumptions, Kramers32 derived an approximate diffusion equation in action (J) space for the oscillator probability distribution p(J, t): ap(J, t)/at= -(a/aJ)j(J, t) = :J {D(J) [:J + (JW(J)J} p(J, t) (3.14) The flux in action is j(J, t); the action itself is the inte gral J(E) = (21Tt1 f dx P (3.15) of the momentum over an orbit at fixed energy. The oscillator frequency is w(J) = aE/aJ; the action diffusion coefficient is (3.16) The rapid reactive step proper is accounted for by the absorbing boundary condition (Be) p(Jd, t) = 0, (3.17) where Jd = J(Ed) is the action at dissociation. We will consider two oscillator models. The first is a truncated harmonic oscillator (HO), for which W = Wo = constant, E(J) = wr# "" Ed • (3.18) J. Chern. PhY5., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions2720 R. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II The second and more realistic model is the anharmonic Morse oscillator (MO), for which36,31 w(J)=wo[1 -(woJ/2Ed»)' E(J) = Ed[1 -(W/wO)2 . (3.19) Kramers obtained approximate rate constants for the well escape problem for a high threshold f3Ed» 1 32,38-40: Kr=?;f3Ede-SEa (HO) , (3.20a) KY=2?;f3Eae-SEd (MO) (3.20b) These provide a useful comparison for our own results. 2. Dissociation rate constants Reaction from the threshold energy Ed severely de pletes the population of high energy oscillators. This nonequilibrium suppression (Fig. 4) can extend to en ergies significantly below Ea. Well away from Ea, how ever, the energy distribution is essentially its equilib rium form. For large threshold energy (f3Ed» 1), this leaves a significant energy range where internal equilib rium holds. Together with the irreversible flux condi tion across the surface E = Ed' this exactly matches the SSP viewpoint. We therefore consider the barrier rate constant for dissociation (3.21) [cf. Eq. (1. 1) J. The intermediate region I is defined by the threshold action Jd(Ed) and the action Js(Es) at the reactant stable state surface (cL Fig. 3). The action flux tcf integral equation (3.21) can be evaluated by methods developed by Northrup and Hynes4; the details are sketched in Appendix B. Our result for the rate constant is plpeq 0.5 3 9 15 {3E FIG. 4. Ratio p!p.Q of the Morse oscillator internal energy distribution to its equilibrium value versus reduced energy {3E. Leftmost curve: {3E,=10.0; rightmost curve; (3E,=15. These curves were'found by an approximate steady state solution of Eqs. (3.14)-(3.17). TABLE I. Dissociation rate constants, divided by t e-SEa, for the harmonic oscillator (HO) and Morse oscillator (MO) for various dimensionless dissociation energies {JEd' (3Ed Kfa HO 10 9.1 15 13.9 20 18.9 MO 10 13.2 15 21. 7 20 30.6 aCalculated from Eq. (3.22). bFrom Ref. 39. cCalculated from Eq. (3.20). dCalculated from Eq. (3.30). num b Kf 8.9 13.9 18.9 12.9 31. 3 Kf= [Q(Es) JJd dJR(J)r . Js KrC k d Kf f 10.0 8.9 15.0 13.9 20.0 18.9 20.0 12.8 30.0 21. 7 40.0 30.6 Here R(J) is a local "resistance" to action flow R(J) = [D(J) e-8E(J »)-1 , and the reactant partition function is (3.22) (3.23) (3.24) For each oscillator model and for a given Ed' the stable state reactant surface Js' and thus Es' was varied to minimize Kf.4 We find that Kf(Es) has a very wide Es range for which Kf assumes very nearly its minimum value, which we adopt. The results are shown in Table I. Our results are in excellent agreement with those of Bak and Andersen, 39 who numerically integrated the dif fusion equations (3.14) and (3.17). Kramers' Eq. (3. 20a) is quite good for the (unrealistic) harmonic case (5%-10% error). In contrast, Kramers' result [Eq. (3. 20b)} is in error by 30%-50% for the nonlinear Morse oscillator. A major source of this error can be seen from Eqs. (3.22) and (3.23). The integral over region I is proportional to f~! dE rl(E) eBE• If J(E} is approxi mated by its threshold value Jd=2Ed/WO' then Eq. (3.20b) follows. Evidently, the E dependence of J over all of region I is important and is correctly handled in the SSP result Kf• 3. Nonequilibrium in the stable state We now consider the low threshold case (f3Etf"S 5). Nonequilibrium suppression of the energy distribution by reactive loss at Ed now spreads very far down into the oscillator stable state R. Thus, internal equilibrium no longer holds in R. According to the SSP, the rate constant for dissociation will now depend both on (a) the barrier rate constant Kf [Eq. (3.22)] and (b) the internal equilibration rate within R. The latter is governed by (cf. Sec. I and Sec. IV of I) KR = {p;!(Js} f'" dt[pO(Js, t IJs) -p.,,.(Js} If (3.25) This internal energy diffusion rate constant depends upon J. Chern. Phys .. Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsR. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II 2721 TABLE II. Dissociation rate constants, divided by 1; e-BEd, for the harmonic and Morse oscillator models. [jEd Kja (HO) 2 3.7 5 4.8 10 9.1 (MO) 2 4.5 5 6.4 10 13.2 "Calculated from Eq. (3.22). bCalculated from Eq. (3.26). cFrom Ref. 39. dcalculated from Eq. (3.20>. k/ k~umc ~rd j 2.4 2.4 2.0 4.2 4.1 5.0 8.9 8.9 10.0 2.7 2.8 4.0 5.4 5.1 10.0 13.0 12.9 20.0 the approach of the distribution pO(Js, t I Js) at Js to its equilibrium value if J = Js initially and no reaction is al lowed. The total dissociation rate constant is then [cf. Eq. (1. 4)] (3.26) The tcf expression (3.25) for KR is approximately evalu ated in Appendix B as jJs jJ K;l= dJe-SE(J) dJ'R(J') . ° ° (3.27) Thus, K~1 is a weighted and integrated resistance to en- ergy flow up to Js' the upper boundary of the stable state. Our numerical results for the HO and MO are given in Table II. Aside from the excellent agreement with the results of Bak and Andersen, 39 the main point of Table II is the following: For small f3Ed' Kj begins to severely overestimate the rate, due to the neglect of the slow en ergyequilibration, governed by KR, in the stable state. This energy flow is not slow enough to be rate limiting in the overall flow to Ed' but does exert an important influence. (When f3Ed=2, Kf/KR",0.6; internal energy relaxation is only about twice as fast as flow through [. In contrast, Kj/KR '" 102 when f3Ed=10.) The SSP equa tion (3.26) oorrectly accounts for this effect. 4. Alternate stable state definition An important characteristic of the SSP is its flexi bility. In the preceding subsections, we have defined the reactant stable state by energies ° .;; E .;; Es such that internal equilibrium applied when f3Ed» 1. Suppose that we redefine the stable state to be the entire energy range from zero up to the dissociation energy Ed. Then the discussion of Appendix E of I and Eq. (1. 4) show that we can write the rate constant as (3.28) Here Kj is the rate constant for the rapid dissociative step at Ed and [cf. Eq. (3.25)] (3.29) is just the internal energy diffusion rate constant for the entire stable state 0.;; E .;; Ed' Since the model prescrip tion of Sec. IDB1 tells us that Kj»KR" Eq. (3.29) simplifies. We can find the rate constant k, = KR, by evaluating KR" i. e., by letting Js Jd in Eq. (3.27) for KR• We have then41,42 [fJd fJ ]-1 kj= ° dJ e-SE(J) ° dJ'R(J') . (3.30) In this version of the SSP, the entire rate constant is determined by nonequilibrium energy flow within the stable state R'. This is, of course, a consequence of our definition of R' to include both R and the intermediate region [. For both oscillator cases, shown in Table I, the agreement of Eq. (3.30) with our previous stable states result Kf is excellent. Indeed, examination of the domi nant contributions to Eq. (3.22) shows that, for f3Ed » 1, 43 kfRjKj= [~Jd dJe-SE(J) ~J dJ'R(J,)]"1 (3.31) As in Eq. (3.27), this involves the equilibrium average of the integrated resistance to energy flow. C. Generalizations The results presented above can be generalized in several directions. We begin with the diffusion-con troled regime (Sec. III B). First, unimolecular disso ciation need not always be rapid compared to internal energy diffUSion, especially at higher buffer gas densi ties. If Ki denotes the intrinsic unimolecular rate con stant for this dissociation at Ed' then the overall rate constant is [cf. Eq. (1.4)] (3.32) where Kf is given by either Eqs. (3.22), (3.30), or (3.31) if f3Ed» 1. Second, if the reaction is a reversible isomerization (Fig. 5), then the overall rate constant is [cf. Eq. (1. 5) and Sec. IV or I] R p FIG. 5. Schematic potential energy ,and level diagram for an isomerization. The reaction coordinate is unspecified but might, for example, correspond to an angle. Only a few levels are shown. Note that the levels above the barrier top are prop erly included in region I. With reference to Eq. (3.36), E, is the minimum product energy and Eo is the lowest I region energy. J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions2722 R. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II (3.33) Here KR and Kp are the internal rate constants for the reactant and product; Ki and K~ are the intrinsic forward and reverse unimolecular rate constants for intramolec ular interconversion at threshold, respectively. Equa tion (3.33) shows the characteristic dependence on both forward and reverse reactive processes discussed at length in I. In the low density limit, Ki and K~ greatly exceed KR and Kp, respectively. Then Kf is determined only by K 11> K p, and the equilibrium constant K OQ = Ki/~. If the isomerization double well potential is symmetriC, then Keq= 1 and1,3,4 (3.34) Thus, kf is reduced by a factor of 2 compared to the re actant rate constant KR• The physical origin of this re duction is simple: The rate is determined by slow ac tivation up to the threshold and slow deactivation down into the product well. 1,3.4 We limit our comments on the strong coupling case (Sec. lIT A) to the generalization to an isomerization in the low density limit. Then, for example, Eq. (3.13) is replaced by ki1 = 4 + (K"qK~r1 , (3.35) where Kr is the reverse rate constant (cf. Fig. 5) (3.36) and K"q is the equilibrium constant. The importance of stable product formation is again emphasized by Eq. (3.35). At low density, the forward rate is determined not by passage back and forth at energy Eo between the well "tops" but by promotion out of the reactant well and stabilization in the product well (Fig. 5). Finally, we have considered senSible but simplified models of the dynamiCS in both the strong and weak cou pling limits. It is possible (and important) to apply our SSP formulas for more complex and realistic dynamical models.25.27.33 Our current efforts12 are focused on rate constants for "sticking" on the surfaces of solids34(b) and gas phase clusters. 29 IV. REACTIONS IN SOLUTION: BARRIER CROSSING MODELS We now apply the SSP and our rate constant formulas to models of chemical reactions in liquids. We will focus on the reaction viewed as passage over a mean po tential barrier between two wells. The classic investi gation of Kramers32 still provides a standard reference pOint for this problem. We therefore briefly review some salient aspects of Kramers' approach and results. Kramers viewed the (forward) reaction as a one di mensional passage of an effective "particle" over a po tential barrier located between reactant and product potential wells. This motion was described by the sto chastic Langevin equation (LE) iJ. dv/dt=F -I;v + F* (4.1) for the acceleration of the particle of mass jJ.. The force ariSing from the potential is F; F* is a Gaussian random force.44 The net effect of the "collisions, " 1. e., dy namical interactions, between particle and solvent is thus approximately accounted for by the frictional or damping force F a= -I;v, where I; is a friction constant. The potential in the barrier region was taken as an in verted parabola with a frequency Wb related to the bar rier curvature Ikbl by wb=(lkbl/jJ.)1f2. Kramers analyzed the steady state Fokker-Planck equation associated with the LE (4.1) to find the rate constantz' 32, 45 KKr = KTST{[(1;/2p.wb)2 + 1]1/2 _ (1;/2p.wb)} • (4.2) (In this section, we drop the subscript f denoting "for ward. ") Kramers' result KKr predicts a reduction, due to collisions, of the rate constant from its TST value (4.3) Here wR is the frequency of the reactant well bottom and Eo is the activation energy ({3Eo» 1). This TST result is, in turn, based on two key assumptions: (a) full internal equilibrium holds on the entire reac tant side of the barrier for particles passing to products and (b) each and every passage across the barrier top maximum leads to products without an intervening re crOSSing, 1. e., free or inertial streaming holds across the barrier top. In fact, Kramers' result shows that collisions occurring in the barrier region lead to at least a partial, and sometimes an extreme, breakdown of both assumptions. Thus, at low and intermediate friction (I;/wbl-l «1), in tercepting collisions can return a particle to the reac tant side of the barrier before recognizable product is formed. Under high friction conditions (I;/WbiJ.» 1), continuous collisional buffeting leads to repeated barrier top crossing and recrossing. The memory, or correla tion, time (/:/p.r1 of the particle's directed velocity is short compared to the time spent on or near the barrier top. Here velocity equilibrium applies but there is sig nificant spatial nonequilibrium. Then KKr ex 1;-1, thus characterizing a "diffusion-controlled" passage across the barrier region. Two central assumptions made by Kramers in finding Eq. (4.2) are of major importance. These are32 (a) in ternal equilibrium holds "near" the well bottom asso ciated with the reactants, i. e., away from the barrier top; (b) at some point past the barrier top, passage into the product well is essentially certain. (Note the differ ence between these and the TST assumptions. ) The above two assumptions coincide precisely with those of our SSP (cf. Sec. I). Indeed, Northrup and Hynes2 obtained Kramers' equation (4.2) by evaluation of a Green's function version of the tcf expression for the SSP barrier rate constant [cf. Eq. (1.1)J (4.4) We now discuss the evaluation of Eq. (4.4) for more realistic models of the dynamics. J. Chem. Phys., Vol. 73. No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsR. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II 2723 -0 o FIG. 6. Schematic model for barrier crossing reactions in solution. (a) Mean potential energy surface indicating the reac tion coordinate". Solvent collisions lead to various motions on this surface not dictated exclusively by the surface itself. (bl One dimensional cut along x indicating the stable state divid ing surfaces. The extended potential is shown (---I (see the text>. For an isomerization, there would be potential wells in the stable reactant and product regions. A. Model for barrier crossing Our model for solution reactions is shown in Fig. 6(a). The "reaction coordinate" (a) leads from reactants to products and (b) passes through a saddle point region. This picture would apply, for example, to atom transfer reactions23,46(&) or structural isomerizations. 23,46 We next isolate this reaction coordinate and reduce the prob lem to motion of some effective particle of mass IJ. in the one dimensional potential of Fig. 6(b). (This step is made for simplicity; as discussed in Sec. IVE, it need not be taken. ) In the SSP, the stable reactant and product state sur faces are located away from the barrier top. These and the intermediate or barrier region I are shown in Fig. 6(b). We make the SSP assumptions that the continual interactions with solvent molecules are sufficient (a) to maintain internal equilibrium in the reactants R and (b) to stabilize incipient products P so that passage through S p is irreversible on the time scale that determines the rate constant. By conservative estimate (see below), these conditions will hold for activation energies (jEo ~ 5. Then the rate constant is given by Eq. (4.4). . We ~ust now specify a model for the dynamics along the reaction coordinate in the presence of the solvent. We assume that a generalized Langevin equation (GLE) dv ft IJ.dt(t)=F(t)- 0 dr1;(r)v(t-r)+F*(t) (4.5) holds in the intermediate, or barrier, region. 47 We also assume that the static potential is an inverted parab ola in this region so that F=IJ.W~x. (4.6) The time dependent friction 1;(t) is the solvent-aver aged tcf of the fluctuating forces exerted on the effective particle along the reaction coordinate 1;(t) = f3F*F*(t) . (4.7) We assume that F*(t) is a Gaussian random force. 44 These fluctuating forces arise only from the dynamical interaction with the solvent; any average or mean solvent potential forces are accounted for by F. 48 In the sim plest case where the reaction coordinate x is a relative spatial or angle difference, F* is to be computed along this coordinate. In an atom-transfer reaction A + BC -AB +C, F* is along a combination of vibrational nor mal coordinates. 24,46 In any event, the tcf 1:(t) asso ciated with the barrier region I will generally be quite different from that appropriate in the reactant and prod uct regions. We will determine K for any given 1;(t) and then examine a few representative models of 1;(t). The non-Markovian GLE (4.5) reduces to the Mar kovian LE (4.1) if the random force correlations decay rapidly while the velocity hardly changes. Then { dr 1; (r)v(t -r) ~ 1;v(t) o (4.8) and the friction only enters through the "zero-frequency" component (4.9) of the Laplace transform ~(E) = f'O dt e-ef1;(t) of the time dependent friction 1;(t). This reduction (GLE -LE) is usually a very poor approximation. For single molecule motion in liquids, it completely misses important ve locity correlation features such as "caging. ,,49 It is, however, often argued that the LE is satisfactory for reactions. Supposedly, information is required only on long time scales of order K-1 a:: e6Eo; short time details such as those incorporated in 1;(t) are presumed to be irrelevant. The (incorrect) argument just given misses the essen tial point: Dynamics that determine K are often associ ated with high frequency motion in the barrier region where strong forces operate. ConSider, for example, passage over a typically sharp, high frequency (wb) barrier when the friction is not too high. The barrier crossing rate for a particle will depend upon the short time "nonadiabatic" friction experienced during the passage and will involve high frequency components hE) of the spectral resolution of 1; (t). The successful pas sage to stable product can be made long before the "adiabatic" approximation (4.8) holds; the GLE must be retained. Our argument above has been Simplified for emphasis. Its major point, i. e., that K should depend on the de tails of the dynamiCS reflected in 1;(t), will be verified below. J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions2724 R. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II B. Rate constant evaluation 1. Simplification The tcf in Eq. (4.4) for K can be considerably simpli fied by the following device to replace the BC's at SR = -a and S p = a 4: The true potential is artificially ex tended beyond -a and a [Fig. 6(b)]. These artificial forces play the same role as the BC' s in providing es sentially complete absorption past -a and a. Now, however, we can accurately use unrestricted dynamics on the full extended potential. 50.51 With the above replacement, we need not require solely incoming and outgoing fluxes at S R = -a and S p = a, respectively. The dynamics itself will take care of this; any recrossing trajectories will cancel their own contribution. Therefore we can write the rate con stant as 50.51 (4.10) Many alternate forms of Eq. (4.10) analogous to those of Sec. II could be used in the following; they all will yield the same result. 51 2. Time-dependent probability density We begin the evaluation of Eq. (4.10) by writing out the integrand in terms of the probability distribution p(x, v, tlxo, vo). This governs the "particle's" position x (with respect to the barrier top) and velocity v at time t, given the initial values Xo and vo' We obtain K = I~ dt f dx dv f dxodvo<P.q(vo)l/!eq(xo) o x vo6(xo + a)v6(x -a)p(x, v, t Ixo, vo) , which simplifies to Here <P.q(v) is the Maxwellian and l/!.q( -a) = e-BU(-al /QR = e-BEo exp[fl(/lw~/2)a2]/QR (4.11) (4.12) ( 4.13) is the equilibrium probability for x = -a. The activa tion energy is Eo. Since the "random" force in the GLE (4. 5) is assumed to be Gaussian, the probability distribution of relative positions and velocities at time t is also Gaussian44•52: p(x, v, tlxo, vo) =1T-lldetQ-tI1l2exp(_yT. Q-l. y). (4.14) The vector y represents the fluctuations in x and v from their (time-dependent) average values: ( 4.15) The matrix Q is proportional to the second moments of the distribution: (4.16) We willl'equire explicit expressions for the average dis-placement xxo.vo(t) and velocity vxo.vo(t). By formally solving the GLE (4.5) by Laplace transformation as an initial value problem and noting that the average F* xo.vo(t} vanishes, we find that xxo.vo(t) = Ctt (t)xo + C12(t)vo , 1:iXo•vo(t) = C21(t)XO + C22(t)VO • (4. 17a) (4. 17b) The time dependent coefficients Cilt) are found to be related to one of them [C21(t);: C(t)] according to C11 = C(t) , C12 = C2it) , (4. 18a) with initial values C11(O) = 1, CI2(0) = 0, C21(0) = 0, and C22(0) = 1. According to Eq. (4.17b) the coefficient C21(t}=C(t} determines the average velocity at time t if the particle starts at rest at position Xo with respect to the barrier top: C(t)=vxo.O(t)/xo _ (4. 18b) The formal solution to this initial value problem shows that the Laplace transform of C(t) is C(E)=wHE2_W~+E/l-l~(d]-I. (4.19) Here the frequency-dependent friction is the transform (4.20) of the time dependent friction. 3. Reactive frequency and " The time dependence of C(t) [Eq. (4.18)] has one spe cial feature crucial to our development: There is a posi tive reactive eigenvalue, or frequency, in its spectrum. This frequency arises from the unstable, or divergent, motion in the barrier region. Therefore, the reactive frequency exclusively governs the long time behavior of C(t). To see this, consider first the case of no friction. Then, according to Eq. (4.19), we have C(t) = (wb/2)(eWbl _ e-Wbf) • (4.21) In this trivial case, the actual barrier frequency Wb in the diverging exponential is the reactive frequency or eigenvalue. In the general case when friction is present, we can formally represent C(t) as C(t) = L Cn e-~nf + Cr e~rt • (4.22) n The frequencies An> 0 govern modes of stable relaxation which need not concern us here. The positive reactive frequency (eigenvalue) Ar reflects the unstable reactive motion in the barrier region. 53 Our concern with AT will now be justified. According to our tcf formula (4.12) and the distribution (4.14), the rate constant K is K = l/!.q( -a) f~ dt f dv f dVo<Poq(VO)VOV1T-t o x IdetQ(t) 1-112 exp[-yT(t) • Q-l(t) • y(t)] , (4.23) where Xo = -a and x = a are to be taken. In Appendix C, J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsR. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II 2725 this formidable expression is exactly evaluated to give the exceedingly simple result (4.24) Thus, the rate constant is just the TST rate constant KTST = [(21f!3j..1.)1/2QR]-1 e-BEo= (wR/21T) e-BEo (4.25) times the ratio of the reactive frequency Ar and the bar rier frequency wb• To determine the reactive frequency Ar, we consider Eq. ( 4. 19) in time language aC(t)/at=w~1T(t) +w~ It dT1T(T)C(t-T) , (4.26) ° where 1T(t) is the inverse transform of n-(E) = [E + E(E)/ j..I.]"l. We can find Ar if we look for a long time solution C(t)-CreArt. Substitution into Eq. (4.26) and passage to the long time limit gives [1T(t) -0 as t -00 ] (4.27) W2 Ar = A b • Ar + I; (Ar)/ j..I. (4.28) We will therefore generally need to know the behavior of I;(t) at all times through '(Ar); our long time limit was only a method of extracting Eq. (4.28). c. Discussion of the rate constant Equations (4.24) and (4.28) are our key results. They show that (a) K is determined by Ar and that (b) the reac tive frequency Ar is determined both by the barrier fre quency Wb and by the frequency component of the time dependent friction (4.29) at the reactive frequency Ar• The self-consistent equation (4.28) for Ar can be writ ten in an instructive form when we realize that n-(E) = [E +'(E)/j..I.]-l is the Laplace transform of the velocity tcf49 1T(t) = (v2)-1(vv(t) (4.30) (4.31) for the motion of a hypothetical particle of mass j..I. obey ing the GLE j..I. dv(t)/dt= -II dTI;(T)V(t -T) + F*(t) (4.32) ° in the absence of the barrier, but experiencing the actual friction. Thus, Eq. (4.28) can be written as Ar = w~ I'" dt e-Art (v2)"1(vv(t) o = w~ f'" dt e-Arl1T(t) . o (4.33) Many qualitative aspects of the rate constant K follow from Eq. (4.33). If the friction is very weak [f(Ar)/ J..L «Arl, the velocity correlations die very slowly compared (0) \ \ \ (b) OI-----=----+----.--~'--.3._;;::::;;oo-i FIG. 7. Illustration of the time behavior of exp(-Art) (-) and the velocity tcf tr(t) (---) in the cases of (a) low friction and (b) high friction. to e-Ar t. Trajectories across the barrier are negligibly perturbed by collisions with solvent molecules [cf. Fig. 7(a)] and Eq. (4.33) gives A ~ w2 I'" dt e-Ar t1T( t -0) -W2/A r b - - b r o (4.34) or Ar ~ wb• Thus, the reactive frequency is just the bar rier frequency and the TST result (4.25) is obtained. In the opposite limit of large friction [~(Ar)/ j..I. »Ar], barrier region trajectories are strongly and continually perturbed by solvent collisions. The reactive frequency will then be much less than wb; "shuttling" back and forth across the barrier top occurs before final stabilization in either the reactant or product stable states [cf. Fig. 7(b)]. Velocity correlation, or memory, dies rapidly compared to e-Art and Eq. (4.33) gives Ar ~ w~ 1'" dt 1T(t) = j..I.W~/~(O) . o (4.35) We thus obtain the diffusion limit result of Kramers4,32 (4.36) which is inversely proportional to the zero-frequency, or steady state, friction ,(0) = 1;. This is the only limit in which the friction constant provides a satisfactory de scription of the interaction with the solvent. In general, we require a knowledge of the full dynamics in I;(t). We consider this next. D. Influence of time-dependent friction We now examine our rate constant result [Eqs. (4.24) and (4.28)] for several interesting models of the time de pendent friction I;(t) in the barrier region. For purposes of comparison, we begin with the Kramers assumption. 1. Delta friction The reduction of the GLE to the LE with constant fric tion is equivalent to the delta function assumption (4.37) Thus, the full force of the integrated friction I; = fa dt1;(t) is effective instantaneously. This gives '(Ar) J. Chern. PhY5., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions2726 R. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. " ==1; and Ar=={[(1;/2j..LW b)2 + 1]1/2 -(1;/2j..Lwb)}Wb from Eq. (4.28). Since K == KTST(Ar/Wb)' this yields the Kramers result (4.2). 2. Gaussian friction A reasonable model for I;(t) for short and intermediate times is the Gaussian l;(t)=l;oe-r2t2. (4.38) This form can give an oscillatory velocity tcf (4.31), i. e., aspects of dense fluid "caging" of the velocity. 49 The required transform E(Ar) is E(Ar) =1; exp(Ar/r)2erfc(Ar/r) ; (4.39) erfc is the complementary error function and I; = I; 017i/ r is the zero frequency friction. With Eq. (4.39) for ~(Ar)' Eq. (4.28) can be solved numerically to determine Ar; this then gives the rate constant by Eq. (4.24). Our results are shown in Fig. 8 for representative values of the key parameter r/wb, the ratio of the fric tion decay rate to the barrier frequency. Except for broad barriers, reasonable estimates give r/wb values of 0(1) or less. The effective friction E(Ar) can then be dramatically less than the zero frequency value 1;. When, for example, Wb '" 1014 sec-t and r'" 10t3 sec-t, the effective friction is quite low; K is very close to its TST value. This suggests very strongly that many solution phase rate constants will be well approximated by the TST prediction; the constant friction approximation KKr can often drastically overestimate the solvent's influence. As is evident from Fig. 8, this overestimate becomes less severe as the barrier becomes broader. 3. Oscillatory friction Our final model friction is oscillatory in time: I;(t) =1:0 e-·tI2[cos(wtt) + (y/2wt)sin(wtt)] . (4.40) ----------- 0.5 KIK TST .... 2 FIG. 8. Ratio K/KTST of the rate constant to its transition state value, for the Gaussian friction model. The reduced zero fre quency friction is t/pwb• Legend: (-), zero frequency fric tion resultEq. (4.2); ( ... ), r/Wb= 1. 0; (---), r/wb,=O.l. [As 1: approaches 0, K should approach zero; the SSP in energy space space should be used instead. This behavior is limited to the extremely small t region on our scale (Sec. IV. E ). J 0.5 , ...... , ..... , ..... , ...... " ...... , , , , , , " "- "- , " " .... K IK TST (a) "- "- "- "-.... .... .... .... "-.... --- 6 10 O~----~----~----~-----L----~ 2 \ . 05 \ ". \ \ \ .... .... ~/pwb (b) .... --- -.:... --=..:.. a_a--= FIG. 9. Ratio K/KTST of the rate constant to its transition state value, for the oscillatory friction model, versus reduced zero frequency friction. (a) w/wb = 0.1; (--), w1h = 0.5; (---), wlh=o.1. (b) Wj/wb=lO.O; ("'), w1h=10; (---), w1h=50. In both (a) and (b), the solid line is the zero frequency friction result (4.2), which is independent of wl/wb and w/i' at fixed reduced friction. Here y measures the overall decay rate and wt gauges the oscillatory behavior. This model appears sensible when the solvent is solid54(a) or very strong forces (e. g. , electrostatic) restrict the solvent molecules. 54(b) The required transform is (4.41) and the zero frequenc y value is I: = 41; oy /( i + 4wi). The results for K shown in Fig. 9 display an interest ing variety. In Fig. 9(a), the barrier frequency Wb ex ceeds the oscillatory frequency wt of the friction. The oscillations in dt) are largely irrelevant; they do not develop before successful barrier passage occurs. In all cases, I: = E(O) is a· significant overestimate of the effective friction and K> KKr, as in the Gaussian case. The opposite limit where I;(t) oscillates on a time scale fast compared to a (broad) barrier frequency is shown in Fig. 9(b). t(Ar) can be much greater than t(O); K rapidly drops below both KTST and KKr. This behavior J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsR. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II 2727 Ely FIG. 10. Ratio f(E)/t of the frequency dependent friction to its zero frequency value t in the oscillatory friction case versus reduced frequency E/-Y. Values of the parameter wlh are in dicated. is most easily explained via Fig. 10, which gives the frequency dependence of ~(E). When w/wb is large, there is a significant range of high frequencies € "" WI where ~(€) far exceeds t; at low frequencies, integrated oscillations cancel and lead to a low friction value. If the oscillations are weakly damped, the enhanced fric tion effect is pronounced and a dramatic reduction in K versus t/IlWb results. Note that this depression of K occurs for slow damping of t(t); this is just the opposite of the usual Markovian assumption t(t) = to(t). 4. Some generalities The above results show that the details of the time de pendent friction playa significant role in the magnitude of the rate constant. This makes generalizations dan gerous, but a few ca~ be stated. First, Eqs..: (4.24) and (4.28) tell us that if t(A) < t, then K> KKr; if t(A) > t, then K< KKr. The oscillatory t(t) example, however, shows that a transition between these regimes can oc cur. Finally, in a number of cases the friction can likely be adequately represented as (4.42) i. e., a combination of fast (on the scale of w~l) and more slowly varying contributions. Then E(A) ",,~/o) + ts(A) determines the rate constant K, whose value will typi cally lie between the values predicted by assuming either t(A) ""E,(O) or t(A) ""E,(O) +UO). Clearly, the time de pendent friction 1;(t) in barrier regions deserves exten sive study. E. Extensions and limitations There is no real need to restrict the derivation of Eq. (4.24) for K to one dimension (lD). Suppose that degrees of freedom perpendicular to the reaction coordinate (cf. Sec. IV A) remain in equilibrium throughout the barrier passage. Then Eq. (4.4) can be extended to 3D by in cluding in KTST certain well known equilibrium factors related to saddle point and reactant well harmonic fre quencies. 55 In general, however, dynamic coupling drives the perpendicular degrees of freedom out of equilibrium during the passage. 11.46 Fortunately, this can be accounted for in Eq. (4.5) by inclusion of the cou pling effects in the friction t(t). This will be discussed elsewhere for atom -transfer reactions and structural isomerizations.46 We have only examined very simple models for the friction 1;(t). One can clearly do a better job by analysiS of suitable molecular expressions for t(t). 48.56 Further more, in addition to the "internal" couplings mentioned above, couplings to appropriate time scale relaxation phenomena in the solvent can influence t(t). Thus, di electric57 and ionic 58 relaxation can influence reactions involving ionic and/or dipolar species59 in solution. These are currently under study. An important concern is the validity of our Gaussian assumption (GA) [Eq. (4.14)]. This is difficult to es tablish precisely. Some favorable evidence is available for single molecule motion in dense fluids. First, the GA is exact for short times60(al and harmonic forces. 60(bl second, the GA gives a satisfactory account for a wide range of dynamical quantities. These include atomic position correlations, 60(01 correlations of molecular lin ear and angular velocity, 60(dl and possibly internal angu lar motion in protein models. 60(81 For motion in barrier regions, we believe that the GA remains reasonable. A likely exception is the motion of a very light effective particle surrounded by massive solvent molecules, a situation recently modeled by Skinner and Wolynes. 31 Finally, the GA only holds for massive particles in light solvents if the interactions are modeled as impulsive, e. g., hard sphere collisIons. 60(f 1 Real forces are con tinuous, however. Their legitimate approximation as impulsive forces remains to be established for reaction problems where high frequency phenomena are often in volved (as in vibrational energy transfer24 (a I). Our re sults suggest that this approximation should only be used with caution. The SSP internal equilibrium assumption discussed in Sec. IV A may break down under certain conditions. For very low barrier reactions in the diffusion limit, there can be Significant nonequilibrium in the stable states; this has been treated elsewhere. 4 For very low friction, the stable states should be de fined by surfaces of energy and not of position as in this section. The appropriate variable is energy (and pos sibly phase) and not position and momentum. 32 Indeed, this was our approach in Sec. III for the low density case. It may be possible to construct interpolation for mulas connecting the two regimes, 31.61 but this requires caution. First, the reaction mechanism may change be tween the limits. 62 Second, it is not clear that 1D reac tion models which are valid for high density remain valid at low density. DimenSionality and multiple degrees of freedom can playa dramatic role in, for example, low density isomerizations. They can significantly reduce nonequilibrium effects associated with repeated passage across a barrier top prior to energy stabilization. 63 Such aspects should be examined before reliable inter polation formulas can be constructed. Finally, when a reactive step is sufficiently fast, as in electron and proton transfer, the rate constant de- J. Chem. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions2728 R. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II pends upon static solvent fluctuations64 in contrast to the dynamical fluctuations treated here. A different ap proach for these cases is required. 64 v. DIFFUSION REGIME REACTIONS IN SOLUTION In Sec. IV, we focused on the "inertial" regime for reactions in solution. Here we turn to the limit where velocity relaxation is rapid (high friction) and spatial dif fusion plays a key or even dominant role. We discuss diffusive structural isomerization and diffusion-influ enced pseudobimolecular solution reactions. As the SSP approach to these cases has been extensively de scribed elsewhere, 4,5 we briefly focus only on those aspects deserving of special attention. A. Diffusion·"controlled" reactions Many chemical reactions and energy transfer pro cesses in solution depend upon the rate of diffusive ap proach of the reactants prior to the nondiffusive, short range reactive step. In the SSP, the stable chemical states correspond t05 (a) an outer spatial region, i. e., reactant separation beyond the range of short range chemical forces responsible for reaction, and (b) an inner region characterized by a potential well of the products. The intermediate zone will often, but need not, be characterized by a potential barrier. 5.62 The SSP expresses the overall rate constant as5•65 (5.1) Here K D is the rate constant for the relative reactant diffusive motion up to the stable reactant-intermediate region interface Sn (often a spatial separation'" a). It accounts for the spatial nonequilibrium in the stable re actant and has been extensively discussed elsewhere. 5 The barrier rate constant Keq = f~ dt(jj(Sn)j:(Sp, t))n o (5.2) governs the actual reactive step of formation of products starting from the stable reactant surface S n = a (wher e the reactants are caged5). The molecular expression (5.2) replaces empirical parameters used in the past for solution reactions. 5 It can be studied12 by analytic and numerical simulation techniques focused solely on the dynamics of reactants at small separation where reaction occurs; this avoids the necessity of simultaneously monitoring the slow and excursive reactant motion at larger separations. 62 This important feature renders molecular dynamics computer simulation of Keq tractable. B. Structural isomerization The high friction diffusive limit for barrier crossing reactions in solution is not easily attained (cf. Sec. IV). Isomerizations involving motion of large, massive groups over a broad barrier in viscous molecular sol vents appear to be the best candidates for this special limit. 4.66 Northrup and Hynes showed that the SSP result for the diffusive isomerization rate constant is the spatial flux tcf expression4 TABLE III. Ratio of the true rate constant to that assuming constant D. D(x-)a K/Kf{Dn) K/Kf [D(O)] D.(x) 1. 25 0.625 Djx) 0.42 0.625 D.(2x) 1. 09 0.543 Dj2x) 0.36 0.543 "Compare Eq. (5.5). The asymp totic values of D. are Do as x --00 and 3Do as x_oo; these are reversed for D_. These values are reached to within 50% at x = ± (2/,61-' w~)1/2 for D.(x-) and to within 25% at the same locations for D.(2x). = [Qn jSP dx e8U (X)D-1(x)r1 sn 'J (5.3) Here D(x) is the diffusion coefficient along the reaction coordinate x and U(x) is the potential energy. To sim plify Eq. (5.3), we assume that u(x) is symmetric about the barrier top x = 0 and that the reactant well and bar rier potentials are harmonic to give, for (3Eo~ 5, Kf = ({3/-LWV21T)1I2 e-8Eo x [f: dXD-l(x)exp{-{3(/-L/2)W~x2}Jl (5.4) The point we wish to stress has already arisen in Sees. III and IV: The rate constant depends on activation and deactivation on both sides of the barrier top. Assuming that Kf is determined only by events on the nominal re actant side (x < 0) of the barrier clearly leads to an error by a factor of 2 if D = constant. 67 The SSP result cor rectly accounts for the feature that the entire interme diate region determines Kf• The intermediate region symmetry can be broken in an interesting way when D = D(x); such x dependence can arise when the molecule changes shape in the reaction. 68 We can examine this via Eq. (5.4) for simple models which correspond, respectively, to increaSing and de creasing D as x increases: D(x) =D.(x) "'Do{2 +tanh[({3/-Lw~/2)J1/2x} , D(x) = D-<x) = Do{2 -tanh[({3/-Lw~/2)j1/2x} . (5.5a) (5.5b) Table III shows the results compared to those based on a constant D value taken either as (a) that of the stable re actant (= Dn) or (b) the value at the barrier top x = 0 ['" D(O) J. (Both are common approximations.) The im portance of the intermediate region, as opposed to just the barrier top x = 0, is again clearly revealed. The Table III entry KAD-<x)l!Kf(D n) is particularly instruc tive: Kf is dramatically overestimated by KADn). Ac cording to the SSP result [Eq. (5.4)], the forward rate constant Kf is Significantly reduced and mainly deter mined by the slow diffusion on the side of the barrier towards products. VI. SUMMARY The power and scope of the SSP approach to reaction rate constants has been demonstrated by application to a J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsR. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II 2729 broad range of chemical reaction models. Here we sum marize our results. For gas phase bimolecular reactions, the SSP tcf rate constant formulas were shown to be dynamical imple mentations of reaction trajectory calculations; the ad vantages of the SSP results were discussed (Sec. II A). Computationally efficient tcf formulas based on saddle point initial conditions were also presented (Sec. lIB) and the connection to transition state theory was made (Sec. IIC). For gas phase unimolecular reactions, we showed that the SSP approach can be readily exploited to find rate constants in the limits of large (Sec. III A) and small (Sec. IIIB) energy transfer per collision. In both cases, the SSP includes the important role of dynamics occurring away from the reaction threshold. In the en ergy diffusion limit (Sec. III B), this leads to a correc tion by a factor of 2 of a standard result. In both this limit and in the large energy transfer regime, the SSP results display clearly the often overlooked feature that a rate constant is typically determined by dynamics on both the reactant and product sides of a barrier or en ergy threshold (Sec. mC). For barrier crossing reactions in solution, the SSP approach was applied to a non-Markovian generalized Langevin description of the dynamics (Sec. IV A). The major result was obtained that the rate constant depends on the frequency component of the dynamical friction at the reactive frequency [Eqs. (4. 24) and (4. 28)]. This result frees solution phase reaction rate theory from the unrealistic confines of a macroscopic Brownian motion description. Various realistic friction models were shown to lead to dramatic departure from the standard constant friction predictions of Kramers (Sec. IV D). When spatial diffusion plays a key role in solution re actions, the SSP approach provides both a natural sepa ration of the slow diffusive step from the short range, nondiffusive intrinsic reaction step and a molecular rate constant expression for the laUer (Sec. V A). For dif fusive barrier crossings, the SSP formulas naturally reflect the importance of dynamics on both the reactant and product sides of the barrier top. This was illus trated for a model case where the forward rate constant is dominated by slow diffusion on the nominal product side of the barrier (Sec. VB). APPENDIX A Here we obtain Eq. (3.7). The P production rate con stant in dpp/dt= Kf Pp is Eq. (3.6). We must therefore evaluate (A1) where pi(tl m} is the probability at time t of being in level l of I, given that state m of I was initially occupied. No transitions into I are allowed in the dynamics of pT(tlm}. Here W~~=LI Rp~qwlm' where pr=e-S.I/QR' To determine f'O dt pi(tl m}, we integrate the appro priate kinetic equations */ z * "Z eq * * ap, at= -PI + ~ PI Pi -WIPI , p;(t= O} = 0lm to find JEI (A2) X [Olm+ZP~ £'" dtP;(tlm}] , (A3) where pi(tl m} is the sum of pi'(tl m} over all states 1 in I. Summation of Eq. (A3) over 1 states in I gives f'" dtpi(tlm}= [(z +wmp}(l-y}]-t , o (A4) where we have defined y = LZP~q(Z + Wjp)"t . (A5) JEI Insertion of Eq. (A4) into Eq. (A3) gives f'Odtpi(tlm), which when inserted into Eq. (A1) yields, after some algebra, Kf=(l-y)"t K; , (A6) where K; is the RRKM result [Eq. (3. 6)]. If KI is the equilibrium constant for the formal reaction R=I, Eq. (A5) shows that, for (3Eo» 1, y .;; LP~ = (1 + KI)"tKI« 1 JEI and Eq. (A6) reduces to Eq. (3.6). (A7) As noted in I, the population tcf can also be used to . determine the rate constant. For the present model, it is found that Eq. (3.6) holds when (3Eo» 1. 64(b) APPENDIX B Here we sketch the derivation of the rate constant re sults quoted in Sec. IIIB. Equation (3. 22) for Kf may be obtained by direct evaluation of Eq. (3.21) by the meth ods of Ref. 4. A Simpler approach4 is to integrate the action flux j = -D[(a/aJ} + (3w]p from the lower (J,) to the upper (Jd) stable state surfaces and impose the ab sorbing BC p(Jd) = O. Since j is constant with respect to J in the steady state, we find [f"d ]-t j(t} = dJ R(J} e8Edp(J, t} , "I (B1) where R-1(J} =D(J} e-8E(.T). The stable state assumption is p(Js, t} = P~q(JS}PR(t}. With Eq. (B1), this gives the rate law j(t}= KfPR(t}, with Kf given by Eq. (3.22). We next consider the derivation of Eq. (3.27). By Eq. (3.25), Kil equals [p~q(Js)]-lopO(Js,Js;E=O}. The Laplace transform of the deviation opo of the distribution P from its equilibrium value p.q is to be calculated from Eq. (3.14) with a delta function source at J=Js and re flecting wall BC's applied at J = Js and J = O. To sim plify the problem, we replace the lower reflecting BC by the assumption of equilibrium: p(J = O} = p~(J}. This will be an excellent approximation except when (3Ed:52. (If required, the exact solution can be found.) The prob lem of finding K R is now identical to that solved in Ref. J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions2730 R. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II 4 for spatial diffusion; Eq. (3.27) results. Note that Eq. (3.30) is also the rate constant deter mined from the population tcf. 3,4, 64(b) APPENDIX C The first step in the evaluation of Eq. (4.10) is the elimination of the time integral. Since vp(a, v, tl -a, vo,O) is the flux into the stable product state, the inte grated flux £~ dtf dv v p(a, v, t 1-a, vo, 0) = limP(x> a, t 1-a, vo) '" limP RX(t 1-a, vo) (C1) t-flO t-oo is the total probability of reaction, given the initial con ditions (xo = -a, vo). The reaction probability P RX(t I -a, vo) can be obtained directly by integration of the full dis tribution (4.14) to give PRX(t 1-a, vo) = f~ dx f~ dv p(a, v, t 1-a, vo) a -~ = f~ dX(7TQ11r1l2exp(-yt/Q11) ' (C2) a which works out to be ( 1 ) 1 ra(1 + C11) -voc121 PRX t -a, Vo ="2 erfc l IQU J . (C3) Here erfc is the complementary error function, Q11 is twice the mean square fluctuation in the displacement (4.16), and C11 and C12 are defined in Eq. (4.17). With Eq. (C3), the integration over Vo in Eq. (4.14) can be performed; we find that K = (.fi {3J.L Q Rtl e-BEo X!~~ {CI2CPexprJ.L~2W~ _a2cp2(1 +C11)2J} (C4) where cp-l=VQ11 + (2/{3J.L)cI2' Equation (C4) can be considerably simplified by noting that IQU, C12' and C11 all diverge in the same way as t-00 (cf. Sec. IV B 2). Since K is independent of a (cf. Sec. IVB 1), we conclude from Eq. (C4) that . ({3J.LW2)tl2 limcp=lim\T cil, (C5) and Eq. (C4) simplifies to K = Qjl e-BEO(w;/27T{3J.L)1/2lim(CI2/C11) (C6) To complete the derivation, we first note from Eq. (4.18) and L'Hospital's rule that lim(CI2/C11) = lim(C22/C21) = wb2lim[dlnCW/dt] . (C7) As discussed in Sec. IVB3, C(t)-Cre.\rtas t_oo, where Ar is the reactive eigenvalue. Therefore, Eqs. (C6) and (C7) give (C8) which is just Eq. (4.24) with Eq. (4.25). It is note worthy that a similar derivation shows that the average kinetic energy for reacting particles is just the thermal value. IS. H. Northrup and J. T. Hynes, J. Chern. Phys. 73, 2700 (1980), preceding paper. 2S. H. NorthrupandJ. T. Hynes, J. Chern. Phys. 68,3203 (1978). 3S• H. Northrup and J. T. Hynes, Chern. Phys. Lett. 54, 248 (1978). 4S. H. Northrup and J. T. Hynes, J. Chern. Phys. 69, 5246, 5261 (1978). Professor K. Schulten has pointed out to us an error in the first paper in the evaluation of the internal rate constants KR and Kp. This will be corrected elsewhere; it does not influence the results of the present paper. 5S. H. Northrup and J. T. Hynes, J. Chern. Phys. 71, 871, 884 (1979); Chern. Phys. Lett. 54, 244 (1978). GEquations 0.4) and (1.5) have certain conditions for their validity which are discussed at length in I. We will restrict their application in this paper to such conditions in Sees. (Ill. B), (III. C), and (V. B). 7See, for example, P. J. Kuntz, in Dynamics of Molecular Collisions, Part B, edited by W. H. Miller (Plenum, New York, 1976). 8This is closely related to Eq. (A5) of T. Yamamoto, J. Chern. Phys. 33, 281 (1960). The upper limit At in that equation should be interpreted as any time after the collision duration. S J. T. Hynes (unpublished). lOW. H. Miller, J. Chern. Phys. 61, 1823 (1974); 62, 1899 (1975); P. Pechukas, in Ref. 7; P. Pechukas and F. J. McLafferty, J. Chern. Phys. 58, 1622 (1973). tlR. A. Marcus, J. Chern. Phys. 45, 4493, 4500 (1966); 49, 2610 (1968); S. F. Fischer, G. L. Hofacker, and R. Seiler, ibid. 51, 3951 (1969); N. H. Hijazi and K. J. Laidler, ibid. 58, 349 (1973). 12R. Grote and J. T. Hynes (work in progress). 13J. C. Keck, Adv. Chern. Phys. 13, 85 (1967); Adv. At. Mol. Phys. 8, 39 (1972). For recent developments see B. C. Garrett and D. G. Truhlar, J. Phys. Chern. 83, 1052 (1979) and references therein. 14J. B. Anderson, J. Chern. Phys. 58, 4684 (1973); 62, 2446 (1975). A related development is the determination of reac tion coordinates starting from 5s; see, for example, C. Leforestier, J. Chern. Phys. 68, 4406 (1978). 15C• H. Bennett, inAlgorithms for Chemical Computation, edited by R. E. Christofferson (American Chemical Society, Wash ington, D.C., 1977); J. A. Montgomery, D. Chandler, and B. J. Berne, J. Chern. Phys. 70, 4056 (1979). 160ur Eq. (2.13) has already been applied to diffusion limit reactions in S. H. Northrup and J. A. McCammon, J. Chern. Phys. 72, 4569 (1980). 17S. Glasstone, K. J. Laidler, and H. Eyring, The Theory of Rate Processes (McGraw-Hill, New York, 1941); E. Wig ner, Trans. Faraday Soc. 34, 29 (1938); J. Horiuti, Bull. Chern. Soc. Jpn. 13, 210 (1938). 16This form can often be convenient for calculation as the fluxes are both across the saddle point surface S s' Unfortunately, state-to-state rate constants cannot be simply related to the total flux at S s' 19Consider a one dimensional model where x = 0 is the saddle point of energy U(O). The integral of the tcf for any t> 0 is < j(O) 8 (vt» R' where 8 is the step function. This is just Qile-BU(O) (v)+, which is the TST answer. (The + notation means a positive velocity average.) 20Quoted in R. Rosenstein, Ber. Bunsenges. Phys. Chern. 77, 493 (1973). 21R• Kapral, J. Chern. Phys. 56, 1842 (1972); F. Garisto and R. Kapral, ibid. 58, 3129 (1973); S. Hudson and J. Ross, J. Stat. Phys. 14, 469 (1976); D. Chandler, J. Chern. Phys. 68, 2959 (1978). 221n the gas phase bimolecular reaction case, nonequilibrium J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissionsR. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. " 2731 effects arise from finite translational and vibrational equili bration rates of the stable reactants. Trajectories into 1 are in fact sampled from nonequilibrium reactant distributions. These effects are typi~ally negligible in the bimolecular case; see, for example, B. Shizgal and M. Karplus, J. Chern. Phys, 52, 4262 (1970); 54, 4345 (1971); B. Shizgal, ibid. 57. 3915 (1972). Nonequilibrium effects can often be regarded as equivalent to surface recrossing effects. See Ref. 13 for discussion. 23H. S. Johnston, Gas Phase Reaction Rate Theory (Ronald, New York, 1966). 24(a) E. E. Nikitin, Theory of Elementary Atomic and Molecu lar Processes in Gases (Clarendon, Oxford, 1974); (b) W. Forst, Theory of Unimolecular Reactions (Academic, New York, 1973l. 25For recent discussions of models of reactive transition prob abilities" see: for example, N. C. Blais and D. G. Truhlar, J. Chern. Phys. 65, 5335 (1977); 70, 2962 (1979); W. Forst andA. P. Penner, ibid. 72, 1435 (1980). 26B. Widom, J. Chem. Phys. 55. 44 (1971); 61, 672 (1972); H. O. Pritchard, in Specialist Periodical Reports, Reaction Kinetics (Chemical Society, London, 1975), Vol. 1; R. K. Boyd, Chem. Rev. 77, 93 (1977). 27critical discussion of the strong collision limit has been given in J. Troe, J. Chern. Phys. 66,4745,4759 (1977); and J. Troe and H. Gg. Wagner, in Physical Chemistry of Fast Re actions (Plenum, New York, 1973). 28The steady state assumption for i inI gives, with Eq. (3.4), pj(t)/L;jPj(t)=Z(w ip+Z}-1 and Pj=pr in R. 29The "M" in RRKM refers23,24 toa specific model of the w/P' which we have not specified here. Work in this direction is under way for the cluster sticking probabilities determined numerically in J. W. Brady, J. D. Doll, and D. L. Thomp son, J. Chem. Phys. 71, 2467 (1979). 30,-. A. Bak and J. L. Lebowitz, Phys. Rev. 131, 1138 (1963); S. E. Nielsen and T. A. Bak, J. Chern. Phys. 41, 665 (1964). See also E. E. Nikitin, Discuss. Faraday Soc. 33, 288 (1962). 31J. L. Skinner and P. G. Wolynes, J. Chern. Phys. 72, 4913 (1980); 69, 2143 (1978). Important non-Gaussian effects may also arise when solvent motion involves barrier jumping. 32H. A. Kramers, Physica (The Hague) 7, 284 (1940). 33See, for example, Ref. 24(a) and J. Keck and G. Carrier, J. Chern. Phys. 43, 2284 (1965). The role of rotation is quite important in diatomic dissociation; it can lead to an order of magnitude increase in the rate constant in the diffusion limit [M. N. Safaryan, E. V. Stupochenko, and N. M. Pruchkina. Theor. Exp. Chern. 5, 115 (1969)J. We neglect rotation here only for simplicity; the SSP approach can also handle its in clusion. 34(a) S. A. Landon and J. C. Keck, J. Chern. Phys. 48, 374 (1968); M. R. Flannery, Ann. Phys. (N. Y.) 67, 376 (1971); (b) P. J. Pagni and J. C. Keck, J. Chern. Phys. 58, 1162 (1973). 35A. N. Kaufmann, Phys. Rev. Lett. 27, 376 (1971); G. M. Zaslavskii and B. V. Chirikov, Sov. Phys. Usp. 14, 549 (1972); J. O. Eaves and J. T. Hynes (work in progress). 3SSee, for example, D. W. Oxtoby and S. A. Rice, J. Chern. Phys. 65, 1676 (1976). 37Assumption (1) in Sec. III. B.l is more difficult to satisfy for the MO, since w-0 as E -Ed' One should really derive a generalized diffusion equation including phase memory to re place Eq. (3.14). See also J. C. Light, J. Chern. Phys. 36, 1016 (1962); P. Mazur, Physica (Utrecht) 25, 149 (1959). 38Equations (3.20a) and (3.20b) have also been obtained in Refs. 39 and 40 below when {3Ed» 1. Equation (3. 20b) does not appear in Ref. 32 but is implicit in the development. 39T• A. Bak and K. Andersen, Mater. Fys. Medd. Dansk. Vidensk. Selsk. 33, 109 (1975). 4OW. Brenig, H. Mililer, and R. Sedlmeier, Phys. Lett. A 54, 109 (1975). 41EQuation (3.30) i.s the inverse mean first passage time for dissociation, given an equilibrium distribution in the osciUa tor.l,3 It is also equivalent to a result of Brenig et al. 40 found by smallest eigenvalue' analysis. 42For the HO, Eq. (3.30) gives kit = [Ei(.BE,,) -In(.BE,,) -rl, where Ei is the exponential integral and r is Euler's constant. A similar formula has been found by P. B. Visscher, Phys. Rev. B 13, 3272 (1976). 43Another route to this result is mentioned in Appendix B. 44See, for example, articles by S. Chandrasekhar and M. Wang and G. E. Uhlenbeck, in Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954). 45Kramers' result [Eq. (4.2)1 has been repeatedly found to give the rate constant accurately in reaction computer simulations based on the LE equation (4. 1). Some recent examples are Ref. 4; R. M. Levy, M. Karplus, and J. A. McCammon, Chern. Phys. Lett. 65, 4 (1979); M. R. Pear and J. H. Wein er, J. Chern. Phys. 69, 785 (1978); E. Helfand, J. Chern. Phys. 69, 1010 (1978); J. S. McCaskill and R. G. Gilbert, Chern. Phys. 44, 389 (1979). An interesting case where Eq. (4.2) fails with LE simulated dynamics is M. P. Allen, "Brownian Dynamics Simulation of a Chemical Reaction in Solution" (preprint); the reaction studied is multidimensional and energy controlled and Eq. (4.2) does not apply. 46R. Grote and J. T. Hynes, (a) "Atom Transfer Reactions in Solution"; (b) "Structural Isomerization Reactions in Solu tion" (to be submitted.) 47GLE -type equations are well known in a variety of nonreactive transport and relaxation problems; see, for example, J. T. Hynes andJ. M. Deutch, in Physical Chemistry, An Advanced Treatise (Academic, New York, 1975), Vol. IIB; S. A. Adelman and J. D. Doll, Acc. Chern. Res. 10, 378 (1977). The GLE can only be regarded as a model in the absence of any rigorous deri vation; see Ref. 48. 48J. M. Deutch and I. Oppenheim, J. Chern. Phys. 54.3547 (1971); P. Ullersma and J. A. Tjon, Physica (Utrecht) 71, 294 (1974). 49See, for example, J. T. Hynes, Annu. Rev. Phys. Chern. 28, 301 (1977) and references therein. 500n the true potential, there would be ultimate return from re gions Rand P into region I. However, this is irrelevant for the evaluation of Eq. (4.4), which already contains the infor mation that K is determined by dynamics in region I. 51These statements have been verified by direct numerical examination in the constant friction case and by analytic methods in general. In the former case, the validity holds down to about {3E 0'" 3 when internal nonequilibrium becomes important. Caution is required at high friction; the absorbing BC's in Eq. (4.4) should be interpreted as irreversible pas sage beyond surfaces of small but finite width of the order of the distance traveled during a velocity correlation time ("'" /01 It>. 52S. Adelman, J. Chern. Phys. 64, 124 (1976); R. M. Mazo, in in Stochastic Proce~ses in Nonequilibrium Systems, edited by L. Garrido, P. Seglar, and P. J. Shepherd (Springer, New York, 1978). The GLE is not equivalent to the so-called "retarded" Fokker-Planck equation; cf. E. L. Chang, R. M. Mazo, and J. T. Hynes, Mol. Phys. 28, 997 (1974) and the references above. 53This should be carefully distinguished from the" smallest eigenvalue" often used in kinetics; the latter is the rate con stant itself. 54(a) J. M. Dickey and A. Paskin, Phys. Rev. 188, 1407 (1969); S. A. Adelman and B. J. Garrison, J. Chern. Phys. 65, 3751 (1976); A. Rahman, M. J. Mandell, and J. P. McTague, ibid. 64, 1564 (1976); M. shugard, J. C. Tully, and A. Nit zan, ibid. 66, 2534 (1977); (b) F. H. Stillinger and A. Rah man, J. Chern. Phys. 60, 1545 (1974); W. F. Edgell, inln frared and Raman Spectroscopy, edited by E. G. Brame and J. Grasselli (Dekker, New York, 1976), Vol. 1A. 55H. C. Brinkman, Physica (utrecht) 22, 149 (1956); R. J. Donnelly and P. H. Roberts, Proc. R. Soc. (London) Ser. A 312, 519 (1969). J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions2732 R. F. Grote and J. T. Hynes: Stable states picture of chemical reactions. II 56R. 1. Cukier, R. Kapral, J. R. Mehaffey, and K. J. Shin, J. Chern. Phys. 72, 1844 (1980); S. A. Adelman, ibid. 71, 4471 (1979). 57see, for example, P. G. Wolynes, Annu. Rev. Phys. Chern. 31, (1980) (in press) and references therein. 58J• M. Schurr, Chern. Phys. 45, 119 (1980); s. Harris, Mol. Phys. 26, 953 (1973); M. J. Stephen, J. Chern. Phys. 55, 3878 (1971). 59See, for example, E. M. Kosower, An Introduction to Physi cal Organic Chemistry (Wiley, New York, 1968). 60(a) P. A. Schofield, "Inelastic Scattering of Neutrons," Pro ceedings of the Symposium in Vienna 1960, Vol. I, p. 39; (b) J. T. Hynes, R. Kapral, and M. Weinberg, J. stat. Phys. 13, 427 (1975); (c) A. Rahman, Phys. Rev. Sect. A 136, 405 (1964); (d) G. D. Harp and B. J. Berne, J. Chern. Phys. 49, 1249 (1968) (we thank Professor P. G. Wolynes for recalling this reference to our attention); (a) J. A. McCammon, P. G. Wolynes, and M. Karplus, Biochemistry 18, 927 (1979); (0 M. Pagitsas, J. T. Hynes, and R. Kapral, J. Chern. Phys. 71, 4492 (1979). 61J. Troe, in Physical Chemistry, An Advanced Treatise (Aca demic, New York, 1975), Vol. VIB; also in High Pressure Chemistry, edited by H. KeIrn (Reidel, Dordrecht, 1978); P. B. Visscher, Phys. Rev. B 14, 347 (1976). 62J. T. Hynes, R. Kapral, and G. M. Torrie, J. Chern. Phys. 72, 177 (1980). 63X. Chapuisat and Y. Jean, J. Am. Chern. Soc. 97, 6325 (1975); A. warshiel and M. Karplus, Chern. Phys. Lett. 32, 11 (1975). G4(a) See, for example, J. Ulstrup, Charge Transfer Processes in Condensed Media (Springer, Berlin, 1979) and references therein; (b) J. T. Hynes and S. H. Northrup (in preparation). 65 A simple derivation along the lines of Sec. IV of I also gives Eq. (5.1). Consider the reaction scheme A+B~[A+Bla-C, with rate constants KD, K_D' and Ki, involving the "caged" re actant pair at separation cr and at "bulk" separations. Steady state analysis on [A + B la gives Kf as Kf= (Kj + K_D)-l Kj KD• The ratio KD/K_D is the "equilibrium constant" for separation cr and for any typical bulk separation. This is the equilibrium pair distribution g(cr) , so that KD=g(cr)K_ D• Together with the identity Keq =g(cr)Ki, 5 this gives Eq. (5.1). 66This limit has been adopted for several simulations of poly mer dynamics; see, for example, M. Fixman, J. Chern. Phys. 69, 1527(1978); G. T. EvansandD. C. Knauss, ibid. 72, 1504 (1980); M. J. Pear and J. H. Weiner, ibid. 71, 212 (1979). E. Helfand, Z. R. Wasserman, and T. A. Weber, ibid. 70, 2016 (1979). 67This feature appears not to be generally appreciated; see, for example, 1. Procaccia and J. Ross, J. Chern. Phys. 67, 5565 (1977). 68D. C. Knauss and G. T. Evans, J. Chern. Phys. 72, 1499 (1980). J. Chern. Phys., Vol. 73, No.6, 15 September 1980 Downloaded 24 Aug 2013 to 137.99.26.43. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jcp.aip.org/about/rights_and_permissions

5.0022315.pdf

Appl. Phys. Lett. 117, 243902 (2020); https://doi.org/10.1063/5.0022315 117, 243902 © 2020 Author(s).Flywheel-based traveling-wave thermoacoustic engine Cite as: Appl. Phys. Lett. 117, 243902 (2020); https://doi.org/10.1063/5.0022315 Submitted: 20 July 2020 . Accepted: 25 November 2020 . Published Online: 14 December 2020 T. Biwa , T. Watanabe , and G. Penelet COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Observation of quantum interference conductance fluctuations in metal rings with strong spin–orbit coupling Applied Physics Letters 117, 242402 (2020); https://doi.org/10.1063/5.0031708 A perspective on semiconductor-based superconducting qubits Applied Physics Letters 117, 240501 (2020); https://doi.org/10.1063/5.0024124 Charge-spin conversion signal in WTe 2 van der Waals hybrid devices with a geometrical design Applied Physics Letters 117, 242401 (2020); https://doi.org/10.1063/5.0029071Flywheel-based traveling-wave thermoacoustic engine Cite as: Appl. Phys. Lett. 117, 243902 (2020); doi: 10.1063/5.0022315 Submitted: 20 July 2020 .Accepted: 25 November 2020 . Published Online: 14 December 2020 T.Biwa,1,a) T.Watanabe,1and G. Penelet2 AFFILIATIONS 1Department of Mechanical Systems Engineering, Tohoku University, Sendai 980-8579, Japan 2Laboratoire d’Acoustique de l’Universit /C19e du Mans, Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France a)Author to whom correspondence should be addressed: tbiwa@tohoku.ac.jp ABSTRACT Motivated by the success of a traveling-wave thermoacoustic engine with a linear load, we built a thermoacoustic engine with a flywheel and a reciprocal piston. The engine has a looped tube with a regenerator and heat exchangers at the ends. When the temperature difference between the ends of the regenerator is increased above a threshold, the flywheel undergoes steady rotation. From simultaneous measurementsof the pressure and velocity of the working gas, we demonstrated experimentally that this flywheel-based thermoacoustic engine is a kine-matic Stirling engine. Published under license by AIP Publishing. https://doi.org/10.1063/5.0022315 Ceperley’s proposal for a traveling-wave acoustic engine 1trig- gered the reinvention of Stirling engines.2His idea was based on the in-phase oscillations of pressure and velocity inherent to a traveling acoustic wave. Because this phasing, which is called traveling-wave phasing, is essentially the same as that of the working gas in the regen- erator in a Stirling engine, when a traveling wave propagates through the regenerator, it executes a thermodynamic cycle that is equivalent to the Stirling thermodynamic cycle.3–5Here, the regenerator is com- posed of a porous material that has tiny flow channels, which enhance the thermal contact between the gas and the channel walls. The regen-erator is an indispensable component in a Stirling engine because the Stirling cycle requires that the thermodynamic processes are isother- mal. Note that a traveling-wave engine differs from a standing-wave acoustic engine . 6In the latter, the irreversible thermodynamic cycle is due to the gas oscillating in a stack region through imperfect thermal contact with the flow channel walls.7 The first traveling-wave engine utilized thermoacoustic spontane- ous oscillations of a gas column confined in a looped tube.3When there is a sufficiently large temperature difference across the regenera- tor of the engine, the gas column starts to oscillate with the natural fre- quency of the most unstable longitudinal acoustic mode.8,9Increasing the temperature difference increases the amplitude of the oscillations, resulting in acoustic shock waves and chaotic oscillations when the engine is unloaded,10–12and to an increase in the output power of the engine when loaded.5,13A traveling-wave acoustic-engine electric gen- erator was built by connecting a linear alternator with the looped tubevia a branch tube.14It can be thought of as a free-piston Stirling engine generator whose displacer piston has been replaced by acoustic waves. Various types of electric generators have been developed by connect- ing or installing a linear alternator, such as a voice-coil alternator, in the looped tube,15–18or by using a bi-directional turbine in place of the linear alternator.19 In this work, we reinvented a kinematic Stirling engine . The kine- matic Stirling engine has its roots in Stirling’s original patent issued in 1816. In contrast to a free-piston Stirling engine, the power piston and the displacer piston are linked with a flywheel in a kinematic Stirling engine.2Therefore, the amplitudes of the oscillations of the pistons are mechanically fixed, but the operating frequency speeds up with the input heat power. We built a kinematic Stirling engine without a displacer piston and with only a power piston connected to a flywheel to convert the reciprocal motion of the piston to rotational motion. The obvious advantage of such a flywheel-based traveling-wave engine over a kinematic Stirling engine is the simplicity of the mechanicalstructure due to the absence of a solid displacer facing the hot side of the regenerator. The rotational motion allows us to use a conventional alternator, which is more widely available than a linear alternator. 20 Therefore, if the flywheel-based traveling-wave electric generator isbuilt using a rotational alternator, it will find practical applications that are different from those of thermoacoustic electric generators using linear alternators. An essential aspect of a traveling-wave engine is the reversible thermodynamic cycle, which realizes mutual energy conversion Appl. Phys. Lett. 117, 243902 (2020); doi: 10.1063/5.0022315 117, 243902-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplbetween two energy flows,21,22heat flow Q andwork flow I ,s u s t a i n e d by the oscillating gas flow. Here, Q¼AqmTmhSuiandI¼Ahpui, where qmandTmare the temporal mean density and temperature of the gas and S,p,a n d uare the entropy oscillation, pressure oscillation, and velocity oscillation of the gas. Ais the cross-sectional area of the gas flow channel. The angular brackets and over bar represent temporaland cross-sectional averages, respectively. In a traveling-wave engine, thework flow I, which is called the acoustic power in acoustics, is amplified as the wave moves up the temperature gradient from the cold to the hotside of the regenerator. 3,10,23,24In a standing-wave engine, however, Iis emitted from the cold and hot sides of the stack.6In this case, Igoes to zero at a certain position inside the stack region where the flow directionchanges. At this particular position, the phasing between the pressureand velocity is the same as standing-wave phasing ( 6p=2). Thus, an engine is classified as a traveling-wave engine if Ienters the cold side of the regenerator, whereas it is a standing-wave engine ifIgoes out from the cold side. Note that a pulse-tube engine, 26which has a piston connected to a flywheel and a tube with stacked meshscreens, was identified as a standing-wave engine after direct observa-tions of I. This result indicates that to identify whether an engine is a traveling-wave engine or a standing-wave one, the flow pattern of I needs to be determined, as the classification does not depend onhaving a regenerator or a stack. In this study, we employed a loop configuration to make the feed- back of Ipossible. Thus, Iflows in one way in the regenerator. The looped tube is connected to a conventional reciprocal piston mechanismwith a flywheel and a connecting rod. We demonstrated the steadyoperation of the engine and gathered experimental evidence to show that it is a traveling-wave engine by observing the flow direction of I. Figure 1 shows a schematic illustration of our prototype thermo- acoustic looped tube engine loaded with a piston-flywheel assembly.The engine was filled with the working gas, which was air at ambient temperature and pressure. The thermoacoustic looped tube engine has a 40-mm-diameter main tube and a 15-mm-diameter feedback tube.Parts of the tubing were replaced with transparent acrylic tubes of 36and 10 mm in diameter to allow optical measurements of the gasvelocity. The average length of the loop was 646 mm. The regenerator and heat exchangers were contained in the main tube. The regenerator was made of a 35-mm-long stack of screenmeshes with a mesh number of 30 and a wire diameter of 0.18 mm.The cold and hot heat exchangers had parallel fins evenly spaced1.0 mm apart. The fin thickness and length were 0.5 and 15 mm. Thecold heat exchanger was cooled by running water whose temperaturewas kept at 19 61 /C14C. The hot heat exchanger was heated by feeding a dc current to a heating wire wound around it. A secondary ambientheat exchanger was installed at the bottom of the loop to maintain thewall temperature at the temperature of the cooling water. The second-ary heat exchanger did not have internal fins. A sheathed thermocou-ple was inserted into the loop to monitor the temperature T Hat the hot side of the regenerator. We used a glass cylinder-piston assembly (Tsubasa Industry), an aluminum connecting rod, and a brass flywheel. The diameter andlength of the piston were 35 and 158 mm. The height of the top deadcenter of the piston was aligned with the top of the cylinder. The con-necting rod was sufficiently long compared with the nominal pistonstroke (peak-to-peak displacement) of 2 rso that motion of the piston was almost sinusoidal. The piston stroke was set to 2 r¼10 mm. Thetotal mass of the piston and the connecting rod was M¼0.174 kg. The flywheel was a cylindrical disk with a thickness of 7 mm and a radiusof 35 mm. The mass moment of inertia of the flywheel was J¼1:44 /C210 /C04kg m2, including the contribution of the rotating shaft. First, the operation of the engine was investigated by monitoring its internal pressure p(t). A small pressure transducer (JTEKT DD104) was connected to a short duct mounted on the tee at the junction of t h em a i nt u b ea n dt h ef e e d b a c kt u b e ,a ss h o w ni n Fig. 1 .T h et r a n s - ducer signal was measured with a spectrum analyzer (Onosokki DS-2000) at a sampling frequency of 2048 Hz. After establishing a thermally steady state with a constant T Hby feeding a constant electrical current to the heater wire of the hot heat exchanger, we started up the engine by manually rotating the flywheel. Then, we started recording the pressure oscillations p(t) in the freely running state of the engine. We confirmed that the dynamic behaviorof the engine was independent of the direction of the initial rotation. The observed pressure oscillations p(t) were quasi-sinusoidal waves with almost constant amplitude and with frequency gradually changing with time. We detected the time t iof the ith maximum (or minimum) of p(t)a ss h o w ni nt h ei n s e to f Fig. 2 . From the time inter- valDt¼tiþ1/C0tibetween neighboring maxima (or minima), we determined the time-averaged frequency /C22fat/C22t¼ðtiþtiþ1Þ=2a s defined by FIG. 1. Experimental setup. The main tube of the loop has heat exchangers (cold and hot HEXs) and a regenerator. Oscillations of the axial velocity and pressure ofthe working gas were measured with a laser Doppler velocimeter (LDV) and pres-sure transducers. For the LDV measurements, parts of the loop were made of transparent acrylic tubing. The cylinder and axis of the flywheel were fixed to a sup- port frame. The direction of the gravitational acceleration gis shown by the arrow.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 243902 (2020); doi: 10.1063/5.0022315 117, 243902-2 Published under license by AIP Publishing/C22f¼1 Dt: (1) Figure 2(a) shows the temporal change of /C22fwhen THwas set to 19/C14C( u n h e a t e d ) ,1 4 0/C14C, or 234/C14C. When TH¼19/C14C, the frequency /C22frapidly decreased from the initial value, but at the minimum of /C22f, therotation switched to libration in which the flywheel acted as a pen- dulum under gravity.27The libration was also observed at other tem- peratures TH. The frequency /C22fapproached a constant value of 2.4 Hz before the flywheel stopped at its equilibrium position. The constant value of /C22fduring libration is related to the frequency of the small- amplitude oscillations of a pendulum under gravity. Consider a simple pendulum expressed by J€hþMgrsinh¼0; (2) where his the angle of the pendulum, Jis the mass moment of inertia of the flywheel, Mis the mass of the piston, gis the gravitational accel- eration, and ris the crank radius. The natural frequency of small- amplitude oscillations is given by f¼1 2pffiffiffiffiffiffiffiffi ffi Mgr Jr ; (3) which was evaluated as f¼1.2 Hz for the present experimental setup. Since the maximum pressure occurs twice in one cycle of the libration, /C22f=f¼2 indicates good agreement. When THwas increased to 140/C14C,/C22fdecreased at a slower rate. A further increase to TH¼234/C14C resulted in the rotations lasting more than 5 min. The mean frequency /C22fobserved with even higher THis shown in Fig. 2(b) . The engine reached a steady rotation state with/C22f¼6:1 Hz as it slowed down from its initial state. When the initial frequency was lower than the final value, the engine reached either thesteady rotation state or the rest state via libration. The threshold initial rotation frequency was in the range from 4.2 to 4.4 Hz, when T H¼245/C14C. The steady rotation frequency increased to 9.1 Hz for TH¼288/C14C, whereas the threshold frequency decreased to about 3.2 Hz. To determine the work flow I, we measured the axial velocity ucðtÞin the central axis of the tube using a laser Doppler velocimeter (LDV) in the transparent acrylic tube region. We used cigarette smoke as seeding particles. We maintained a steady rotation state with f¼9:1H z b y k e e p i n g TH¼288/C14C while measuring I.T h ev e l o c i t y and the pressure p(t) were measured simultaneously at a given axial position. For these measurements, the pressure was recorded via a low- pass filter (Model 3611, NF Corporation) whose cutoff frequency wasset to 1 kHz. The measured velocity and pressure were fitted to sinusoidal waves: u cðtÞ¼Ucsinð2pftþ/ucÞ; (4) pðtÞ¼Psinð2pftþ/pÞ: (5) These were used to determine the amplitudes UcandPofucandpand their initial phases /j(j¼uc;p). Because the tube diameter is much smaller than the characteristic length c=x(cis the sound speed of the working gas), the pressure was independent of the radial coordinate. Therefore, the work flow can be written as I¼A 2PVcosð/pvÞ; (6) where Vis the amplitude of the cross-sectional mean velocity v(t)a n d /pv¼/p/C0/vi st h ep h a s ed i f f e r e n c eb e t w e e n p(t)a n d v(t).Ais the cross-sectional area of the tube. We estimated the amplitude and phase ofv(t) from those of the central velocity ucðtÞusing the theoretical sol- utions for a laminar oscillating flow of a viscous fluid.6,22,28,29Namely, the conversion from ucðtÞtov(t) was made with the following equa- tions: V¼U=1:04 and /v¼/ucþ2:35/C14i nt h em a i nt u b ea n d V¼U=1:15 and /v¼/ucþ8:99/C14in the feedback tube. To deter- mine the phase /pof the pressure accurately, we considered the phase delay (2 :73/C14) introduced by the low-pass filter used to measure the pressure. Figure 3 shows the measured work flow Ias a function of the axial coordinate Xnnormalized by the average length of the loop. The flow direction of Iis represented by its sign. A positive value of I means that it flows clockwise in the loop in Fig. 1 , which is in the positive direction of Xn. In contrast, a negative value corresponds to a counterclockwise flow in the loop. The work flow measured in the feedback tube was consistently positive, meaning that Iflows into the regenerator from the cold side. In other words, Fig. 3 provides the experimental evidence that this flywheel-based thermoacoustic engine has traveling waves. To capture an overall picture for I, we considered the change of I in the regenerator. In a traveling-wave engine, the regenerator acts likea power amplifier. If the viscous loss is negligibly small and good thermal contact is maintained between the gas and the solids in the regenerator, the power gain Gis given by G¼T H=TC.N a m e l y , Gis equal to the ratio of the end temperatures of the regenerator.1,23,25TheFIG. 2. Mean frequency /C22fas a function of time /C22twhen (a) TH¼19/C14C, 140/C14C, or 234/C14C and when (b) TH¼245/C14C or 288/C14C. The inset shows p(t) where /C22f¼1=Dt. In (b), the three blue curves are obtained for the same THof 245/C14C, but for three different values of the initial rotating frequency, while the same information is plotted with black or gray curves for THof 288/C14C.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 243902 (2020); doi: 10.1063/5.0022315 117, 243902-3 Published under license by AIP Publishingpresent regenerator was made of a pile of screen meshes whose effec- tive flow channel radius was R¼0.2 mm.30The thermal relaxation timesfor the gas to achieve equilibrium over the cross section of the flow channel is expressed by s¼R2=ð2aÞ,w h e r e Ris the radius of the flow channel and ais the thermal diffusivity of the working gas.22If we evaluate afor air at the mean temperature of the regenerator [ðTHþTCÞ=2], we obtain a¼4:3/C210/C05mm2/s and, hence, s¼0:4 ms. The angular frequency x¼2pf.T h e r e f o r e ,w eh a v e xs¼0:03 for our experimental conditions, which ensures good ther- mal contact in the regenerator region.22Based on this estimate, we assumed a power gain of G¼TH=TC¼1:9t od r a w Ioutside the measurement section in Fig. 3 .A ss h o w nb yt h el i n e s , Iemitted from the hot side of the regenerator separates at the junction of the loop and the branch to the piston. The difference DIat the junction is the acoustic power entering the piston. A rough estimate yields DI¼11 mW, which is consumed by the energy dissipated by the pis- ton-crank-flywheel assembly. If the temperature ratio is further increased, then additional output power is produced in the regenera- tor, which increases the rotation frequency of the flywheel. Figure 4 shows the frequency /C22fas a function of the hot-end tem- perature TH. On increasing THto 413/C14C,/C22fincreased to 12.5 Hz. This increase in /C22fis attributable to an increase in the gain of Iin the regen- erator. In the temperature range TH/C21380/C14C ,t h ep e r i o d i cl i b r a t i o n state was also stable. /C22ffor the stable libration state is also shown in Fig. 4 . The frequency of this state decreased with an increase in TH, whereas the oscillation amplitude increased. The coexistence of a sta- ble rotation state and a stable libration state, in addition to a stable rest state, is clearly different from a thermoacoustic engine with a linear alternator, in which the limit-cycle oscillation state appears via a Hopf bifurcation. In our future study, we aim to account fully for the nonlinear dynamics of this flywheel-based traveling-wave thermo- acoustic engine. In summary, we have designed, built, and tested a flywheel-based traveling-wave thermoacoustic engine. By making simultaneous mea- surements of the pressure and velocity, we demonstrated the feedback ofI. Therefore, the present engine is a kinematic Stirling engine in which the displacer piston has been replaced by oscillating gas columns. Further research study is needed to assess the potential ofthe engine from an engineering perspective, as well as to understand the nonlinear dynamics of the engine associated with the mechanics ofthe piston-flywheel assembly. T.B. would like to thank the Laboratoire d’Acoustique de l’Universit /C19e du Mans for providing an opportunity to visit Le Mans Universit /C19e via the Acoustics Hub program. Also, the authors would like to thank Professor Hamaguchi of Meisei University for his help in constructing the piston-flywheel assembly. This study was partly supported financially by the Thermal & Electric Energy Technology Foundation. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1P. 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1.4959816.pdf

Dynamic magnetic hysteresis and nonlinear susceptibility of antiferromagnetic nanoparticles Yuri P. Kalmykov , Bachir Ouari , and Serguey V. Titov Citation: J. Appl. Phys. 120, 053901 (2016); doi: 10.1063/1.4959816 View online: http://dx.doi.org/10.1063/1.4959816 View Table of Contents: http://aip.scitation.org/toc/jap/120/5 Published by the American Institute of Physics Dynamic magnetic hysteresis and nonlinear susceptibility of antiferromagnetic nanoparticles Yuri P. Kalmykov,1Bachir Ouari,2and Serguey V. Titov3 1Laboratoire de Math /C19ematiques et Physique (LAMPS, EA 4217), Universit /C19e de Perpignan Via Domitia, F-66860, Perpignan, France 2Physics Department, University of Tlemcen, BP 119 Chetouane, Tlemcen, Algeria 3Kotelnikov’s Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino, Moscow Region, 141190, Russian Federation (Received 11 February 2016; accepted 14 July 2016; published online 1 August 2016) The nonlinear ac stationary response of antiferromagnetic nanoparticles subjected to both external ac and dc fields of arbitrary strength and orientation is investigated using Brown’s continuous diffusion model. The nonlinear complex susceptibility and dynamic magnetic hysteresis (DMH) loops of an individual antiferromagnetic nanoparticle are evaluated and compared with the linear regime forextensive ranges of the anisotropy, the ac and dc magnetic fields, damping, and the specific antiferromagnetic parameter. It is shown that the shape and area of the DMH loops of antiferromagnetic particles are substantially altered by applying a dc field that permits tuning of thespecific magnetic power loss in the nanoparticles. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4959816 ] I. INTRODUCTION The physical properties of ferromagnets and antiferro- magnets are drastically modified when their dimensions are reduced to the nanometric scale. This fact has prompted both the fabrication of and various studies of the behavior of fer- romagnetic and antiferromagnetic nanoparticles with the aimof seeking new properties and applications, especially in data storage 1,2and medicine.3–6Ferromagnetic nanoparticles are characterized by instability of the magnetization due to ther- mal agitation causing spontaneous change of particle orienta- tion from one metastable state to another resulting in the phenomenon of superparamagnetism.7,8Furthermore, due to the large magnetic dipole moment of ferromagnetic nanopar- ticles ( /C24104–106lB), the Zeeman energy is large even in rel- atively weak external magnetic fields causing nonlinear effects in the dynamic susceptibility, stochastic resonance, dynamic magnetic hysteresis (DMH), etc. In the case of anti- ferromagnetic nanoparticles, their magnetization dynamics may differ in many respects from those of ferromagnetic ones because of the intrinsic properties of antiferromagnetic materials. Moreover, the magnetic behavior of antiferromag- netic nanoparticles can be quite different from that observed in the bulk, e.g., enhanced magnetic moment and coercivity, exchange bias, increase in magnetic moment with tempera- ture, decrease in the susceptibility with temperature below the ordering (N /C19eel) temperature TN, and its enhancement compared to that of the bulk.9Moreover, due to thermal agi- tation, antiferromagnetic nanoparticles should become super- paramagnetic at finite temperatures just as ferromagnetic nanoparticles (the so-called superantiferromagnetism10). The initial theory of the thermal fluctuations in magnetic nanoparticles due to N /C19eel7was further developed by Brown11,12 and is consequently known as the N /C19eel-Brown model. His treatment utilizes the classical theory of the Brownian motion in conjunction with the Landau- Lifshitz-Gilbert equation13,14augmented by random fields regarded as the magnetic Langevin equation governing the thermoactivated transitionsof the magnetic moment in a nanoparticle in the coherent rotation approximation. This model may also be adapted to antiferromagnetic nanoparticles. An ideal antiferromagnetcan be divided into two sublattices, with equal and opposite magnetic moments. If the numbers N 1andN2of ions in these sublattices are equal, we say, the sample is “compensated.”In fine antiferromagnetic particles, total magnetic compensa-tion of the sublattices is impossible for a number of reasons (unequal numbers of spins in crystal planes, spin frustration near the surface, lattice defects, etc.) so that N 16¼N2result- ing in the effective spontaneous magnetization in such par- ticles. If we will assume the ionic magnetic moments m remain the same and N1>N2, so the total moments of the sublattices m1¼N1muandm2¼/C0N2muare not equal. Here, uis the antiferromagnetic vector, which specifies the decompensation magnetic moment l¼m1þm2¼ul, where l¼mðN1/C0N2Þ. In an antiferromagnetic nanoparti- cle, an external dc magnetic field H0directed along uwill not tend to rotate the moments m1andm2, but a field normal touwill. Furthermore, an antiferromagnetic nanoparticle possesses a considerable transverse magnetic susceptibility vAcharacterizing the induced magnetic moment of the parti- cle which can be written as vvA½H0/C0uðu/C1H0Þ/C138=2. Thus, the magnetic moments of two sublattices of the particle withallowance for decompensation can be presented as 9 m1;2¼6vMSþl 2/C0vvA 2H0/C1uðÞ/C20/C21 u; (1) where MS¼mðN1þN2Þ=ð2vÞis the bulk sublattice magne- tization and vis the particle volume. Due to a specific super- antiferromagnetic effect, vAexceeds that of a bulk crystal by a factor of 2 or 3 and attains a typical value of order 10/C04 0021-8979/2016/120(5)/053901/12/$30.00 Published by AIP Publishing. 120, 053901-1JOURNAL OF APPLIED PHYSICS 120, 053901 (2016) (Refs. 9and 10), while the magnitude of the magnetic moment lcan be estimated as l/C24lBðN1/C0N2ÞðN1þN2Þ1=2,10 where lBis the Bohr magneton, so that for a particle with a diameter /C2410nm and N/C24105–106,lmay vary from a few lBto few hundred lB15–20(that is of the same order of mag- nitude as the magnetization of weak ferromagnets). At tem- peratures below TN, as long as the external magnetic fields are much weaker than the exchange field, one may assume9 that the decompensation magnetic moment is constant in magnitude so that the unit vector ucan only rotate but cannot change its length. Thus, just as in nanoscale ferromagnets,the magnetization dynamics of antiferromagnetic nanopar- ticles in the presence of thermal agitation can be described using Brown’s diffusion model of a magnetic moment (clas- sical spin) via a Fokker-Planck equation for the distribution function Wðu;tÞof magnetic moment orientations on a unit sphere, viz., 11,12 @W @t¼1 2sNr2Wþbr/C1WrV ðÞ þb au/C1rV/C2rW ½/C138ðÞ/C26/C27 ; (2) where Vis the free-energy of the particle comprising the magnetic anisotropy and Zeeman energy densities, sN ¼s0ðaþa/C01Þis the characteristic free diffusion time, s0 ¼bl0l=ð2cÞ,l0¼4p/C110/C07JA/C02m/C01is the permeability of free space in SI units, b¼1=ðkTÞ,kis the Boltzmann’s con- stant, Tis the absolute temperature, cis gyromagnetic ratio, andais the damping parameter. In spherical polar coordi- nates basis ðer;e#;euÞ,21r¼ @=@u¼e#@#þeucsc#@uis the gradient operator on the unit sphere, where #anduare the polar and azimuthal angles, respectively, and u¼er ¼ðsin#cosu;sin#sinu;cos#Þ. Without the third (gyro- magnetic) term in the right hand side of Eq. (2), the Fokker–Planck equation has the same mathematical form as the diffusion equation for the noninertial rotational Brownian motion of a particle in a mean field potential.9,22 In the simplest case of a uniaxial antiferromagnetic particle, if the easy axis of the nanoparticle coincides with the Z-axis of the laboratory coordinate system, its free energy Vð#;u;tÞ in superimposed magnetic dc and ac fields HðtÞ¼H0 þH1cosxtis given by9,10 V#;u;t ðÞ ¼ vKsin2#/C0l0lHtðÞcosH /C01 2vl0vAH2tðÞsin2H: (3) Here, HðtÞ¼H0þH1cosxtsince the vectors H0andH1 are assumed parallel, His the angle between the unit vectors uandh¼H0=H0so that cosH¼ðu/C1hÞ¼c1sin#cosuþc2sin#sinuþc3cos#; c1;c2;c3are the direction cosines of H0, and Kis the anisot- ropy constant with an archetypal order of a few 104J/m3for ferritin.16ForvA¼0, Eq. (3)yields the free energy for uni- axial ferromagnetic nanoparticles with the anisotropy and Zeeman terms. The last term in the right-hand side of Eq. (3)is a contribution due to the induced moment non-existent for ferromagnetic nanoparticles. This term affects the dynamicsof the magnetic moment in the presence of an ac driving fieldand determines main features of the nonlinear response of nanoscale antiferromagnets. Thus, even moderate external magnetic fields can produce nonlinear effects in the particlemagnetization dynamics that may differ from those observedin ferromagnetic nanoparticles. Using Brown’s continuous diffusion model, Raikher and Stepanov 9have evaluated characteristic relaxation times and the linear dynamic susceptibility of a suspension of antiferro-magnetic nanoparticles in the particular case of dc and ac mag- netic fields parallel to the easy axis of the particle. Their results were extended in Ref. 23for the general case when a dc magnetic field is applied at an arbitrary angle to the easyaxis. Moreover, the magnetization reversal time sof antiferro- magnetic nanoparticles has been evaluated in Ref. 24via the Kramers escape rate theory adapted to magnetic nanopar-ticles; 22,25the analytic equations for sso obtained agree favor- ably with the numerical solution of the Fokker-Planck’sequation (2). 23,24The primary goal of this paper is to give a detailed investigation of the magnetization, nonlinear dynamicmagnetic susceptibility, and dynamic magnetic hysteresis (DMH) of antiferromagnetic nanoparticles and to demonstrate that in superimposed dc and ac fields these magnetic character-istics change substantially leading to interesting nonlineareffects. We remark that the theoretical treatment of nonlinearresponse phenomena inherently poses a complicated mathe-matical problem because no unique response function govern- ing the transient and ac stationary responses, unlike in linearresponse, exists. However, these difficulties may be overcomeby extending the method developed in Refs. 26–32to the dynamic nonlinear stationary response of ferromagnetic nano-particles driven by an ac magnetic field. The essential feature of this method is that it allows one to evaluate both the linear and nonlinear ac stationary responses in a wide range of damp-ing and at all frequencies of interest from the very low fre-quencies up to the very high (GHz) frequencies. If the field H 0 is applied at an arbitrary angle to the easy axis of the particle, a strong intrinsic dependence of magnetic characteristics (suchas the reversal time, complex magnetic susceptibility, etc.)on the damping aarising from coupling of the longitudinal and transverse modes of the magnetization exists. As shownby Garc /C19ıa-Palacios and Svedlindh, 26the nonlinear dynamical response of nanomagnets in the underdamped regime, a<1, is very sensitive to the damping due to the coupling induced by the driving field between the precession of the magneticmoment and its thermoactivated reversal. The large dampingdependence of the nonlinear response can be used to determinethe damping coefficient a. 26 II. BASIC EQUATIONS For convenience, we introduce the following dimension- less variables: r¼vbK,n0¼bl0lH0, and n¼bl0lH1are, respectively, the anisotropy, dc, and ac applied field parame- ters, and f¼vvA=ðbl0l2Þis the “antiferromagnetic” param- eter. We remark that due to the decompensation origin ofthe magnetic moment, the field parameters n 0andnfor053901-2 Kalmykov, Ouari, and Titov J. Appl. Phys. 120, 053901 (2016) antiferromagnetic nanoparticles are about two orders of mag- nitude smaller than those for ferromagnetic nanoparticles of the same size /C2410 nm.9Nonlinear effects (saturation, etc.) in the ac stationary response of antiferromagnetic nanoparticles become pronounced at n>1; however, their main features can be studied at n/C281. On seeking the solution of the Fokker-Planck equation (2), where the free energy potential defined by Eq. (3), as a series of spherical harmonics Ylmð#;uÞ;viz.,Wð#;u;tÞ¼X1 l¼0Xl m¼/C0lclmðtÞY/C3 lmð#;uÞ; (4) the task of calculating the nonlinear ac stationary response of an antiferromagnetic nanoparticle to an external driving field can be reduced22to the solution of an infinite hierarchy of 25- term differential-recurrence relation for the statistical moments(the expectation values of the spherical harmonics) c lmðtÞ sNd dtcnmtðÞ¼v/C0/C0 nmcn/C02m/C02tðÞþv/C0 nmcn/C02m/C01tðÞþvnmcn/C02mtðÞþvþ nmcn/C02mþ1tðÞþvþþ nmcn/C02mþ2tðÞ þw/C0/C0 nmcn/C01m/C02tðÞþw/C0 nmtðÞcn/C01m/C01tðÞþwnmtðÞcn/C01mtðÞþwþ nmtðÞcn/C01mþ1tðÞþwþþ nmcn/C01mþ2tðÞ þx/C0/C0 nmcnm/C02tðÞþx/C0 nmtðÞcnm/C01tðÞþxnmtðÞcnmtðÞþxþ nmtðÞcnmþ1tðÞþxþþ nmcnmþ2tðÞ þy/C0/C0 nmcnþ1m/C02þy/C0 nmtðÞcnþ1m/C01tðÞþynmtðÞcnþ1mtðÞþyþ nmtðÞcnþ1mþ1tðÞþyþþ nmcnþ1mþ2tðÞ þz/C0/C0 nmcnþ2m/C02tðÞþz/C0 nmcnþ2m/C01tðÞþznmcnþ2mtðÞþzþ nmcnþ2mþ1tðÞþzþþ nmcnþ2mþ2tðÞ: (5) Here, the asterisks designate the complex conjugate, and the spherical harmonics Ylmð#;uÞare defined as21 Ylm#;uðÞ ¼ /C0 1ðÞmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2lþ1ðÞ l/C0mðÞ ! 4plþmðÞ !s eimuPm lcos#ðÞ ; Pm lðxÞare the associated Legendre functions21 clmðtÞ¼h YlmiðtÞ¼ð2p 0ðp 0Ylmð#;uÞWð#;u;tÞsin#d#du;(6) and the time-dependent coefficients vnmðtÞ,wnmðtÞ,xnmðtÞ, etc., are given in the Appendix . Since the ac stationary response is independent of the initial conditions, we need only the steady state solution of Eq. (5)so that clmðtÞcan be expanded in a Fourier series as clmðtÞ¼X1 k¼/C01ck lmðxÞeikxt: (7) The Fourier coefficients ck lmðxÞcan be evaluated using matrix continued fractions (see the Appendix ). Consequently, the average magnetic moment of the particle in the direction of the ac driving field lHðtÞ¼l½hcosHiðtÞþfðn0þncosxtÞhsin2HiðtÞ/C138;(8) which is expressed via the statistical moments hYlmiðtÞ(see Eqs. (A5) and(A6) in the Appendix ), can also be presented as a Fourier series lHðtÞ¼lX1 k¼/C01mk 1ðxÞeikxt; (9) where mk 1ðxÞis the amplitude of the kth harmonic in the non- linear response given by Eq. (A4) in the AppendixIn order to illustrate the nonlinear effects induced by the ac field, we focus on the time-independent or dc component of the magnetization Mndefined by the mean value Mn¼x 2plð2p=x 0lHtðÞdt¼m0 1xðÞ; (10) which we note is entirely real yielding in the limits of van- ishing ac field ,n!0, the static dc magnetization given by lim n!0Mn¼M0¼hcosHi0þfn0ð1/C0hcos2Hi0Þ;(11) where the angular brackets hi0denote equilibrium ensem- ble averaging defined as hcosnHi0¼1 Zð2p 0ðp 0cosnHe/C0E0#;uðÞsin#d#du; Z¼ð2p 0ðp 0e/C0E0ð#;uÞsin#d#du; is the partition function, and the free energy E0ð#;uÞis given by E0#;uðÞ ¼ rsin2#/C0n0cosH/C0fn2 0 2sin2H: (12) The free energy potential, Eq. (12), has, in general, a bista- ble structure with two minima separated by a potential bar-rier with a saddle point. 23Henceforth, we shall assume without loss of generality that the dc magnetic field vectorH 0lies in the XZ-plane of the laboratory coordinate system making an angle wwith respect to the easy axis so that the direction cosines of the vector H0arec1¼sinw;c2¼0; andc3¼coswyielding053901-3 Kalmykov, Ouari, and Titov J. Appl. Phys. 120, 053901 (2016) cosH¼coswcos#þsinwsin#cosu: (13) Furthermore, we evaluate the spectrum of the fundamental component of the magnetization defined by vxðÞ¼6m1 1xðÞ n; (14) which represents the linear and lowest order nonlinear sus- ceptibility. Furthermore, via m1 1ðxÞ, we can calculate the dimensionless area of the DMH loop An, which is the energy loss per particle and per cycle of the ac field, defined as33–35 An¼1 4lHþ lHtðÞdH tðÞ¼/C0p 2Imm1 1xðÞ/C2/C3 (15) (the phenomenon of the DMH in single-domain magnetically isotropic nanoparticles was discovered by Ignatchenko and Gekht36). The DMH loop represents a parametric plot of the steady-state time-dependent magnetization as a function ofthe ac field, i.e., M HðtÞ¼lHðtÞ=vvs.HðtÞ¼Hcosxt.A l l other harmonic components mk 1ðxÞwith k>1 may be inves- tigated in a similar way. III. RESULTS AND DISCUSSION In order to illustrate the nonlinear effects in the time- independent but frequency-dependent magnetization, MnðxÞ is plotted in Fig. 1as a function of the dc field and anisotropy (or the inverse temperature) parameter rfor various ac field magnitudes nand antiferromagnetic parameter f. The imagi- nary v00ðxÞ¼/C0 Im½vðxÞ/C138and the real v0ðxÞ¼Re½vðxÞ/C138 parts of the susceptibility as functions of the anisotropyparameter rand frequency xare shown in Figs. 2and3, respectively, for various antiferromagnetic parameters fboth in the linear ðn/C281Þand nonlinear ðn/C211Þregimes. (The condition n!0 corresponds to the linear response, where l HðtÞ=H1is independent of the ac field strength.) The calcu- lations indicate that a marked dependence of vðxÞon the antiferromagnetic parameter f, anisotropy r, ac field n,d cfield n0, damping a, and angle wexists. For the special case of linear response, n/C281, the results agree with the indepen- dent numerical calculations.24In strong ac fields, n/C211;pro- nounced nonlinear effects occur; see Figs. 1–3illustrating the dependence of the nonlinear response on the ac fieldparameter n. Just as for ferromagnetic nanoparticles, 27–31 three peaks appear in the spectra of the magnetic loss v00ðxÞ in the neighborhood of frequencies where v0ðxÞchanges (see Fig. 3). The characteristic frequencies of these peaks, i.e., where v00ðxÞreaches local maxima, are s/C01,axpr, and xpr, where sis the switching (reversal) time of the magnetic moment between two minima of the free energy separated by a potential barrier with a saddle point, and xpr¼cHefis the precession frequency of the magnetic moment in theeffective magnetic field H ef:The low-frequency behavior of v0ðxÞandv00ðxÞis dominated by the barrier-crossing relaxa- tion mode. Here, the reversal time of the magnetization scan be evaluated from the characteristic frequency xmax, whereFIG. 1. Dimensionless magnetization Mnvs. the dc bias field ((a) and (b)) and anisotropy (inverse temperature) parameters ((c) and (d)) for various acfields and antiferromagnetic parame- ters at damping a¼1, oblique angle w¼0, and frequency xs N¼0:001. FIG. 2. Real and imaginary parts of the dynamic susceptibility vs. r/C24T/C01 for various antiferromagnetic parameters fat dc field parameter n0¼2, fre- quency xsN¼0:1, damping a¼1, angle w¼p=4, and ac field parameters n¼0:001 (linear response) and n¼2 (nonlinear response). Dashed and solid lines: the linear and nonlinear response, respectively.053901-4 Kalmykov, Ouari, and Titov J. Appl. Phys. 120, 053901 (2016) v00ðxÞreaches a maximum, and/or the half-width Dxof the spectrum of v0ðxÞas s/C25x/C01 max/C25Dx/C01: (16) In weak ac fields, n/C281, the magnetization reversal time s can be associated with the inverse of the smallest nonvanish-ing eigenvalue k 1of the Fokker-Planck operator in Eq. (2).22 In this case, comparison of sas extracted from the spectra v0ðxÞandv00ðxÞvia Eq. (16) with s¼k/C01 1calculated using the independent method22viak1of the Fokker-Planck opera- tor shows that both methods yield identical results.Furthermore, our calculations indicate that when the dc fieldparameter n 0is increased, the magnitude of the low- frequency peak decreases due to the depletion of the popula-tion in the shallowest potential well of the free-energy den-sityE 0ð#;uÞ, Eq. (12); this effect is signified by the virtual disappearance of the low-frequency peak in the magneticloss spectrum v 00ðxÞ(see Fig. 3(d)). For weak dc bias field, n0<0:3, the low-frequency peak shifts monotonically to higher frequencies as the ac field amplitude nis increased. For strong dc bias field, n0>1, the low frequency peak shifts to lower frequencies reaching a maximum at n/C24n0, thereafter decreasing rapidly with increasing n. In other words, as the dc field increases, the reversal time of the mag-netization initially increases and then having attained its maximum at some critical value n/C24n 0decreases. In addi- tion, a second relaxation peak in v00ðxÞand the correspond- ing dispersion of v0ðxÞappearing at higher frequencies /C24axprare due to the near degenerate “intrawell” relaxation modes, which are virtually indistinguishable in the frequencyspectra. Now for n 0=r<0:1, the amplitude of this peak is far weaker than that of the low-frequency one. However, forn 0=r>0:4, this peak may come to dominate the spectra because as h0increases, the magnitude of the low-frequency peak drastically decreases (see Fig. 3(d)). Figure 3(d) also illustrates the inherent dependence of vðxÞon the damping parameter aarising from the coupling of the longitudinal and transverse relaxation modes.26This coupling appears in the dynamical equation of motion of the magnetic moment lðtÞ,where its longitudinal component is entangled with the trans- verse components resulting in the appearance of a third anti-ferromagnetic resonance (AFMR) peak in the spectrum of v 00ðxÞdue to excitation of transverse modes with characteris- tic frequencies close to the precession frequency xprin the effective magnetic field Hef(see Fig. 3). The AFMR peak appears only for low damping ða/C281Þand strongly mani- fests itself at high frequencies x/C24xpr(see Fig. 3(c); a fea- ture which is absent for a>1). As seen in Fig. 3, with increasing ac field n, the magnitude of the AFMR peak decreases and the peak half-width broadens showing pro- nounced nonlinear effects. In addition, a second weak reso- nance peak owing to parametric resonance of the nonlinear oscillatory (precessional) motion of the magnetic moment lðtÞappears at frequencies /C24xpr=2 (Fig. 3(a)), while for very low damping, a<0:01;resonance peaks with charac- teristic frequencies nxpr,n¼2, 3,… due to the high- frequency resonant modes are discernible in the spectrum (only the peak with n¼2 is visible in Figs. 3(b)–3(d) ). These peaks virtually disappear, however, for w¼0. Such nonlin- ear effects always exist in nonlinear oscillator systems driven by an ac external force.37 The DMH loops, i.e., mðtÞ¼lHðtÞ=lvs.hðtÞ¼HðtÞ=H ¼cosxt, for various antiferromagnetic fand inverse temper- ature r/1=Tparameters are presented in Figs. 4and5for a wide range of other model parameters (damping, oblique angle, etc.). At finite temperatures due to thermal motion, the particle magnetic moment is never completely saturated wan-dering between the “up” and “down” states. Furthermore, the shape of the DMH loops for given values of anisotropy parameter r, antiferromagnetic parameter f, damping a,o b l i - que angle w, and dc field n 0depends on the amplitude nand frequency xof the ac field. Figures 4(a) and 4(b) show, respectively, that the shape and area of the DMH loops strongly depend on the anisotropy rand the dc field parame- tersn0.I np a r t i c u l a r ,F i g . 4(b) illustrates that the area of the DMH loops decreases with increasing the dc field parameter n0. Also, with decreasing anisotropy parameter r, i.e., with increasing temperature, the DMH loops become narrow (Fig. 4(a)), which implies that a small amount of energy is used upFIG. 3. Imaginary part of the dynamic susceptibility /C0Im½vðxÞ/C138vs. the dimen- sionless frequency xs0for (a) various anisotropy parameters r, (b) angles w,( c ) damping a, and (d) dc field parameters n0a tt h ea cfi e l da m p l i t u d e s n¼0:001 (linear response: dashed lines) and 2 (nonlinear response: solid lines).053901-5 Kalmykov, Ouari, and Titov J. Appl. Phys. 120, 053901 (2016) in reversing the magnetization. Figure 4(c) illustrates the dependence of the shape and area of the DMH loops on theoblique angle w, while in Fig. 4(d), the DMH loops are pre- sented for very low and intermediate frequencies. At low fre-quencies, xs N/C2410/C04/C010/C03, the loops are large while at intermediate frequencies, xsN/C241, the shape of loops becomes elliptic with small area. At low frequencies, wherechanges of the ac field are quasi-adiabatic, the magnetizationdynamics represent the so-called switching regime meaningthat the magnetization may reverse due to the cooperativeshuttling action of thermal agitation and applied field. Here, the DMH loop area decreases as the antiferromagnetic param- eterfincreases. Moreover, the coercivity, the remanent mag- netization, and the saturation magnetization strongly dependonrso that considerable variations in the area and shape of loops exist at low frequencies. At high frequencies, xs N/C291, the DMH is mainly due to the absorption and dispersion inthe “intrawell” and AFMR modes. Thus, the DMH arisingfrom a high-frequency periodic signal may be evaluated per-mitting quantitative analysis of ultrafast switching of the mag-netization in nanoscale antiferromagnets. At x/C24x pr,t h e DMH occurs due to the resonant behavior of the nonlinear response, and under such conditions, the switching may betermed “resonant,” leading naturally to the concept of reso- nant switching of the magnetization . 28–31Since the resonant DMH occurs at very high (GHz) frequencies, the magnetiza-tion switching is, for the most part, governed by the frequencyof the external driving field. Hence, the magnetization may beadvantageously switched in this situation because the fieldneeded to reverse it is then much smaller than the quasi-staticcoercive force. Here, the phase difference dbetween l HðtÞ and HðtÞ;governing loop orientation , may undergo a pro- nounced variation. In particular, the phase difference dmay exceed p=2 as typical of a resonant process.28–31Obviously,this effect does not exist at low and intermediate frequencies, where relaxation processes dominate and dis always less than p=2: The shape and area of the DMH loops alter as the antifer- romagnetic parameter fvaries (see Figs. 5and6). In particu- lar, as seen in Fig. 6,Anstrongly depends on temperature, namely, on increasing r, i.e., decreasing temperature, the loop area initially increases, reaches a maximum, and then decreases. Furthermore, at low r(high temperatures), the behavior of Anfor antiferromagnetic and ferromagnetic nano- particles is very similar, while for large r>10, it can differ substantially. Figure 7shows the behavior of the area An, Eq. (15), as a function of the ac field nfor various anisotropy (inverse temperature) parameters r/C24T/C01. For a weak ac field, the DMH loops are ellipses with area Angiven by Eq. (15); the behavior of An/C24/C0Imðm1 1Þbeing similar to that ofFIG. 4. DMH loops for (a) various anisotropy parameters r¼6;10;12, (b) dc fields n0¼0;1;2;3, (c) angles w¼p=6;p=4;p=3, and (d) frequency xsN¼10/C04;10/C03;10/C02;1. FIG. 5. DMH loops for various antiferromagnetic parameters fatxsN¼0:1, r¼5,a¼2,w¼p=6,n¼3, and n0¼0.053901-6 Kalmykov, Ouari, and Titov J. Appl. Phys. 120, 053901 (2016) the magnetic loss /C0Im½vðxÞ/C138[cf. Eq. (14)]. In strong ac fields, n>1, the DMH area alters substantially (see Fig. 7); nevertheless, Anis still determined by /C0Imðm1 1Þ[cf. Eq. (15)] with maxima appearing at strong fields, n/C212. IV. CONCLUSION The nonlinear forced ac stationary response of antiferro- magnetic nanoparticles in superimposed ac and dc bias exter- nal magnetic fields is studied via continuous diffusionmodel. 5It is shown that the nonlinear dynamic susceptibility and DMH in an ac field applied at an angle to the easy axis of the particle, so that the axial symmetry is broken, are verysensitive to both the ac field orientation and amplitude owing to the coupling induced by the symmetry breaking driving field between the precession of the magnetic moment and itsthermally activated reversal. This explains why the nonlinearac stationary response of antiferromagnetic nanoparticles isvery sensitive to damping representing a signature of the coupling between the longitudinal and precessional modes ofthe magnetization just as nanoscale ferromagnets. 26–32 Furthermore, it is found that under appropriate conditions a small (in comparison with that of internal anisotropy) bias dcfield can strongly affect the shape of the DMH loops in anti-ferromagnetic nanoparticles. This result implies that by vary- ing the dc bias field strength, one may control the heat production (specific power loss) in a nanoparticle. Since theresults are valid for ac fields of arbitrary strength and orien- tation, they provide a rigorous basis for the treatment of thenonlinear ac stationary response of antiferromagnetic nano-particles in strong ac fields, where perturbation theory is nolonger valid. Our calculations of the nonlinear response of anindividual antiferromagnetic nanoparticle can be generalizedto calculate the average magnetic moment of an assembly ofrandomly oriented noninteracting nanoparticles by averagingover particle easy axis orientations as described in detail inRef. 27. We have also assumed that all the particles are iden- tical and interparticle interactions are negligible. In order toaccount for the polydispersity, it is necessary to average overthe appropriate distribution function over the particle vol-umes. The neglect of interparticle interactions in the presentmodel suggests that the results apply only to systems, whereinteractions are ignored, such as individual nanoparticles anddilute solid suspensions of nanoparticles. ACKNOWLEDGMENTS We thank W. T. Coffey, W. Dowling, and P. M. D/C19ejardin for a critical reading of the paper and useful comments and suggestions. We would like to thank FP7-PEOPLE-Marie Curie Actions-International Research StaffExchange Scheme (Project No. 295196 DMH) for financialsupport. S. V. Titov also acknowledges the financial supportof the program “Mechnikov” (the Embassy of France inRussia). APPENDIX: MATRIX CONTINUED FRACTION SOLUTION OF EQ. (5)FOR THE AC STATIONARY RESPONSE The time-dependent coefficients wnmðtÞ,xnmðtÞ, etc., in Eq.(5)can be presented as wnmðtÞ¼w0 nmþw1 nmðeixtþe/C0ixtÞþw2 nmðe2ixtþe/C02ixtÞ; xnmðtÞ¼x0 nmþx1 nmðeixtþe/C0ixtÞþx2 nmðe2ixtþe/C02ixtÞ; ynmðtÞ¼y0 nmþy1 nmðeixtþe/C0ixtÞþy2 nmðe2ixtþe/C02ixtÞ; etc., where the various time-independent coefficients wi nm, xi nm, etc., are given by x0 nm¼/C0nnþ1ðÞ 2/C0in0mc3 2a þf 82n2 0þn2/C16/C17 1/C03c2 3/C0/C1 þr/C20/C21 nnþ1ðÞ /C03m2 2n/C01 ðÞ 2nþ3 ðÞ;FIG. 7. Area Anof the DMH loop vs. the ac field n: (a) for various barrier parameters r¼8;10;12;14 with the antiferromagnetic parameter f¼0:4, dc field parameter n0¼0, frequency xsN¼10/C03, and damping a¼2;(b) angles w¼p=6;p=4;p=3.FIG. 6. Area Anof the DMH loop vs. the anisotropy (inverse temperature) parameter r/C24T/C01for various antiferromagnetic parameters fatn¼3, a¼1, and n0¼0.053901-7 Kalmykov, Ouari, and Titov J. Appl. Phys. 120, 053901 (2016) x1 nm¼/C0inmc3 4aþfn0n 41/C03c2 3/C0/C1 nnþ1ðÞ /C03m2 2n/C01 ðÞ 2nþ3 ðÞ; x2 nm¼fn21/C03c2 3/C0/C1 nnþ1ðÞ /C03m2/C2/C3 16 2 n/C01 ðÞ 2nþ3 ðÞ; x/C00 nm¼c1þic2 ðÞ32m/C01 ðÞ f2n2 0þn2/C16/C17 c3 82n/C01 ðÞ 2nþ3 ðÞ/C0in0 4a2 43 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ nþmðÞq ; x/C01 nm¼c1þic2 ðÞ32m/C01 ðÞ fn0nc3 42n/C01 ðÞ 2nþ3 ðÞ/C0in 8a"#ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ nþmðÞq ; x/C02 nm¼c1þic2 ðÞ32m/C01 ðÞ fn2c3 16 2 n/C01 ðÞ 2nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ nþmðÞq ; x/C0/C00 nm¼3f2n2 0þn2/C16/C17 c1þic2 ðÞ2 16 2 n/C01 ðÞ 2nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ 2þn/C0m ðÞ /C01þnþm ðÞ nþmðÞq ; x/C0/C01 nm¼3fnn 0c1þic2 ðÞ2 82n/C01 ðÞ 2nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ 2þn/C0m ðÞ /C01þnþm ðÞ nþmðÞq ; x/C0/C02 nm¼3fn2c1þic2 ðÞ2 32 2 n/C01 ðÞ 2nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ 2þn/C0m ðÞ /C01þnþm ðÞ nþmðÞq ; y0 nm¼/C0n0 2c3nþim arþf2n2 0þn2/C16/C17 1/C03c2 3/C0/C1 8"# ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1ðÞ2/C0m2 2nþ1 ðÞ 2nþ3 ðÞs ; y1 nm¼/C0n 4nc3þimfn01/C03c2 3/C0/C1 a/C20/C21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1ðÞ2/C0m2 2nþ1 ðÞ 2nþ3 ðÞs ; y2 nm¼/C0imfn21/C03c2 3/C0/C1 16affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1ðÞ2/C0m2 2nþ1 ðÞ 2nþ3 ðÞs ; y/C00 nm¼c1þic2 ðÞinþ2m ðÞ f2n2 0þn2/C16/C17 c3 8a/C0nn0 4"#ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ 2þn/C0m ðÞ 1þ2n ðÞ 3þ2n ðÞs ; y/C01 nm¼c1þic2 ðÞ2ifn0nc3 4anþ2m ðÞ /C0nn 8/C20/C21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ 2þn/C0m ðÞ 1þ2n ðÞ 3þ2n ðÞs ; y/C02 nm¼ifn2c3c1þic2 ðÞ nþ2m ðÞ 16affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ 2þn/C0m ðÞ 1þ2n ðÞ 3þ2n ðÞs ; y/C0/C00 nm¼if2n2 0þn2/C16/C17 c1þic2 ðÞ2 16affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ 2þn/C0m ðÞ 3þn/C0m ðÞ nþmðÞ 1þ2n ðÞ 3þ2n ðÞs ; y/C0/C01 nm¼ifn0nc1þic2 ðÞ2 8affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ 2þn/C0m ðÞ 3þn/C0m ðÞ nþmðÞ 1þ2n ðÞ 3þ2n ðÞs ; y/C0/C02 nm¼ifn2c1þic2 ðÞ2 32affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þn/C0m ðÞ 2þn/C0m ðÞ 3þn/C0m ðÞ nþmðÞ 1þ2n ðÞ 3þ2n ðÞs ; w0 nm¼c3n0 2nþ1ðÞ /C0im af2n2 0þn2/C16/C17 1/C03c2 3/C0/C1 8þr"# !ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2/C0m2 4n2/C01r ;053901-8 Kalmykov, Ouari, and Titov J. Appl. Phys. 120, 053901 (2016) w1 nm¼n 4c3nþ1ðÞ /C0imn0f a1/C03c2 3/C0/C1/C20/C21 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2/C0m2 4n2/C01r ; w2 nm¼/C0imfn21/C03c2 3/C0/C1 16affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2/C0m2 4n2/C01r ; w/C00 nm¼/C0 c1þic2 ðÞ n0nþ1 4/C0i1/C02mþn ðÞ f2n2 0þn2/C16/C17 c3 8a"#ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþmðÞ nþm/C01 ðÞ 4n2/C01r ; w/C01 nm¼/C0 c1þic2 ðÞ nnþ1 8/C0i1/C02mþn ðÞ fn0nc3 4a/C20/C21 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþmðÞ nþm/C01 ðÞ 4n2/C01r ; w/C02 nm¼i1/C02mþn ðÞ fn2c3c1þic2 ðÞ 16affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþmðÞ nþm/C01 ðÞ 4n2/C01r ; w/C0/C00 nm¼/C0if2n2 0þn2/C16/C17 c1þic2 ðÞ2 16affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþm/C02 ðÞ nþm/C01 ðÞ 1þn/C0m ðÞ nþmðÞ 4n2/C01r ; w/C0/C01 nm¼/C0ifnn 0c1þic2 ðÞ2 8affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþm/C02 ðÞ nþm/C01 ðÞ 1þn/C0m ðÞ nþmðÞ 4n2/C01r ; w/C0/C02 nm¼/C0ifn2c1þic2 ðÞ2 32affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþm/C02 ðÞ nþm/C01 ðÞ 1þn/C0m ðÞ nþmðÞ 4n2/C01r ; z0 nm¼/C0f2n2 0þn2/C16/C17 81/C03c2 3/C0/C1 þr"# n 2nþ3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1ðÞ2/C0m2hi nþ2ðÞ2/C0m2hi 2nþ1 ðÞ 2nþ5 ðÞvuut; z1 nm¼/C0nfn0n1/C03c2 3/C0/C1 42nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1ðÞ2/C0m2hi nþ2ðÞ2/C0m2hi 2nþ1 ðÞ 2nþ5 ðÞvuut; z2 nm¼nfn23c2 3/C01/C0/C1 16 2 nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1ðÞ2/C0m2hi nþ2ðÞ2/C0m2hi 2nþ1 ðÞ 2nþ5 ðÞvuut; z/C00 nm¼n2n2 0þn2/C16/C17 fc1þic2 ðÞ c3 42nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1ðÞ2/C0m2hi n/C0mþ2 ðÞ 3þn/C0m ðÞ 2nþ1 ðÞ 2nþ5 ðÞvuut; z/C01 nm¼nnn0fc1þic2 ðÞ c3 22nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1ðÞ2/C0m2hi n/C0mþ2 ðÞ 3þn/C0m ðÞ 2nþ1 ðÞ 2nþ5 ðÞvuut; z/C02 nm¼nfn2c1þic2 ðÞ c3 82nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1ðÞ2/C0m2hi n/C0mþ2 ðÞ 3þn/C0m ðÞ 2nþ1 ðÞ 2nþ5 ðÞvuut; z/C0/C00 nm¼nf2n2 0þn2/C16/C17 c1þic2 ðÞ2 16 2 nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1/C0m ðÞ 2þn/C0m ðÞ 3þn/C0m ðÞ 4þn/C0m ðÞ 2nþ1 ðÞ 2nþ5 ðÞs ; z/C0/C01 nm¼nnn0fc1þic2 ðÞ2 82nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1/C0m ðÞ 2þn/C0m ðÞ 3þn/C0m ðÞ 4þn/C0m ðÞ 2nþ1 ðÞ 2nþ5 ðÞs ; z/C0/C02 nm¼nfn2c1þic2 ðÞ2 32 2 nþ3 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1/C0m ðÞ 2þn/C0m ðÞ 3þn/C0m ðÞ 4þn/C0m ðÞ 2nþ1 ðÞ 2nþ5 ðÞs ;053901-9 Kalmykov, Ouari, and Titov J. Appl. Phys. 120, 053901 (2016) v0 nm¼f2n2 0þn2/C16/C17 81/C03c2 3/C0/C1 þr"# nþ1 2n/C01ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n/C01ðÞ2/C0m2hi n2/C0m2 ðÞ 2nþ1 ðÞ 2n/C03 ðÞvuut; v1 nm¼nþ1ðÞ fn0n1/C03c2 3/C0/C1 42n/C01 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n/C01ðÞ2/C0m2hi n2/C0m2 ðÞ 2nþ1 ðÞ 2n/C03 ðÞvuut; v2 nm¼nþ1ðÞ fn21/C03c2 3/C0/C1 16 2 n/C01 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n/C01ðÞ2/C0m2hi n2/C0m2 ðÞ 2nþ1 ðÞ 2n/C03 ðÞvuut; v/C00 nm¼nþ1ðÞ f2n2 0þn2/C16/C17 c1þic2 ðÞ c3 42n/C01 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþm/C02 ðÞ nþm/C01 ðÞ n2/C0m2 ðÞ 2nþ1 ðÞ 2n/C03 ðÞs ; v/C01 nm¼nþ1ðÞ fn0nc1þic2 ðÞ c3 22n/C01 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþm/C02 ðÞ nþm/C01 ðÞ n2/C0m2 ðÞ 2nþ1 ðÞ 2n/C03 ðÞs ; v/C02 nm¼nþ1ðÞ fn2c1þic2 ðÞ c3 82n/C01 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþm/C02 ðÞ nþm/C01 ðÞ n2/C0m2 ðÞ 2nþ1 ðÞ 2n/C03 ðÞs ; v/C0/C00 nm¼/C0nþ1ðÞ f2n2 0þn2/C16/C17 c1þic2 ðÞ2 16 2 n/C01 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþm/C03 ðÞ nþm/C02 ðÞ /C01þnþm ðÞ nþmðÞ 2nþ1 ðÞ 2n/C03 ðÞs ; v/C0/C01 nm¼/C0nþ1ðÞ fn0nc1þic2 ðÞ2 42n/C01 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþm/C03 ðÞ nþm/C02 ðÞ /C01þnþm ðÞ nþmðÞ 2nþ1 ðÞ 2n/C03 ðÞs ; v/C0/C02 nm¼/C0nþ1ðÞ fn2c1þic2 ðÞ2 32 2 n/C01 ðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþm/C03 ðÞ nþm/C02 ðÞ /C01þnþm ðÞ nþmðÞ 2nþ1 ðÞ 2n/C03 ðÞs : The coefficients with the superscripts þandþþcan be cal- culated according to the rule: aþ nm¼ða/C0 n/C0mÞ/C3and aþþ nm ¼ða/C0/C0 n/C0mÞ/C3. The stationary ac response can be calculated from the formally exact matrix continued fraction solution because Eq.(5)can be rearranged as matrix three-term recurrence relations Q1C1ðxÞþQþ 1C2ðxÞ¼R; (A1) QnðxÞCnðxÞþQþ nCnþ1ðxÞþQ/C0 nCn/C01ðxÞ¼0;n¼2;3;:::; (A2)C1¼... c/C02 nðxÞ c/C01 nðxÞ c0 nðxÞ c1 nðxÞ c2 nðxÞ ...0 BBBBBBBBBBBBBB@1 CCCCCCCCCCCCCCA;c k nðxÞ¼ck 2n/C02nðxÞ ... ck 2n2nðxÞ ck 2n/C01/C02nþ1ðxÞ ... ck 2n/C012n/C01ðxÞ0 BBBBBBBBBBB@1 CCCCCCCCCCCA; withc 0ðxÞ¼ð c0 00Þ. Here, ck lmðxÞare the Fourier coefficients in the Fourier series in the time. The elements of the three- diagonal supermatrixes QnandQ6 nare defined as053901-10 Kalmykov, Ouari, and Titov J. Appl. Phys. 120, 053901 (2016) ½Q6 n/C138lm¼dl/C02mr6 nþdl/C01mp6 nþdlmq6 nþdlþ1mp6 n þdlþ2mr6 n; ½Qn/C138lm¼dl/C02mrnþdl/C01mpnþdlmðqn/C0imsNxIÞþdlþ1mpn þdlþ2mrn; and R¼/C0Q/C0 1C0¼/C01ffiffiffiffiffiffi 4pp... 0 r/C0 1 p/C01 q/C01 p/C01 r/C01 0 ...0 BBBBBBBBBBBBBBBBBB@1 CCCCCCCCCCCCCCCCCCA; where d lmis Kronecker’s delta; the supermatrixes q/C0 n;qn;qþ n,p/C0 n;pn;pþ n,r/C0 n;rn;rþ nare defined in Ref. 27, and the column vectors ck 1ðxÞ,p/C0 1,q/C0 1, and r/C0 1are given by p/C0 1¼/C0ffiffiffiffiffiffiffiffiffiffi 3=40p fn0nðc1/C0ic2Þ2 /C0ffiffiffiffiffiffiffiffiffiffi 3=10p fn0nðc1/C0ic2Þc3 fn0nð1/C03c2 3Þ=ffiffiffiffiffi 20p ffiffiffiffiffiffiffiffiffiffi 3=10p fn0nðc1þic2Þc3 /C0ffiffiffiffiffiffiffiffiffiffi 3=40p fn0nðc1þic2Þ2 ffiffiffiffiffiffiffiffiffiffi 1=24p nðc1/C0ic2Þr nc3=ffiffiffiffiffi 12p /C0ffiffiffiffiffiffiffiffiffiffi 1=24p nðc1þic2Þ0 BBBBBBBBBBBBBBBBB@1 CCCCCCCCCCCCCCCCCA;q /C0 1¼/C0ffiffiffiffiffiffiffiffiffiffiffiffiffi 3=160p fð2n2 0þn2Þðc1/C0ic2Þ2 /C0ffiffiffiffiffiffiffiffiffiffi 3=40p fð2n2 0þn2Þðc1/C0ic2Þc3 ½2rþfð2n2 0þn2Þð1/C03c2 3Þ=4/C138=ffiffiffi 5p ffiffiffiffiffiffiffiffiffiffi 3=40p fð2n2 0þn2Þðc1þic2Þc3 /C0ffiffiffiffiffiffiffiffiffiffiffiffiffi 3=160p fð2n2 0þn2Þðc1þic2Þ2 ffiffiffiffiffiffiffiffi 1=6p n0ðc1/C0ic2Þ n0c3=ffiffiffi 3p /C0ffiffiffiffiffiffiffiffi 1=6p n0ðc1þic2Þ0 BBBBBBBBBBBBBBBB@1 CCCCCCCCCCCCCCCCA; r /C0 1¼n2f 16ffiffiffi 6 5r/C0c1/C0ic2 ðÞ2 /C02c1/C0ic2 ðÞ c3 21/C03c2 3/C0/C1 =ffiffiffi 6p 2c1þic2 ðÞ c3 /C0c1þic2 ðÞ2 0 0 00 BBBBBBBBBBBBBB@1 CCCCCCCCCCCCCCA: The exact solution of Eqs. (A1) and(A2) is then rendered by the matrix continued fraction C 1¼/C0S1Q/C0 1C0; (A3) where the infinite matrix continued fraction S1is defined by the recurrence equation Sn¼/C0 ½ QnþQþ nSnþ1Q/C0 nþ1/C138/C01: Having calculated the Fourier amplitudes ck lmðxÞfrom Eq. (A3), we can evaluate the Fourier amplitudes mk 1ðxÞin Eq. (9)for the average dipole moment lHðtÞ, Eq. (8),a s mk 1xðÞ¼2fn0 3d0kþfn 6d1kþd/C01k ðÞ þffiffiffiffiffiffi 2p 3rffiffiffi 2p coswck 10xðÞþsinwck 1/C01xðÞ/C0ck 11xðÞ/C2/C3no /C0fffiffiffiffiffiffi 2p 15r(ffiffiffi 2 3r 3 cos2w/C01/C0/C1 n0ck 20xðÞþn 2ck/C01 20xðÞþckþ1 20xðÞ/C2/C3/C18/C19 þsin 2wn 0ck 2/C01xðÞ/C0ck 21xðÞ/C2/C3 þn 2ck/C01 2/C01xðÞþckþ1 2/C01xðÞ/C0ck/C01 21xðÞ/C0ckþ1 21xðÞ/C2/C3/C18/C19 þsin2wn 0ck 22xðÞþck 2/C02xðÞ/C2/C3 þn 2ck/C01 22xðÞþckþ1 22xðÞþck/C01 2/C02xðÞþckþ1 2/C02xðÞ/C2/C3/C18/C19 ) : (A4) Here, we have assumed without loss of generality that the direction cosines of the vector H0arec1¼sinw;c2¼0;c3¼ coswand have used the known definitions of the spherical harmonics of the first and second rank and equations for hcosHiðtÞ andhcos2HiðtÞexpressed via the statistical moments hY1miðtÞandhY2miðtÞas053901-11 Kalmykov, Ouari, and Titov J. Appl. 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Investigation of the annealing temperature dependence of the spin pumping in Co20Fe60B20/Pt systems M. Belmeguenai , K. Aitoukaci , F. Zighem , M. S. Gabor , T. Petrisor , R. B. Mos , and C. Tiusan Citation: Journal of Applied Physics 123, 113905 (2018); doi: 10.1063/1.5011111 View online: https://doi.org/10.1063/1.5011111 View Table of Contents: http://aip.scitation.org/toc/jap/123/11 Published by the American Institute of PhysicsInvestigation of the annealing temperature dependence of the spin pumping in Co 20Fe60B20/Pt systems M.Belmeguenai,1,a)K.Aitoukaci,1F.Zighem,1M. S. Gabor,2,b)T.Petrisor, Jr.,2R. B. Mos,2 and C. Tiusan2,3 1LSPM (CNRS-UPR 3407), Universit /C19e Paris 13, 99 Avenue Jean-Baptiste Cl /C19ement, 93430 Villetaneuse, France 2Center for Superconductivity, Spintronics and Surface Science, Technical University of Cluj-Napoca, Str. Memorandumului No. 28, RO-400114 Cluj-Napoca, Romania 3Institut Jean Lamour, CNRS, Universit /C19e de Nancy, BP 70239, F–54506 Vandœuvre, France (Received 30 October 2017; accepted 1 March 2018; published online 20 March 2018) Co20Fe60B20/Pt systems with variable thicknesses of Co 20Fe60B20a n do fP th a v eb e e ns p u t t e r e da n d then annealed at various temperatures ( Ta)u pt o3 0 0/C14C. Microstrip line ferromagnetic resonance ( M S - F M R )h a sb e e nu s e dt oi n v e s t i g a t eC o 20Fe60B20and Pt thickness dependencies of the magnetic damping enhancement due to the spin pumping. Using diffusion and ballistic models for spinpumping, the spin mixing conductance and the spin diffusion length have been deduced from the Co 20Fe60B20and the Pt thickness dependencies of the Gilbert damping parameter aof the Co20Fe60B20/Pt heterostructures, respectively. Within the ballistic simple model, both the spin mixing conductance at the CoFeB/Pt interface and the spin-diffusion length of Pt increase with the increasing annealing temperature and show a strong enhancement at 300/C14C annealing temperature. In contrast, the spin mixing conductance, which increases with Ta, shows a different trend to the spin diffusion length when using the diffusion model. Moreover, MS-FMR measurements revealed that the effective magnetization varies linearly with the Co 20Fe60B20inverse thickness due to the perpendicular interface anisotropy, which is found to decrease as the annealing temperature increases. It also revealed that theangular dependence of the resonance field is governed by small uniaxial anisotropy which is found to vary linearly with the Co 20Fe60B20inverse thickness of the annealed films, in contrast to that of the as grown ones. Published by AIP Publishing. https://doi.org/10.1063/1.5011111 I. INTRODUCTION Nowadays, several phenomena are known to occur espe- cially in heavy metal/ferromagnet (FM) systems such asthe spin Hall effect (SHE), 1,2spin orbit torques,3the Dzyaloshinskii-Moriya interaction,4,5and the spin pumping.6,7 The latter mechanism is considered to be an efficient route to generate a spin current in non-magnetic materials (NMs), which is one of the pillars of modern spintronics.8 Ferromagnetic resonance (FMR) induced spin pumping is anemerging process for dynamically injecting a pure spin currentinto a NM without the need of charge flowing, in contrast toSHE. This involves significant potential impact on the energyefficiency of the spintronic devices. In this process, the reso-nant precession of the FM magnetization pumps a spin currentinto the NM, which decays on a length scale called the spin-diffusion length ( k SD). The value of this latter quantity is of great interest, since it could allow the increase of the spin cur- rent injection efficiency in the FM/NM bilayer by the optimi- zation of the NM thickness.9The spin injection process into the NM is accompanied by an angular momentum loss in theFM leading to a broadening of the FMR linewidth, which isdirectly linked to the Gilbert damping parameter ( a). The line- width broadening is more pronounced when the NM is aheavy metal having a high spin-orbit coupling (SOC), due tothe increased rate of spin scattering events, owing to the SOC.This opens an interesting possibility to tune the damping value depending on the desired technological application. From another side, ferromagnetic CoFeB alloys have attracted an intense attention due to their high spin polariza- tion and low damping properties. 10–12Today, CoFeB thin films are considered to be among the most promising candi-dates for magnetic tunnel junction (MTJ) electrodes due totheir ability to provide very large magnetoresistance ratios at room temperature (RT), in conjunction with crystalline MgO tunnel barriers. 13,14The CoFeB/MgO based MTJs are widely used in spintronic devices such as magnetic random accessmemories (MRAMs), magnetic read heads, and magneticsensors. 15,16It is worth mentioning that, unlike the thickness range of CoFeB investigated in this paper, ultrathin CoFeB- based MTJs with a perpendicular magnetic anisotropy due tothe interfacial contribution are highly needed for spin trans-fer torque MRAMs. 17The magnetic properties of the CoFeB thin films are strongly influenced by their thickness and interfacial effects. It is well known that the room temperaturedeposited CoFeB films are amorphous and in order to inducetheir crystallization, an annealing process is required. Therefore, it is of great interest for both fundamental and technological reasons to point out the annealing temperaturedependencies of the interfacial anisotropy, the spin diffusionlength, and the spin pumping efficiency in Co 20Fe60B20/Pt systems. The experimental strategy used in this paper con- sists thus to use the ferromagnetic resonance in microstripline (MS-FMR) under in-plane and perpendicular applieda)belmeguenai.mohamed@univ-paris13.fr b)mihai.gabor@phys.utcluj.ro 0021-8979/2018/123(11)/113905/9/$30.00 Published by AIP Publishing. 123, 113905-1JOURNAL OF APPLIED PHYSICS 123, 113905 (2018) magnetic fields combined with the vibrating sample magne- tometry technique. By investigating the Gilbert damping parameter dependence as a function of the thickness of the Co20Fe60B20and the Pt layers, the spin mixing conductance andkSDin Co 20Fe60B20/Pt heterostructures have been stud- ied as a function of the annealing temperature. We demon- strate that spin mixing conductance is drastically enhanced for samples annealed at 300/C14C. Moreover, this work shows the presence of a perpendicular surface magnetic anisotropy,which decreases as the annealing temperature increases. II. SAMPLE PREPARATION AND EXPERIMENTAL METHODS Co20Fe60B20/Pt bilayers were grown at room temperature (RT) onto thermally oxidized Si substrates using a magnetron sputtering system having a base pressure lower than 2 /C210/C08Torr. Two sets of samples have been considered: (i) Co20Fe60B20films with variable thicknesses ( tCFB¼10, 8, 6, 4, and 3 nm) capped by a 10 nm thick Pt layer and (ii) 4 nm thick Co 20Fe60B20capped by a Pt layer of various thicknesses (tPt¼1, 2, 3, 4, 6, 8, 10, and 12 nm). After the growth of the stack, the structures were ex situ annealed at different temper- atures ( Ta¼RT, 200/C14C, 250/C14C, and 300/C14C) for 60 min in vacuum (with a pressure lower than 2 /C210/C06Torr). The first (second) set of samples serves for determining the spin mix- ing conductance (spin diffusion length) from the investigation of the Gilbert damping parameter of FM/NM heterostructures versus tCFB(versus tPt). The structural properties of the sam- ples have been determined by x-ray diffraction (XRD) experi-ments using a four-circle diffractometer. The static magnetic characteristics were investigated by using a vibrating sample magnetometer (VSM). For the dynamic measurements, we used the microstrip ferromagnetic resonance (MS-FMR) 18 technique (in the sweep-field mode), where the externalapplied magnetic field (up to 1.9 T) is modulated at 170 Hz by a small (4 Oe) alternating magnetic field and the measured signal is proportional to the field first derivative of the absorbed power. During the measurement, the external mag- netic field Hwas applied perpendicular to the sample plane or in-plane in various directions with respect to the sample edges. All the measurements presented here have been per- formed at room temperature. III. RESULTS AND DISCUSSIONS Figure 1shows the x-ray 2 h-x(out of plane) diffraction patterns for the CoFeB (6 nm)/Pt (10 nm) measured in a 2 h angle window around the expected positions of the (111) reflection of Pt and the (110) reflection of CoFeB. The pat- terns show only the (111) Pt diffraction peaks. Laue oscilla- tions are also observable which indicate a good crystalline quality for the Pt films. It should be mentioned that patterns measured on a wider 2 hrange did not show the presence of other reflections except for the substrate ones. This indicates that in our samples, Pt has a strong (111) out-of-plane textur- ing and that the CoFeB films are most likely amorphous or nano-crystalline. This has also been suggested by the lack of a clear in-plane magnetocrystalline anisotropy, confirmed by the VSM measurements. The hysteresis loops for appliedmagnetic fields at various orientations with respect to the sub- strate edges are illustrated in Fig. 2(a) for CoFeB (4 nm)/Pt (4 nm), indicating a slightly in-plane uniaxial character of themagnetic anisotropy. To test this, we have performed angularremanence magnetization (ARM) measurements, by measur-ing the remanence magnetization after saturation at differentin-plane angles. Figure 2(b) shows the typical normalized ARM curve in our samples presented here for the CoFeB (6 nm)/Pt(10 nm) annealed at 300 /C14C .T h es h a p eo ft h eA R M curve is not perfectly isotropic confirming the uniaxial charac-ter. This uniaxial character is not surprising for sputtered films,and it is due to the presence of a residual magnetic field of themagnetron sputter sources during growth 19and it is not con- nected with in-plane crystalline anisotropy. It is worth men- tioning that it was shown that the crystallization of CoFeB and the annealing temperature for which the crystallization occursare capping material dependent, 20,21for instance, Co 40Fe40B20 crystallization occurs at an annealing temperature of 375/C14C when MgO is used as the capping layer.22This suggests that in the case of our Pt capped CoFeB films, higher annealing tem- peratures are required in order to achieve crystallization. The magnetization at saturation should be precisely evaluated in order to determine the spin mixing conductance.For this, the thickness dependence of the saturation magneticmoment per unit area has been measured by VSM [shown inFig.2(c) only for the as grown and 300 /C14C annealed CoFeB films capped by the 10 nm thick Pt layer, for clarity] and then used to determine the magnetization at saturation andthe magnetic dead layer: the slope gives the magnetization atsaturation ( M s), while the horizontal axis intercept gives the extent of the magnetic dead layer. As indicated in Fig. 2(c), the magnetic dead layer thickness is nearly zero for these systems whatever the annealing temperature. The magnetiza- tion at saturation is determined to be 960 650, 970 650, 990650, and 1070 670 emu/cm3. Within the error bars, it shows a slight increase for the samples annealed at 300 C.For as grown films, the obtained value of the magnetizationat saturation is lower than that ( M s/C251200 emu/cm3) of W/CoFeB/MgO23and that ( Ms¼1100 emu/cm3) of Ru/ CoFeB/Ta and MgO/CoFeB/Ta.24However, it is also com- parable to the one ( Ms¼800–1100 emu/cm3) obtained byFIG. 1. X-ray 2 h-x(out of plane) diffraction patterns for Co 20Fe60B20(6 nm)/ Pt(10 nm) thin films annealed at various temperatures.113905-2 Belmeguenai et al. J. Appl. Phys. 123, 113905 (2018)Wang et al.25for Ru/CoFeB/Ru and Ta/CoFeB/Ta. Similar comments can be made for the annealed films at 300/C14C. Therefore, we conclude on the possibility of absence/weakproximity induced magnetization (PIM) in Pt. Since the mag- netization of the thin film strongly depends on its interfaces and crystallization degrees of CoFeB, the Msenhancement with annealing could be attributed to the crystallization of CoFeB at higher annealing temperature. However, since no XRD peaks have been detected for CoFeB films, the increaseinM sis most likely attributed to the boron diffusion away from CoFeB towards the interfaces, which can trigger a tran- sition to a nano-crystalline sate and therefore increase themagnetic moment. 22,26It is worth mentioning that it is very difficult to have a clear and detectable XRD signal from the nano-crystalline film with very small grain sizes. The MS-FMR spectra display a single line, identified with the uniform precession mode. The typical obtained MS- FMR spectra are represented in Fig. 3(a) for the as grown 6 nm thick CoFeB layer capped by a 10 nm thick Pt layer under an in-plane applied magnetic field. Due to the static magnetic field modulation (allowing lock-in detection toenhance the measured signal), the recorded signal is thus proportional to the field derivative of the absorption power as a function of the applied magnetic field. However, due to FIG. 3. (a) Ferromagnetic resonance spectra representing the amplitude of the field derivative of the absorbed power as a function of the applied mag- netic field for the as grown 6 nm thick CoFeB film capped by a 10 nm thickPt layer, measured at different driven frequencies. The symbols refer to experimental data and the solid lines are fits using Eq. (1). (b) Variation of the uniform precession mode frequency as a function of the perpendicularly applied magnetic field for Co 20Fe60B20(6 nm)/Pt(10 nm) heterostructures annealed at various temperatures Ta. The symbols refer to experimental data and the solid lines are fits using Eq. (2).FIG. 2. VSM hysteresis loops of the as grown Co 20Fe60B20(4 nm)/Pt(4 nm) measured for an in-plane magnetic field applied at an angle uHwith respect to the substrate edge. (b) Polar representation of the angular remanence curve (ARM) for the Co 20Fe60B20(6 nm)/Pt(10 nm) sample annealed at 300/C14C showing a weak uniaxial character. (c) Thickness dependencies of the satura-tion magnetic moment per unit area for the as grown and the 300 /C14C annealed CoFeB films of various thicknesses (t CFB) capped by the 10 nm thick Pt layer. Symbols in (c) refer to experimental data, and solid lines are linear fits.113905-3 Belmeguenai et al. J. Appl. Phys. 123, 113905 (2018)the coupling between the magnetic layer and the microstrip line, the recorded spectra are a mixture of dissipative (sym- metric) and dispersive parts (as obvious from the asymmetric line shape) of the susceptibility. Therefore, the resonance field and the FMR linewidth of this mode are obtained from the best fit of the recorded data assuming a line shape given by Eq. (1), as shown in Fig. 3(a) dPab dH¼A0/C02DHH/C0HR ðÞ cosðdÞþDH2/C0H/C0HR ðÞ2hi sinðdÞ DH2þH/C0HR ðÞ2hi2 þA1; (1) where ddenotes the mixing angle between the dispersive and the dissipative components, A0is the amplitude, A1is an offset value, HRis the resonance field, and DHis the half linewidth at half maximum. Note that the fitted experimental data with Eq. (1)allows deducing DH. However, since the field derivative of the asymmetric absorption signal, which has two peaks [as shown in Fig. 3(a)], is measured, an alter- native definition of the linewidth, namely the peak to peak linewidth ( DHPP) is used. The two linewidths are linked by DHPP¼2ffiffi 3pDH. Therefore, in this paper, only DHPP, deter- mined using the above-mentioned method, is presented. The gvalue, which determines the gyromagnetic factor cis of utmost important in this study since it is involved in the precise determination of the Gilbert damping and the effective magnetization. It is precisely accessible by the MS- FMR technique through the study of the frequency variationversus the amplitude of the applied magnetic field perpendic- ular to the film plane. The typical variations of the resonance frequency versus the perpendicular applied magnetic field are shown in Fig. 3(b) for 6 nm thick CoFeB films annealed at various temperatures T a. Owing to the theoretical varia- tions of the resonance frequency versus the normal to the film plane applied magnetic field given by Eq. (2), the best fits of the experimental data lead to c/2p¼30.13 GHz/T (g¼2.15), which does not show any significant variation versus Ta. This value is in good agreement with that obtained by Devolder12for the as deposited films F?¼c 2p/C18/C19 H/C04pMeff ðÞ ; (2) where 4 pMeff¼4pMs/C0H?17refers to the effective magneti- zation and H?is the perpendicular anisotropy field. In the above expression, the small in-plane anisotropy fields (less than 50 Oe as it will be shown below) have been neglected, since the applied magnetic fields overpass the 9 kOe in the investigated frequency range, as shown in Fig. 3(b). Figure 4(a)shows the angular dependencies of the reso- nance field of the different CoFeB thin films capped by a 10 nm thick Pt layer and annealed at 300/C14C. It shows that the angular behavior is governed by a small uniaxial anisot- ropy with the magnetization easy axis direction depending on the sample. To quantify the uniaxial anisotropy field (Hu), the angular dependence, shown in Fig. 4(a), has been analyzed using Eq. (3), giving the resonance frequency forthe in-plane applied magnetic field at an angle uHwith respect to the sample edges18 F2 ==¼c 2p/C18/C192 HcosðuH/C0uMÞþHucos 2ðuM/C0uuÞ ½/C138 /C2/C20 HcosðuH/C0uMÞþ4pMeff þHu 2ð1þcos 2ðuM/C0uuÞÞ/C21 ; (3) where uM(uH)anduuare angles defining the direction of the magnetization (the applied field) and the planar uniaxial anisotropy easy axis with respect to the substrate edges, respectively. The variations of Huversus the inverse CoFeB thickness (1/tCFB) are shown in Fig. 4(b).W h i l e Huof the as grown film does not show a clear behavior versus the CoFeB inverse thickness, clear linear behavior versus 1/ tCFBcan be observed for the annealed samples as T aincreases, suggesting an interfacial contribution to this the uniaxial anisotropy. Thederived uniaxial surface anisotropy constants are 1.6 /C210 /C03,FIG. 4. (a) Resonance field versus the direction of the in-plane applied field with respect to the sample edge ( uH) measured at 8 GHz driving frequency for Co 20Fe60B20(tCFB)/Pt(10 nm) annealed at 300/C14C. (b) CoFeB thickness (tCFB) dependence of the uniaxial in-plane anisotropy fields ( Hu) deduced the angular dependence of the resonance fields of Co 20Fe60B20(tCFB)/ Pt(10 nm) annealed at various temperatures. The symbols refer to experi- mental data, and the solid lines are the linear fits.113905-4 Belmeguenai et al. J. Appl. Phys. 123, 113905 (2018)3.3/C210/C03, and 3.7 /C210/C03erg/cm2,f o rt h e2 0 0/C14C, 250/C14C, and 300/C14C annealed films, while the volume uniaxial anisot- ropy field is around 30 Oe. The precise origin of this interface uniaxial anisotropy is not clear and a completely satisfactory interpretation of Huand of its Tadependence is still missing. The variation of the uniform precession mode resonance frequency has been measured versus the in-plane magnetic field applied along the easy axis and then fitted using Eq. (3) to extract the effective magnetization for each sample. Figure 5(a)shows the extracted effective magnetization ver- sus the inverse film thickness 1/ tCFBfor the various anneal- ing temperatures. It can be seen that Mefffollows a linear variation. We conclude that the perpendicular anisotropy field includes a surface energy term. Therefore, the effective perpendicular anisotropy constant K?(with H?¼2K?/Ms) could be phenomenologically separated in a volume and an interface contributions and approximately obeys the relation K?¼KvþKs/tCFB.27–29This allows us to derive the perpen- dicular surface anisotropy constants Ks¼1.33, 1.11, 0.97, and 0.74 erg/cm2, respectively, for Ta¼RT, 200/C14C, 250/C14C, and 300/C14C. Similarly, the deduced perpendicular volume constants are found to be Kv¼/C02.12/C2106,/C01.81/C2106,/C01.52/C2106, and /C00.52/C2106erg/cm3, respectively, for Ta¼RT, 200/C14C, 250/C14C, and 300/C14C. Both anisotropy con- stants decrease with increasing annealing temperature.Several explanations can be given for the origin of this linear behavior of the effective magnetization versus the CoFeB thickness and its dependence on T a: (i) magneto-elastic anisotropy contribution, (ii) roughness that creates in-plane demagnetizing fields at the edges of terraces reducing the shape anisotropy and therefore, favors perpendicular magne-tization, and (iii) interdiffusion and mixing which might occur at the interfaces during the deposition of the layers introducing thus, randomness in the magnetic pair bondsaccordingly, which obviously reduces the interface anisot- ropy. 29(iv) Another possible reason for the perpendicular magnetic anisotropy is the interfacial hybridization betweenthe magnetic and the Pt metals. These four mechanisms are annealing temperature dependent. Although, mixing at the interface should be excluded since no magnetic dead layerhas been revealed from the VSM measurement and in con- trast intermixing of Co (or Fe) with Pt forms CoPt or FePt alloys which are well-known to have perpendicular anisot-ropy (increase in anisotropy constants), it is not again obvi- ous to determine which mechanism is responsible of this interface anisotropy. Moreover, according to Ref. 29, the influence of the misfit strain appears as a volume contribu- tion to the anisotropy and can lead to an apparent interface contribution for ultrathin films below some critical thickness(around 3 nm). It is thus most likely that the electron hybridi- zation plays the most important role in dictating this interface magnetic anisotropy. The variation of M effversus the capping Pt layer thickness is shown in Fig. 5(b) for the 4 nm thick CoFeB films. While it fluctuates slightly for the as grown and films annealed at 200/C14C, significant changes can be observed for films annealed at 300/C14C, where significant inter- face changes occur. Its mean value increases with increasing Taconfirming the decrease of perpendicular anisotropy. T h efi e l dp e a kt op e a kl i n e w i d t h( DHPP) has been obtained from the fit [using Eq. (1)]o fM S - F M Rs p e c t r am e a - sured under an in-plane applied field at various directionswith respect to the substrate edges. Since the presence of extrinsic contributions 30to FMR linewidth, which are usually field direction dependent (anisotropic), lead to an overestima-tion of Gilbert damping, the angular dependence of DH PPhas been measured for each sample. The direction giving the min- imalDHPPvalue and thus minimizing the extrinsic contribu- tions to the linewidth has thus been determined. This in-plane direction is found to be sample dependent similar to the direc- tion of the in-plane anisotropy easy axis. The frequencydependencies of this linewidth have then been measured for the magnetic field applied along this direction as shown in Fig.6(a)for CoFeB(6 nm)/Pt(10 nm) annealed at various tem- peratures. It is clearly seen for all samples that DH PPvaries linearly with frequency, leading to the conclusion that the damping is Gilbert type and not caused by any other pro-cesses. Therefore, the used measurement scheme, consisting of determining the direction of the applied field giving the minimal value of DH PP, seems to be efficient to avoid any extrinsic contributions to damping since extrinsic contribu- tions to damping should lead to non-linear variation forFIG. 5. (a) CoFeB thickness dependence of the effective magnetization (4pMeff) extracted from the fit of FMR measurements for CoFeB thin films of thickness tCFBcapped with a 10 nm thick Pt layer and annealed at Ta. The symbols refer to experimental data, and the solid lines are linear fits. Pt thickness dependence of 4 pMefffor CoFeB(4 nm)/Pt( tPt) films annealed at Ta.113905-5 Belmeguenai et al. J. Appl. Phys. 123, 113905 (2018)DHPP. Moreover, as we mentioned above, the uniaxial anisot- ropy fields in our samples are very weak. Therefore, these fields are very low compared to the resonance fields at the driven frequencies used in this work (3–18 GHz). Indeed, in this condition, the magnetization direction will be almostparallel to the applied magnetic field ( uM¼uM). Therefore, experimental data of the frequency dependence of DHPPhave been fitted using the simple equation (4)leading to the deter- mination of the damping parameter a30,31 DHPP¼DH0þ2ffiffiffi 3pa c2pf; (4) where fis the driven frequency and DH0is the inhomoge- neous residual peak to peak linewidth, which is frequencyindependent. The multiplying factor of 1ffiffi 3p, in Eq. (4), is the correction of the difference between the full width at halfmaximum (FWHM) and the peak to peak linewidth for theline shape of Lorentzian. 31 The obtained results from the fit of experimental data, using Eq. (4), are shown in Figs. 6(b) and6(c)as a function of the CoFeB and the Pt thicknesses, respectively. Figure6(b) reveals that the damping constant increases linearly with 1/( t CFB). The dependence of aversus the Pt thickness, presented in Fig. 6(c)for the 4 nm thick CoFeB layer, shows an exponential behavior. Due to the linear variation of DHPP versus the frequency (Gilbert type damping) and since the extrinsic contributions to damping have been minimized, theenhancement of the damping as t CFBdecreases and its depen- dence on Pt thickness are attributed to the spin pumping cur-rent induced in Pt by the FMR precession of magnetization.In fact, according to the theory, 32if this additional damping (Da) is caused by spin pumping, then the FMR linewidth which contains aCFBþDa(where aCFB is the intrinsic Gilbert damping constant of CoFeB), should vary linearlyversus the microwave frequency as shown in Fig. 6(a).T h i s amount of the spin pumping is closely related to the SOCthrough the spin flip relaxation time and the spin mixing con-ductance that we aim to determine below. Moreover, it should be mentioned that the inhomogeneity contribution to the line- width broadening DH 0decreases as the CoFeB thickness increases and it increases with the annealing temperature asshown in the inset of Fig. 6(b): a linear dependence versus 1/ t CFB, suggesting an interface contribution, can be observed. To analyze this behavior of damping, we will consider two models (ballistic and diffusive models). Both modelsconsider that the net spin current ( I s) through the interface FM/NM is given by Is¼Ipump/C0Iback, where IpumpandIback are the spin pumped and the backflow spin currents, respec- tively, but differ in the evaluation of Iback. In the ballistic limit,33kSDis much lower than the mean free path of elec- trons. Therefore, the NM is considered as a perfect conductorand thus the I backvaries exponentially with the NM thick- ness.33Within this simple model, the thickness dependencies of the total damping are given by Eqs. (5)33,34and(6)33,35 a¼aCFBþglB 4pMstCFBg"#; (5) a¼aCFBþglB 4pMstCFBg"#1/C0e/C02tPt kSDhi ; (6) where lBis the Bohr magneton and g"#is the intrinsic spin mixing conductance of the interface CoFeB/Pt. Note thatEq.(5)is only valid for thick enough NM layers with respectFIG. 6. (a) Peak to peak FMR field linewidth versus the driven frequency measured for the magnetic field applied in the direction where DHPPis mini- mal and (b) CoFeB thickness dependence of the Gilbert damping parameter, deduced from the frequency dependence of the linewidth using Eq. (4),o f the CoFeB thin films of a thickness tCFBcapped by a 10 nm Pt layer and annealed at different temperatures. The inset shows the evolution of the inhomogeneous FMR linewidth versus the CoFeB thickness for various Ta. (c) Pt thickness dependence of the Gilbert damping parameter, deduced from the frequency dependence of the linewidth using Eq. (4), of the 4 nm thick CoFeB thin films capped with the Pt layer of thickness ( tPt) and annealed at different temperatures. The symbols refer to experimental data, and the solid lines in (b) and (c) are fits using the ballistic model described in the paper and the parameters summarized in Table I.113905-6 Belmeguenai et al. J. Appl. Phys. 123, 113905 (2018)tokSD, where no reflection of the spin current takes place at interfaces. It can thus be obtained from Eq. (6), which has been obtained by assuming that kSDis much lower than the mean free path of electrons and considering that the backflow spin current varies exponentially with the NM thickness. By fitting experimental data of Fig. 6with Eqs. (5)and(6), the determined values of aCFB,kSD,a n d g"#are summarized in Table I. The intrinsic damping of the CoFeB increases with the annealing temperature similar to the reported behavior by Conca et al.22and by Bilzer et al.36The increase in aCFBwith increasing annealing temperature could be caused by severe atomic intermixing between the magnetic and the adjacent nonmagnetic layers and/or by the develop- ment of polycrystalline grains. However, since no magnetic dead layer has been detected from the VSM measurements, the increase in aCFBis probably due the development of polycrys- talline grains after annealing. Note also the enhancement of the spin mixing conductance and spin diffusion length as the annealing temperature increases. An important phenomenon that accompanies the magne- tization precession induced spin injection is the back diffu- sion (backflow) of injected spins to the interface, which effectively reduces the spin current injection. This backflow current has been approximatively modelled within the ballis- tic approach [Eq. (6)]. Therefore, an alternative model (dif- fusive model)37where the spin accumulation at the FM/NM interface, generated by the injected spins, is taken into account when calculating the backflow spin current has been introduced. Within this model, the additional damping due to the spin pumping is given by Da¼glB 4pMstCFBg"# 1þg"# gext; (7) where gextis the electrical conductance of Pt and rPtis given bygext¼h e2rPt kSDtanhtPt kSD/C16/C17 ,37for a simple FM/NM interface. Note that in contrast to the ballistic model, this model takes into account the finite electrical resistance of the NM mate- rial, which depends on the preparation conditions and mayinclude interface contribution. Indeed, if the conductivity of a NM layer is assumed to be constant, but in reality it decreases with the decreasing layer thickness (as we willshow below), and the fitted spin diffusion length will incur a systematic inaccuracy. 38Thus, fits of spin pumping data with a model that ignores the thickness dependence of the conduc-tivity provide, at best, an upper bound. 38Therefore, to analyze the thickness dependence of damping through this model, thesheet resistance ( R Sheet) of the entire CoFeB(4 nm)/Pt(t Pt)h a s been measured as a function of the Pt thickness using a four-probe method with an in-plane current. Figure 7(a)shows the typical behavior for samples annealed at 200 and 300 /C14C for clarity. By assuming that the CoFeB layer resistivity is con-stant in CoFeB (4 nm)/Pt( t Pt) and that this CoFeB layer together with the Pt layer act as parallel resistors, the sheetresistance can be written as 38 RSheet¼tCFB qCFBþtPt qPt/C18/C19 ; (8) where qPt,qCFBare the Pt and the CoFeB resistivity and tCFB¼4 nm is the thickness of the CoFeB layer. The fit of experimental data of Fig. 7(a) with Eq. (8)revealed that the resistivities of Pt scale inversely with its thickness, sugges-ting an interfacial contribution due to interface scattering ofconduction electrons, 38,39and allowed to isolate both volume and interface contributions. While the volume resistivity ( qv) of Pt is found to be insensitive to the annealing temperature (qv¼1.7/C210/C07Xm and qv¼1.8/C210/C07Xm for Ta¼RT and 300/C14C, respectively), the interface resistivity ( qs) increases drastically with the annealing temperature (forinstance, q s¼9/C210/C016Xm2andqs¼16/C210/C016Xm2for Ta¼RT and 300/C14C, respectively), suggesting an enhanced interface scattering of charges, most likely due to B diffusionat the interface. The obtained values of the volume resistivi-ties are similar to the previously reported measured values(1.6–2 /C210 /C07Xm).38,40,41We used the function of the thick- ness dependent of qPt, obtained from the fit of RSheet mea- surements, with Eq. (8)to fit the experimental data of both the thickness dependencies of the damping as shown in Figs.7(b) and7(c). The fitting parameters are k SD,g"#, and aCFB. The obtained values are tabulated in Table I. Apart from the similar values of aCFB, the diffusive model gives higher (lower) values of g"#(kSD) compared to the ballistic model. The higher value of g"#in the diffusive model is a conse- quence of the underestimation of the backflow spin current in the ballistic model. Therefore, to obtain the same net spin current, the spin mixing conductance should be higher forthe diffusive model. The similar trend versus the annealingtemperature of the obtained values of k SDandg"#using the ballistic model is unexpected. Indeed, the increase in bothk SDandg"#is not compatible. Since the increase in the spin mixing conductance implies more losses of the magneticmoment in the NM material, one expects a decrease in thespin diffusion length, in accordance with the behavior of k SD and g"#deduced from fit using the diffusive model. The neglect of the electrical resistivity, especially when it isthickness dependent and of the spin accumulation in the bal-listic model could lead to wrong conclusion and interpreta-tion of the obtained results. The use of the spin diffusivemodel incorporating the thickness dependence of the Ptresistivity leads to an accurate determination of the intrinsicTABLE I. Parameters obtained from the best fit of the thickness dependen- cies of damping of CoFeB/Pt systems annealed at various temperaturesusing two different models (Ballistic and diffusive), described in the text. n.m refers to “not measured.” T a(/C14C) Model aCFB(/C210/C03) g"#(nm/C02) kSD(nm) RT Ballistic 3.4 38.6 1.7 Diffusive 3.4 42.1 0.37 200 Ballistic 3.23 41 1.75 Diffusive 3.23 45.3 0.37 250 Ballistic 4.2 42.2 n.m Diffusive n.m n.m n.m 300 Ballistic 6.4 52.4 3 Diffusive 6.5 65.5 0.28113905-7 Belmeguenai et al. J. Appl. Phys. 123, 113905 (2018)spin-mixing conductance and the spin diffusion length. It is worth mentioning that due to the lack of experimental proof that spin transport can proceed via ballistic channels inmetallic multilayer structures even when thicknesses are less than the mean free path, 42the diffusive model remains more realistic for experimental data analysis. We should mentionthat the increase in the intrinsic damping, the intrinsic spin mixing conductance, the surface resistivity of Pt, and the decrease in the spin diffusion length and the perpendicularanisotropy as the annealing temperature increases are coher- ent and could be linked to the evolution of the CoFeB/Pt interface with annealing. However, the exact reason for thisbehaviour remains unclear and only speculation can be given. The efficiency of spin current injection across CoFeB/ Pt interfaces and thus the spin mixing conductance dependson the crystalline ordering of CoFeB, as well as the smooth- ness and cleanliness of the interface. In fact, according to first-principles calculation 43,44the interface roughness as well as the disorder generally enhances spin mixing conductance and hence the s-d exchange coupling. Surface/interface rough- ness and disorder in thin films contribute to electron scatteringand thus to film electrical resistivity. 45,46Therefore, since a drastic increase in the surface contribution to the electric resistivity for samples annealed at 300/C14C has been observed, we speculate that the interface roughness and/or disorder increases with the annealing temperature. The enhancement of the spin mixing conductance of the CoFeB/Pt interfacewith the annealing temperature is thus most probably due to the increase in the disorder and/or interface roughness. This is in good agreement with the observed decrease in the interfaceanisotropy as the annealing temperature increases. Note the discrepancy of the published values of the spin diffusion length (due to the interface quality and conductivity differ-ence which are growth dependent) for Pt, ranging from 0.5 to 10 nm. 38Lower values have been also obtained38and there- fore, the obtained values here of kSDusing the two models are within this range. Finally, we should comment on the effect of the proxim- ity induced magnetization on the damping. In FM/Pt systems,it is usually expected that the ferromagnetism may extend beyond the physical structure of the interface and results in proximity induced magnetization (PIM) in Pt. This producesextra damping as it has been observed by Sun et al. 47in yttrium iron garnet (YIG)/Pt and affects the spin mixing con- ductance. According to Sun et al. , the extra damping origi- nates from the ferromagnetic ordering in Pt atomic layers near the FM/Pt interface and the dynamic exchange coupling between the ordered Pt spins and spins in the FM film.47 Indeed, Sun claimed that because of the presence of the ferro- magnetic Pt (FM Pt), the conventional spin pumping from the YIG film (in his case) to the Pt film does not occur. However,there exists a spin pumping from the FM Pt into the paramag- netic Pt, which contributes to the damping of the FM Pt. The dynamic YIG-Pt coupling allows for the transfer of a part ofthe damping of the FM Pt to the YIG film. This extra damping cannot be described by existing models and therefore new theoretical models on damping in FM/NM are needed. Thisrole has been addressed experimentally by Caminale et al. 48 who showed that PIM yields a nearly linear dependence ofFIG. 7. (a) Pt thickness dependence of the four-probes measured sheet resistance of CoFeB(4 nm)/Pt( tPt) films annealed at 200 and 300/C14C. The symbols refer to the measurements, and the solid lines are fits using Eq. (8). (b) CoFeB thickness dependence of the Gilbert damping parameter, deduced from the frequency dependence of the linewidth using Eq. (4),o f CoFeB thin films of thickness tCFB capped by a 10 nm Pt layer and annealed at different temperatures. (c) Pt thickness dependence of the Gilbert damping parameter, deduced from the frequency dependence of the linewidth using Eq. (4), of the 4 nm thick CoFeB thin films capped with the Pt layer of thickness ( tPt) and annealed at different temperatures. The symbols refer to experimental data, and the solid lines in (b) and (c)are fits using the diffusive model described in the paper and the parameters summarized in Table I.113905-8 Belmeguenai et al. J. Appl. Phys. 123, 113905 (2018)the interface-related Gilbert damping enhancement on the heavy metal layer thicknesses for low thickness values (below 2 nm for Pt). In the absence of x-ray resonant magnetic reflec-tivity measurements (XRMR), we cannot conclude on the absence/presence of PIM in our sample although the lower measured M swith respect to other systems including CoFeB. Therefore, we cannot estimate the contribution of PIM to spin pumping damping. XRMR combined to MS-FMR measure- ments will be used in forthcoming papers to address the linkbetween PIM and spin pumping damping. IV. CONCLUSION The thickness dependencies of Gilbert damping constant have been used to measure both the spin diffusion length andthe spin mixing conductance in CoFeB/Pt systems via the ferromagnetic resonance induced spin pumping. Special interest has been given to the effect of the annealing temper-ature on damping, spin mixing conductance, spin diffusion length, and magnetic anisotropy. The obtained results on damping demonstrate the efficiency of the possibility of tun-ing the Gilbert damping constant by a judicial choice of the non-magnetic layer and the annealing temperature, depend- ing on the desired application. The experimental results havebeen analysed considering diffusive and ballistic models to deal with the additional damping induced by the spin pump- ing. A comparison between the two models revealed that thediffusive model enhances the accuracy on the determination of spin mixing conductance and spin diffusion length, espe- cially when the electrical resistivity of the NM is thicknessdependent. We also showed that the perpendicular anisot- ropy in CoFeB/Pt systems includes an interface contribution that increases with decreasing annealing temperatures. ACKNOWLEDGMENTS This work has been partially supported by the Conseil R/C19egional, ^Ile-de-France through the DIM NanoK (BIDUL project). 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1.3561744.pdf

Thermal transpiration flow: A circular cross-section microtube submitted to a temperature gradient Marcos Rojas Cardenas, Irina Graur, Pierre Perrier, and J. Gilbert Meolans IUSTI UMR 6595, Université de Provence–Polytech’ Marseille, 5, rue Enrico Fermi, 13453 Marseille cedex 13, France /H20849Received 12 November 2010; accepted 13 February 2011; published online 9 March 2011 /H20850 Thermal transpiration is the macroscopic movement of rarefied gas molecules induced by a temperature gradient. The gas moves from the lower to the higher temperature zone. An originalmethod is proposed here to measure the mean macroscopic movement of gas in the case of a longcircular cross-section glass microtube onto which a gradient of temperature is applied. The massflow rate and the thermomolecular pressure difference have been measured by monitoring theabsolute pressure evolution in time at both ends of the capillary using high-speed response pressuregauges. Two gases, nitrogen and helium, are studied and three different temperature differences of50, 60, and 70 °C are applied to the tube. The analyzed gas rarefaction conditions vary fromtransitional to slip regime. © 2011 American Institute of Physics ./H20851doi:10.1063/1.3561744 /H20852 It is well known that applying a disequilibrium of tem- perature to a tube filled with a rarefied gas without any initial difference of pressure or any difference in chemical consti-tution the gas will macroscopically move from the lower tothe higher temperature zone. This phenomenon was namedthermal transpiration by Reynolds 1in 1878. In an investiga- tion with a plaster-of-Paris plug separating two regions /H208491 and 2 /H20850maintained at different temperatures, he showed that at very low densities the equilibrium pressures on the twosides are related by the law p 2/p1=/H20849T2/T1/H208501/2. Maxwell2 closely followed this investigation up by mathematically analyzing the phenomenon using, at that time, the still con-troversial kinetic theory. Later on, in 1909, Knudsen 3proved Reynolds law for the case of a tube by using both theoreticaland experimental research. Knudsen’s research led to thesupposition that the above cited law had validity only at azero-flow final equilibrium state, which followed a transi-tional stage of gas displacement. Later, Liang 4in 1951 and Takaishi et al.5in 1963 proposed semiempirical formulas that predict how the thermomolecular pressure ratio p2/p1at the final equilibrium zero-flow state varies in function of thegas rarefaction, the gas physical properties, and the appliedtemperature difference to the tube. Consequentially, severalexperimental studies were realized: Edmonds et al. 6in 1965 compared their results with the semiempirical equation ofLiang for thin wall apertures finding a good agreement forhelium; Watkins et al. 7in 1966 compared their results with the semiempirical equation of Weber et al. and found data with a satisfying qualitative agreement; Annis8in 1972 suc- cessfully compared his thermomolecular pressure difference/H20849tpd/H20850p 2−p1experimental results with the numerical results of Loyalka et al. ,9who studied thermal transpiration on a cylindrical tube by using the Bhatnagar–Gross–Krook model. Finally, in 1978, Porodnov et al.10emphasized that despite the great number of experimental works, still no cor-rect theory existed for the tpd effect at arbitrary gas rarefac-tion conditions. This work, 10in combination with the workof Storvick et al. ,11gave a final good characterization of the thermomolecular pressure difference for different gases atdifferent rarefaction conditions by applying different tem-perature gradients to the tube. In recent times, the advent ofmicroelectromechanical systems made way for new perspec-tives on thermal transpiration. The possibility of using the pumping effect of thermal transpiration to create a micro-compressor without moving parts led to the experimentalworks of Vargo et al. 12in 1999 and Han et al.13in 2007, in which the attention is mainly focused on the pressure in-crease due to the application of a temperature gradient alonga channel. This final work of Han et al. 13together with the work of York et al.14in 1999 gave a first characterization of the transitional stage of gas displacement by measuring thepressure variation in time before reaching the final zero-flowstate. At the moment, no real efforts have been made to mea-sure the mass flow rate induced by thermal transpiration. In this letter, an original method for thermal transpiration mass flow rate /H20849mfr/H20850measurements is proposed using Knud- sen’s intuition of a transitional gas displacement stage. Herewe present the first results of the investigation conducted viameasuring in situ the pressure evolution in time at both ends of the tube using two high-speed response pressure gauges. The rarefaction parameter delta /H9254=1 2pD/H20849/H9262/H208812RT/H20850−1indicates the rarefaction state of the gas and depends on thermody- namic parameters as pressure pand temperature T, the diam- eter D, which is the characteristic length of the system, and the gas physical properties as the viscosity /H9262and the specific gas constant R. Due to the tube’s characteristic length and the applied pressure working conditions, which vary between13.3 and 465.5 Pa for nitrogen and between 13.3 and1330 Pa for helium, the gas rarefaction conditions vary fromtransitional to slip regime: helium /H9254=0.15–15.3; nitrogen /H9254=0.45–15.6. The long circular cross-section glass microtube /H20849D=490/H110061/H9262m;L=3.053 /H110060.01 cm /H20850is connected to two reservoirs /H20849Fig.1/H20850. One reservoir is heated while the other is maintained at room temperature. The imposed temperaturePHYSICS OF FLUIDS 23, 031702 /H208492011 /H20850 1070-6631/2011/23 /H208493/H20850/031702/4/$30.00 © 2011 American Institute of Physics 23, 031702-1difference between both reservoirs creates a steep tempera- ture gradient along the microtube. The temperature gradientwas proved to be linear by monitoring the temperature dis-tribution on the tube’s surface using an infrared camera.Each reservoir is coupled to a high-speed response time/H2084930 ms /H20850capacitance diaphragm gauge /H20849CDG /H20850. The hot-side CDG is especially performative for high gas-temperatureworking conditions. The temperature is monitored at bothends of the microtube and at the hot-side reservoir usingthree different thermocouples. The temperature is maintainedstable during the whole duration of the experiment. The internal ring of the system where microtube and reservoirs are positioned has two main functions: to connectthe reservoirs twice, through the microtube and through a bigdiameter tube junction equipped with a below-sealed valve/H20849valve A /H20850, and to dampen any pressure oscillations produced by the vacuum system having, consequentially, the functionof a stabilizing chamber. The external open ring from oneside supplies pure gas to the internal ring by opening theregulation butterfly-valves of two external high-pressuretanks containing helium and nitrogen; from the other side, itjoints the internal ring a vacuum pump which can vacuumthe system until 0.1 Pa. The pressure inside the internal ringis regulated by means of a below-sealed regulation valve/H20849valve B /H20850. The methodology of the experiment is defined by four main stages /H20849Fig. 2/H20850. After imposing inside the stabilizing chamber the experimental gas pressure conditions, valves Band C are closed /H20849stage 1 /H20850. These two valves will remain closed for the whole duration of the experience. At this mo-ment, the internal ring starts damping all the residual oscil-lations of pressure until a pressure equilibrium stage isreached. In order to have no pressure difference in the reser-voirs, valve A is kept open: this allows the pressure to dis-tribute itself in a continuous and equal manner in the system.In this first stage, the thermal transpiration phenomenon isalready present. A stationary macroscopic gas displacementalready exists along the tube due to the imposed temperaturegradient. In order to understand the physical basic mecha-nism causing the movement of the gas, which is initially atrest when a temperature difference is still not imposed, it isnecessary to regard the boundary conditions. It is possible toshow that the balance of momentum exchange between anelementary wall surface and the molecules of a fluid particleresults in a force which is applied to the gas in the tempera-ture gradient direction. 15In other respects, the imposed gra- dient of temperature creates a gradient of density in directionof the tube’s axis. The gas motion tends to reduce the density gradient: the gas flows from the higher to the lower densityregion id est from the cold-side to the hot-side reservoir. It is possible to consider the cold-side reservoir as a gas sourcevolume, while the hot-side reservoir can be considered as astorage volume. At time t=0, valve A is closed. At time t =0 +, the transitional stage /H20849stages 2 and 3 /H20850of the experiment starts: the pressure variates in time increasing in the outletreservoir while decreasing in the inlet reservoir. This is thestarting point of the experience and the pressure gaugesmonitor the pressure variation in time p/H20849t/H20850. In the first phase of the transitional stage /H20849stage 2 /H20850, the pressure variation in time is linear and thus the phenomenon is considered to bestationary. Subsequently, the pressure-variation speed /H20849pvs/H20850 decreases /H20849stage 3 /H20850: this second phase of the transitional stage is characterized by a nonlinear pressure variation intime and thus the phenomenon is considered to be nonsta-tionary. The pvs decrease directly depends on the continuousincrease of the pressure difference between the reservoirscreated by the gas displacement. The pressure difference cre-ates a flow which is directed contrarily to the thermal tran-spiration flow. At time t=0 +/H20849stage 2 /H20850, this counterflow is zero and is negligible during the first phase of the transitionalstage. Consequentially, in the nonstationary phase of the ex-periment /H20849stage 3 /H20850, the flow driven by the created pressure difference is not negligible and modifies the mean mass flowrate intensity. The counterflow will completely balance thethermal transpiration flow at the final stage of the experiment/H20849stage 4 /H20850. When the pressure difference arrives to its maxi- mum, the mean mass flow rate is zero. This is the final equi-librium stage known as the zero-flow state where the final f on initial ipressure ratio for the hot-side reservoir his p fh/pi/H110221, while for the cold-side reservoir cispfc/pi/H110211. The here adopted notations pfhandpfcsubstitute the initially cited notations of Reynolds law p2andp1, respectively. The pressure variation inside the reservoirs ph/H20849t/H20850and pc/H20849t/H20850are exponential and are well defined by the functions fh/H20849t/H20850=A/H208511−e−/H20849t//H9270h/H20850/H20852and fc/H20849t/H20850=B/H20851e−/H20849t//H9270c/H20850−1/H20852for the hot-side and cold-side, respectively, where A,B, and/H9270are adjustable parameters. The average standard error of the pressure-variation fitting is within /H110061%. The two functions f h/H20849t/H20850and fc/H20849t/H20850are not perfectly mirror-symmetric with respect to the zero pressure axis, since the volumes of the two reservoirs differ Vh/Vc=0.81 and, consequentially, the pressure varia- tion in time in the reservoirs is different. The pressure-variation speed is given by df h/H20849t/H20850/dt=/H20849A//H9270h/H20850e−/H20849t//H9270h/H20850and dfc/H20849t/H20850/dt=−/H20849B//H9270c/H20850e−/H20849t//H9270c/H20850. The pressure variation and theN2 Hevalve AVacuum pump microtube H O TC O L D CDGHeaterthermocouples CDGCDG 1 2 3 D.A.Q.>>valve Cvalve B FIG. 1. Experimental setup scheme: circular cross-section microtube con- nected to two reservoirs, vacuum pump, heater, acquisition system, pressuregauges /H20849CDG /H20850, three thermocouples, regulation valves, and high-pressure gas tanks.1.451.461.471.481.491.5 0 10 20 30 40 50 60Pressure [torr] Time [s]Pressure variation inside cold-side reservoir Pressure variation inside hot-side reservoir2 1 3 4 FIG. 2. /H20849Color online /H20850Helium Th−Tc=60 °C. /H208491/H20850Initial pressure equilib- rium stage. /H208492/H20850Stationary transitional stage. /H208493/H20850Nonstationary transitional stage. /H208494/H20850Final pressure equilibrium stage.031702-2 Rojas Cardenas et al. Phys. Fluids 23, 031702 /H208492011 /H20850pressure-variation speed behaviors depend on the gas physi- cal properties, the applied temperature difference, and thegas rarefaction conditions. In Fig. 3, the fitting functions of the pressure evolution in time f hand fcand the respective pvsdfh/H20849t/H20850/dtanddfc/H20849t/H20850/dtare shown for different conditions of the gas rarefaction which is indicated in function of delta . The values of the rarefaction parameter delta slightly varyalong the axis of the tube due to the imposed temperaturedifference and slightly vary during the duration of the experi-ment due to the pressure variation in time: here the rarefac-tion parameter is used to characterize each experiment atdifferent gas rarefaction conditions. Thus, an average of deltais given considering the variation of the thermodynamic pa-rameters along the axis at the first stage of the experiment/H20849stage 1 /H20850. The pressure-variation speed increases with increasing values of the rarefaction parameter and seems to tend asymp-totically to a maximum value for gas rarefaction conditionsof /H9254/H1101513.2. The linearity of the pressure variation in the first phase of the transitional stage is proven as shown in Fig.3/H20849d/H20850: the pvs is almost constant in the first second for rar- efaction parameter values inferior to /H9254/H110156.6. For higher val- ues, the pressure-variation linearity is still a good approxi-mation if considered at time t=0 +. Due to the different molar mass /H20849MN2/H110157MHe/H20850, helium has a higher pressure-variation speed with respect to the one of nitrogen. The classical approach to thermal transpiration advises us to look at the zero-flow equilibrium pressure difference atthe end of the experience. This creates a comprehensiveoverview on the properties of the phenomenon in function ofthe gas physical properties, the applied temperature differ-ences, and the gas rarefaction conditions. In Fig. 4, the tpd for nitrogen and helium are shown. At the beginning of the transitional regime /H20849 /H9254/H110150.44 /H20850, the pressure difference starts at a minimum value and then rapidly increases for higher values of delta. It is possible toobserve how the thermal transpiration pumping effecttouches a maximum value in the transitional regime. As ex-pected, the intensity of this maximum is greater for higherapplied temperature differences. Helium’s maximum thermo- molecular pressure difference is /H110113 times greater than nitro- gen’s maximal values. The maximum values are to be foundfor both gases at the same rarefaction conditions /H9254/H110154–5. The tpd strongly decreases while exiting the transitional re-gime and entering the slip regime region. In this work, it is shown that the adopted original mea- sure mechanism leads to a time-dependent phenomenonsince after time t=0 +, the pressure-variation speed decreases with time. If the microtube’s ends were connected to infinitevolumes, the pressure-variation speed will stay constant forthe whole duration of the process leading to a constant fluxof gas through the tube. Taking into account the initial sta-tionary not-perturbed state of the flow, before time t=0, the mass flow rate can also be considered stationary at timet=0 +after the valve’s closure. Here the mean mass flow rate is identified to be at its maximum intensity and has not yetbeen reduced by the created pressure difference. In the res-ervoirs, the stationary mass flow rate is easily related to aconstant pressure-variation speed, 16which here is given by /H20849dfc/dt/H20850/H20841t→0+. Thus from Ref. 16we obtain Qc=Vc RTc·/H20849−B/H20850 /H9270c. /H208491/H20850 Equation /H208491/H20850gives the mean mass flow rate of the gas at the microtube’s inlet, leaving the cold-side reservoir: this res-ervoir is considered to be a gas source volume, thus notedwith a minus sign. From the law of conservation of mass, thesame mean mass flow rate intensity is registered at the mi-crotube’s outlet in the hot-side reservoir: this reservoir isconsidered to be a gas storage volume. In Fig. 5, the mass flow rate results for helium and nitrogen are shown for awide spectrum of gas rarefaction conditions and for threedifferent applied temperature differences. As expected, an increase of the mass flow rate intensity is achieved by increasing the rarefaction parameter valuesshifting them from transitional to slip regime conditions.Higher temperature differences lead to increased values ofthe mass flow rate intensity. The gas physical properties giveas well different values of the mfr, which are higher fornitrogen in transitional regime. In slip regime conditions, thetendency seems to change at least for the higher applied tem-perature difference: helium’s mass flow rates are slightlyhigher than nitrogen’s. It is to be noticed that the pressureworking conditions of the gases are different: it is possible toobtain higher mfr intensities for nitrogen if the same pres-sures are applied to the system /H20851Fig. 5/H20849c/H20850/H20852. Helium instead FIG. 3. /H20849Color online /H20850Nitrogen Th−Tc=60 °C. /H20849a/H20850Example of the pressure variation in time for /H9254=3.5 and superposed the corresponding numerical fit. /H20849b/H20850Pressure variation in time fitting curves; each curve represents a different gas rarefaction condition. /H20849c/H20850Pressure-variation speed curves in time. /H20849d/H20850 Pressure-variation speed curves logarithmic scale in time. FIG. 4. /H20849Color online /H20850Thermomolecular pressure difference pfh−pfcin function of the rarefaction parameter delta for three imposed temperaturedifferences T h−Tc=50, 60, and 70 °C. /H20849a/H20850Helium. /H20849b/H20850Nitrogen.031702-3 Thermal transpiration flow Phys. Fluids 23, 031702 /H208492011 /H20850can be used at higher pressures in order to obtain same order mfr intensities than with respect to nitrogen. The measure-ments have a relative error of /H110062.3%. In this letter, we have proven that by the use of this original method, it is possible to measure the mass flow rateinduced through a tube by thermal transpiration. The ana-lyzed mass flow rate intensities are very small due to thesmall diameter of the here used microtube and the relativelylow temperature differences applied to the tube. The heremeasured values of the mass flow rate vary from 5.8/H1100310 −13/H20851kg /s/H20852for helium at a temperature difference of 50 °C in transitional regime conditions /H20849/H9254=0.18 /H20850to 2.2 /H1100310−11/H20851kg /s/H20852for helium at a temperature difference of 70 °C in slip regime conditions /H20849/H9254=14.7 /H20850. This means that in function of the applied temperature and the rarefaction work- ing conditions, the mass flow rate intensity can vary by /H1101138 times. When lower pressures are applied to the system, largerdiameters dimensions can be used to obtain the same gasrarefaction conditions and thus increase the mass flow rateintensity. Thermal transpiration is prevalent as well in ambi-ent pressure working devices with nanoscale characteristiclength dimensions. This leads to the possibility of using ther-mal transpiration in a vast area of devices and working con-ditions which do not have to be necessarily of extremely small scale or work at extremely low pressure. No movingparts are needed to create a pressure difference between theends of the device or to create a mass flow rate through thetube. Thermal transpiration has potential for applications inmass spectrometers, gas chromatographs, air-pollution mea-suring systems, and all systems requiring precise control ofgas flow. The research leading to these results has received fund- ing from the European Community’s Seventh FrameworkProgram /H20849FP7/2007–2013 under Grant Agreement No. 215504 /H20850. 1O. Reynolds, “On certain dimensional properties of matter in the gaseous state. Part I and Part II,” Proc. R. Soc. Lond. 28, 303 /H208491878 /H20850. 2J. C. Maxwell, “On stresses in rarified gases arising from inequalities of temperature,” Philos. Trans. R. Soc. Lond. 170, 231 /H208491879 /H20850. 3M. Knudsen, “Eine revision der gleichgewichtsbedingung der gase. Ther- mische molekularstromung,” Ann. Phys. 336, 205 /H208491909 /H20850. 4S. C. Liang, “Some measurements of thermal transpiration,” J. Appl. Phys. 22, 148 /H208491951 /H20850. 5T. Takaishi and Y . Sensui, “Thermal transpiration effect of hydrogen, rare gases and methane,” Trans. Faraday Soc. 59, 2503 /H208491963 /H20850. 6T. Edmonds and J. P. Hobson, “A study of thermal transpiration using ultrahigh-vacuum techniques,” J. Vac. Sci. Technol. 2,1 8 2 /H208491965 /H20850. 7R. A. Watkins, W. L. Taylor, and W. J. Haubach, “Thermomolecular pres- sure difference measurements for precision helium-3 and helium-4 vapor- pressure thermometry,” J. Chem. Phys. 46, 1007 /H208491967 /H20850. 8B. K. Annis, “Thermal creep in gases,” J. Chem. Phys. 57, 2898 /H208491972 /H20850. 9S. K. Loyalka and J. W. Cipolla, Jr., “Thermal creep slip with arbitrary accommodation at the surface,” Phys. Fluids 14, 1656 /H208491971 /H20850. 10B. T. Porodnov, A. N. Kulev, and F. T. Tuchvetov, “Thermal transpiration in a circular capillary with a small temperature difference,” J. Fluid Mech. 88, 609 /H208491978 /H20850. 11T. S. Storvick, H. S. Park, and S. K. Loyalka, “Thermal transpiration: A comparison of experiment and theory,” J. Vac. Sci. Technol. 15, 1844 /H208491978 /H20850. 12S. Vargo, E. P. Muntz, G. R. Shiflett, and W. C. Tang, “Knudsen compres- sor as a micro- and macroscale vacuum pump without moving parts orfluids,” J. Vac. Sci. Technol. A 17, 2308 /H208491999 /H20850. 13Y . Han, E. P. Muntz, A. Alexeenko, and M. Young, “Experimental and computational studies of temperature gradient-driven molecular transportin gas flows through nano/microscale channels,” Nanoscale Microscale Thermophys. Eng. 11, 151 /H208492007 /H20850. 14D. C. York, A. Chambers, A. D. Chew, and A. P. Troup, “Measurement of thermal transpiration across an array of parallel capillaries using a differ-ential capacitance manometer,” Vacuum 55, 133 /H208491999 /H20850. 15T. Ewart, P. Perrier, I. Graur, and J. G. Méolans, “Mass flow rate mea- surements in gas micro flow,” Exp. Fluids 41,4 8 7 /H208492006 /H20850. 16Y . Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications /H20849Birkhauser, Berlin, 2007 /H20850, pp. 237–238. FIG. 5. /H20849Color online /H20850Mass flow rate for three imposed temperature differ- ences Th−Tc=50, 60, and 70 °C. /H20849a/H20850Helium in function of the rarefaction parameter /H9254./H20849b/H20850Nitrogen in function of the rarefaction parameter /H9254./H20849c/H20850 Helium and nitrogen in function of the pressure.031702-4 Rojas Cardenas et al. Phys. Fluids 23, 031702 /H208492011 /H20850Physics of Fluids is copyrighted by the American Institute of Physics (AIP). Redistribution of journal material is subject to the AIP online journal license and/or AIP copyright. For more information, see http://ojps.aip.org/phf/phfcr.jsp

1.4960705.pdf

Low Gilbert damping in Co 2FeSi and Fe 2CoSi films Christian Sterwerf, , Soumalya Paul , Behrouz Khodadadi , Markus Meinert , Jan-Michael Schmalhorst , Mathias Buchmeier , Claudia K. A. Mewes , Tim Mewes , and Günter Reiss Citation: J. Appl. Phys. 120, 083904 (2016); doi: 10.1063/1.4960705 View online: http://dx.doi.org/10.1063/1.4960705 View Table of Contents: http://aip.scitation.org/toc/jap/120/8 Published by the American Institute of Physics Articles you may be interested in Voltage-controlled magnetization switching in MRAMs in conjunction with spin-transfer torque and applied magnetic field J. Appl. Phys. 120, 203902203902 (2016); 10.1063/1.4968543 Origin of low Gilbert damping in half metals J. Appl. Phys. 95, 022509022509 (2009); 10.1063/1.3157267 Low Gilbert damping in Co 2FeSi and Fe 2CoSi films Christian Sterwerf,1,a)Soumalya Paul,2Behrouz Khodadadi,2Markus Meinert,1 Jan-Michael Schmalhorst,1Mathias Buchmeier,2Claudia K. A. Mewes,2TimMewes,2 and G €unter Reiss1 1Center for Spinelectronic Materials and Devices, Physics Department, Bielefeld University, 33615 Bielefeld, Germany 2Department of Physics and Astronomy/MINT Center, The University of Alabama, Tuscaloosa, Alabama 35487, USA (Received 30 June 2016; accepted 29 July 2016; published online 26 August 2016) Thin highly textured Fe 1þxCo2–xSi (0/C20x/C201) films were prepared on MgO (001) substrates by magnetron co-sputtering. Magneto-optic Kerr effect (MOKE) and ferromagnetic resonance (FMR)measurements were used to investigate the composition dependence of the magnetization, the mag- netic anisotropy, the gyromagnetic ratio, and the relaxation of the films. Both MOKE and FMR measurements reveal a pronounced fourfold anisotropy for all films. In addition, we found a stronginfluence of the stoichiometry on the anisotropy as the cubic anisotropy strongly increases with increasing Fe concentration. The gyromagnetic ratio is only weakly dependent on the composition. We find low Gilbert damping parameters for all films with values down to 0 :00126 0:0007 0:0001 for Fe1.75Co1.25Si. The effective damping parameter for Co 2FeSi is found to be 0 :001860:0034 0:0004. We also find a pronounced anisotropic relaxation, which indicates significant contributions of two-magnon scattering processes that is strongest along the easy axes of the films. This makes thinFe 1þxCo2–xSi films ideal materials for the application in spin transfer-torque magnetic RAM (STT- MRAM) devices. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4960705 ] I. INTRODUCTION Half-metallic ferromagnets have attracted great interest during the past few years because they promise to boost theperformance of spintronic devices. High spin polarization at the Fermi level can generate high tunnel magnetoresistance (TMR) ratios. A TMR effect can be measured in a magnetictunnel junction (MTJ) that consists of two ferromagneticfilms separated by a thin insulator. The same structures can also be utilized to spin transfer torque induced magnetization switching; 1however, in this case, a low switching current density is desirable. Thus, low magnetic damping and a high spin polarization are frequently required for spin transfer tor- que based devices.2A high spin polarization can be found in half-metals where one spin band structure is semiconducting while the other spin band structure is metallic. Co- and Fe- based Heusler compounds are good candidates for materialswith high Curie temperatures and half-metallic behavior. Full Heusler compounds have the formula X 2YZ, where X and Y are transition metals and Z is a main group element. There are two different ordered structures: the L2 1structure and the X astructure with a different occupation sequence. Both structures consist of a four-atom basis and an fcc lat-tice. The prototype of the L2 1structure is Cu 2MnAl (space group Fm /C223m) with the occupation sequence X-Y-X-Z.3The prototypes for the X astructure are Hg 2CuTi and Li 2AgSb with an occupation sequence Y-X-X-Z, with the two X-atoms at inequivalent positions in the lattice.4,5In this work, we investigate the magnetic properties of a stoichio-metric series ranging from Co 2FeSi to Fe 2CoSi, where Co2FeSi crystalizes in the L2 1structure and Fe 2CoSi in theXastructure, respectively. Both compounds should have a (pseudo-)gap in the minority states as predicted by first prin-ciple calculations. By substituting Co and Fe atoms the num-ber of electrons varies and the Fermi level is expected to beshifted to lower energies when the Fe concentration is increased. As we reported previously, magnetic tunnel junc- tions based on the Fe 1þxCo2–xSi films exhibit very high TMR ratios for all stoichiometries.6At 15 K a maximum TMR ratio of 262% was found for the intermediate stoichi-ometry Fe 1.75Co1.25Si, while the Co 2FeSi and Fe 2CoSi based MTJs showed a TMR ratio of 167% and 227%, respectively.One possible explanation for the high TMR ratio is that for Fe 1.75Co1.25Si the Fermi energy is shifted inside the pseudo- gap. In this work, we present results of the magnetic proper-ties for the magnetization dynamics, in particular, includinganisotropy and the Gilbert damping parameter of theFe 1þxCo2–xSi films, as the intrinsic relaxation is expected to be low for half-metals.7 II. PREPARATION AND CHARACTERIZATION TECHNIQUES Thin Fe 1þxCo2–xSi (x¼0, 0.25, 0.5, 0.75, 1) films were fabricated using co-sputtering in an UHV sputtering systemwith a base pressure of 1 /C210 /C09mbar. The Ar pressure dur- ing sputtering was 2 /C210/C03mbar. The films were grown by dc- and rf-magnetron sputtering from elemental targets ontoMgO (001) substrates. Additional MgO and Cr seed layerswere used to accommodate small lattice mismatches and to promote coherent and epitaxial growth, as the Cr seed layer grows in 45 /C14direction on the MgO layer, which has a lattice parameter of 4.212 A ˚. The lattice mismatch between two unit cells of Cr (2 /C22:885 A ˚at 20/C14C (Ref. 8)) and one unita)Electronic mail: csterwerf@physik.uni-bielefeld.de 0021-8979/2016/120(8)/083904/6/$30.00 Published by AIP Publishing. 120, 083904-1JOURNAL OF APPLIED PHYSICS 120, 083904 (2016) cell of Co 2FeSi (5.64 A ˚(Ref. 9)) or Fe 2CoSi (5.645 A ˚(Ref. 10)) is about 2%. The 5 nm thick MgO and Cr films were in- situ annealed at 700/C14C to obtain smooth surfaces. Fe1þxCo2–xSi films with a thickness of 20 nm were deposited at room temperature and ex-situ vacuum annealed at 500/C14C. A 2 nm thick MgO capping layer was used to prevent oxida-tion of the films. To determine the stoichiometry and to adjustthe sputtering powers, x-ray fluo rescence measurements were carried out. To obtain information about the magnetizationdynamics, in-plane ferromagne tic resonance (FMR) measure- ments were performed using a broadband coplanar waveguide setup with a maximum frequency of 40 GHz. Least square fitsof the raw data using a first derivative of a Lorentzian lineshape were done to precisely determine the resonance field andthe peak-to-peak linewidth DH. 11,12For the FMR in-plane angle dependent measurements, the samples were mounted on a rotating stage and the resonance spectra were measured at afrequency of 30 GHz while the in-plane angle was changed in5 /C14steps. In addition quasistatic magnetization reversal meas- urements were carried out using the magneto-optic Kerr effect (MOKE) in a vector MOKE setup with an s-polarized laser with a wavelength of 488 nm. Anisotropy measurements werecarried out using a rotating sample holder. The magnetic fieldwas applied in the plane of the films. III. CRYSTALLOGRAPHIC PROPERTIES X-ray diffraction measurements were used to investigate the crystallographic properties of the Fe 1þxCo2–xSi films. X-ray diffraction pattern for all films are presented in Figure 1. The (002) and (004) peaks of the Heusler film are located at 32/C14and 66.6/C14, respectively. A (002) peak of the Cr buffer can be found at 65/C14. There are no differences in the pattern for the Co 2FeSi films in L2 1structure and Fe 2CoSi films in Xastructure, as the atomic scattering factors for Co and Fe are nearly the same for this energy. Ordering parameters,based on the intermixing between the Co and Si as well as the Fe and Si atoms, determined from x-ray diffraction, were already discussed in our previous work 6and found to be high for Co 2FeSi and decrease when going to Fe 2CoSi. In order to test the films for crystallographic symmetry uscans are per- formed on the (220) planes of the Fe 1þxCo2–xSi films. Figure 2shows the results together with the (220) plane of the MgO(001) substrate. The result shows that the (100) Heusler plane is rotated by 45/C14with respect to the MgO (100) plane. The fourfold symmetry of the u-scans clearly verifies the highly textured growth of all Fe 1þxCo2–xSi films of this study. IV. MAGNETIZATION DYNAMICS In this section, we present in-plane broadband FMR measurements for the Fe 1þxCo2–xSi samples to obtain infor- mation about the magnetic properties of the films. The Landau-Lifshitz-Gilbert equation describes the dynamics of the magnetization vector ~Min the presence of an effective field ~Heff, which contains both dc and ac fields. It is given by13 d~M dt¼/C0c~M/C2~Heffþa M~M/C2d~M dt/C18/C19 ; (1) where cis the gyromagnetic ratio and ais the Gilbert damp- ing parameter. According to the Landau-Lifshitz-GilbertEquation (1), the resonance condition can be expressed in terms of the second derivatives of the free-energy density E by the Smit-Beljers formula 14 f c0/C18/C192 ¼1 Msinh ðÞ2@2E @h2@2E @u2/C0@2E @h@u !22 43 5/C12/C12/C12/C12/C12/C12 h0;u0;(2) where c0¼c=2p,handuare the polar and azimuthal angles of the magnetization ~M,a n d h0andu0the corresponding equilibrium values. Measurements of the magnetic fielddependent resonance frequency were carried out in two differ-ent orientations of the sample: in [100] and [110] direction of the Fe 1þxCo2–xSi Heusler alloy, as the [100] direction is the magnetic easy axis and the [110] direction is the magnetichard axis, respectively. Figure 3shows the exemplary Kittel plots along [100] and [110] directions for the Fe 2CoSi sample. The experimental data were fitted simultaneously using the Kittel equations for both easy and hard configurations15 f¼c0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hres–ha/C0H4 ðÞ Hres–haþH4 2þ4pMeff/C18/C19s ;(3) FIG. 1. X-ray diffraction pattern for all x in Fe 1þxCo2–xSi. The (002) and (004) peaks of the Heusler films can be found at 32/C14and 66.6/C14, respectively.FIG. 2. u-scans of the (220) Fe 1þxCo2–xSi peak and (220) MgO substrate peak showing the fourfold symmetry of the films.083904-2 Sterwerf et al. J. Appl. Phys. 120, 083904 (2016) f¼c0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðHres–eaþH4ÞðHres–eaþH4þ4pMeffÞp ; (4) where Meff;c0, and H4are shared fit parameters. H4describes the magnitude of the in-plane fourfold anisotropy field.H res–haandHres–eadenote the resonance field along the mag- netic hard and the magnetic easy axis, respectively. Theresults of these fits for the gyromagnetic ratio c 0are pre- sented in Fig. 6(a) for all x in Fe 1þxCo2–xSi. Within the error bars, it is nearly linear for x /C210.25 and slightly smaller for Co2FeSi. The fitted effective magnetization, which includes any perpendicular anisotropy present in the films, is shownin Fig. 4for the Fe 1þxCo2–xSi samples. The error bars origi- nate from fitting of the Kittel equations and the determina- tion of the volume of the unit cell. For bulk Co 2FeSi and Fe2CoSi the experimentally determined magnetizations are 5:95lB/f.u.9and 4 :99lB/f.u.,10respectively, which match the expected magnetizations according to the Slater-Paulingrule (visualized by the dashed line in Fig. 4on the right axis). The deviation from the expected values might be attributed to residual atomic disorder in the films or the pres-ence of a perpendicular anisotropy caused by a small tetrago-nal distortion in the [001] direction. The frequencydependence of the linewidth of the ferromagnetic resonance absorption provides direct information about the magneticrelaxation. The frequency dependence of the linewidth 16,17 can under certain conditions be characterized by an inhomo- geneous residual linewidth at zero field DH0and an intrinsic contribution18 DH¼DH0þ2ffiffiffi 3paeff c0f: (5) For correct determination of the effective damping parameter, it is necessary to measure the linewidth over a wide frequencyrange to determine the slope. It is not sufficient to measure DHat a fixed frequency, because a non-zero extrinsic line- width DH 0results in an overestimated damping parameter aeff.F i g u r e 5shows the peak-to-peak linewidth DHfor all fre- quencies and all x. The measurements were performed in the direction of the magnetic hard axis of the Heusler films. The experimental data were fitted by Equation (5)to determine the effective damping parameters. The slope at higher fre-quencies was used to determine the damping parameters. The corresponding fit functions are presented in Figure 5as well. The inhomogeneous residual linewidth at zero field DH 0is presented in Fig. 6(b) for all stoichiometries. The upper error margin result from the different slopes in the DHvs.fcurves, while the lower limit is based on the assumption of a vanish- ing residual linewidth. The residual linewidth decreases as theFe concentration increases and reaches its lowest value ofDH 0¼12 Oe for Fe 2CoSi. McMichael et al.19found that small grain size distributions can lead to low inhomogeneous line broadening. The effective Gilbert damping parameter aeffis shown in Fig. 6(c).C o 2FeSi exhibits a damping parameter of 0.0018. For the estimation of the upper limit of the error margins, one might assume that the damping parameter issolely caused by Gilbert type damping at 40 GHz. This leadsto an error margin with an upper limit of 0.0034. The lowerlimit of the error margin is 0.0004 and is caused by different slopes of the curve. In the following, these asymmetric error margins are expressed as 0 :00186 0:0034 0:0004.F e 2CoSi films show a slightly larger value of 0 :001960:0007 0:0001. Kasatani et al. found damping parameters from 0.0023 to 0.0061 for Co 2FeSi films and 0.002 for Fe 2CoSi.20In general, the Gilbert damp- ing is expected to be low in half-metallic materials, where spin-flip processes are suppressed.7,21–23The small dampingFIG. 3. Resonance frequency versus magnetic field (Kittel plot) along the in-plane magnetic hard [110] and the magnetic easy [100] axis for Fe 2CoSi. The experimental data are fitted using a combined fit (Equations (3)and(4)) to determine Meffandc0. FIG. 4. Fitted effective magnetic moment per formula unit for the Fe1þxCo2–xSi films (left axis). The dashed line shows the interpolated expected magnetic moments according to the Slater-Pauling rule (right axis).FIG. 5. Frequency dependent FMR linewidth for all samples measured along the magnetic hard axis [110] of the Fe 1þxCo2–xSi films together with the fit functions according to Equation (5).083904-3 Sterwerf et al. J. Appl. Phys. 120, 083904 (2016) parameters of the metallic films show that a pseudo-gap as present in the Fe 1þxCo2–xSi system is sufficient to give rise to a low Gilbert damping. The magnetic measurements do not give any indication that the phase transition from L2 1to X aoccurs abruptly. The effective magnetization, presented in Figure 4, as well as the damping parameter, shown in Figure 6(c), shows no pronounced changes as a function of stoichiometry. Oneobserves a linear behavior for c’,DH 0, and K4for the stoi- chiometries between Fe 1.5Co1.5Si and Fe 2CoSi, respectively. Consequently, it seems that the structural transition takesplace gradually at stoichiometries with a high amount of Fe (x/C210:5). This is different from the behavior close to Co 2FeSi, where c’ andDH0show more distinctive variations between two stoichiometries, while K 4remains constant. This is also consistent with the ordering parameters pre-sented in an earlier publication. 6 Figure 7shows the frequency dependent linewidth along easy and hard axes for the Fe 2CoSi. The sample with the lowest damping, Fe 1.75Co1.25Si, is presented as a compari- son. The linewidth of the Fe 2CoSi data exhibits an inflection point around 19 GHz along the hard axis. We observed non-linear behavior in the linewidth vs. frequency response forall samples. This nonlinear dependence of the FMR line- width on frequency is a typical observation when twomagnon scattering contributes significantly to the relaxa- tion. 24,25Two-magnon scattering is an extrinsic relaxation mechanism and can be induced by means of different scatter-ing centers such as voids or pores, 26surface roughness,24and grain size27or by network of misfit dislocations32which causes scattering of the FMR mode (k ¼0) into propagating spin waves (k 6¼0). A. FMR in-plane rotation measurements To obtain further information about the magnetic aniso- tropies and magnetic relaxation additional FMR measure-ments were carried out as a function of the in-plane angle ofthe applied field with respect to the Fe 1þxCo2–xSi [110] axis. The operating frequency for the rotation measurements was 30 GHz. At this frequency, the resonance fields are highenough to saturate the magnetization along the easy and hardaxes. All measurements were performed at room temperature. A fourfold symmetry is observed in the in-plane angle dependence of the ferromagnetic resonance field for allsamples. Figure 8(a) exemplarily shows the ferromagnetic resonance field H resversus the in-plane rotation angle for Fe2CoSi. The dependence of the resonance field on the in- plane angle was simulated numerically using Equation (2), assuming a cubic magnetic anisotropy contribution to the Gibbs free energy28,29 Ecubic¼/C01 2K4a4 1þa4 2þa4 3/C0/C1 ; (6) where K4is the cubic magnetic anisotropy constant and a1, a2,a n d a3are the directional cosines with respect to the cubic principal axes. The experimentally determined in-plane angledependent H resdata were fitted with the numerical solution (red line in Fig. 8(a)) to determine the cubic anisotropy con- stant. Figure 8(b) shows the corresponding linewidth data, which also shows a clear fourfold symmetry. The linewidthexhibits maxima along the easy axes and minima along thehard axes of the cubic magnetic anisotropy. Randomly distrib-uted crystalline or surface roughness defects oriented alongthe in-plane principal crystallographic axis 30,31or a fourfold distribution in misfit dislocations32which induce the same symmetry on the strength of two magnon scattering can explain the observed anisotropic relaxation.FIG. 7. FMR linewidth for Fe 2CoSi measured along both the magnetic hard [110] and magnetic easy [100] axis along with a measurement for Fe1.75Co1.25Si, which showed the lowest damping parameter. FIG. 6. (a) Gyromagnetic ratio c0, (b) Extrinsic contribution to the linewidth DH0of the FMR spectra, (c) effective Gilbert damping parameter, and (d) cubic magnetic anisotropy constant K4for Fe 1þxCo2–xSi films with x ¼0, 0.25, 0.5, 0.75, 1.083904-4 Sterwerf et al. J. Appl. Phys. 120, 083904 (2016) The magnetic fourfold symmetry matches the crystallo- graphic symmetry of the highly textured Fe 1þxCo2–xSi films mentioned before. A polar plot of the MOKE squareness ver-sus the rotational angle for Fe 2CoSi is presented in Fig. 9. This measurement confirms the cubic anisotropy present inthe films as seen in the FMR measurement. The magneticeasy axis is located along the [100] crystallographic axis,and the magnetic hard axis is located along the [110] crystal-lographic axis. A cubic anisotropy with the easy magneticaxis in the Heusler [100] direction is found for all samples.The cubic magnetic anisotropy constant K 4obtained from the FMR measurements changes significantly in this series from 55 :8kerg cm3for Fe 2CoSi to 16 :6kerg cm3for Co 2FeSi, respec- tively. The cubic anisotropy constants for all stoichiometries are presented in Fig. 6(d). Hashimoto et al. found a similarcubic anisotropy constant of 18kerg cm3for crystalline Co 2FeSi with a film thickness of 18.5 nm.33For some samples in this series we observed an additional uniaxial anisotropy, whichcan originate from miscut substrates. 34,35 V. CONCLUSION In summary, we found very small damping parameters for the half-metallic Fe 1þxCo2–xSi films varying from 0:001260:0007 0:0001to 0:001960:0007 0:0001.C o 2FeSi exhibits a damping parameter of 0 :001860:0034 0:0004. FMR and MOKE measurements reveal a fourfold magnetocrystalline anisotropy for all films in accordance with the fourfold crystalline symmetry in thehighly textured films. ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from Bundesministerium f €ur Bildung und Forschung (BMBF) and Deutsche Forschungsgemeinschaft (DFG,Contract No. RE 1052/32-1) as well as support through theMINT Center summer program. S. Paul, B. Khodadadi, andT. Mewes would like to acknowledge support by the NSF-CAREER Award No. 0952929, C. K. A. Mewes would liketo acknowledge support by the NSF-CAREER Award No.1452670. 1L. Berger, Phys. Rev. B 54, 9353 (1996). 2J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 3A. J. Bradley and J. W. Rodgers, Proc. R. Soc. London, Ser. A 144, 340 (1934). 4M. Puselj and Z. Ban, Croat. Chem. Acta 41, 79 (1969). 5H. Pauly, A. Weiss, and H. Witte, Z. Metallkd. 59, 47 (1968). 6C. Sterwerf, M. Meinert, J.-M. Schmalhorst, and G. Reiss, IEEE Trans. Magn. 49, 4386 (2013). 7C. Liu, C. Mewes, M. Chshiev, and T. Mewes, Appl. Phys. Lett. 95, 022509 (2009). 8M. E. Straumanis and C. C. Weng, Acta Crystallogr. 8,3 6 7 (1955). 9S. Wurmehl, G. Fecher, H. Kandpal, V. Ksenofontov, C. Felser, H.-J. Lin,and J. Morais, Phys. Rev. B 72, 184434 (2005). 10H. Luo, Z. Zhu, L. Ma, S. Xu, H. Liu, J. Qu, Y. Li, and G. Wu, J. Phys. D: Appl. Phys. 40, 7121 (2007). 11M. E. Schabes, H. Zhou, and H. N. Bertram, J. Appl. Phys. 87, 5666 (2000). 12N. Pachauri, B. Khodadadi, and M. Althammer, J. Appl. Phys. 117, 233907 (2015). 13B. Heinrich and J. F. Cochran, Adv. Phys. 42, 523 (1993). 14H. Suhl, Phys. Rev. 97, 555 (1955). 15X. Liu, Y. Sasaki, and J. K. Furdyna, P h y s .R e v .B 67, 205204 (2003). 16B. Heinrich, J. F. Cochran, and R. Hasegawa, J. Appl. Phys. 57, 3690 (1985). 17H. Lee, L. Wen, M. Pathak, and P. Janssen, J. Phys. D: Appl. Phys. 41, 215001 (2008). 18C. Mewes and T. Mewes, “Relaxation in magnetic materials forspintronics,” in Handbook of Nanomagnetism (Pan Stanford, 2015), p. 74. 19R. D. McMichael, D. J. Twisselmann, and A. Kunz, Phys. Rev. Lett. 90, 227601 (2003). 20Y. Kasatani, S. Yamada, H. Itoh, M. Miyao, K. Hamaya, and Y. Nozaki, Appl. Phys. Express 7, 123001 (2014). 21G. M. M €uller, J. Walowski, M. Djordjevic, and G. X. Miao, Nat. Mater. 8, 56 (2009).FIG. 8. Polar plots of (a) the resonance fields H resand (b) the linewidth DH as a function of the in-plane angle of the applied field with respect to the [110] axis of a 20 nm thick Fe 2CoSi film measured at a microwave fre- quency of 30 GHz. FIG. 9. Polar plots of the squarenessMR MSfor Fe 2CoSi obtained by MOKE measurements.083904-5 Sterwerf et al. J. Appl. Phys. 120, 083904 (2016) 22Y. Cui, B. Khodadadi, S. Sch €afer, T. Mewes, J. Lu, and S. A. Wolf, Appl. Phys. Lett. 102, 162403 (2013). 23Y. Cui, J. Lu, S. Sch €afer, B. Khodadadi, T. Mewes, M. Osofsky, and S. A. Wolf, J. Appl. Phys. 116, 073902 (2014). 24H. Lee, Y. Wang, C. Mewes, and W. H. Butler, Appl. Phys. Lett. 95, 082502 (2009). 25P. Landeros, R. E. Arias, and D. L. Mills, Phys. Rev. B 77, 214405 (2008). 26M. J. Hurben and C. E. Patton, J. Appl. Phys. 83, 4344 (1998). 27R. D. McMichael, M. D. Stiles, and P. J. Chen, J. Appl. Phys. 83, 7037 (1998). 28M. Farle, Rep. Prog. Phys. 61, 755 (1998). 29B. Heinrich and J. Bland, “Radio frequency techniques,” in Ultrathin Magnetic Structures II (Springer, 1994), p. 195.30I. Barsukov, P. Landeros, R. Meckenstock, J. Lindner, D. Spoddig, Z.-A. Li, B. Krumme, H. Wende, D. L. Mills, and M. Farle, Phys. Rev. B 85, 014420 (2012). 31N. Pachauri, B. Khodadadi, A. V. Singh, J. Beik Mohammadi, R. L.Martens, P. R. LeClair, C. K. A. Mewes, T. Mewes, and A. Gupta, J. Magn. Magn. Mater. 417, 137 (2016). 32G. Woltersdorf and B. Heinrich, Phys. Rev. B 69, 184417 (2004). 33M. Hashimoto, J. Herfort, H. P. Schonherr, and K. H. Ploog, Appl. Phys. Lett. 87, 102506 (2005). 34M. Rickart, T. Mewes, M. Scheib, S. O. Demokritov, and B. Hillebrands, J. Phys. D: Appl. Phys. 38, 1047 (2005). 35M. Rickart, T. Mewes, M. Scheib, S. O. Demokritov, and B. Hillebrands, Phys. Rev. B. 70, 060408 (2004).083904-6 Sterwerf et al. J. Appl. Phys. 120, 083904 (2016)

1.4813315.pdf

Improvement of the yttrium iron garnet/platinum interface for spin pumping- based applications M. B. Jungfleisch, V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands Citation: Appl. Phys. Lett. 103, 022411 (2013); doi: 10.1063/1.4813315 View online: http://dx.doi.org/10.1063/1.4813315 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v103/i2 Published by the AIP Publishing LLC. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 15 Jul 2013 to 132.74.1.4. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsImprovement of the yttrium iron garnet/platinum interface for spin pumping-based applications M. B. Jungfleisch,a)V. Lauer, R. Neb, A. V. Chumak, and B. Hillebrands Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit €at Kaiserslautern, 67663 Kaiserslautern, Germany (Received 26 February 2013; accepted 20 June 2013; published online 12 July 2013) The dependence of the spin pumping efficiency and the spin mixing conductance on the surface processing of yttrium iron garnet (YIG) be fore the platinum (Pt) deposition has been investigated quantitatively. The ferromagneti c resonance driven spin pumping injects a spin polarized current into the Pt layer, which is tr ansformed into an electromotive force by the inverse spin Hall effect. Our experiments show that the spin pumping effect indeed strongly depends on the YIG/Pt interface condition. We measure an enhancement of the inverse spin Hallvoltage and the spin mixing conductance by more than two orders of magnitude with improved sample preparation. VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4813315 ] In the last decades, there was rapidly increasing interest in the field of spintronics. The promising aim is to exploit the intrinsic spin of electrons to build efficient magnetic stor- age devices and computing units.1An emerging sub-field of spintronics is magnon spintronics, where magnons, the quanta of spin waves (collective excitations of coupled spins in a magnetically ordered solid), are used to carry and pro-cess information. 2Magnons pose a number of advantages to conventional spintronics. One of them is the realization of insulator-based devices with decreased energy consumption:Since spin-wave based spin currents in insulators are not accompanied by charge currents, parasitic heating due to the movement of electrons can be excluded. 3 In order to combine magnon spintronics and charge- based electronics, it is necessary to create effective converters, which transform spin-wave spin currents into conventionalcharge currents. The combination of the spin pumping 4,5and the inverse spin Hall effect (ISHE)6,7turned out to be an excellent candidate for this purpose. Spin pumping refers tothe generation of spin polarized electron currents in metals by the magnetization precession in an adjacent ferromagnetic layer, whereas the ISHE transforms this spin current into aconventional charge current. In the last years, hetero-structures consisting of a mag- netic insulator yttrium iron garnet (YIG) film and an adjacentplatinum (Pt) layer 2attracted considerable attention. Since YIG is an insulator with a band gap of 2.85 eV (Ref. 8), no direct transition of a spin polarized electron current from theYIG into the Pt layer is possible. Thus, spin pumping is the only method applicable in these structures to inject spin cur- rents into the Pt layer. It has been shown that standing 9as well as propagating10magnons in a wide range of wave- lengths from centimeters to hundred nanometers11,12can be efficiently converted into charge currents using a combina-tion of spin pumping and ISHE. Furthermore, the Pt thick- ness dependence on the ISHE voltage from spin pumping 13 and non-linear spin pumping14has been investigated.Since spin pumping is an interface effect, it is of crucial importance to investigate how to control and manipulate the YIG/Pt interface conditions in order to obtain an optimal magnon-to-spin current conversion efficiency. Recently, theinfluence of Ar þion beam etching on the spin pumping effi- ciency in YIG/Au structures was investigated.15It was shown that the spin mixing conductance determined by the Gilbertdamping constant can be increased by a factor of 5 using Ar þ etching. Nevertheless, there are no systematic, quantitativestudies of the influence of the YIG/Pt interface treatment onthe ISHE voltage from spin pumping up to now. In this letter, we present our results on the influence of processing of the YIG film surface before the Pt depositionon the spin pumping efficiency. We measure the ferromag- netic resonance (FMR) spectra using conventional micro- wave techniques, as well as the inverse spin Hall voltage,which allows us to calculate the spin pumping efficiency, defined as the ratio of the detected ISHE charge current to the absorbed microwave power. Our experimental resultsclearly show a significant difference (up to factor of 152) between the different surface treatments. A sketch of the experimental setup is shown in Fig. 1(a). The YIG samples of 2.1 lm and 4.1 lm thickness and a size of 3/C24m m 2were grown by liquid phase epitaxy on both sides of 500 lm thick gadolinium gallium garnet (GGG) sub- strates. Since the GGG substrate between the two YIG layers is rather thick (500 lm), the second YIG layer has no influ- ence on our studies. After a conventional pre-cleaning step, all samples were cleaned by acetone and isopropanol in an ultrasonic bath. In order to provide the same cleanliness of all samples, the purityof each sample was monitored after this first step. Afterwards, the following surface treatments have been applied: /C15Cleaning in “piranha” etch, a mixture of H 2SO4and H2O2. This exothermic reaction at around 120/C14Ci s strongly oxidizing and, thus, removes most organic mat- ter. Among the used surface treatments, “piranha” clean-ing is the only one, which was performed outside the molecular beam epitaxy (MBE) chamber. Thus, we can- not exclude, that the samples are contaminated by a)Electronic mail: jungfleisch@physik.uni-kl.de 0003-6951/2013/103(2)/022411/4/$30.00 VC2013 AIP Publishing LLC 103, 022411-1APPLIED PHYSICS LETTERS 103, 022411 (2013) Downloaded 15 Jul 2013 to 132.74.1.4. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsmicroscopic dirt or a water film due to air exposure before the Pt deposition. /C15Heating at 200/C14C for 30 min in order to remove water from the sample surfaces, performed in situ . /C15Heating at 500/C14C for 5 h performed in situ in order to remove water from the sample surfaces. It might be that for this method lattice misfits in the YIG crystal are annealed as well.16,17 /C15In situ Arþplasma cleaning at energies of 50–100 eV for 10 min. SRIM simulations show that the used energies are below the threshold for sputtering, and thus, thiscleaning method acts mechanically. /C15In situ O þ/Arþplasma cleaning at energies around 50–100 eV for 10 min. In addition to the mechanicalcleaning effect, O þoxidizes organic matter. The used surface processes are summarized in Table I. They fulfill different requirements: First of all, the samplesare cleaned removing microscopic particles, (organic) matter and water. In the case of the samples, which are heated, the crystalline YIG structure might be annealed as well. Anotheraspect of the used treatments might be a modification of the spin pinning conditions (not part of the present studies; requires further deeper investigations). After cleaning, the10 nm thick Pt layer was grown by MBE at a pressure of 5 /C2 10 /C08mbar and a growth rate of 0.01 nm/s. It is important to note that the Pt film was deposited on each of the samples ofone set simultaneously, ensuring identical growth conditions. The measurement of the spin pumping efficiency was performed in the following way. The samples were magne-tized in the film plane by an external magnet field H(see Fig. 1(a)). The magnetization precession was excited at a constant frequency of f¼6:8 GHz by applying microwave signals of power P applied to a 600 lm wide Cu microstripantenna. The Pt layer and the microstrip antenna were elec- trically isolated by a silicon oxide layer of 100 lm thickness in order to avoid overcoupling of the YIG film with the antenna. While sweeping the external magnetic field theinverse spin Hall voltage U ISHE (typical spectrum depicted in Fig. 2(a)) as well as the microwave reflection and trans- mission (Fig. 2(b)) were recorded. The voltage UISHE was measured across the edges of the Pt layer perpendicular to the external magnetic field using a lock-in technique. For this purpose the microwave amplitude was modulated with a frequency of 500 Hz. Changing the external magnetic field to the opposite direction results in an inverted voltage prov-ing the ISHE nature of the observed signal. 2,6,7The compli- cated absorption and reflection spectra depicted in Fig. 2(b) are due to the interference of the electromagnetic signal inthe microstrip line reflected from the YIG sample and the edges of the line. 18This behavior does not influence our studies since we further use only the maximum of UISHEto calculate the spin pumping efficiency (see Fig. 2(a)). Knowing the applied microwave power Papplied and meas- uring microwave reflection Prefland transmission Ptrans(see Fig. 2(b)) enables us to calculate the absorbed power as Pabs¼Papplied /C0ðPreflþPtransÞ:The results are depicted in Fig. 2(c).19At the FMR field HFMR/C25170 mT, energy is transferred most effectively into the magnetic system and, thus, the microwave absorption is maximal (see Fig. 2(c)). In resonance condition, the angle of precession is maximaland spin currents are most efficiently pumped from the YIG into the Pt layer (Fig. 1(b)) and transformed into an electric current by the ISHE. 2,7Subsequently, we measure the maxi- mal ISHE voltage UFMR ISHEatHFMR. The dependence of UFMR ISHE as a function of the applied microwave power Papplied is shown in Fig. 3, left scale (illus- trated is method 7, Table I, discussed below with a rather high enhancement factor of 104). The interface improvement provides the possibility to observe UFMR ISHE over a wide range of applied powers Papplied . We find a linear relation between UFMR ISHE and Papplied over the whole power region of nearly four orders of magnitude. The ISHE voltage UFMR ISHEincreases from 100 nV (for Papplied /C25100lW) to approximately 500 lV (for Papplied /C255 0 0 m W ) .O nt h er i g h ts c a l ei nF i g . 3,t h e absorbed microwave power Pabsis shown as a function of the applied microwave power Papplied .Pabsdepends also linearly onPapplied . In order to investigate the different cleaning methods described above we measure the spectra for three different microwave powers Papplied of 1 mW, 10 mW, and 100 mW. We investigate three sets of samples (2 sets at 2.1 lm, 1 set at 4.1 lm YIG thickness). In order to compare the samples of one set we introduce the spin pumping efficiency as gðPapplied ;nÞ¼UFMR ISHE R/C1PFMR abs; (1) where nis the index number of the sample (i.e., the index number of the cleaning method, see Table I),UFMR ISHE is the maximal ISHE voltage in resonance HFMR,Ris the electric resistance of the Pt layer, Papplied is the applied microwave power, and PFMR absis the absorbed microwave power at HFMR. FIG. 1. (a) Sketch of the sample and the setup geometry. (b) Illustration of spin pumping and inverse spin Hall effect. TABLE I. /C22/C15is the calculated mean value of the /C15-parameter, which is the thickness and power independent spin pumping efficiency (Eq. (2)),r(/C22/C15)i s the standard deviation, spin mixing conductance g"# eff. Sample index n Process /C22/C15 rð/C22/C15Þ[%] g"# eff(/C21019m/C02) 1 Simple cleaning 1 … 0.02 2 Piranha 14 131 0.323 Piranha þ200 /C14C 64 93 1.45 4 Piranha þArþ79 54 1.79 5 Piranha þOþ/Arþ152 61 3.43 6 200/C14CþOþ/Arþ86 52 1.94 7 Piranha þ500/C14C 104 40 2.35022411-2 Jungfleisch et al. Appl. Phys. Lett. 103, 022411 (2013) Downloaded 15 Jul 2013 to 132.74.1.4. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsFurther, we introduce the power and thickness independ- ent parameter /C15by normalizing the efficiency of the n-th sample to the first sample of each set n¼1, which underwent only a simple cleaning process, as /C15ðnÞ¼gðn;PappliedÞ gðn¼1;PappliedÞ: (2) The /C15-parameter is a measure for the enhancement of the spin pumping efficiency due to the surface treatment. Foreach of them we calculate the /C15-values: three microwave powers for each set. The general tendency is the same for all series of measurements. According to the literature, 21the spin pumping efficiency should not depend on the YIG thick- ness for the used samples due to their large thicknesses. We observe an ISHE voltage for the 4.1 lm set to be around 20% of that of the 2.1 lm sets, which we associate with a better quality of the 2.1 lm YIG film. Nevertheless, the gen- eral tendency of the enhancement /C15ðnÞdue to the used sur- face treatments is rather independent on the film thickness: the absolute value of the /C15-parameter for the 4.1 lm set is 80% of that of the 2.1 lm sets. In order to obtain an easily comparable measure for the spin pumping efficiency, we fur- ther introduce the mean value /C22/C15of these /C15-values. The stand- ard deviation is given by rð/C22/C15Þand is a measure of the reliability for the specific surface treatments. The results are summarized in Table I. Our investigations of the YIG/Pt interface improvement on spin pumping show the following trend. The conventional cleaning by acetone and isopropanol in an ultrasonic bath (sample 1) achieves the worst results ( /C22/C15¼1,UISHE¼270 nV for Papplied ¼10 mW). Surface cleaning by “piranha” improves the efficiency by a factor of 14, but this method isnot reliable (standard deviation of rð/C22/C15Þ¼131% is very large). Since this surface treatment takes place outside the MBE chamber and since the sample is in air contact aftercleaning, the sample might be contaminated again by water and possibly by microscopic dirt before the Pt deposition. Heating the samples after “piranha” cleaning in the MBEchamber at 200 /C14C for 30 min (sample 3) removes mainly the water film from the surface and results in a 64 times higher efficiency. However, this cleaning method also does notguarantee a high ISHE voltage, which is reflected in the high standard deviation. Using an Ar þplasma (method 4) is more efficient (see Fig. 2) and particularly more reliable. Method 4 acts mechanically and removes water as well as other dirt from the sample surface. The best results ( /C22/C15¼152) are obtained for the Oþ/Arþplasma (sample 5). The additional advantage of this surface treatment is the removal of organic matter. In order to check the reliability of the Oþ/Arþplasma cleaning (method 5), we substituted the “piranha” cleaningby heating the sample: even without “piranha” etching, but with heating at 200 /C14C and Oþ/Arþplasma (method 6), we achieve a considerable spin pumping efficiency of /C22/C15¼86. This is mainly attributed to the Oþplasma. It is remarkable that purely heating the sample at 500/C14C for 5 h (method 7) results in a comparable high efficiency of /C22/C15¼104. The addi- tional reason might be that the temperature is sufficiently high to anneal crystal defects of the YIG samples.16,17 Since most of the samples exhibit FMR and spin- pumping spectra as shown in Fig. 2, where several non- resolvable modes contribute to the observed signals, it is impossible to obtain the FMR linewidth of these samples. Inorder to determine the spin mixing conductance g "# effof our samples, we choose one particular YIG sample with a very pronounced mode structure (see UISHE spectrum in Fig. 4) and we performed FMR measurements on it using a vector network analyzer (sample 7, with and without Pt layer on the top).20The FMR linewidth DHis related to the Gilbert damp- ing parameter as15,21–23a¼cDH=2x, where cis the gyro- magnetic ratio and x¼2pfis the microwave angular frequency. For the bare YIG sample, we measure DH0¼0:06 mT, which corresponds to a0¼1:2/C210/C04at a frequency f¼6:8 GHz, whereas for the Pt covered YIG sample DHPt ¼0:16 mT, corresponding to aPt¼3:3/C210/C04,r e s p e c t i v e l y (linewidths obtained by the ISHE-spectrum vary between 0.123 mT and 0.377 mT as shown in Fig. 4). We obtain a change of the Gilbert damping constant Da¼2:1/C210/C04. The spin mixing conductance g"# effis related to the change of the Gilbert damping Da¼aPt/C0a0as21–23 g"# eff¼4pMSdF glBDa: (3)FIG. 2. (a) ISHE voltage as a function of the applied magnetic field H. Magnetic field dependence of (b) transmitted and reflected microwavepower P trans,Prefl, and (c) absorbed microwave power Pabs.A p p l i e d microwave power Papplied ¼10 mW, YIG thickness: 2.1 lm. FIG. 3. Maximal inverse spin Hall effect induced voltage UFMR ISHE and absorbed microwave power Pabsat FMR as a function of the applied micro- wave power Papplied . YIG thickness: 2.1 lm, cleaning method 7, “piranha” etch and heating at 500/C14C.022411-3 Jungfleisch et al. Appl. Phys. Lett. 103, 022411 (2013) Downloaded 15 Jul 2013 to 132.74.1.4. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsTaking MS¼140 kA/m and dF¼2:1lm into account we obtain a spin mixing conductance of g"# eff¼2:35/C21019m/C02 for this particular sample. Considering that the intrinsic damp- ing is the same for all samples and since the spin mixing con- ductance g"# effis proportional to the spin pumping efficiency and, thus, consequently to the enhancement parameter /C22/C15,w e can calculate g"# efffor the other surface treatments. The results are summarized in Table I. As it is apparent from Table I, the spin mixing conduct- ance can be varied by treating the YIG surface before the Pt deposition in the range of two orders of magnitude. The larg- est value of the spin mixing conductance is obtained for acombined surface treatment by “piranha” etch and O þ/Arþ plasma, g"# eff¼3.43/C21019m/C02. The obtained values for g"# eff agree with the values reported in the literature.15,23–25Our maximal spin mixing conductance g"# eff¼3.43/C21019m/C02is one order of magnitude larger than the one reported in Refs. 15and23for YIG/Au ( g"# eff¼5/C21018m/C02) and even three orders of magnitude larger than the one estimated in Ref. 2for YIG/Pt ( g"# eff¼3/C21016m/C02). On the other hand, our maximal value is still one order of magnitude smaller compared to theo n er e p o r t e di nR e f . 25for YIG/Pt ( g "# eff¼4.8/C21020m/C02). In conclusion, we have shown a strong dependence of the spin pumping effect on the interface condition of YIG/Ptbilayer structures. We improved the ISHE signal strength by a factor of more than 150 using a combination of “piranha” etch and in situ O þ/Arþplasma treatment in comparison to standard ultrasonic cleaning. The combined cleaning by “piranha” etch and heating at 500/C14C yields a comparable enhancement of the spin pumping efficiency (by a factor of104). The spin mixing conductances for the different surface treatments were calculated. We find a maximal value of g "# eff¼3.43/C21019m/C02. Since the voltage generated by the ISHE scales with the length of the Pt electrode, optimal interface conditions are extremely essential for the utilization of spin pumping and ISHE in micro-scaled devices. Ourresults are also important for studies on the reversed effects: the amplification26and excitation2of spin waves in YIG/Pt structures by a combination of the direct spin Hall and the spin-transfer torque effect.2,27 We thank E. Saitoh and K. Ando for helpful discussions. Financial support by Deutsche Forschungsgemeinschaft (CH1037/1-1) and the Nano-Structur ing Center, TU Kaiserslautern, for technical support, is gratefully acknowledged. 1I./C20Zutic´, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). 2Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). 3A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys. D: Appl. Phys. 43, 264002 (2010). 4Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 5M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der Waal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603 (2006). 6J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 7E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 8X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Europhys. Lett. 96, 17005 (2011). 9M. B. Jungfleisch, A. V. Chumak, V. I. Vasyuchka, A. A. Serga, B. Obry, H. Schultheiss, P. A. Beck, A. D. Karenowska, E. Saitoh, and B. Hillebrands, Appl. Phys. Lett. 99, 182512 (2011). 10A. V. Chumak, A. A. Serga, M. B. Jungfleisch, R. Neb, D. A. Bozhko, V. S. Tiberkevich, and B. Hillebrands, Appl. Phys. Lett. 100, 082405 (2012). 11C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, M. B. Jungfleisch, E. Saitoh, and B. Hillebrands, Phys. Rev. Lett. 106, 216601 (2011). 12H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J. Ferguson, and S. O. Demokritov, Appl. Phys. Lett. 99, 162502 (2011). 13V. Castel, N. Vlietstra, J. B. Youssef, and B. J. van Wees, Appl. Phys. Lett. 101, 132414 (2012). 14K. Ando, T. An, and E. Saitoh, Appl. Phys. Lett. 99, 092510 (2011). 15C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Appl. Phys. Lett. 100, 092403 (2012). 16O. G. Ramer and C. H. Wilts, Phys. Status Solidi B 79, 313 (1977). 17D. Y. Choi and S. J. Chung, J. Cryst. Growth 191, 754 (1998). 18W. Barry, IEEE Trans. Microwave Theory Tech. MTT-34 , 80 (1986). 19In the UISHEsignal as well as in the microwave absorption spectra several modes are visible. They are slightly different for each sample and are iden- tified as higher width modes and perpendicular standing thickness spin-wave modes. Since only the maximal voltage is used to determine the spin pumping efficiency, the mode structures of the different samples are of minor interest for the present study. 20S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P.Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. 99, 093909 (2006). 21H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Takahashi, Y. Kajiwara, K. Uchida, and Y. Fujikawa, and E. Saitoh, Phys. Rev. B 85, 144408 (2012). 22O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, andA. Hoffmann, Phys. Rev. Lett. 104, 046601 (2010). 23B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y.-Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107, 066604 (2011). 24F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Althammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch, W. Limmer, H. Huebl, R. Gross,and S. T. B. Goennenwein, Phys. Rev. Lett. 107, 046601 (2011). 25S. M. Rezende, R. L. Rodrguez-Su /C19arez, M. M. Soares, L. H. Vilela-Le ~ao, D. L. Dom /C19ınguez, and A. Azevedo, Appl. Phys. Lett. 102, 012402 (2013). 26Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin, Phys. Rev. Lett. 107, 146602 (2011). 27J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); L. Berger, Phys. Rev. B 54, 9353 (1996).FIG. 4. Pronounced mode structure of the ISHE voltage signal in the case of the used surface treatment method 7 (piranha þ500/C14C) and corresponding Lorentzian fits. Experimental data as a black solid line, individual Lorentzian fits as solid blue lines and sum of Lorentzian fits as red line.022411-4 Jungfleisch et al. Appl. Phys. Lett. 103, 022411 (2013) Downloaded 15 Jul 2013 to 132.74.1.4. 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1.2837030.pdf

Angular dependence of spin-wave resonance and relaxation in half-metallic films Tetiana Nosach , Gabriella Mullady , Nicole Leifer , Venimadhav Adyam , Qi Li, Steven Greenbaum , and Yuhang Ren Citation: Journal of Applied Physics 103, 07E311 (2008); doi: 10.1063/1.2837030 View online: https://doi.org/10.1063/1.2837030 View Table of Contents: http://aip.scitation.org/toc/jap/103/7 Published by the American Institute of Physics Articles you may be interested in Pulsed laser deposition of epitaxial yttrium iron garnet films with low Gilbert damping and bulk-like magnetization APL Materials 2, 106102 (2014); 10.1063/1.4896936 Magnetic anisotropy, damping, and interfacial spin transport in Pt/LSMO bilayers AIP Advances 6, 055212 (2016); 10.1063/1.4950971 Spin-wave propagation in ultra-thin YIG based waveguides Applied Physics Letters 110, 092408 (2017); 10.1063/1.4976708 Stress-induced magnetic properties of PLD-grown high-quality ultrathin YIG films Journal of Applied Physics 123, 203902 (2018); 10.1063/1.5031198 Inverse spin-Hall effect induced by spin pumping in metallic system Journal of Applied Physics 109, 103913 (2011); 10.1063/1.3587173 Ferromagnetic resonance linewidth in metallic thin films: Comparison of measurement methods Journal of Applied Physics 99, 093909 (2006); 10.1063/1.2197087Angular dependence of spin-wave resonance and relaxation in half-metallic Sr2FeMoO 6films T etiana Nosach,1Gabriella Mullady,1Nicole Leifer,1Venimadhav Adyam,2,a/H20850Qi Li,2 Steven Greenbaum,1and Yuhang Ren1,b/H20850 1Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, New York 10065, USA 2Department of Physics, Pennsylvania State University, University Park, Pennsylvania 16802, USA /H20849Presented on 9 November 2007; received 11 September 2007; accepted 7 November 2007; published online 29 February 2008 /H20850 We investigated the magnetic anisotropic parameters and spin-wave relaxation of thin films of the ferromagnetic half-metallic Sr 2FeMoO 6by ferromagnetic resonance technique. The resonance field and linewidth were recorded as a function of relative angle between applied magnetic field andcrystallographic axes of the sample. The resonance field varies sinusoidally and considerablelinewidth broadening occurs when the applied field is rotated parallel to the sample plane. Theresults are described using higher order components of anisotropy fields. We obtain the values of thecubic anisotropic field, 2 K 4/H20648/M=0.096 11 T, the effective demagnetization field, 4 /H9266M−2K2/H11036/M =0.1216 T, and the planar anisotropic field, 2 K4/H11036/M=−0.108 T. Moreover, we estimated the spin relaxation time /H20849damping factor /H20850from the analysis of the angular dependence of peak-to-peak linewidth, leading to an intrinsic value of /H9251/H110110.000 25 /H20849Gilbert damping /H20850.©2008 American Institute of Physics ./H20851DOI: 10.1063/1.2837030 /H20852 Recently, the ordered double-exchange perovskite Sr2FeMoO 6/H20849SFMO /H20850attracts much attention because of the observation of intrinsic tunneling-type magnetoresistance atroom temperature. 1,2The compound is believed to be a half- metal and has exhibited a fairly high Curie temperature/H20849T C=410–450 K /H20850. The properties make it a potential mate- rial for industrial applications in magnetoresistive devices as well as in spintronics at high temperature. In SFMO, Fe3+ /H208493d5;t2g3eg2,S=5 /2/H20850and Mo5+/H208493d1;t2g1,S=1 /2/H20850couple anti- ferromagnetically via exchange interaction and the down-spin electron of Mo 5+is considered itinerant.3The character of spin ordering at the transition to the ferromagnetic metal-lic state has been shown to have an influence on the mobilityof charge carriers and, correspondingly, on the electronictransport. Therefore, in studying the magnetic properties ofSFMO, it is important to investigate the spin dynamics, asthis is intimately connected with the charge transfer betweendifferent ions. Ferromagnetic resonance /H20849FMR /H20850technique is the resonant magnetoabsorption of microwaves by ferromag-nets and has been proved to be a powerful tool in studying spin and magnetization dynamics in the thin film forms. 4 In this paper, we report on the study of the dynamical magnetic properties of thin films of Sr 2FeMoO 6including magnetic anisotropy parameters and spin relaxation time/H20849damping factor /H20850using FMR technique. We measured the resonance field and linewidth as a function of relative anglebetween applied magnetic field and crystallographic axes ofthe sample. The analysis of the results allows us to calculatethe values of the cubic and uniaxial magnetic anisotropyfields using the Landau-Lifshitz-Gilbert equation. We obtainthe values of the cubic anisotropic field, 2 K 4/H20648/M =0.09611 T, the effective demagnetization field, 4 /H9266M −2K2/H11036/M=0.1216 T, and the perpendicular cubic aniso- tropic field, 2 K4/H11036/M=−0.108 T. Moreover, we estimated the spin relaxation time /H20849damping factor /H20850from the analysis of the angular dependence of peak-to-peak linewidth. Thatgives an intrinsic value of /H9251/H110110.000 25 /H20849Gilbert damping /H20850. SFMO films were prepared by pulsed laser deposition in an oxygen partial pressure of 10−3Torr. From a stoichio- metric target on the /H20849100 /H20850SrTiO 3substrates, the details of which are reported elsewhere.5Magnetization was measured independently at 1 T using a vibrating sample magnetometer.The magnetic moments are found to be /H110113.8 /H9262Bat 10 K and /H110112/H9262Bat room temperature. The low temperature value is slightly less than that of the ideal saturation magnetic mo-ment of 4 /H9262Bfor the 3 d5/H20849Fe3+/H20850:4d1/H20849Mo5+/H20850atomic configu- ration. The Curie temperature, TC/H11011415 K, was determined by the measurement of ac susceptibility. FMR measurementswere carried out at Xband /H20849/H110119.74 GHz /H20850using a Bruker EMX electron paramagnetic resonance spectrometer. The ex- perimental setup and the polar coordinate system used in the subsequent discussion are plotted in the inset of Fig. 1. The dc magnetic field Hwas applied in the horizontal plane and the microwave magnetic field was along the vertical direc-tion. The samples were placed in a quartz tube inserted in themicrowave cavity and were rotated with respect to Hin an orientation between the normal to the layer plane /H20849 /H9258=0° /H20850 and the in plane /H20849/H9258=90° /H20850. The first derivative of the power adsorption /H20849W/H20850was detected as a function of applied field, and the field range was from 0 to 0.7 T. The resonance field is defined as the field where the rate of change of power withrespect to field strength is zero /H20849dW /dH=0/H20850and the peak-to-a/H20850Present address: Cryogenic Engineering Center, Indian Institute of Tech- nology Kharagpur, Kharagpur 721302, India. b/H20850Electronic mail: yre@hunter.cuny.edu.JOURNAL OF APPLIED PHYSICS 103, 07E311 /H208492008 /H20850 0021-8979/2008/103 /H208497/H20850/07E311/3/$23.00 © 2008 American Institute of Physics 103 , 07E311-1peak distance measured is the maxima and minima in the derivative signal. Figure 1shows the FMR spectra for a 400 nm SFMO film with various angles for Hrotating from in-plane orien- tation to normal to the film. The resonance field and linewidth of the resonance peak show clear dependences of therelative angle /H9258between the external magnetic field and the film out of plane: with increasing /H9258the resonance field shifts to a lower value and also considerable linewidth broadeningoccurs. The maxima in the resonance field can be seen at /H9258 =0° and 180°, while the minima are located at 90° and 270°.We plot the resonance field H Ras a function of /H9258in Fig. 2. The resonance field varies sinusoidally with /H9258, and we ob- serve a clear twofold out-of-plane symmetry in the sample.This is consistent with the tetragonal symmetry of perovskite structure. We describe our FMR data of SFMO by the Landau- Lifshitz-Gilbert equation of motion for the magnetizationM. 6For angle dependent analysis, we consider the expres- sion for the free energy density Fin an applied dc magnetic fieldHwith a tetragonal symmetry7 F=−MH /H20851cos/H9258cos/H9258H+ sin/H9258sin/H9258Hcos /H20849/H9272−/H9272H/H20850/H20852 −2/H9266M2sin2/H9258−K2/H11036cos2/H9258−1 2K4/H11036cos4/H9258 −1 2K4/H206481 4/H208493 + cos 4 /H9272/H20850sin4/H9258−K2/H20648sin2/H9258sin2/H20873/H9272−/H9266 4/H20874, /H208491/H20850 where the first term describes the Zeeman energy, the second term is the demagnetizing energy /H20849shape anisotropy /H20850,K2/H11036 and K4/H11036are constants representing the perpendicular uniaxial and cubic anisotropies, respectively, and a uniaxialterm K 2/H20648and a cubic term K4/H20648are constants representing the planar anisotropies. The eigenfrequencies of the magnetiza-tion procession are obtained by minimizing Fwith respect to /H9258and/H9272, /H20873/H9275 /H9253/H208742 =/H20875HRcos /H20849/H9258H−/H9258/H20850+/H20873−4/H9266M+2K2/H11036 M+K4/H11036 M −K4/H20648 2M/H20874cos 2/H9258+/H20873K4/H11036 M+K4/H20648 2M/H20874cos 4/H9258/H20876 /H11003/H20875HRcos /H20849/H9258H−/H9258/H20850+/H20873−4/H9266M+2K2/H11036 M +K4/H20648 M/H20874cos2/H9258+/H208732K4/H11036 M+K4/H20648 M/H20874cos4/H9258−2K4/H20648 M −K2/H20648 M/H20876, /H208492/H20850 with/H9275as the angular frequency of the microwave field and /H9253 as the gyromagnetic ratio. The quantity 4 /H9266Min this equation is defined as the demagnetizing field, where M=M/H20849T,H/H20850is the magnetization of our sample, which, in general, shows a dependence of external magnetic field and temperature.Resonance was observed at the applied magnetic field H R which depends on /H9258,/H9272,/H9258H,/H9272Hand the microwave frequency /H9275/2/H9266. In our configuration, /H9272is along the /H20851100 /H20852direction, /H9272=0. The solid line in the Fig. 2indicates the fitting using Eq. /H208492/H20850. The good consistency between theory and experiment indicates that we need to include magnetic anisotropy energyto describe our FMR data. We determine the values of thecubic and the uniaxial magnetic anisotropy fields as listed inTable I. In particular, the value of /H9253=1.8446 /H11003107Hz Oe−1 /H20849g=2.1 /H20850tells us that, the FMR signal is due to the Fe ions. In addition, 2 K4/H11036/Mand 2 K2/H11036/Mhave negative values, which indicate the easy axis of magnetization Mlying in the plane of the SFMO film /H20849along the /H20851100 /H20852direction /H20850. The results are in consistent with those from our magnetization measure-ments. It is important to note that, to obtain these high- FIG. 1. /H20849Color online /H20850FMR spectra observed for the 400 nm SFMO speci- men at room temperature, at various orientations /H9258HforHin the out-of- plane configuration, where angle /H9272is fixed and angle /H9258is changing from 0° to 360°. The inset shows the configuration of the FMR measurements. Themagnetic field Hwas applied at different angles /H9272Hand/H9258Hwith respect to the symmetry axes and the polar and azimuthal angles of the magnetizationMare /H9272and/H9258. FIG. 2. /H20849Color online /H20850The angular dependence of the ferromagnetic reso- nance signal in the “out-of-plane” configuration. Dots indicate experimentalresults and the solid line represents a theoretical fitting.07E311-2 Nosach et al. J. Appl. Phys. 103 , 07E311 /H208492008 /H20850symmetry results, we have assumed that the magnetization Mis aligned with the applied field H. In fact, the fields at which FMR is observed are sufficiently high to turn Mpar- allel to Heven when the latter is applied parallel to its hard axis. Moreover, with /H20841K2/H11036/H20841/greatermuch/H20841K4/H11036/H20841and /H20841K4/H20648/H20841/greatermuch/H20841K2/H20648/H20841, this as- sumption is expected to hold for the data we discussed here. In addition to the magnetic anisotropy parameters, we are also able to derive spin relaxation properties from theFMR spectra. Figure 3depicts the linewidth /H9004H p.p.of FMR spectra versus the out-of-plane angles /H9258. In contrast to that of the resonance field, the linewidth of the resonance signalshows a minimum at 0° /H20849or 180° /H20850, where the magnetic field was applied perpendicular to the surface of the film. Thepeak-to-peak width shows significant broadening as the mag-netic field is applied in the sample plane. In general, themeasured peak-to-peak FMR linewidth can be expressed as /H9004H p.p.=/H9004Hp.p.ext+/H9004Hp.p.in, /H208493/H20850 where /H9004Hp.p.extis the extrinsic linewidth that is frequency in- dependent and arises from the presence of magnetic inhomo-geneities and /H9004H p.p.inis the intrinsic contribution to linewidth of the resonance signal. The intrinsic linewidth depends onmicrowave frequency and mainly represents a perfect crystal structure. Typically, the peak-to-peak linewidth of the reso-nance signal due to intrinsic damping of magnetization isgiven by the Shul expression, 8,9 /H9004Hp.p.in=1 /H208813/H9251 MS/H20873d2F d/H92582+1 sin2/H9258d2F d/H92722/H20874/H20879d/H20849/H9275//H9253/H20850 dHR/H20879−1 . /H208494/H20850 The solid line in Fig. 3shows the calculated peak-to- peak linewidth of the resonance signal due to the intrinsiccontribution by substituting the magnetic anisotropy param-eters. We estimate the Gilbert damping constant to be /H9251 /H110110.000 25. Meanwhile, we note a significant difference be- tween the experimental value and that expected for the in-trinsic damping contribution, particularly when the magneticfield is rotated to the in-plane orientation of the sample. Thedifference could be due to the existence of magnetic disorderin SFMO films that could induce a pronounced extrinsicdamping. In fact, the role of magnetic and nonmagnetic dis-orders has been discussed thoroughly in SFMO according totheir magnetotransport data. 10,11 In summary, magnetic resonance and spin-wave relax- ation of films of SFMO were studied by ferromagnetic reso-nance technique. Higher order components of anisotropyfields were included to describe the angle dependence of thesinusoidal resonance field and considerable linewidth broad-ening. We obtain the values of the cubic anisotropic field,2K 4/H20648/M=0.096 11 T, the effective demagnetization field, 4/H9266M−2K2/H11036/M=0.1216 T, and the planar anisotropic field, 2K4/H11036/M=−0.108 T. The spin relaxation time /H20849damping fac- tor/H20850takes an intrinsic value of /H9251/H110110.000 25. We gratefully acknowledge supports from the Petroleum Research Fund, the CUNY Collaborative Incentive ResearchGrants Program and PSC-CUNY Award. 1K. I. Kobayashi, T. Kimura, H. Sawada, K. Terakura, and Y. Tokura, Nature /H20849London /H20850395, 677 /H208491998 /H20850. 2H. Q. Yin, J. S. Zhou, J. P. Zhou, R. Dass, J. T. McDevitt, and J. B. Goodenough, Appl. Phys. Lett. 75, 2812 /H208491999 /H20850. 3Z. Klencsár, Z. Németh, A. Vértes, I. Kotsis, M. Nagy, Á. Cziráki, C. Ulhaq-Bouillet, V. Pierron-Bohnes, K. Vad, S. Mészáros, and J. Hakl, J.Magn. Magn. Mater. 281,1 1 5 /H208492004 /H20850. 4X. Liu, Y. Sasaki, and J. K. Furdyna, Phys. Rev. B 67, 205204 /H208492003 /H20850. 5A. Venimadhav, F. Share, P. M. Attfields, and M. G. Blamire, J. Magn. Magn. Mater. 269, 101 /H208492004 /H20850. 6L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 /H208491935 /H20850;T .L . Gilbert, Phys. Rev. 100, 1243 /H208491955 /H20850. 7M. Farle, Rep. Prog. Phys. 61, 755 /H208491998 /H20850. 8H. Shul, Phys. Rev. 97, 555 /H208491955 /H20850. 9Yu. V. Goryunov, N. N. Garif’yanov, G. G. Khallinlin, I. A. Garfullin, L. R. Targirov, F. Schreiner, Th. Muhge, and H. Zabel, Phys. Rev. B 52, 13450 /H208491995 /H20850. 10H. Yanagihara, M. B. Salamon, Y. Lyanda-Geller, Sh. Xu, and Y. Mori- tomo, Phys. Rev. B 64, 214407 /H208492001 /H20850. 11T. T. Fang, Phys. Rev. B 71, 064401 /H208492005 /H20850.TABLE I. Magnetic anisotropic parameters obtained by fitting the angular dependence of the resonant field in the 400 nm SFMO film. Sample 4 /H9266M−2K2/H11036/M /H20849T/H20850 2K4/H20648/M /H20849T/H20850 2K4/H11036/M /H20849T/H20850 Sr2FeMoO6 0.1216 0.09611 −0.108 FIG. 3. /H20849Color online /H20850The linewidth of FMR /H9004Hp.p.in Sr2FeMoO6film as a function of /H9258between the dc magnetic field and the sample plane. The dots represent the measured data from peak to peak; the red line indicates thecalculated linewidth using the Gilbert damping and the black line is a guidefor the eyes.07E311-3 Nosach et al. J. Appl. Phys. 103 , 07E311 /H208492008 /H20850

1.2151800.pdf

Micromagnetic simulations of the magnetization precession induced by a spin- polarized current in a point-contact geometry (Invited) D. V. Berkov and N. L. Gorn Citation: Journal of Applied Physics 99, 08Q701 (2006); doi: 10.1063/1.2151800 View online: http://dx.doi.org/10.1063/1.2151800 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/99/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic simulations of persistent oscillatory modes excited by spin-polarized current in nanoscale exchange-biased spin valves J. Appl. Phys. 105, 07D107 (2009); 10.1063/1.3057912 Micromagnetic modeling of magnetization switching driven by spin-polarized current in magnetic tunnel junctions J. Appl. Phys. 101, 063914 (2007); 10.1063/1.2496202 Micromagnetic investigation of the dynamics of magnetization switching induced by a spin polarized current Appl. Phys. Lett. 88, 132506 (2006); 10.1063/1.2190450 Effect of the classical ampere field in micromagnetic computations of spin polarized current-driven magnetization processes J. Appl. Phys. 97, 10C713 (2005); 10.1063/1.1853291 Spin-polarized current-driven switching in permalloy nanostructures J. Appl. Phys. 97, 10E302 (2005); 10.1063/1.1847292 [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: 136.165.238.131 On: Mon, 22 Dec 2014 06:59:07Micromagnetic simulations of the magnetization precession induced by a spin-polarized current in a point-contact geometry „Invited … D. V. Berkova/H20850and N. L. Gorn Innovent e.V ., Prüessingstrasse 27B, D-07745, Jena, Germany /H20849Presented on 3 November 2005; published online 17 April 2006 /H20850 This paper is devoted to numerical simulations of the magnetization dynamics driven by a spin-polarized current in extended ferromagnetic multilayers when a point-contact setup is used. Wepresent /H20849i/H20850detailed analysis of methodological problems arising by such simulations and /H20849ii/H20850 physical results obtained on a system similar to that studied in Rippard et al. /H20851Phys. Rev. Lett. 92, 027201 /H208492004 /H20850/H20852. We demonstrate that the usage of a standard Slonczewski formalism for the phenomenological treatment of a spin-induced torque leads to a qualitative disagreement betweensimulation results and experimental observations, and discuss possible reasons for this discrepancy.©2006 American Institute of Physics ./H20851DOI: 10.1063/1.2151800 /H20852 I. INTRODUCTION Nearly a decade after prediction1and subsequent experi- mental discovery2of magnetic excitation and magnetization switching induced by a spin-polarized current /H20849SPC /H20850in a thin magnetic film, high-quality experiments providing quantita-tive information concerning the corresponding magnetizationdynamics have been performed /H20849see, e.g., Refs. 3–7 /H20850. For this reason the important problem of the quantitative verification of existing theoretical models for the spin-transfer phenom-ena and SPC-induced magnetization dynamics 8can be ad- dressed. Corresponding research clearly requires full-scalemicromagnetic simulations, because single-domain /H20849mac- rospin /H20850approximation 9by definition cannot incorporate im- portant effects resulting from the inhomogeneity of the mag- netization states. Such strongly nonuniform magnetizationconfigurations during the SPC-induced precession are pre-dicted already for thin-film elements of very small sizes 10 /H20849/H1101130 nm /H20850, which are less than typical dimensions of any actually used experimental samples. Micromagnetic simulations performed up to now deal only with the experiments performed in the so-called colum-nar geometry, 11where an electric current flows through a multilayer magnetic element with very small lateral sizes/H1101110 2nm. Such simulations were able to reproduce many important features of the experimental data obtained on co-lumnar structures, in particular, the existence of regular andquasichaotic oscillation regimes, dependence of the oscilla-tion frequency on the current strength, etc. /H20849However, we mention that even the most sophisticated model failed toreproduce experimental data quantitatively 12./H20850In contrast to this situation, to the best of our knowledge, simulations ofthe point-contact experiments, 6,7which show some important magnetization dynamics features different from those ob-served on columnar structures, have not been carried out. In this paper we present full-scale micromagnetic simu- lations of the SPC-induced magnetization dynamics in thepoint-contact setup. The paper is organized as follows: inSec. II we discuss methodological problems of such simula- tions that are specific for the geometry under study. Next/H20849Sec. III /H20850we present our results, starting from the analysis of the magnetization dynamics without the current-inducedmagnetic field /H20849which enables a much more transparent pre- sentation of several important effects /H20850. Brief comparison with the experimental data is performed in the last section. II. METHODOLOGICAL PROBLEMS OF NUMERICAL SIMULATIONS IN THE POINT-CONTACT SETUP Numerical simulations were carried out with our soft- ware package MicroMagus .13This package uses the modified Bulirsch-Stoer algorithm with the adaptive step-size controlfor integrating the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation to calculate the time evolution of the system magnetizationconfiguration. For the additional torque created by the spin-polarized current /H20849SPC /H20850, we have assumed the symmetric Slonczewski form /H9003=/H20849a J/MS/H20850·/H20851M/H11003/H20849M/H11003S/H20850/H20852 /H20849Sis the spin polarization direction /H20850in order to study the SPC-induced dy- namics in the simplest possible approximation. In the trilayersystem under study /H20849see below /H20850, the spin torque was assumed to act on the magnetization of the “free” layer only. The sitedependence of the spin torque magnitude a J/H20849r/H20850/H20849in the first approximation confined within the point-contact area /H20850is dis- cussed below. To enable a comparison with the experimental results reported in the most advanced quantitative studies of magne-tization oscillations in the point-contact geometry, 6,7we have chosen the system parameters as close as possible to thosereported in Ref. 6. We have simulated a trilayer system con-sisting of two magnetic layers and an interlayer nonmagneticspacer: the lower “fixed” layer /H20849underlayer /H20850with the thick- ness h 1=10 nm and magnetic parameters typical for Co90Fe10/H20849saturation magnetization MS=1500 G, exchange constant A=2/H1100310−6erg/cm /H20850; the upper /H20849thin/H20850Permalloy- like magnetic layer with the thickness h2=5 nm and MS =640 G /H20849as measured in Ref. 6 /H20850and A=1/H1100310−6erg/cm /H20849standard value for Py14/H20850; the spacer thickness was set to hsp=5 nm as for the Cu spacer used in Ref. 6. All results presented below were obtained for an external field H0a/H20850Electronic mail: db@innovent-jena.deJOURNAL OF APPLIED PHYSICS 99, 08Q701 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08Q701/6/$23.00 © 2006 American Institute of Physics 99, 08Q701-1 [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: 136.165.238.131 On: Mon, 22 Dec 2014 06:59:07=1000 Oe. Coordinate axes 0 xand 0 zlie in the film plane, with the xaxis directed along the external field. The “fixed” layer thickness h1=10 nm was taken less than the experimental value6h1exp=20 nm, because proper simulations of such relatively thick layers require not onlytheir in-plane discretization, but also the subdivision into su-blayers, which would lead in this case to prohibitively largecomputational times. The influence of underlayer thicknesson the magnetization dynamics will be discussed elsewhere. We also did not study the effect of the polycrystalline structure of Co 90Fe10underlayer, although the magnetocrys- talline anisotropy of this material is not negligible /H20849cubic anisotropy with K1/H110155.6/H11003105erg/cm3was reported in Ref. 15/H20850. The magnetization dynamics of the system under study turned out to be very complicated already for ideal layers/H20849without taking into account their polycrystalline structure and corresponding random anisotropy /H20850, so we have post- poned the study of the random magnetocrystalline anisotropyeffects. In contrast to micromagnetic study of spin injection ef- fects observed in the columnar structures, 3,4where standard micromagnetic methods can be applied,11simulations of the point-contact experiments encounter serious methodologicalproblems. Here, we discuss two of them: /H20849i/H20850artificial inter- ference effects occurring both for open and periodic bound-ary conditions and /H20849ii/H20850artificial shortwave magnetization os- cillations arising by the usage of a sharp cutoff of charge andspin currents. Artificial interference effects . This problem occurs due to the combination of two circumstances: first, for practical pur-poses magnetic materials with low dissipation rate /H9261 /H110110.01–0.02 are used /H20849to ensure a small linewidth of the oscillation spectrum and a reasonably low excitation thresh-old/H20850, and second, a lateral size of a system available for simulations is much smaller than that used experimentally. Inreal experiments the lateral size of a multilayer is about10 mkm 6, which is definitely far above the value accessible for numerical simulations, especially taking into account thatdynamic simulations require a much finer mesh than quasi-static ones. In particular, we have found out that to obtain amesh-independent result for a system with geometric andmagnetic parameters given above, a lateral discretization asfine as /H9004x/H11003/H9004z=2.5/H110032.5 nm 2is necessary, which limits the simulated lateral system size to /H110111 mkm2/H20851which means more /H20849/H11011105/H20850cells per layer /H20852. In a layer made of a material with low dissipation /H20849see above /H20850, the decay length of the spin wave with the wave vector corresponding to the inverse sizeof the point contact has the same order of magnitude as thesimulated area size. When open boundary conditions /H20849OBC /H20850are used /H20849i.e., a finite-size element is simulated /H20850, this means that the wave emitted by the point contact propagates across the wholeelement, is reflected at its free borders, and returns to thecontact location. The strong interference of this reflectedwave with primary /H20849SPC-induced /H20850magnetization oscillations leads to unphysical artifacts, especially taking into accountthat both waves have the same frequency. Another problemthat arises by using OBC is a complicated pattern of the wave reflection due to the inhomogeneous magnetizationconfiguration on the element edges. For periodic boundary condition /H20849PBC /H20850the magnetiza- tion configuration at the simulated area borders is homoge-neous /H20849which is the main reason to use PBC /H20850, but the pri- mary wave also propagates across the whole simulated areaand due to PBC enters this area from the opposite side, caus-ing the same undesirable interference effects. To eliminatethese effects, a method to absorb the wave near the simula-tion area borders, not affecting the low dissipation at andnear the point contact, is required. To ensure such an absorption, we have embedded in our code an artificial site dependence of the dissipation coeffi-cient/H9261/H20849r/H20850. The function describing this dependence should fulfill several conditions: /H20849i/H20850the dissipation within and nearby the point-contact area should remain equal to itsphysical value /H9261 0to preserve the dynamic properties of the system under study; /H20849ii/H20850the dissipation coefficient far from the point contact /H20849near the simulation area borders /H20850should be large enough to ensure the wave energy absorption; /H20849iii/H20850spa- tial variation /H9261/H20849r/H20850should be smooth enough to prevent the wave reflection from the border between the areas of small and relatively large /H9261/H20849due to the abrupt changes of the media properties /H20850. Following these requirements, we have adopted a site-dependent dissipation /H9261/H20849r/H20850in the form /H9261/H20849r/H20850=/H92610+/H9004/H9261/H208731 + tanhr−R0 /H9268/H9261/H20874. /H208491/H20850 Here, it is assumed that the point contact is located at the coordinate origin. The function /H208491/H20850provides a gradual in- crease of the dissipation parameter above the /H20849small /H20850basic value/H92610, which starts at the distance /H11015/H20849R0−/H9268/H9261/H20850from the point contact and occurs smoothly within a ring of the width /H110152/H9268/H9261. The maximal dissipation value reached outside of this ring is /H9261max=/H92610+/H9004/H9261. We have found out that for the system size Lx/H11003Lz=1/H110031 mkm2and basis dissipation values in the range/H92610=0.01–0.04 the introduction of the additional dissi- pation /H208491/H20850with R0=300 nm, /H9268/H9261=40nm, and /H9004/H9261=0.1 ensured the wave absorption at the simulation area borders, notchanging the magnetization dynamics within and around thepoint contact area. For simulations considered here, we haveused the basis dissipation value /H9261 0=0.02. Spurious magnetization oscillations caused by a sharp spatial cutoff of a point-contact current . By simulations of the columnar geometry the current is usually assumed to bedistributed homogeneously within the layer plane of a nano-element, which does not lead to any methodical problemsbecause the magnetization is also present only inside the areawhere the current flows. In contrast to this simple situation,by simulations of a point-contact setup the naive usage of thesteplike current density in the form j/H20849r/H33355D/2/H20850=j 0and j/H20849r /H11022D/2/H20850=0/H20849rbeing the distance from the contact center, D the contact diameter /H20850lead to the development of artificial magnetization oscillations with the smallest wavelength sup-ported by the given lattice. The reason for these oscillationsis the nonphysical abrupt change of the current density j/H20849r/H20850 atr=D/2, so that its spatial Fourier image and the Fourier image of the current-induced magnetic field /H20849the Oersted08Q701-2 D. V. Berkov and N. L. Gorn J. Appl. Phys. 99, 08Q701 /H208492006 /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: 136.165.238.131 On: Mon, 22 Dec 2014 06:59:07field /H20850HOe/H20849k/H20850exhibits large tails up to the highest values of the wave vectors available for the simulated discrete system. This leads to artificial instabilities for these wave vectors,resulting in the appearance of corresponding magnetizationoscillations. In order to avoid this problem, we have smoothed the spatial distribution of H Oe/H20849r/H20850obtained in the approximation of a sharp electric current cutoff convolving it with the Gaussian kernel exp /H20849−r2/2/H9268H2/H20850. A physically meaningful choice of the smoothing parameter /H9268Hwould require reliable information about the lateral diffusion of the electric currentcarriers to calculate the actual spatial distribution of the cur-rent density. Lacking such knowledge, we have simplyadopted the minimal value /H9268H=2/H9004x/H20849two times larger than the mesh size /H20850, which was sufficient to eliminate the artificial oscillations mentioned above. Further increment of this pa-rameter within a reasonable range /H20851up to /H9268H=/H208494–5 /H20850·/H9004x/H20852had only a minor influence on physical results. A similar problem is caused by the sharp spatial cutoff of thespincurrent density represented by the amplitude aJ/H20849r/H20850of the SPC-induced torque /H9003=/H20849aJ/MS/H20850·/H20851M/H11003/H20849M/H11003S/H20850/H20852, al- though spatial oscillations caused by this cutoff are weaker than discussed in the previous paragraph. Nevertheless, thesame kind of smoothing is also required to solve this prob-lem. The smoothing parameter of the corresponding kernelexp/H20849−r 2/2/H9268S2/H20850is directly related to the spin diffusion length and could in principle be computed from the corresponding theory. In this study, however, we have also simply used thevalue /H9268S=2/H9004xfor the same reasons as explained above for /H9268H. At this point we would like to emphasize that, in contrast to the Oersted field smoothing parameter, the /H9268Svalue sig- nificantly influences the system behavior; in particular, thethreshold value of a Jfor the onset of steady-state microwave oscillations substantially depends on /H9268S. For this reason the problem of calculating the actual distribution of a spin cur- rent in the point-contact geometry deserves special attention. All results presented below were obtained employing the site-dependent damping /H208491/H20850and smoothing of the Oersted field and spin current distribution with parameters givenabove. The in-plane discretization of both magnetic layerswith the mesh size /H9004x/H11003/H9004z=2.5/H110032.5 nm 2and the full size of the simulation area Lx/H11003Lz=1/H110031 mkm2/H20849with PBC /H20850were used. The point-contact diameter was set to D=40 nm. III. NUMERICAL SIMULATIONS: RESULTS AND DISCUSSION In this paper we discuss only simulation results obtained without taking into account the effect of thermal fluctuations/H20849T=0/H20850. Even without these effects the system demonstrates very complicated dynamics, which should be understood be- fore thermal fluctuations are taken into consideration. Dynamics without taking into account the Oersted field . We start with the analysis of the magnetization dynamicswhen the influence of the Oersted field is neglected. We pointout here that in the experimental situation 6the current- induced magnetic field is comparable with the externally ap-plied field: For the total current I/H110114 mA flowing through the point contact with the diameter D/H1101540 nm, the maximalvalue of the Oersted field is H Oemax/H11011400 Oe, whereas the ex- ternal field used in Ref. 6 to present the most detailed resultsisH ext=1000 Oe. For this reason we do not expect the ap- proximation when HOeis neglected to be quantitatively cor- rect, but neglecting the Oersted field simplifies the magneti-zation dynamics of the system under study, preserving mostof its qualitative features, which thus can be demonstratedmore clearly. The major feature of the simulated magnetization dy- namics in the point-contact geometry is the existence of two current regions where the steady-state precession of the mag-netization within and nearby the point-contact area exists/H20849Fig. 1 /H20850. In the first current region before the magnetization in the point-contact area is switched under the SPC influence, i.e.,when the magnetization is /H20849on average /H20850still directed along the external field the magnetization dynamics is relativelysimple. Magnetization configuration of the thin /H20849upper /H20850layer within the point-contact area remains roughly collinear. Spinwaves emitted from the area under the contact are smoothand have a simple elliptical wave front /H20849Fig. 1 /H20850. The limit oscillation cycle of the magnetization m avaveraged over this area represents a slightly bent ellipse /H20849Fig. 2 /H20850, time depen- dencies of the magnetization components are nearly idealharmonic functions, so that oscillation power spectra consist of a single and very narrow peak. As usual for such in-planeoscillations, the frequency of m xavoscillations /H20849the component along the external field Hext/H20850is twice the mzavfrequency /H20849the in-plane component perpendicular to Hext/H20850, because mxavgoes FIG. 1. Dependencies of the oscillation frequency /H20849a/H20850and the total oscilla- tion power /H20849b/H20850ofmxav/H20849circles /H20850andmzav/H20849squares /H20850magnetization projections /H20849averaged over the point-contact area /H20850on the spin current strength given by the Slonczewski torque amplitude aJ.T h e aJvalue, where the magnetization within the contact area switches to the direction opposite to the externalfield, is marked by the vertical dashed line.08Q701-3 D. V. Berkov and N. L. Gorn J. Appl. Phys. 99, 08Q701 /H208492006 /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: 136.165.238.131 On: Mon, 22 Dec 2014 06:59:07back and forth twice during a single oscillation cycle. The oscillation frequency decreases monotonically with increas-ing current /H20849a Jin our formalism /H20850, which is mainly due to the increase of the oscillation amplitude /H20849longer limit cycle /H20850with the current strength. The oscillation power sharply increaseswhen the current exceeds the threshold for the oscillationonset, and then growth smoothly until the magnetization un-der the point-contact area is switched by the SPC. It turns out, however, that in the model simulated here a steady-state precession exists also after the point-contact area switching caused by spin injection. By the transition tothis second regime we observe a large frequency jump—forthe system parameters used in this study the frequency dropsdown from f bef/H110159.8 GHz to faft/H110154.0 GHz and then remains almost current independent. Although the limit cycles in thisregime have a more complicated form and the time depen-dencies of the magnetization components considerably differfrom ideal sinusoids, spectral lines are still so narrow that,within the physical time corresponding to the longest simu-lation run performed /H20849/H1101520 ns /H20850, their width could not be re- solved /H20849/H9004f/H3335550 MHz /H20850. Oscillation amplitude slowly de- creases with current, leading to the corresponding decrease of the oscillation power. Although spectral lines in this second precession mode remain quite narrow, the magnetization configurations ap-pearing during the precession are extremely complicatedeven when the Oersted field is neglected. First, we note thatthe precession frequency is below the frequency of the ho-mogeneous ferromagnetic resonance /H20849FMR /H20850mode for the layer under consideration /H20849f 0=/H20849/H9253/2/H9266/H20850·/H20851H0/H20849H0+4/H9266MS/H20850/H208521/2 /H110158.4 GHz. For this reason the “normal” circular /H20849elliptical /H20850 wave cannot exist in this regime, so that the energy is emit-ted in the form of the solitonlike wave packages, as shown inFig. 3. The magnetization within and nearby the contact areaitself forms vortex/antivortex pairs /H20849the latter state is also sometimes called a crosslike configuration /H20850, the creation and annihilation of which is the basic feature of the magnetiza-tion dynamics in this regime; a typical example of such astructure is shown in Fig. 4. Detailed analysis of these chal- lenging structures will be performed elsewhere. Dynamics with the Oersted field included . Inclusion of the Oersted field requires establishment of the relation be-tween the current strength used in the actual experiment andthe parameter a Jused in simulations to set the amplitude of the SPC-induced torque. Lacking the exact microscopictheory that could provide such a relation, we have used thesame procedure as in Ref. 12, i.e., we assumed that the os-cillation onset threshold a J=2.2 corresponds to the minimal current Imin/H110154 mA, where the magnetization precession is observed experimentally. The current-induced magnetic field HOe, being strongly inhomogeneous, results in a much more complicated magne-tization dynamics than in the absence of H Oe. The most ob- vious change is the appearance of the wave asymmetry in thesteady-state precession regime before switching: in that halfof the layer where the Oersted field H Oeis directed opposite to the external field H0/H20849thus partly compensating it /H20850, the wave amplitude is significantly larger than in the other halfof the film /H20849where the external field is enhanced by H Oe/H20850. The FIG. 2. Steady-state precession before switching of the contact area: 3D trajectory of the magnetization mavaveraged over the contact area /H20849left panel /H20850and snapshots of the waves emitted from the contact area shown as gray-scale maps of the component mx/H20849r/H20850along the external field and my/H20849r/H20850—perpendicular to the layer plane /H20849right panels /H20850. On the my/H20849r/H20850map the superposition of the two waves with the wavelengths corresponding to theprecession frequencies of /H20849i/H20850longitudinal /H20849m x/H20850and /H20849ii/H20850transverse /H20849myormz/H20850 magnetization components is clearly seen. Physical size of the imagesshown on the right is 900 x900 nm 2. FIG. 3. The same as in Fig. 2 for the steady-state precession after switching of the magnetization under the contact. Two compact wave packages emit-ted from the contact area and their propagation direction are marked bywhite arrows. FIG. 4. Magnetization configuration of the central part of the images pre-sented in Fig. 3 shown as arrows, which represent the orientation and mag-nitude of the in-plane magnetization. The physical size corresponding to thisarrow map is /H1101590/H1100390 nm 2/H20849the point-contact diameter is D=40 nm /H20850.08Q701-4 D. V. Berkov and N. L. Gorn J. Appl. Phys. 99, 08Q701 /H208492006 /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: 136.165.238.131 On: Mon, 22 Dec 2014 06:59:07inhomogeneity of HOeleads also to further complication of the magnetization states in the “after-switching” regime: thenumber of vortex-antivortex pairs that might exist simulta-neously increases and the precession trajectory /H20849limit cycle /H20850 of the average magnetization of the point-contact area be-comes quasiperiodic. The complete analysis of the corre-sponding dynamics also will be presented elsewhere. The major effect of the current-induced field is, however,not the quantitative changes in the “before-” and “after- switching” precession modes discussed above, but the ap-pearance of a new intermediate regime in between these twocurrent regions. The corresponding interval is marked in Fig.5 by the legend “complicated magnetization dynamics.” Forcurrents within this interval /H208492.45/H11021a J/H110212.9 for parameters used in our simulations /H20850, the xprojection of the magnetiza- tion under the contact area exhibits relatively rare transitionsbetween the values close to the maximal possible valuem xmax=1 and values close to mxav=0/H20851Fig. 6 /H20849a/H20850/H20852. Correspond- ingly, its power spectrum /H20851Fig. 6 /H20849b/H20850/H20852has a large component at/H20849relatively /H20850low frequencies. The zprojection of the mag- netization /H20849in-plane projection perpendicular to the external field /H20850oscillates with very different frequencies depending on themxvalue, i.e., on the magnetization configuration in the point-contact region: for nearly homogeneous magnetizationstate /H20849m xavclose to 1 /H20850the oscillation frequency of mzavis much higher than for a strongly inhomogeneous configuration/H20849small values of m xav/H20850. Oscillation power spectrum of mzav consists of several relatively broad lines, the quantitative analysis of which requires better simulation statistics /H20849longer runs /H20850than those that could be carried out up to now. IV. COMPARISON WITH EXPERIMENTAL DATA In this last section we briefly compare our simulation data with experimental results from Ref. 6. First, we notethat several important features of experimentally observedmagnetization precession in the point-contact setup could bereproduced by our simulations. In particular, in the oscilla-tion regime before switching we have obtained, in accor-dance with Ref. 6, very narrow spectral lines /H20849in our simula- tions the linewidth was /H9004f/H1102150 MHz /H20850and nearly linear decay of the oscillation frequency with increasing current/H20851see Figs. 1 /H20849a/H20850and 5 /H20852. Such extremely small linewidths can be naturally explained by a smooth variation /H20849in space /H20850of the magnetization configuration, which is due to the absence ofstrong demagnetizing field effects characteristic for nanoele- FIG. 5. The same as in Fig. 1, but for the magnetization dynamics simulated including the current-induced magnetic field. In the intermediate interval ofthe spin torque amplitudes a Jbetween two dashed lines, the magnetization under the point-contact area demonstrates a very complicated dynamics thatcannot be described as a steady-state precession with a simple closed limitcycle /H20849see Fig. 6 /H20850. FIG. 6. Typical time dependencies of mxav/H20849a/H20850andmzav /H20849c/H20850magnetization projections /H20849averaged over the point- contact area /H20850ataJ=2.8 when the Oersted field is taken into account. Corresponding oscillation power spectraare shown in panels /H20849b/H20850and /H20849d/H20850.08Q701-5 D. V. Berkov and N. L. Gorn J. Appl. Phys. 99, 08Q701 /H208492006 /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: 136.165.238.131 On: Mon, 22 Dec 2014 06:59:07ments in the columnar geometry /H20849see, e.g., Refs. 3, 4, and 11/H20850. The decrease of the oscillation frequency with increas- ing current is a consequence of the growing precession am-plitude when the current /H20849and hence the spin torque magni- tude /H20850is increased, which for nonlinear oscillations results in the larger precession period. More important, however, are the disagreements be- tween simulated and measured data, among which the quali- tative one—the existence of at least two oscillation regimes for the simulated precession /H20849whereby only one precession regime was observed experimentally /H20850—is the major problem. There are several possible reasons for this discrepancy,among them /H20849i/H20850the simplicity of the torque term used in simulations /H20849for more complicated forms that should be tested next, see Refs. 16 and 17 /H20850;/H20849ii/H20850influence of the SPC- induced effective field recently measured in Ref. 18; /H20849iii/H20850 exchange weakening within magnetic layers resulting fromthe local Joule heating /H20849in the point-contact geometry this effect may be especially pronounced due to high current den-sities required to induce magnetization oscillations /H20850; and /H20849iv/H20850 substantial contribution of the layer regions outside thepoint-contact area to the measured microwave oscillationspectra. In particular, for the last reason mentioned, the sig-nal from these regions would be present in the regime beforeswitching, but nearly absent for the after-switching mode /H20849as it can be seen from the comparison of the right panels ofFigs. 2 and 3 /H20850, thus strongly enhancing the oscillation power in the regime before switching compared to the second re- gime. However, to clarify whether it is really the case, cal-culations of the current distribution for the concrete experi-mental setup are required. Another interesting problem is the existence of a strong second harmonic in the experimentally measured spectrum.Its presence could be caused by the local /H20849under the point contact /H20850deviation of the underlayer magnetization from the external field direction. This would lead to the contributionsfrom the oscillations of both longitudinal /H20849m x/H20850and transverse /H20849mz/H20850in-plane magnetization components, thus providing the required second harmonic /H20849mx/H20850and basic /H20849mz/H20850oscillation frequency.12Such a magnetization deviation from the H0di- rection could exist due, e.g., to the random magnetic aniso-tropy of Co 90Fe10crystallites. Another explanation of the second harmonic presence could be the above-mentionedcontribution of the area around the point contact because,due to the local conservation of the magnetic moment mag-nitude, the waves of both m xand mzcomponents contain both frequencies /H20849see Fig. 2, right panels /H20850.V. CONCLUSION In conclusion, we point out that full-scale micromagnetic simulations /H20849performed in frames of the Slonczewski formal- ism/H20850of the magnetization dynamics induced by a spin- polarized current in the point-contact geometry recover sev-eral important features of experimental observations, likevery narrow spectral lines and current dependence of theoscillation frequency. However, if we assume that the mea-sured signal comes solely from the magnetization oscillationunder the point-contact area, simulation results exhibit seri-ous qualitative disagreements with experimental data, themain of which is the existence of at least two precessionmodes in the two current intervals corresponding to /H20849i/H20850the precession around the external field direction /H20849before- switching mode /H20850and /H20849ii/H20850around the direction opposite to H 0 /H20849after-switching mode /H20850. This disagreement clearly shows that further refinement of theoretical models is required for un-derstanding of the spin torque-induced magnetization excita-tions in point-contact experiments. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 , /H208491996 /H20850; L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850 2J. Z. Sun, J. Magn. Magn. Mater. 202, 157 /H208491999 /H20850; M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev.Lett. 80, 4281 /H208491998 /H20850. 3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoe- lkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850425,3 8 0 /H208492003 /H20850. 4S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, M. Rinkoski, C. Perez, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 93, 036601 /H208492004 /H20850. 5W. H. Rippard, M. R. Pufall, and T. J. Silva, Appl. Phys. Lett. 82, 1260 /H208492003 /H20850. 6W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. J. Silva, Phys. Rev. Lett. 92, 027201 /H208492004 /H20850. 7W. H. Rippard, M. R. Pufall, S. Kaka, T. J. Silva, and S. E. Russek, Phys. Rev. B 70, 100406 /H20849R/H20850/H208492004 /H20850. 8X. Waintal, E. B. Myers, P. W. Brouwer, and D. C. Ralph, Phys. Rev. B 62, 12317 /H208492000 /H20850; M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 /H208492002 /H20850; M. D. Stiles and A. Zangwill, J. Appl. Phys. 91,6 8 1 2 /H208492002 /H20850. 9J. Z. Sun, Phys. Rev. B 62, 570 /H208492000 /H20850. 10D. V. Berkov and N. L. Gorn, Phys. Rev. B 71, 052403 /H208492005 /H20850. 11J. Miltat, G. Albuquerque, A. Thiaville, and C. Vouille, J. Appl. Phys. 89, 6982 /H208492001 /H20850; Z. Li and S. Zhang, Phys. Rev. B 68, 024404-1 /H208492003 /H20850; J.-G. Zhu, X. Zhu, IEEE Trans. Magn. 40, 182 /H208492004 /H20850; X. Zhu, J.-G. Zhu, and R. M. White, J. Appl. Phys. 95, 6630 /H208492004 /H20850;K .J .L e e ,A .D e a c ,O . Redon, J. P. Nozieres, and B. Dieny, Nat. Mater. 3, 877 /H208492004 /H20850. 12D. V. Berkov and N. L. Gorn, Phys. Rev. B 72, 094401 /H208492005 /H20850. 13D. V. Berkov and N. L. Gorn, MicroMagus —package for micromagnetic simulations; http:\\www.micromagus.de 14W. Doering, in Micromagnetismus , Handbook of Physics, edited by S. Flügge /H20849Springer, Berlin, 1966 /H20850. 15J. Pelzl, R. Meckenstock, D. Spoddig, F. Schreiber, J. Pflaum, and Z. Frait, J. Phys.: Condens. Matter 15, S451 /H208492002 /H20850. 16J. C. Slonczewski, J. Magn. Magn. Mater. 247,3 2 4 /H208492002 /H20850. 17J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70, 172405 /H208492004 /H20850. 18M. A. Zimmler, B. Özyilmaz, W. Chen, A. D. Kent, J. Z. Sun, M. J. Rooks, and R. H. Koch, Phys. Rev. B 70, 184438 /H208492004 /H20850.08Q701-6 D. V. Berkov and N. L. Gorn J. Appl. Phys. 99, 08Q701 /H208492006 /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. 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5.0045584.pdf

Appl. Phys. Lett. 118, 162601 (2021); https://doi.org/10.1063/5.0045584 118, 162601 © 2021 Author(s).Vortex ordering and dynamics on Santa Fe artificial ice pinning arrays Cite as: Appl. Phys. Lett. 118, 162601 (2021); https://doi.org/10.1063/5.0045584 Submitted: 27 January 2021 . Accepted: 26 March 2021 . Published Online: 21 April 2021 Wenzhao Li , C. J. O. Reichhardt , B. Jankó , and C. Reichhardt COLLECTIONS Paper published as part of the special topic on Mesoscopic Magnetic Systems: From Fundamental Properties to Devices ARTICLES YOU MAY BE INTERESTED IN Artificial spin ice: Paths forward Applied Physics Letters 118, 110501 (2021); https://doi.org/10.1063/5.0044790 Microscopic metallic air-bridge arrays for connecting quantum devices Applied Physics Letters 118, 162108 (2021); https://doi.org/10.1063/5.0045557 Perspectives on spintronic diodes Applied Physics Letters 118, 160502 (2021); https://doi.org/10.1063/5.0048947Vortex ordering and dynamics on Santa Fe artificial ice pinning arrays Cite as: Appl. Phys. Lett. 118, 162601 (2021); doi: 10.1063/5.0045584 Submitted: 27 January 2021 .Accepted: 26 March 2021 . Published Online: 21 April 2021 Wenzhao Li,1C. J. O. Reichhardt,2,a) B.Jank /C19o,1and C. Reichhardt2 AFFILIATIONS 1Department of Physics, University of Notre Dame, Notre Dame, Indiana 46656, USA 2Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA Note: This paper is part of the APL Special Collection on Mesoscopic Magnetic Systems: From Fundamental Properties to Devices. a)Author to whom correspondence should be addressed: cjrx@lanl.gov ABSTRACT We numerically examine the ordering, pinning, and flow of superconducting vortices interacting with a Santa Fe artificial ice pinning array. We find that as a function of magnetic field and pinning density, a wide variety of vortex states occur, including ice rule obeying states andlabyrinthine patterns. In contrast to square pinning arrays, we find no sharp peaks in the critical current due to the inherent frustration effect imposed by the Santa Fe ice geometry; however, there are some smoothed peaks when the number of vortices matches the number of pinning sites. For some fillings, the Santa Fe array exhibits stronger pinning than the square array due to the suppression of one-dimensionalflow channels when the vortex motion in the Santa Fe lattice occurs through the formation of both longitudinal and transverse flowchannels. Published under license by AIP Publishing. https://doi.org/10.1063/5.0045584 In an artificial spin ice (ASI) system, 1,2the states can be effectively described as spin-like degrees of freedom, which can obey the same ice rules found for the ordering of protons in water ice3or of atomic spins in certain materials.4,5One of the first artificial spin ice systems was con- structed from coupled magnetic islands in which the magnetic moment of each island can be described as a single classical spin.1,6In this sys- tem, for specific arrangements of the effective spins at the vertices, the ground state obeys the ice rules and a vertex at which four spins meet has two spins pointing “in” and two spins pointing “out.” Configurations that obey the ice rule can have long range order, such as in square ice,1,6–8or they can be disordered, such as in kagom /C19ei c e .1,9A wide range of additional geometries beyond square and kagom /C19ei c e have been proposed2,10and realized,11–15including mixed geometries that force the formation of excited vertices.10,11,16 In addition to studies in magnetic ASI systems, there are several particle-based realizations of ASI,17where a collection of interacting particles is coupled to some form of substrate to create states that obey the ice rules. Such systems have been studied for colloidal particles coupled to ordered substrates,18–22magnetic skyrmions,23and vortices in type-II superconductors with nanostructured pinning arrays.24–30 The ice rule obeying states often arise through different mechanismsin the particle-based systems compared to magnetic ice systems since the particle ices minimize Coulomb energy rather than vertexenergy. 21,31In this work, we consider vortices interacting with a varia- tion of the ASI geometry, which is called a Santa Fe spin ice. Santa Fe (SF) spin ice, shown in Fig. 1(a) , can contain both frus- trated and non-frustrated vertices,10,16,32forcing some vertices to be in an excited state. In the particle-based model, this would mean thatsome fraction of particles are close together, whereas the ground stateof a square ice array does not include such close particle spacings. In superconducting ASI systems with non-frustrated ground states, a series of peaks appear in the critical current at the fields correspondingto ice rule obeying states as well as at higher matching fields. 25,26,28 In this paper, we use numerical simulations to investigate the vortex configurations, pinning, and flow patterns in a system with aSanta Fe ASI geometry. We find numerous distinct vortex patterns for increasing magnetic fields or different pinning densities. At the half matching field, the vortex configurations are close to those expectedfor the ground state of the SF ice. Compared to the square pinning lat-tice, in the SF ice, we find only weak or smeared peaks in the critical depinning current, and we show that the vortex flow patterns are much more disordered. For dense pinning arrays, the ice rule obeyingstates vanish, but the vortices form a labyrinthine pattern. We consider a two-dimensional system of N vvortices interacting with an ordered array of pinning sites, which are placed either in aSanta Fe artificial spin ice pattern or in a square lattice. The system Appl. Phys. Lett. 118, 162601 (2021); doi: 10.1063/5.0045584 118, 162601-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplcontains Nppinning sites. The number of vortices is proportional to the applied magnetic field B,a n d Nv¼Npcorresponds to the matching condition B=B/¼1:0, where B/is the matching field. There are peri- odic boundary conditions in the xand y-directions, and the equation of motion for a vortex iis given by gdR dt¼Fvv iþFp iþFd: (1) The damping constant gis set to unity. The vortex-vortex interactions are repulsive and given by Fvv i¼PF0K1ðRij=kÞ^Rij,w h e r e K1is the modified Bessel function, Rijis the distance between vortex iand vortex j,a n d F0¼/2 0=2pl0k3. We set the penetration depth to k¼1:8. In the absence of pinning sites, the vortices form a triangular lattice. A uniform driving force Fd¼Fd^xis applied to all the vortices, and the system is considered depinned when the average steady-statevortex velocity is larger than a non-trivial value. The pinning sites are modeled as localized traps of radius r pwith the form Fp i¼/C0PNp k¼1FpRikexpð/C0R2 ik=r2 pÞ^Rik,w h e r e Rikis the dis- tance between vortex iand pin k, and we set rp¼0.6. We match our system geometry to the experiments on square vortex ice systems,25 where dis the distance between two pinning sites. In Fig. 1(a) ,w e show an example of a Santa Fe pinning array containing four cells. Each cell is divided into eight elementary rectangular plaquettes in which the pinning sites can be grouped into pairs that are spaced a dis- tance d¼0.825 apart. The vortex configurations are obtained by simu- lated annealing from a high temperature. InFig. 1(b) , we show the vortex configurations in the SF geome- try for a system with d¼0.825 at B=B/¼1=2. Since the vortices arerepulsive, they move as far away from each other as possible; however, when pinning is present, there is a competing pinning energy thatfavors having the vortices occupy the pinning sites. At B=B /¼1=2, two neighboring pinning sites can be regarded as a single double welltrap. An individual vortex can occupy one end of this double trap,determining the direction of the effective spin. For the three-leg or z¼3 vertices, the lowest energy per vertex would have two effective spins pointing “out” (away from the vertex) and one effective spinpointing “in” (toward the vertex); however, due to the geometric con-straints, there must be some vertices with two spins in and one spinout, giving an energy higher than the ground state. A single cell in theSF ice contains rectangular plaquettes that surround an inner square,as shown in Fig. 1(a) .I nFig. 1(b) , most of the vortices in the z¼3v e r - tices can form the two-out, one-in ground state; however, due to the geometric constraints, some vertices adopt the two-in, one-out excitedconfiguration. In addition, the inner square contains several locationswhere two vortices are close together in neighboring pins, creating ahigh energy excitation. These excitations arise during the annealingprocess. The overall configuration is close to the expected ice rule obeying state with forced excitations, as predicted for the magnetic version of the SF spin ice. 10,32InFig. 1(c) , we illustrate the vortex con- figurations for B=B/¼1:0 at the commensurate field. Figure 1(d) shows the configurations at B=B/¼1:5 ,w h e r et h e r ea r en u m e r o u s instances of individual pinning sites capturing two vortices to form avortex dimer state, along with several cases where vortices are locatedin the interstitial regions in the middle of the rectangular plaquettes. In this case, there is no long range order. For B=B /¼2:0 and 2.5 (not shown), the system remains disordered. InFig. 2(a) , we plot the critical depinning force Fc,w h i c hi sp r o - portional to the critical current as a function of B=B/for the SF system inFig. 1 and for a square pinning lattice. The xaxis is normalized by the matching field B/for the SF lattice, and in these units, the match- ing field of the square lattice falls at 1 :286B/. The square lattice exhib- its a pronounced peak in Fcat the matching field similar to that found in other studies of vortex pinning on square substrates.33,34In the SF lattice, there is instead a broadened peak in Fcaround the first match- ing field. For a square ice system at B=B/¼1=2, different from the square lattice shown in Fig. 2(a) , previous work25has shown that a FIG. 1. (a) Blue dots indicate the pinning sites arranged in a Santa Fe ASI geome- try with d¼0.825. The entire panel represents one cell of the Santa Fe ASI. (b) The vortex positions (black dots) and pinning locations (blue dots) for the system inpanel (a) at B=B /¼1=2, where the ice rule is mostly obeyed but there are scat- tered excitations present (red dots). (c) B=B/¼1:0. (d) B=B/¼1:5. FIG. 2. (a) The critical depinning force FcvsB=B/for the SF system in Fig. 1(a) (blue curve) with d¼0.825 and for a square pinning array (red curve). The match- ing field B/is for the SF array; in these units, the matching field for the square array is at 1 :286B/. The labels a and b indicate the value of B=B/at which the images in Fig. 3 were obtained. (b) The critical depinning force FcvsB=B/for SF systems with different densities of d¼0.4 (green curve) and d¼1.8 (blue curve).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 162601 (2021); doi: 10.1063/5.0045584 118, 162601-2 Published under license by AIP Publishingpeak corresponding to the ice rule obeying the ground state appears in the critical current, that is as large as the matching peak atB=B /¼1:0. For the SF array, there is no peak at B=B/¼1=2d u et o the high energy excitations that are forced to exist by the SF geometry.Figure 2(a) shows that the overall pinning strength of the SF lattice is generally smaller than that of the square lattice due to the fact thatthere are fewer pins; however, there are several regimes where the depinning force for the SF geometry is higher than that of the squarearray, particularly for B=B />1:0. This is due to the tendency for the vortices in the square lattice to form easy flow one-dimensional chan-nels along the symmetry axis of the pinning array. To more clearlyillustrate this effect, in Figs. 3(a) and3(b), we show the vortex trajecto- ries at B=B /¼1:67 for the SF and square arrays, respectively, from Fig. 2(a) just above depinning. For the square array, the vortex motion follows one-dimensional interstitial channels between the vorticestrapped at the pinning sites, while for the SF array, the motion occursthrough a combination of longitudinal and transverse flow channels,so that some vortices move perpendicular to the direction of drive attimes. Both the reduction of the critical current by easy vortex flowalong symmetry directions of the pinning array 35–37and the enhance- ment of the critical current when easy flow channels are absent38–52 have been described in previous work. The results in Fig. 2(a) show that the critical depinning currents for both the SF lattice and thesquare lattice almost coincide for most of the range of fields we con-sider. Future work will focus on the case of lattices with equal overallpinning density rather than equal lattice constants. In nanomagnetic spin ice systems, the ice rules are lost as the dis- tance between the magnets is increased due to the reduction in thepairwise interactions between neighboring islands. 6In the supercon- ducting vortex system, the effective vortex interaction can be tuned bychanging the distance between adjacent pinning sites. In Fig. 2(b) ,w e plot the depinning force F cvsB=B/for two different pinning FIG. 3. The vortex trajectories for the system in Fig. 2(a) with d¼0.825 at B=B/¼1:67. (a) The SF lattice exhibits winding labyrinthine flow channels. (b) The square lattice has easy flow one-dimensional channels. The different colorscorrespond to different times. Multimedia View shows the motion corresponding tothese two panels. Multimedia views: https://doi.org/10.1063/5.0045584.1 ;https:// doi.org/10.1063/5.0045584.2 FIG. 4. (a)–(c) The pinning site locations (blue dots) and vortex positions (black dots) for the system in Fig. 2(b) atd¼1.8 with weak vortex interactions where the ice rule is lost. (d)–(f) The same for the system in Fig. 2(b) withd¼0.4, where the pinning sites begin to overlap, creating labyrinthine vortex states. (a) and (d) B=B/¼0:5. (b) and (e) B=B/¼1:0. (c) and (f) B=B/¼2:0.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 162601 (2021); doi: 10.1063/5.0045584 118, 162601-3 Published under license by AIP Publishingdistances, the much larger value d¼1.8, which corresponds to weak vortex interactions, and the much smaller value d¼0.4, which produ- ces strong vortex interactions. At d¼1.8, the vortices are far enough apart that the pinning dominates their dynamical behavior, and theoverall depinning threshold is much higher. Additionally, there is nopeak at B=B /¼1:0, but instead, there is a downward step in the criti- cal current. In Figs. 4(a)–4(c) , we show the vortex configurations for thed¼1.8 sample at B=B/¼0:5, 1.5, and 2.0. For B=B/¼1=2, the vertex populations are random and do not form ice rule obeying states.For B=B /¼1:5 and 2.0, there is a combination of doubly occupied sites and interstitial vortices. For a denser pinning lattice with d¼0.4, there are no peaks at B=B/¼1=2 or 1.0 and the overall pinning force is reduced. Since the pinning radius is fixed, at the smaller d,t h ep i n - ning sites begin to overlap, creating paths of low potential along whichthe vortices can flow, thereby reducing the effectiveness of the pinning.This also produces increasingly labyrinthine vortex configurations, asshown in Figs. 4(d)–4(f) forB=B /¼1=2, 1.5, and 2.0. For higher fi e l d sa tt h i sv a l u eo f d, the labyrinthine pattern persists. In an actual superconducting material in this regime, the vortices could merge toform multi-quanta states, which would have to be studied using amodel different from the point-like vortex model we consider here. We have numerically investigated vortex configurations, pinning, and dynamics in a system with a Santa Fe artificial ice pinning sitearrangement. This pinning geometry forces some vertices to occupyhigher energy states. At half filling, the vortex configurations that weobserve are close to the ground state expected for the magnetic SantaFe ice, with most of the vertices in low energy ice rule obeying statesbut with a small number of high energy vertices present. The criticaldepinning currents do not show a peak at the half matching field, butexhibit a rounded peak near the first matching field. For certain fil-lings, the Santa Fe ice has higher pinning than a square pinning arraybecause the vortex flow in the Santa Fe ice runs both transverse andparallel to the driving direction, as opposed to the strictly parallel flowthat occurs in the square array. For dense pinning arrays where thepinning sites begin to overlap, we find that the vortices can form anintricate labyrinthine state in the Santa Fe ice. We gratefully acknowledge the support of the U.S. Department of Energy through the LANL/LDRD program for this work. Thiswork was supported by the U.S. Department of Energy through theLos Alamos National Laboratory. The Los Alamos NationalLaboratory is operated by Triad National Security, LLC, for theNational Nuclear Security Administration of the U.S. Departmentof Energy (Contract No. 892333218NCA000001). W.L. and B.J.were supported in part by No. NSF DMR-1952841. 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Kwok, “Reconfigurable pinwheel artifi-cial-spin-ice and superconductor hybrid device,” Nano Lett. 20, 8933– 8939 (2020). 31C. Nisoli, “Unexpected phenomenology in particle-based ice absent in magnetic spin ice,” Phys. Rev. Lett. 120, 167205 (2018). 32X. Zhang, A. Duzgun, Y. Lao, N. S. Bingham, J. Sklenar, H. Saglam, S. Subzwari, J. T. Batley, J. D. Watts, D. Bromley, C. Leighton, L. O’Brien, C. Nisoli, and P. Schiffer, “Excitation strings and topological surgery in artificialspin ice,” e-print arXiv:2008.07571 (2020). 33M. Baert, V. V. Metlushko, R. Jonckheere, V. V. Moshchalkov, and Y. Bruynseraede, “Composite flux-line lattices stabilized in superconducting films by a regular array of artificial defects,” Phys. Rev. Lett. 74, 3269–3272 (1995). 34C. Reichhardt, J. Groth, C. J. Olson, S. B. Field, and F. Nori, “Spatiotemporal dynamics and plastic flow of vortices in superconductors with periodic arrays of pinning sites,” Phys. Rev. B 54, 16108–16115 (1996). 35C. Reichhardt and F. Nori, “Phase locking, devil’s staircases, Farey trees, and Arnold tongues in driven vortex lattices with periodic pinning,” Phys. Rev. Lett. 82, 414–417 (1999). 36A. V. Silhanek, L. Van Look, S. Raedts, R. Jonckheere, and V. V. Moshchalkov, “Guided vortex motion in superconductors with a square antidot array,” Phys. Rev. B 68, 214504 (2003). 37J. E. Villegas, E. M. Gonzalez, M. I. Montero, I. K. Schuller, and J. L. Vicent, “Directional vortex motion guided by artificially induced mesoscopic potentials,” Phys. Rev. B 68, 224504 (2003). 38D. Ray, C. J. Olson Reichhardt, B. Jank /C19o, and C. Reichhardt, “Strongly enhanced pinning of magnetic vortices in type-II superconductors by confor- mal crystal arrays,” Phys. Rev. Lett. 110, 267001 (2013). 39S. Gu /C19enon, Y. J. Rosen, A. C. Basaran, and I. K. Schuller, “Highly effective superconducting vortex pinning in conformal crystals,” Appl. Phys. Lett. 102, 252602 (2013). 40Y. L. Wang, M. L. Latimer, Z. L. Xiao, R. Divan, L. E. Ocola, G. W. Crabtree,and W. K. Kwok, “Enhancing the critical current of a superconducting film ina wide range of magnetic fields with a conformal array of nanoscale holes,” Phys. Rev. B 87, 220501 (2013). 41Q. Le Thien, D. McDermott, C. J. O. Reichhardt, and C. Reichhardt, “Enhanced pinning for vortices in hyperuniform pinning arrays and emergent hyperuni-form vortex configurations with quenched disorder,” Phys. Rev. B 96, 094516 (2017). 42I. A. Sadovskyy, Y. L. Wang, Z.-L. Xiao, W.-K. Kwok, and A. Glatz, “Effect of hexagonal patterned arrays and defect geometry on the critical current ofsuperconducting films,” Phys. Rev. B 95, 075303 (2017). 43K. Tanaka, I. Robel, and B. Janko, “Electronic structure of multiquantum giant vortex states in mesoscopic superconducting disks,” Proc. Natl. Acad. Sci. U. S. A.99, 5233–5236 (2002). 44G. Carneiro, “Dynamical phase diagrams for moving vortices interacting with periodic pinning,” Phys. Rev. B 66, 054523 (2002). 45C. Reichhardt and C. J. Olson Reichhardt, “Dynamical ordering and directional locking for particles moving over quasicrystalline substrates,” Phys. Rev. Lett. 106, 060603 (2011). 46C. Reichhardt and C. J. O. Reichhardt, “Structural transitions and dynamical regimes for directional locking of vortices and colloids driven over periodicsubstrates,” J. Phys.: Condens. Matter 24, 225702 (2012). 47M. Kemmler, C. G €urlich, A. Sterck, H. P €ohler, M. Neuhaus, M. Siegel, R. Kleiner, and D. Koelle, “Commensurability effects in superconducting Nb films with quasiperiodic pinning arrays,” Phys. Rev. Lett. 97, 147003 (2006). 48V. R. Misko, D. Bothner, M. Kemmler, R. Kleiner, D. Koelle, F. M. Peeters, and F. Nori, “Enhancing the critical current in quasiperiodic pinning arrays below and above the matching magnetic flux,” Phys. Rev. B 82, 184512 (2010). 49C. Reichhardt and C. J. O. Reichhardt, “Devil’s staircase and disordering transi-tions in sliding vortices and Wigner crystals on random substrates with trans- verse driving,” Phys. Rev. B 76, 214305 (2007). 50V. R. Misko and F. Nori, “Magnetic flux pinning in superconductors with hyperbolic-tessellation arrays of pinning sites,” P h y s .R e v .B 85, 184506 (2012). 51D. Ray, C. Reichhardt, and C. J. O. Reichhardt, “Pinning, ordering, and dynam-ics of vortices in conformal crystal and gradient pinning arrays,” Phys. Rev. B 90, 094502 (2014). 52Y. L. Wang, L. R. Thoutam, Z. L. Xiao, B. Shen, J. E. Pearson, R. Divan, L. E. Ocola, G. W. Crabtree, and W. K. Kwok, “Enhancing superconducting critical current by randomness,” Phys. Rev. B 93, 045111 (2016).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 162601 (2021); doi: 10.1063/5.0045584 118, 162601-5 Published under license by AIP Publishing

1.2162038.pdf

Domain-wall pinning by steplike thickness change in magnetic thin film M. Takezawa, K. Ejiri, J. Yamasaki, H. Asada, and T. Koyanagi Citation: Journal of Applied Physics 99, 08B701 (2006); doi: 10.1063/1.2162038 View online: http://dx.doi.org/10.1063/1.2162038 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/99/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reversible magnetic domain-wall motion under an electric field in a magnetoelectric thin film Appl. Phys. Lett. 92, 112509 (2008); 10.1063/1.2900886 Micromagnetic study of domain-wall pinning characteristics with grooves in thin films J. Appl. Phys. 97, 10E317 (2005); 10.1063/1.1857652 Thickness dependent wall mobility in thin Permalloy films J. Appl. Phys. 91, 7547 (2002); 10.1063/1.1456403 Quantitative imaging of magnetic domain walls in thin films using Lorentz and magnetic force microscopies J. Appl. Phys. 90, 5220 (2001); 10.1063/1.1412829 Domain and domain wall contributions to optical second harmonic generation in thin magnetic films J. Appl. Phys. 81, 5668 (1997); 10.1063/1.364690 [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: 131.111.164.128 On: Sat, 20 Dec 2014 22:59:29Domain-wall pinning by steplike thickness change in magnetic thin film M. T akezawa,a/H20850K. Ejiri, and J. Yamasaki Department of Applied Science for Integrated System Engineering, Graduate School of Engineering, Kyushu Institute of Technology, 1-1 Sensui-cho, Tobata-ku, Kitakyushu, Fukuoka 804-8550, Japan H. Asada and T . Koyanagi Department of Symbiotic Environmental System Engineering, Graduate School of Science and Engineering,Yamaguchi University, 2-16-1 Tokiwadai, Ube 755-8611, Japan /H20849Presented on 2 November 2005; published online 18 April 2006 /H20850 A thin-film element with a steplike thickness change has been fabricated to investigate experimentally a pinning effect of domain walls by a shape control of thin-film devices. Using aKerr microscope, domain observation has been done to measure pinning characteristics of theelement. It has been shown that 40% steplike thickness change of the film thickness can realize awall pinning, and a pinning field of 2.53 Oe is obtained. The pinning field increases with increasingsteplike thickness change ratio. © 2006 American Institute of Physics ./H20851DOI: 10.1063/1.2162038 /H20852 I. INTRODUCTION Artificial wall pinning is effective in controlling a depin- ning field in sensor applications utilizing large Barkhausenjumps 1and improving properties of high-frequency material applications, such as magnetic-field sensors and cores, due tosuppression of wall motion. 2Etched grooves in a garnet film having a perpendicular anisotropy were used for stabilizingstripe domains in a Bloch line memory. 3,4In a narrow track single-pole head, grooves across the track can control thedomain structure of a main-pole film and suppress the 90°wall motion of closure domains at the film edges when themagnetic field is applied along the longitudinal direction ofthe film. 5 In the previous work, micromagnetic simulation has re- ported that steplike thickness change along a domain wallcan produce wall pinning in an in-plane magnetization thinfilm. 6,7The simulation results have indicated that a bidirec- tional pinning effect for magnetic fields applied along themagnetic domain was obtained at the step. In the present work, thin-film elements with a steplike thickness change have been fabricated, and domain observa-tion has been performed to investigate the pinning character-istics experimentally. II. SIMULATION Numerical simulations were performed by integrating the Landau-Lifshitz-Gilbert equation by an explicit schemeof the modified Dufort-Frankel method. 6–9As illustrated in Fig. 1, a steplike thickness change /H20849/H9004h/H20850along the domain wall /H20849xdirection /H20850is assumed to clarify the wall-pinning char- acteristics with thickness change. The cross section normalto the film plane /H20849y-zplane /H20850containing the thickness change is taken to be the computation region, which is discretizedinto a two-dimensional array. Boundary conditions on thecomputation region are such that the wall is the x-zplane and infinite in extent in the xdirection. Material parameters usedin the simulation are as follows: saturation induction 4 /H9266Ms =8000 G, uniaxial anisotropy constant Ku=3200 ergs/cm3, exchange constant A=10−6erg/cm, gyromagnetic ratio /H9253 =1.76/H11003107/H20849sO e /H20850−1, and damping constant /H9251=0.5.6,7The grid element spacing is 5 nm for the film thickness h /H33355300 nm and 10 nm for h/H11022300 nm, respectively. The easy axis is along the xdirection and magnetic fields /H20849Hp/H20850are applied along the magnetic domain. The time transient of the orthogonal component of an effective field was used for de-termining the depinning field. 10 The simulation result of the dependence of the depinning field on thickness change ratio in a 150-nm-thick film isshown in Fig. 2. The result shows that the depinning fieldincreases with increasing thickness change ratio. The pinningeffect is created by the following mechanism. Although thesteplike thickness change is in the pinned wall area, the ro-tation of magnetization in the pinned wall is more gradualcompared to the wall existing in the thin-film region due tothe magnetostatic coupling between the spins near the step.As a result, the exchange energy of the pinned wall largelydecreases due to the decrease of the wall area by the steplikethickness change. Moreover, the depinning field for the negative applied fields, wherein the wall moved in the thick-film region /H20849left- hand side in Fig. 1 /H20850, is considerably larger than that for the positive ones. This is because the wall energy per unit lengthin the thick-film region, that is the energy difference from thepinned wall, is larger than that in the thin-film region. Thedependence of the depinning field on the thickness changeratio would also be reflected in this character. a/H20850Author to whom correspondence should be addressed; electronic mail: take@ele.kyutech.ac.jp FIG. 1. A simulation model with a steplike thickness change in a thin film.JOURNAL OF APPLIED PHYSICS 99, 08B701 /H208492006 /H20850 0021-8979/2006/99 /H208498/H20850/08B701/3/$23.00 © 2006 American Institute of Physics 99, 08B701-1 [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: 131.111.164.128 On: Sat, 20 Dec 2014 22:59:29III. EXPERIMENT Ni80Fe20films with a thickness of 150 nm and with an in-plane uniaxial anisotropy are deposited by rf sputtering. To obtain a uniaxial anisotropy, a dc field of 30 Oe is appliedto the thin film during rf sputtering. A magnetization curve ofthe film without step is measured with a vibrating samplemagnetometer /H20849VSM /H20850. The thin-film elements with steplike thickness changes of 15 and 60 nm are fabricated by a lift-offprocess, as shown in Fig. 3. The thickness change ratios are0.1 and 0.4, respectively. Magnetic domains of the elementsare observed by using a Kerr microscope. After applying a dcfield of −30 Oe, enough to configure a saturation state, the dcfield to the domain-wall direction is reduced to zero andincreases to positive direction in order to observe pinningand depinning near the step of the film. IV. RESULTS AND DISCUSSION The coercivity and the anisotropy field of the film with- out a step are about 0.5 and 5 Oe, respectively. Figure 4shows the domain patterns of the 15-nm-step element. Thedark and bright domains have magnetizations pointing in up-ward and downward directions, respectively. In the satura-tion state, magnetization direction is fully upward at a dcfield of −30 Oe. In decreasing the field to −0.63 Oe, reversaldomains having downward magnetization component nucle-ate at the edge of the film, as shown in Fig. 4 /H20849b/H20850. These reversal domains grow and move toward the step position asthe negative field decreases, as seen in Fig. 4 /H20849c/H20850. When the field reaches −0.30 Oe in Fig. 4 /H20849d/H20850, a domain wall moves easily over the step from the thick-film region to the thin-film region. It was found that wall pinning is not observed atthe step in the 15-nm-step /H208490.1 thickness change ratio /H20850ele- ment. Figure 5 shows the domain patterns of the 60-nm-step element. Nucleation of reversal domains and growth of thedomains also occur when a dc field decreases after a satura-tion state. The wall pinning at the step is observed at the fieldof +0.64 Oe, as shown in Fig. 5 /H20849c/H20850. After that, the rapid wall motion due to the depinning of the pinning wall occurs whenthe applied field is increased to +2.53 Oe, as shown in Fig.5/H20849d/H20850. The result shows that the 60-nm-step element having 0.4 thickness change ratio can realize the wall pinning by thesteplike thickness change as predicted in the computer simu-lation. It is clear that the depinning field increases with in-creasing step height and the positive depinning field + H pis +2.53 Oe in this case. The intensity of the depinning field isapproximately 2.5% for the simulation result, however. Itseems that a shape of broad step edge causes the small de-pinning field compared with the value of the simulation. Thesimulation results performed by varying the shape of thick-ness change have suggested that the depinning field de- creased as a slope becomes gentler. 6 Although observation of a negative wall depinning from FIG. 2. Dependence of depinning fields for positive and negative magnetic fields on thickness change ratio in a 150-nm-thick film. FIG. 3. A schematic view of a fabricated element with a steplike thicknesschange. FIG. 4. Domain patterns of the element with 15 nm step: /H20849a/H20850ad cfi e l do f −30 Oe, /H20849b/H20850−0.63 Oe, /H20849c/H20850−0.35 Oe, and /H20849d/H20850−0.30 Oe.08B701-2 T akezawa et al. J. Appl. Phys. 99, 08B701 /H208492006 /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: 131.111.164.128 On: Sat, 20 Dec 2014 22:59:29thin-film region to thick-film region was tried, nucleation and growth of another reversal domain occurred and the wallreached the step before depinning the pinned wall at the step.The negative depinning field − H pcannot be measured in thisstudy. However, the negative depinning field is at least larger than the positive one as predicted in the simulation, becausethe negative field as another wall reaches the step is largerthan the positive depinning field. V. CONCLUSION In the present work, a thin-film element with a steplike thickness change has been fabricated to investigate experi-mentally a pinning effect of domain walls by a shape controlof thin-film devices. It has been shown that 40% steplikethickness change of the film thickness can realize a wallpinning and a positive pinning field of +2.53 Oe is obtained.The intensity of the depinning field is very small and ap-proximately 2.5% for the simulation result, however. Thiswill necessitate future investigations about the relation be-tween a step shape and the depinning field. Nonetheless, thesimply fabricated thin film with the steplike thickness changewill allow us to control high-frequency magnetic propertiesdue to domain-wall pinning, and it is notable that the com-puter simulation can at least predict qualitatively the pinningfield. 1J. Yamasaki, K. Mohri, K. Watari, and K. Narita, IEEE Trans. Magn. 20, 1855 /H208491984 /H20850. 2M. Takezawa and J. Yamasaki, IEEE Trans. Magn. 37,2 0 3 4 /H208492001 /H20850. 3D. Klein and J. Engemann, J. Magn. Magn. Mater. 45,3 8 9 /H208491984 /H20850. 4T. Suzuki et al. , IEEE Trans. Magn. 22,7 8 4 /H208491986 /H20850. 5K. Ise and Y. Nakamura, J. Magn. Soc. Jpn. 15, 167 /H208491991 /H20850. 6H. Asada, Y. Hyodo, J. Yamasaki, M. Takezawa, and T. Koyanagi, IEEE Trans. Magn. 40,2 1 1 0 /H208492004 /H20850. 7H. Asada, H. Ii, J. Yamasaki, M. Takezawa, and T. Koyanagi, J. Appl. Phys. 97, 10E317 /H208492005 /H20850. 8S. Konishi, K. Matsuyama, N. Yoshimatsu, and K. Sakai, IEEE Trans. Magn. 24,3 0 3 6 /H208491988 /H20850. 9G. Ronan, K. Matsuyama, E. Fujita, M. Ohbo, S. Kubota, and S. Konishi, IEEE Trans. Magn. 21, 2680 /H208491985 /H20850. 10H. Asada, K. Matsuyama, M. Gamachi, and K. Taniguchi, J. Appl. Phys. 75,6 0 8 9 /H208491994 /H20850. FIG. 5. Domain patterns of the element with 60 nm step: /H20849a/H20850a dc field of −30 Oe, /H20849b/H20850+0.10 Oe, /H20849c/H20850+0.64 Oe, and /H20849d/H20850+2.53 Oe.08B701-3 T akezawa et al. J. Appl. Phys. 99, 08B701 /H208492006 /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: 131.111.164.128 On: Sat, 20 Dec 2014 22:59:29

1.4818723.pdf

Current-driven domain wall motion along high perpendicular anisotropy multilayers: The role of the Rashba field, the spin Hall effect, and the Dzyaloshinskii-Moriya interaction Eduardo Martinez, Satoru Emori, and Geoffrey S. D. Beach Citation: Appl. Phys. Lett. 103, 072406 (2013); doi: 10.1063/1.4818723 View online: http://dx.doi.org/10.1063/1.4818723 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v103/i7 Published by the AIP Publishing LLC. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 25 Aug 2013 to 129.11.21.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsCurrent-driven domain wall motion along high perpendicular anisotropy multilayers: The role of the Rashba field, the spin Hall effect, and theDzyaloshinskii-Moriya interaction Eduardo Martinez,1,a)Satoru Emori,2and Geoffrey S. D. Beach2 1Dpto. Fisica Aplicada, Universidad de Salamanca, Plaza de los Caidos s/n, E-38008, Salamanca, Spain 2Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 30 May 2013; accepted 29 July 2013; published online 15 August 2013) The current-induced domain wall motion along a thin cobalt ferromagnetic strip sandwiched in a multilayer (Pt/Co/AlO) is theoretically studied with emphasis on the roles of the Rashba field, the spin Hall effect, and the Dzyaloshinskii-Moriya interaction. The results point out that these ingredients, originated from the spin-orbit coupling, are consistent with recent experimentalobservations in three different scenarios. With the aim of clarifying which is the most plausible the influence of in-plane longitudinal and transversal fields is evaluated. VC2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4818723 ] The current-induced domain wall motion (CIDWM) along thin ferromagnetic layers with high perpendicular magnetoscrystalline anisotropy sandwiched between a heavy metal and an oxide has been demonstrated to be veryefficient, 1–4and it promises unprecedented opportunities for developing spintronic devices.5Apart from its technological interest, the CIDWM along these asymmetric stacks is alsoof fundamental relevance because it is related to interesting physics phenomena. The CIDWM is often explained in terms of the standard adiabatic and non-adiabatic spin-transfer tor-ques (STTs). 6,7However, the domain wall (DW) moves along the current (against the electron flow) in Pt/Co/AlO (Ref. 1) and in Pt/CoFe/MgO (Ref. 3) stacks, an observation which is contrary to the standard STT unless the polarization factor Por the non-adiabatic parameter nare negative. Moreover, the addressed high velocity in these asymmetricstacks is not consistent with the STT, and recent experimental observations 1,8pointed out that in the presence of structural inversion asymmetry and/or heavy metals like Pt,1,2,9strong spin-orbit coupling (SOC) can lead to additional spin-orbit torques (SOTs) qualitatively different from the STTs. These SOTs could, at least, be originated by two phenomena: theRashba effect due to the large SOC and structure inversion asymmetry at the two different heavy-metal/ferromagnet and ferromagnet/oxide interfaces 10–14and/or the spin Hall current generated from the heavy metal layer and injected in the thin ferromagnet.15–21On the other hand, a thin ferromagnetic layer in contact with a heavy-metal with strong SOC isexpected to experience an interfacial anisotropic exchange due to the Dzyaloshinskii-Moriya interaction (DMI). 22–29The DMI is a chiral spin-orbit interaction originating from relativ-istic effects that occur due to the lack of inversion symmetry of the atomic structure, and it can result in topologically rich magnetization patterns such as spiral, skyrmions 25,27,28or chi- ral domain walls.29In particular, it has been recently pointed out that in a thin ferromagnetic layer sandwiched between aheavy-metal and an oxide, the DMI stabilizes chiral DWs of Neel type which are efficiently driven by the spin Hall effect (SHE).3,30Given the broad interest on the CIDWM in these heavy-metal/ferromagnet/oxide heterostructures, it is crucialto reveal the underlaying physics of all these SOC effects. In this paper, the experimental data by Miron et al. 1for the CIDWM in a Pt/Co/AlO stack are taken as reference toprovide different explanations which could be theoretically consistent with. Based on the experimental available works and by using the one-dimensional model, we find anddescribe three possible scenarios consistent with this highly efficient CIDWM along the current by considering different combinations of STTs, Rashba and spin Hall SOTs, and DMI. In order to mimic the experimental results by Miron and co-workers 1for a Co strip with a cross section of Ly/C2Lz ¼500 nm /C20:6 nm sandwiched between Pt and AlO layers, the following parameters were adopted:1saturation magnetiza- tionMs¼1:09/C2106A=m, exchange constant A¼10/C011J=m, uniaxial anisotropy constant Ku¼1:19/C2106J=m3, and damping a¼0:2.33Under instantaneous injection of a spa- tially uniform current density along the x-axis ~ja¼ja~ux, the magnetization dynamics is governed by the augmentedLandau-Lifshitz Gilbert equation d~m dt¼/C0c0~m/C2~Hef fþa~m/C2d~m dt/C18/C19 þ~sSTþ~sSO;(1) where ~mð~r;tÞ¼~Mð~r;tÞ=Msis the normalized local magnet- ization, c0is the gyromagnetic ratio, and athe Gilbert damp- ing parameter. ~Hef fis effective field, which apart from the standard exchange, magnetostatic, uniaxial anisotropy and Zeeman contributions also includes the DMI22–24 ~HDMI¼/C01 l0Msd/C15DMI d~m; (2) where /C15DMIis the DMI energy density24given by30 /C15DMI¼D½mzr/C1~m/C0ð~m/C1r Þmz/C138 (3)a)Author to whom correspondence should be addressed. Electronic mail: edumartinez@usal.es 0003-6951/2013/103(7)/072406/5/$30.00 VC2013 AIP Publishing LLC 103, 072406-1APPLIED PHYSICS LETTERS 103, 072406 (2013) Downloaded 25 Aug 2013 to 129.11.21.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsand Dis the DMI parameter describing its intensity. The STT~sSTis given by6,7 ~sST¼bJð~ux/C1r Þ~m/C0nbJ~m/C2ð~ux/C1r Þ~m; (4) where bJ¼jalBP eMswith lBthe Bohr magneton and e<0 the electron’s charge. Finally, ~sSOdescribes the SOTs, which includes Rashba and spin Hall contributions ~sSO¼/C0c0~m/C2~HRþgc0n~m/C2ð~m/C2~HRÞþc0~m /C2ð~m/C2HSH~uyÞ; (5) where two contributions from the Rashba effect (1st and 2nd terms in Eq. (5)) and one from the spin Hall effect (3rd term in Eq. (5)) can be identified. In the presence of the Rashba interaction, the charge current flowing in the thin ferromagnetic layer in a direction parallel to theinterfaces generates a spin accumulation that can interact with the local magnetization via an exchange coupling mediated by a Rashba effective field ~H R¼HR~uygiven by1,11,12 ~HR¼aRP l0lBMsð~uz/C2~jaÞ¼aRPja l0lBMs~uy; (6) with aRbeing the Rashba parameter. Other Rashba SOT could also arise either from the spin diffusion inside themagnetic layer or from a spin current associated to Rashba interaction at the interfaces with the spin-orbit metal. 14These phenomena have been predicted to con- tribute to the SOT by means of an additional non- adiabatic contribution to the Rashba SOT,13,14which is proportional to the non-adiabatic parameter n(2nd terms in Eq. (5)). Another possible source of SOT originates from the SHE.15,16In a typical multilayer stack, a spin current can be generated by the SHE in the heavy non-magnetic metal layers such as Pt. This spin current can be injected into the ferromagnetic layer, resulting in an additional SOT (3rd term in Eq. (5)), with amplitude H SHgiven by18–21 HSH¼/C22hhSHja l02eMsLz¼lBhSHja c0eMsLz; (7) where Lzis the thickness of the ferromagnetic layer, hSHis the Spin Hall angle, which is defined as the ra- tio between the spin current and the charge current den- sities. Here, the factor gwas considered to account (g¼1) or not ( g¼0) the Slonczewski-like torque due to the Rashba effect. On the other hand, the SHE results in a Slonczewski-like torque (3rd term at the rhsin Eq. (5)). The one-dimensional model (1DM), assumes that (i) the magnetization varies only in the direction of the strip(here x-axis, ~mðx;tÞ) and that (ii) the static DW profile is essentially preserved during its motion. In this 1DM, the extended LLG Eq. (1)can be integrated over the static DW profile, 3,6,7,30and therefore, the CIDWM, including STTs, SOTs, and DMI, is described by two coupled equations_X D¼ac0 0H/C0c0 0HK 2sinð2UÞþð1þanÞ 1þa2bJ D þp 2c0 0½/C0ð1þangÞHRþaHSH/C0Hy/C138cosðUÞ þp 2c0 0½HDþHx/C138sinðUÞ; (8) _U¼c0 0Hþac0 0HK 2sinð2UÞþðn/C0aÞ 1þa2bJ D þp 2c0 0½/C0ðgn/C0aÞHRþHSHþaHy/C138cosðUÞ /C0ap 2c0 0½HDþHx/C138sinðUÞ; (9) where c0 0¼c0=ð1þa2Þ,X¼X(t) is the DW position, and U¼UðtÞis the DW angle, which is defined as the in-plane (x-y) angle with respect to the positive x-axis: Uð0Þ¼0;p for Neel DW, and Uð0Þ¼p=2;3p=2 for Bloch DW configu- rations. Positive current ( ja>0) is along the positive x-axis, Dis the DW width, and HKis the hard-axis anisotropy field of magnetostatic origin. The DW width is estimated to beD¼ffiffiffiffiffiffiffiffiffiffiffi A=K up /C253 nm, and the shape anisotropy field is given HK¼NxMs, where Nxis the magnetostatic factor given by34 Nx¼LzLogð2Þ=ðpDÞ¼0:044. HD¼D=ðl0MsDÞis the DMI effective field pointing along the x-axis inside the DW.30The applied field has Cartesian components ðHx;Hy;HzÞ. The total field H¼HzþHpðXÞþHthðtÞ includes (i) the applied magnetic field along the easy z-axis (Hz), (ii) the spatial dependent pinning field ( HpðXÞ), which accounts for local imperfections and can be derived from aneffective spatial-dependent pinning potential V pinðXÞas HpðXÞ¼/C01 2l0MsLyLz@VpinðXÞ @X, and (iii) the thermal field ( HthðtÞ), which describes the effect of thermal fluctuations.31,32 In order to explain the experimental results by Miron and co-workers,1several combinations of STTs, SOTs, and DMI have been evaluated. In the following discussion, thepossible ones consistent with the highly efficient CIDWM along the current are described. Scenario 1. The experimental observations by Miron and co-workers 1indicates a DW motion (CIDWM) along the current, reaching velocities around 400 m/s for ja/C253 /C21012A=m2. This high efficiency was interpreted by the authors by suggesting that a strong Rashba field stabilizes the Bloch DW configuration and supports the standard STT with both a high polarization factor ( P/C241) and non- adiabaticity ( n/C241). However, the standard STT considers that both P>0 and n>0 are positive quantities predicting a DWM against the current (along the electron flow). The ex-perimental observations could be in principle consistent with this scenario (strong Rashba field supporting the STT) if one of them ( Porn) is a negative value. For example, it has theoretically suggested that in very narrow walls, the non-adiabaticity could change its sign. 35,36Therefore, here we have explored this scenario by considering a similarvalue of the Rashba parameter as suggested in Ref. 38 (a R¼10/C010eVm with g¼0) along with the standard STTs with P¼0.5 and n¼/C01. The 1DM predictions for the DW velocity and the terminal DW angle are shown in Figs. 1(a) and1(b) and compared to the experimental data by Miron and co-workers1(blue squares). The experimental results for the DW velocity depict a low-current creep regime and a072406-2 Martinez, Emori, and Beach Appl. Phys. Lett. 103, 072406 (2013) Downloaded 25 Aug 2013 to 129.11.21.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionshigh-current flow regime. The first one is dominated by the balance between the driving force (STT) and the local pin-ning potential due to the imperfections which oppose to the free DW motion. For very low currents, the DW does not move because the driving current is still very small to over-come the local energy barrier induced by local pinning. As the current increases, the DW motion is thermally activated, and the DW velocity increases exponentially, a typicalbehavior of the thermally activated DW motion in the creep regime. 37For very high current, the DW reaches the flow re- gime where pinning, and thermal effects play a negligiblerole. 37Therefore, the DW velocity in this high-current flow regime can be fitted by the 1DM in the absence of pinning and thermal effects if the key parameters ( P,n, and aR) are properly chosen, as it is the case of the black squares in Fig. 1(a), where Vpin¼0 and T¼0. As it is shown in Fig. 1(b), the DW configuration is of Bloch type: the internal DW mag-netization points along the positive transversal y-axis for positive currents ( U/C25þp=2) or along the negative transver- saly-axis for negative currents ( U/C253p=2) as it is expected from Eq. (6). The inclusion of pinning ( V pin6¼0) in the 1DM (with V0¼1:8/C210/C020J and p¼30 nm), both at zero tem- perature ( T¼0, open circles in Fig. 1(a)) and at room tem- perature ( T¼300 K, solid red line in Fig. 1(a)), provides a more realistic description of the full experimental results. Note that the flow regime, which is the relevant one toextract the key parameters, does not change substantially with respect to the perfect case. Scenario 2. Although it has been theoretically predicted that the non-adiabatic parameter could be negative in narrow walls, 35,36the experimental verification has not been estab- lished. Moreover, it has been experimentally demonstratedthat the SHE-driven spin accumulation at the heavy-metal/ ferromagnet interface generates a Slonczewski-like torque strong enough to switch uniformly magnetized films. 18,19 Apart from the Slonczewski-like torque due to the SHE, the theoretical work by Wang and Manchon14indicates that alsothe Rashba field could contribute to the Slonczewski-like tor- que, which enters as a correction proportional to the non-adiabaticity. 13Here, it has been verified that considering both the field-like and Slonczewski-like torques due to the Rashba ( g¼1) along with the Slonczewski-like torque due to the SHE ( hSH¼0:13), the experimental results1can be also reproduced if a small and positive non-adiabatic param- eter ( n¼þ0:1) is taken into account (Fig. 1(c)). Note that the deduced value for the spin Hall angle ( hSH¼0:13) is in good agreement with experimental measurements.17–19Also in this scenario, the internal DW adopts an internal magnet-ization close to the Bloch type which again is mainly related to the strong Rashba field-like torque (Fig. 1(d)). Although now the non-adiabaticity is positive and considerably smallthan in the former scenario, a high value of the Rashba pa- rameter ( a R¼10/C010eVm) is still required to achieve the fit. However, several experimental works3,19have pointed out that the Rashba field is indeed around two orders of magni- tude smaller than the used here ( aR¼10/C010eVm (Ref. 38)). Note also that although it is not depicted here, it was verifiedthat by reducing the Rashba parameter by one order of mag- nitude ( a R¼10/C011eVm), the direction of DW motion reverses being along the electron flow, in contradiction to theexperiments (for such a low Rashba field, the STT dominates for all tested values of 0 <n<20aand 0 <h SH<0:2). Scenario 3. Apart from the high Rashba parameter required by previous scenarios, they are only in agreement with the experimental observations in the presence of the standard STT. However, current-induced DW motion isabsent in symmetric Pt/Co/Pt stacks, 2,39–41and semi- classical transport calculations41suggest that spin-polarized current in the ultrathin ( <1 nm) Co layer is vanishingly small. A recent work by Tanigawa et al.42has also shown the vanishing polarization for thinner Co layer in a Co/Ni system. Therefore, in the absence of STT, the Rashba field-like torque (1st term in Eq. (5)) only stabilizes the Bloch DW configuration, but it lacks the correct symmetry to drive FIG. 1. DW velocity and DW angle as a function of the applied density current in three different scenarios: (a), (b) STT with P¼0.5 and n¼/C01 and Rashba field with aR¼10/C010eVm and g¼0; (c), (d) STT with P¼0.5 and n¼þ0:1, Rashba field with aR¼10/C010eVm and g¼1, and SHE with hSH¼0:13; (e), (f) DMI with D¼/C02:4m J=m2and SHE with hSH¼0:08. Blue squares correspond to the experimental data by Miron.1Black squares are the 1DM predictions for a perfect strip at zero temperature. Open circles are the 1DM results considering a rough sample ( Vpin6¼0) at zero temperature, and red lines are the 1DM results considering a rough sample ( Vpin6¼0) at room temperature ( T¼300 K). The simulated pinning potential is given by VpinðXÞ¼V0sin2ðpX=pÞ. The peri- odicity of the pinning landscape is p¼30 nm, and the amplitude is V0¼1:8/C210/C020J;V0¼6/C210/C020J, and V0¼50/C210/C020J for the cases (a), (b), and (c), respectively. A positive velocity indicates a DW motion along the x>0 axis that is along the direction of the positive current.072406-3 Martinez, Emori, and Beach Appl. Phys. Lett. 103, 072406 (2013) Downloaded 25 Aug 2013 to 129.11.21.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsDWs directly. Note also that if the polarization factor is indeed close to zero ( P/C250) for thin Co layers, both field- like and Slonczewski-like torque contributions due to the Rashba field will be also vanishingly small or null. On thecontrary, the SHE is not proportional to P, but its Slonczewski-like torque (3rd term in Eq. (5)) would result to be zero for a perfect Bloch DW configuration. 43Due to mag- netostatics considerations,44the Bloch configuration is expected to be the energetically favored state in most of the experimental studies. Although deviations from the pure Bloch state could be induced by field misalignments, small contributions from the STT or from shape anisotropy in nar-row wires, 8up-down and down-up DWs would be driven in opposite directions. Therefore, SHE alone cannot drive trains of DWs in the same direction,8and it alone is not capable to explain the current-driven DW motion in Pt/Co/AlO and similar stacks. As recently pointed out by Emori et al. ,3the additional ingredient is the DMI, which has been theoreticalshown to promote chiral Neel DWs as a consequence of the anisotropy exchange between the magnetic moments and the interfacial atoms with high SOC. 22–27,29Indeed, the current- driven DW motion in heavy-metal/ferromagnet/oxide struc- tures is naturally explained by the combination of the SHE, which produces the main current-induced torque, and theDMI, which stabilizes chiral Neel DWs whose symmetry permits uniform motion with very high efficiency. 3Here we show that this scenario (SHE along with DMI) is indeedquantitatively consistent with the experimental results by Miron. 1The 1DM results for hSH¼0:08 and D¼ /C02:4m J=m2are compared to the experimental values in Fig. 1(e). Contrary to former scenarios, now the Rashba and the STT are zero. The up-down DW configuration under zero current is of Neel type ( U¼p) with its internal magnetiza- tion pointing mainly along the negative x-axis due to the negative value of the DMI (Fig. 1(f)). For finite currents, the DW deviates from the perfect Neel state, and it tends toreach an intermediate state between Bloch and Neel states for very high currents: U!p=4 for very high positive cur- rents and U!5p=4 for negative currents. The results of Fig. 1indicate that in principle the three scenarios could be consistent with the experimental results. In order to elucidate if one of them is indeed describing thephysics governing the current-driven DW motion in these asymmetric stacks, the influence of in-plane fields, along the longitudinal x-axis or along the transversal y-axis, have been studied by the 1DM considering a perfect sample at zero temperature. The transversal field ( ~B y¼l0Hy~uy) points inthe same direction than the Rashba field ( ~BR¼l0HR~uy), supporting or opposing to it depending on its sign. The longi- tudinal field ( ~Bx¼l0Hx~ux) points in the same direction than DMI ( ~BD¼l0HD~ux). The results for the three scenarios are depicted in Fig. 2under a fixed current of ja¼1012A=m2. Due to the strong Rashba field considered in the scenario 1 (BR/C25791 mT), the current-driven DW velocity is not modi- fied by the in-plane fields (Fig. 2(a)). When the SHE is taken into account along with the STT and the Rashba field (sce- nario 2, the DW reaches a saturation velocity with different signs under strong longitudinal fields with opposite polarities (see Fig. 2(b)). These strong longitudinal field promote the Neel DW configuration which is mainly driven by the SHE. Under transversal fields, the DW velocity also experiences a change of sign around By/C2560 mT which is approximately the value of the SHE effective field HSHgiven by Eq. (7). For strong transversal fields the DW adopts a Bloch state and the velocity tends to vanish because of the low non-adiabaticity. Finally, under the only action of SHE and DMI (scenario 3), the negative (positive) longitudinal field sup- ports (opposes) the DMI field (see Fig. 2(c)). In the absence of in-plane field ( B x¼By¼0), the DW state is of Neel type with internal magnetization pointing to the left ( U/C25p) due to the negative value of the DMI. The DW velocity saturatesunder strong negative longitudinal fields, but it decreases under positive longitudinal fields which oppose to the DMI. Under very strong B x(not shown), the internal DW magnet- ization reverses pointing to the right ( U/C250), and this change in the DW chirality produces also a reversal on the DW motion (see supplementary material in Ref. 3). Transversal fields also modify the DW velocity in this sce- nario. For transversal fields with jByj>200 mT, the DW ve- locity decreases monotonously because the internal DWmagnetization starts to deviate from the pure Neel state. Under very strong transversal fields (not shown) the DW ve- locity tends to zero because the DW configuration adopts aBloch state which cannot be driven by the SHE. It is worthy to note that although our former experimental study was con- ducted in the thermally activated regime in a slightly differ-ent material system, 3it qualitatively shows the same behavior than the predicted results of the scenario 3 (Fig. 2(c)), and it is definitely not consistent with Fig. 2(a) (sce- nario 1) or Fig. 2(b)(scenario 2) studied here. Some preliminary micromagnetic simulations have been also carried out for the same experimental cross section(with L y¼500 nm and Lz¼0:6 nm), indicating that under strong in-plane fields not only the internal DW moment is FIG. 2. Current-driven DW velocity as a function of the in-plane fields (black squares Bx, open circles By) predicted by the 1DM for a perfect sample at zero temperature. Three different scenarios are studied: (a) scenario 1: STT with P¼0.5 and n¼/C01, and Rashba field with aR¼10/C010eVm and g¼0; (b) sce- nario 2: STT with P¼0.5 and n¼þ0:1, Rashba field with aR¼10/C010eVm and g¼1, and SHE with hSH¼0:13; (c) scenario 3: DMI with D¼/C02:4m J=m2 and SHE with hSH¼0:08. The applied current is ja¼1012A=m2in all cases.072406-4 Martinez, Emori, and Beach Appl. Phys. Lett. 103, 072406 (2013) Downloaded 25 Aug 2013 to 129.11.21.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsrotated toward to the field but also the DW plane is eventu- ally tilted and the magnetization in the domains slightly deviates from the z-axis (not shown here). These two last issues cannot be accounted by the simple 1DM used here,and therefore full micromagnetic simulations are need to completely describe the influence of in-plane fields on the current-induced DW motion by the SHE in the presence ofDMI (scenario 3). However, such a full micromagnetic anal- ysis of these issues will require a substantial computational effort and is beyond the scope of the present work. This study is currently being performed, and the results will be addressed elsewhere. 45In the meanwhile, the 1DM results of Fig.2have to be considered as a first approach valid for the low-field range. Indeed, the 1DM results reproduce quite accurately the full micromagnetic simulations in strips withreduced width (see supplementary material for a compared micromagnetic and 1DM analysis). 46 In summary, three different scenarios seem to be con- sistent with recent experimental observations in the high- current flow regime, where the DW propagates along the current with high efficiency. In the first case, a strongRashba field stabilizes the Bloch configuration which is propagated by the spin transfer torque if a negative non- adiabaticity is considered. Similar results are also obtainedfor positive non-adiabaticity if both Rashba and spin Hall contributions to the Slonczewski-like torque are included along with the Rashba field-like torque. The third possibilityindicates that, even in the absence of both Rashba and spin-transfer torques, the DW can be driven along the current by the Slonczewski-like spin Hall torque if the Neel DWconfiguration with a given chirality is adopted as due to the Dzyaloshinskii-Moriya interaction. From our fitting, a strong Dzyaloshinskii-Moriya interaction was inferred(D¼/C02:4m J=m 2) considering similar spin Hall angle as the one directly measured in switching experiments.3With the aim of providing other test for the experiments, the influ-ence of in-plane field on the current-driven DW velocity has been also analyzed. This study could be useful to elucidate which are the real and dominant mechanisms governing theunderlying physics behind the current-driven DW motion along asymmetric stacks. The authors would like to thank M. Miron and G. Gaudin for providing their experimental data. 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Beach, “Spin Hall torque magnetometry of Dzyaloshinskii do- main walls,” arXiv:1308.1432. 46See supplementary material at http://dx.doi.org/10.1063/1.4818723 for the comparison of one-dimensional model results (1DM) to full micromag- netic simulations ( lM) for the scenario 3, when a left-handed Neel domain wall is stabilized by the DMI and driven by the SHE in the absence of the Rashba and spin transfer torques.072406-5 Martinez, Emori, and Beach Appl. Phys. Lett. 103, 072406 (2013) Downloaded 25 Aug 2013 to 129.11.21.2. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions

1.1857752.pdf

Reversal modes of simulated iron nanopillars in an obliquely oriented field S. H. Thompson, G. Brown, and P. A. Rikvold Citation: Journal of Applied Physics 97, 10E520 (2005); doi: 10.1063/1.1857752 View online: http://dx.doi.org/10.1063/1.1857752 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effects of vortex chirality and shape anisotropy on magnetization reversal of Co nanorings (invited) J. Appl. Phys. 107, 09D307 (2010); 10.1063/1.3358233 Six-fold configurational anisotropy and magnetic reversal in nanoscale Permalloy triangles J. Appl. Phys. 106, 063902 (2009); 10.1063/1.3223311 Micromagnetic simulations of current-induced magnetization switching in Co ∕ Cu ∕ Co nanopillars J. Appl. Phys. 102, 093907 (2007); 10.1063/1.2800999 Micromagnetic simulations of nanosecond magnetization reversal processes in magnetic nanopillar J. Appl. Phys. 99, 08G522 (2006); 10.1063/1.2177049 Simulated magnetization reversal in Fe nanopillar Chaos 15, 041106 (2005); 10.1063/1.2130688 [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: 84.88.136.149 On: Fri, 05 Dec 2014 10:16:20Reversal modes of simulated iron nanopillars in an obliquely oriented field S. H. Thompson Department of Physics, School of Computational Science, and Center for Materials Science and Technology, Florida State University, Tallahassee, Florida 32306-4350 G. Brown Center for Computational Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6164 P. A. Rikvold School of Computational Science, Center for Materials Science and Technology and Department of Physics,Florida State University, Tallahassee, Florida 32306-4120, and National High Magnetic FieldLaboratory, Tallahassee, Florida 32310-3706 sPresented on 10 November 2004; published online 10 May 2005 d Stochastic micromagnetic simulations are employed to study switching in three-dimensional magnetic nanopillars exposed to highly misaligned fields.The switching appears to proceed throughtwo different decay modes, characterized by very different average lifetimes and different averagevalues of the transverse magnetization components. © 2005 American Institute of Physics . fDOI: 10.1063/1.1857752 g I. INTRODUCTION Nanoparticles with strong uniaxial anisotropy are fre- quently employed in technologies where the orientation ofthe particle magnetization corresponds to a binary digit. Theswitching process of the magnetization in an applied fieldtherefore becomes a technologically important process, espe-cially for fields that vary on a time scale of nanoseconds.Micromagnetic simulations have been essential for under-standing the properties of systems that evolve over suchshort time scales. In this paper we examine a model nanopillar where the anisotropy is due solely to its shape, and subject it to a re-versal field that is highly misaligned with respect to the easyaxis of the pillar. Using parameters consistent with bulk iron,we find that several interesting features emerge as the mag-netization reverses from an unstable configuration, saturatedopposite the applied field, to the true, stable state. Most no-ticeable is that this highly misaligned system possesses twodistinct decay paths that evolve from the same initial condi-tions. Figure 1 is an example snapshot of a slowly reversingdecay path, at a time when the average zcomponent M zof the system magnetization corresponds to a relatively flat re-gion of the free-energy landscape. As seen in other simula-tions with applied fields along the easy axis of the pillar, 1the end caps display a region of high vorticity due to pole avoid-ance at the end faces of the pillar. This is shown in the leftpartofFig.1.Withanappliedfieldnearlyparalleltotheeasyaxis, the flux lines continue down the volume of the pillar,mostly parallel to the easy axis. The highly misaligned casediscussed in this paper, however, exhibits magnetic-flux linesthat penetrate faces parallel to the yzplane near the midsec- tion, as shown in the right part of Fig. 1. Many nanomagnets have lateral dimensions on the order of the exchange length and can be sufficiently represented bythe Stoner–Wolhfarth mechanism of coherent rotation. 2 Larger systems that span several exchange lengths such asthe one discussed in this paper, however, are not adequatelydescribed by coherent rotation and consequently required the application of full micromagnetics simulations to allow theproper dynamics to emerge. To investigate the dynamics ofthese nanopillars under the influence of an applied field, weuse a micromagnetic simulation performing numerical solu-tions of the Landau–Lifshitz–Gilbert sLLG dequation. 3,4For our model, the pillar’s real dimension of 10 3103150 nm3 is mapped onto a 6 36390 computational lattice, and the FIG. 1. Streamline rendering of a nanopillar before reversing to the stable state.Mzis denoted by the color of the streamlines: + zis dark gray and − z is light gray sred/green for color online d. The applied field represented by the arrow in this right-handed coordinate system is oriented 75° from the − z axis, parallel to the zxplane spointing to the left and out of the page d. The left part shows the top of a pillar; the right part, the same pillar further downthe easy axis.JOURNAL OF APPLIED PHYSICS 97, 10E520 s2005 d 0021-8979/2005/97 ~10!/10E520/3/$22.50 © 2005 American Institute of Physics 97, 10E520-1 [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: 84.88.136.149 On: Fri, 05 Dec 2014 10:16:20LLG equation is evaluated at each site in the presence of a local field, Hsrid. The net saturation magnetization Msis introduced in the LLG equation, dM˜sri˜d dt=g0 1+aHMsridFHsrid−a MsMsridHsridGJ,s1d as a constant magnitude of every unit magnetization vector Msridin the lattice. Also included in the equation are the constant electron gyromagnetic ratio g0=1.67 3107Hz/Oe and a phenomenological damping parameter, whose valuewe take to be a=0.1. The local field Hsridthat determines the dynamics at each site iis a sum of all participating fields, including applied, dipole, and exchange. The finite tempera-ture is introduced by a Gaussian-distributed thermal fieldH tsridthat obeys the fluctuation-dissipation theorem, kHitstdHjtst8dl=dijdst−t8dS2akBT Ms2VD, s2d wherekBis Boltzmann’s constant, Vis the volume of an individual computational lattice cell, and Tis the tempera- ture, here set to 20 K. In this simulation the applied field wasset to 3160 Oe at an angle of 75° to the easy axis, which isnear the coercive field for the slower decay mode. Initially, the nanopillar was allowed to relax in the pres- ence of an unreversed field before sinusoidally reversing thefield at 180° to drive the reversal process. Measurementswere taken at regular intervals during the simulation as theLLG equation was temporally integrated using a Euler inte-gration scheme. II. RESULTS AND CONCLUSIONS The first evidence for the existence of two separate de- cay modes in the nanopillar comes from the distribution ofthe lifetimes t, determined by the time it took the zcompo- nent of the total magnetization, Mz, to reach a value of −0.75. The lifetimes of decay processes such as this one arestrongly determined by the shape of the free-energy land-scape that the system evolves through. For free-energy bar-riers that are much larger than the temperature s@k BTd,a n exponential distribution of the lifetimes should be observed. Regions with negligible barriers lead to a Gaussian distribu-tion of the lifetimes, corresponding to perturbations around adeterministic behavior. In Fig. 2 the cumulative distributionfor all our 17 simulations is shown. The graph is clearlydivided into two regions. The first region corresponds to arapid increase in the distribution for times less than about2 ns, and the second to a much slower increase that is spreadacross tens of nanoseconds. For the faster switches, the evi-dence presented below suggests that there is no free-energybarrier. In that case, we expect the faster switching times tohave an approximately Gaussian distribution. It is not clearwhat form the distribution for the slower switches has. Another useful method that reveals the different reversal modes consists of recording the total magnetization swhich we normalize to be unity when all spins are aligned dof the nanopillar at regular time steps and constructing a phase plotof the evolution. Two examples, with t=1.6 and 17.6 ns, respectively, are shown in Fig. 3 for the xandycomponentsof the total magnetization. Both configurations begin their evolution near the middle of the plot and evolve throughtheir respective transition states, ending up close to the topright corner. There is a difference in the regions where thesetwo examples spend most of the time before reversal, andthis difference is seen for all simulations. This behavior isconsistent with a local minimum in the free energy for theslower mode, and a much shallower sor nonexistent dmini- mum for the faster mode. However, from the projective dy-namics evidence to be presented next, it is not clear that atrue minimum exists, even for the slower mode. Projective dynamics 5–7gives information contrasting the two modes in this nanopillar simulation. By selecting a slowvariable, growth and shrinkage probabilities are accumulatedin bins along that variable and can be used to further eluci-date the properties of the free-energy landscape. Specifically,the difference between the growth and shrinkage probabili-ties is a measure of the shape of the free energy along thatvariable. In this instance, M zwas chosen for binning, and transition probabilities were constructed based on the num-ber of times M zjumped to an adjacent bin. The results of the projective dynamics analysis are shown in Fig. 4. For theslower reversal mode, the growth and shrinkage probabilities FIG. 2. Cumulative distribution of the lifetimes in nanoseconds. The sepa- ration into a faster and a slower mode is clearly seen. FIG. 3. Plot of the MxMyphase plane for the slower sgraydand faster sblack dmodes. Simulations begin close to the center of the graph, evolve through the metastable region, and end near the top right corner.10E520-2 Thompson, Brown, and Rikvold J. Appl. Phys. 97, 10E520 ~2005 ! [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: 84.88.136.149 On: Fri, 05 Dec 2014 10:16:20practically coincide in the region around Mz<0.7. This indi- cates that the free energy is almost flat in this region and thatthere is no strongly preferred direction of motion for M z.I n comparison, the growth probabilities always exceed theshrinkage probabilities for the faster mode, indicating a ten-dency for the magnetization to continuously evolve towardthe stable state. Since neither of these graphs exhibits an areawhere the shrinkage probability exceeds the growth probabil-ity, it is possible that a significant barrier in the free energydoes not exist. It is also possible that M zis not a good choice for the slow variable, e.g., it may not coincide well with thereversal path. The statistics that were collected to constructthe projective dynamics plots were sorted based on the life-time of the particular run. Due to the approximate classifica-tion of the modes, it is likely that there is some mixing be-tween the data in these plots, although not enough to changethe general features observed here.In conclusion, finite-temperature simulations were per- formed on iron nanopillars in a strongly misaligned appliedfield. Sophisticated visualization tools were used to give adetailed picture of the magnetization configurations duringthe decay. Two distinct reversal modes were found despiteidentical initial conditions. Three different methods wereused to characterize the differences between these twomodes: the distribution of the lifetimes, phase plots of themagnetization transverse to the easy axis, and projective dy-namics. Still, our results do not yet provide a very clearpicture of the shape of the free-energy landscape. Fully char-acterizing the free energy will be important for designingnanomagnets that switch in a reliable manner under givenexperimental conditions, and will be a topic of future re-search. ACKNOWLEDGMENTS This work was supported in part by U.S. NSF Grant No. DMR-0120310 and by Florida State University through theCenter for Materials Research and Technology and theSchool of Computational Science. 1G. Brown, M. A. Novotny, and P. A. Rikvold, Phys. Rev. B 64, 134422 s2001 d. 2R. Hertel and J. Kirschner, Physica B 343,2 0 6 s2004 d. 3W. F. Brown, Micromagnetics sWiley, New York, 1963 d. 4A. Aharoni, Introduction to the Theory of Ferromagnetism sClarendon, Oxford, 1996 d. 5M. Kolesik, M. A. Novotny, P. A. Rikvold, and D. M. Townsley, in Com- puter Simulation Studies in Condensed Matter Physics X ,e d i t e db yD .P . Landau, K. K. Mon, and H. B. Schüttler sSpringer, Berlin, 1998 d,p .2 4 6 . 6G. Brown, M. A. Novotny, and P. A. Rikvold, Physica B 343, 195 s2003 d. 7S. H. Thompson, G. Brown, and P. A. Rikvold, in Computer Simulation Studies in Condensed Matter Physics XVII , edited by D. P. Landau, S. P. Lewis, and H. B. Schüttler sSpringer, Berlin, in press d, e-print cond-mat/ 0404198. FIG. 4. Growth and shrinkage probabilities distinguishing the two decaymodes in the simulation. The location and depth of the metastable well inthe free energy depend on the reversal mode. Upper panel: faster mode.Lower panel: slower mode.10E520-3 Thompson, Brown, and Rikvold J. Appl. Phys. 97, 10E520 ~2005 ! [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: 84.88.136.149 On: Fri, 05 Dec 2014 10:16:20

1.3109790.pdf

Reduction in critical current of current induced switching in an inhomogeneous nanomagnet M. C. Wu, A. Aziz, M. Ali, C. H. Marrows, B. J. Hickey, Z. H. Barber, and M. G. Blamire Citation: Applied Physics Letters 94, 122511 (2009); doi: 10.1063/1.3109790 View online: http://dx.doi.org/10.1063/1.3109790 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/94/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Controllable vortex polarity switching by spin polarized current J. Appl. Phys. 105, 013906 (2009); 10.1063/1.3054305 Exchange biased spin polarizer with an embedded nano-oxide layer for a substantially lower switching current density Appl. Phys. Lett. 89, 094103 (2006); 10.1063/1.2337532 Reducing the critical current for short-pulse spin-transfer switching of nanomagnets Appl. Phys. Lett. 87, 112507 (2005); 10.1063/1.2045552 Distinctive current-induced magnetization switching in a current-perpendicular-to-plane giant- magnetoresistance nanopillar with a synthetic antiferromagnet free layer Appl. Phys. Lett. 86, 242506 (2005); 10.1063/1.1949709 Spin-polarized current-driven switching in permalloy nanostructures J. Appl. Phys. 97, 10E302 (2005); 10.1063/1.1847292 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: 138.251.14.35 On: Sun, 21 Dec 2014 01:08:59Reduction in critical current of current induced switching in an inhomogeneous nanomagnet M. C. Wu,1,a/H20850A. Aziz,1M. Ali,2C. H. Marrows,2B. J. Hickey,2Z. H. Barber,1and M. G. Blamire1 1Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, United Kingdom 2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom /H20849Received 20 February 2009; accepted 9 March 2009; published online 27 March 2009 /H20850 We report low current-density switching of pseudospin valve nanopillars fabricated by a three-dimensional focused ion beam lithography. Simulations using the Landau–Liftshitz–Gilbertequation with Slonczewski spin transfer torque term demonstrate that magnetostatic couplingcombined with an in-plane applied field results in a strongly inhomogeneous magnetization, whichis close to the point of switching in both parallel and antiparallel configurations and so significantlyreduced switching currents are possible. © 2009 American Institute of Physics . /H20851DOI: 10.1063/1.3109790 /H20852 Current induced magnetization switching /H20849CIMS /H20850in spin valves or magnetic tunnel junctions has stimulated great in-terest not only because of the important basic physics butalso the range of potential applications. 1–6To integrate spin- current-driven devices with existing semiconductor technolo-gies, however, lower critical switching currents are vital. Thetheoretical prediction of the critical current density /H20849J c/H20850for magnetization switching in the macrospin approximation model can be expressed as Jc=/H9251 /H9257/H208732e /H6036/H20874/H20849lmHkMs/H20850/H208731+2/H9266Ms Hk+H Hk/H20874, /H208491/H20850 where /H9257is the spin polarization of the current, /H9251is the Gil- bert damping coefficient, Hkis the uniaxial-anisotropy field, MSis the saturation magnetization, and lmcan be viewed as the thickness of the magnetic switching layer.7As noted in Eq. /H208491/H20850it is the easy plane anisotropy of thin films, which is reflected in the 2 /H9266Ms/Hkterm, which makes torque-induced switching more problematic than field-induced switching.Several approaches to reducing J chave been reported: for example, reducing MSby using CoFeB as the free layer;8 increasing /H9257by using a double spin-filter structure,9an anti- ferromagnetic pinning structure,10or inserting a Ru spin scattering layer;11or by reducing lm.12Another approach re- lated to the anisotropy field is to use local injection of aspin-polarized current into a local nanomagnet, for example,inserting a nanoaperture inside a nanopillar, 13imposing an out-of-plane anisotropy,14or using a composite free layer consisting of a nanocurrent channel.15,16Here, we discuss a different approach, using magnetostatic coupling betweenthe layers in a nanopillar to induce an inhomogeneous mag-netization state which includes a localized out-of-plane com-ponent on which a spin-polarized current can act to over-come the damping force. We demonstrate a substantialreduction in J cin such devices. The nanopillar devices were fabricated from Si wafers coated with SiO 2/Ta /H208495/H20850/Cu /H20849200 /H20850/CoFe /H208493/H20850/Cu /H208496/H20850/CoFe /H2084912/H20850/Cu /H20849200 /H20850/Ta /H208495/H20850/H20849layer thicknesses in nanometers /H20850pseu- dospin valve heterostructures using a 3D focused ion beamlithography technique.17–20The dynamic resistance /H20849dV /dI/H20850of the nanopillars was measured at room temperature using a four terminal ac lock-in technique using an ac current exci-tation of 200 /H9262A rms at 77 Hz. dV /dIwas measured as a function of the in-plane magnetic field /H20849applied along the geometric easy axis /H20850and DC bias /H20849positive direction corre- sponding to current flowing from the fixed to the free layer /H20850. Figure 1/H20849a/H20850shows magnetoresistance /H20849MR /H20850loops for two pseudospin valve nanopillars of similar size at room tem- perature. The behavior of the MR loops is similar and is a/H20850Electronic mail: mcw42@cam.ac.uk. FIG. 1. /H20849Color online /H20850/H20849a/H20850Experimental MR of the nanopillars; the dashed line shows minor loops for the 160 /H11003200 nm2sample from positive /H20849nega- tive /H20850high field to zero and back to positive /H20849negative /H20850high field. /H20849b/H20850Simu- lated MR of the nanopillar /H20849size 170 /H11003190 nm2/H20850for full loop /H20849solid line /H20850 and minor loop /H20849dash line /H20850cycles. The insets show the orientation of mag- netization of the free layer and fixed layer at different fields, and the arrowsshow the directions of the applied field.APPLIED PHYSICS LETTERS 94, 122511 /H208492009 /H20850 0003-6951/2009/94 /H2084912/H20850/122511/3/$25.00 © 2009 American Institute of Physics 94, 122511-1 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: 138.251.14.35 On: Sun, 21 Dec 2014 01:08:59dominated by the strong magnetostatic coupling so that an antiparallel /H20849AP/H20850high resistance state /H20849RH/H20850is stable at low fields, and a strong field of /H1101195 mT is required to align the magnetization of the CoFe layers yielding a low resistancestate /H20849R L/H20850. Figure 1/H20849a/H20850also shows minor loops: these are essentially antisymmetric and characterized by two switch- ing fields separated by a gradual change in resistance. Thelatter is interpreted as arising from a gradual change of therelative magnetization direction of the CoFe layers. Theoverall magnetostatic field is estimated as /H1101155 mT, which is larger than the coercive field of free layer /H20849/H1101141 mT /H20850. As shown in Fig. 2,dV /dIvsImeasurements show a typical parabolic background which is due to Joule heatingon which are superimposed CIMS steps. Taking the 170/H11003190 nm 2nanopillar as an example, the corresponding changes in resistance at different fields /H20851Figs. 2/H20849a/H20850–2/H20849c/H20850/H20852are all equivalent to the resistance changes in the MR loops inFig. 1/H20849a/H20850, confirming that the magnetization is fully reversed by the spin-polarized current injection. Since the strong magnetostatic field dominates the be- havior of the free layer, applying a compensating externalfield might be expected to shift the switching currents tolower values. It can be seen that when 52 mT is applied, bothswitching currents remain both negative, but at 60 mTI RL→RHbecomes positive /H20851see Fig. 2/H20849b/H20850/H20852. A still larger field /H2084968 mT /H20850reduces both currents to /H110021.5 and 0.5 mA, respec- tively /H20851see Fig. 2/H20849c/H20850/H20852, corresponding to Jcof −4.6 /H11003106and 1.6/H11003106A/cm2, which are substantially lower than previ- ous reports.3,4,8–16At a still higher field /H2084974 mT /H20850theRLstate is energetically more favorable, and so only irreversibleswitching from R HtoRLcan be observed. Similarly low values are obtained for other devices /H20849see Fig. 2inset /H20850, al- though the magnetostatic coupling will be different due tothe different nanopillar shapes and so different compensatingfields are required. The nanopillars are also tested in severalrepeating cycles and the difference of switching current isvery small /H20849see Fig. 2inset, upper curve /H20850. These results imply that an appropriate field not only compensates the dipole field experienced by the free layerbut also reduces the difference between the forward and re-verse switching currents. This behavior cannot be predictedon the basis of Eq. /H208491/H20850in which His merely the net field acting on the free layer. Thus the magnetostatic field addedto the appropriate compensating field results in a configura-tion in which the free layer is close to the point switching in both directions. This configuration must therefore go beyondthe macrospin approximation and so the internal magneticstates of the nanopillars need to be understood. Micromagnetic simulations of the free layer and fixed layer were performed using the three-dimensional /H208493D/H20850ob- ject oriented micromagnetic framework /H20849 OOMMF /H20850software.21 The nanopillar was divided into 5 /H110035/H110033n m3cells. The exchange stiffness was set to 3 /H1100310−11Jm−1.MSand uniaxial-anisotropy constant for CoFe were set to 1.3/H1100310 6Am−1and 5/H11003104Jm−3, respectively, and the damp- ing coefficient was set to 0.5 to obtain rapid convergence.The interlayer exchange energy was set to zero. To comparethe simulations with the experimental results, the relativemagnetization alignment of the cells in the fixed and the freelayer was derived from the micromagnetic simulations, andthe local MR was calculated by MR = /H208731−mfreemfixed /H20841mfree/H20841/H20841mfixed /H20841/H20874/H208822, /H208492/H20850 where mfreeandmfixedare the local magnetization of the free layer and the fixed layer, respectively. The normalized MR ofthe nanopillar is then calculated by summing over these localMR values. The simulated MR result /H20851Fig. 1/H20849b/H20850/H20852shows good agree- ment with the experimental result. The small difference be-tween the simulation and the experimental values may beattributed to the interfacial roughness and edge roughness ofthe nanopillar which was not modeled here. As shown in theinsets to Fig. 1/H20849b/H20850, the magnetic state is strongly inhomoge- neous and a C-state orientation of the free layer can be ob- served in the simulations. It can be found that the ycompo- nent of the magnetization increases with applied field and theC-state will be in a more inhomogeneous state, such as the V-state orientation, if the field keeps increasing. To model the effect of the current induced torque, the Landau–Liftshitz–Gilbert equation with Slonczewski spintransfer torque term was used for modeling the detailed dy-namic switching of the nanopillar, using /H9251=0.014 /H20849Ref. 22/H20850 and/H9257=0.35. The effective field used in the model here in- cludes the external field, the anisotropy field, the demagne-tization field, the exchange field, and the Oersted field. Theinitial magnetization state was calculated by a zero-current,field-switching simulation from the Landau–Liftshitz–Gilbert equation as we have modeled in Fig. 1/H20849b/H20850. Figure 3 shows the simulated evolution of magnetization of free layerat certain applied fields once a current density similar to thatused in the experiments is applied. At low fields, when the initial magnetization is in a less inhomogeneous state, CIMS does not occur /H20851Fig. 3/H20849a/H20850/H20852and the free layer finally settles to a static equilibrium withoutswitching. At fields corresponding to the lowest experimentalswitching currents, the initial magnetization is in a more in-homogeneous state and CIMS occurs as shown in Fig. 3/H20849b/H20850. It can be seen that nucleation of the switching is triggered atthe edge and the highly inhomogeneous state resolves to adomain-wall-like feature, which then moves toward the de-vice edge. The switching from parallel /H20849P/H20850-to-AP configura- tion has also been considered and the CIMS can be observedas shown in Fig. 3/H20849c/H20850. Figure 3/H20849d/H20850shows the static inhomo- geneous magnetization in the x,y, and zcomponents at fields corresponding to the lowest experimental switching currents. FIG. 2. /H20849Color online /H20850Dynamic resistance of the nanopillar as a function of DC bias measured at room temperature with different external fields: /H20849a/H20850 /H1100252 mT, /H20849b/H20850/H1100260 mT, and /H20849c/H20850/H1100268 mT. The curves are offset for clarity. The inset shows dynamic resistance for nanopillar size of 160 /H11003200 nm2at/H1100264 mT /H20851upper /H20849blue /H20850/H20852and nanopillar size of 170 /H11003190 nm2at/H1100268 mT /H20851/H20849lower /H20849red/H20850/H20852.122511-2 Wu et al. Appl. Phys. Lett. 94, 122511 /H208492009 /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: 138.251.14.35 On: Sun, 21 Dec 2014 01:08:59The localized out-of-plane magnetization at the device edge in both AP and P configurations provides the nucleation forlow current-density switching. Although the different switch-ing mechanisms are observed in our simulation for AP-to-Pswitching and P-to-AP switching, the currents and fields re-quired are consistent with the experimental results. The sub-nanosecond switching time /H20849/H110210.5 ns /H20850in our simulation also agreed well with the previous reports. 23,24The reversible switching is observed in these repeating cycle measurementswhich suggest that the same magnetic state is recovered oncycling and the change in magnetization induced by thermalfluctuation is small. The switching angle of magnetization of the free layer from the R HtoRLstate and from the RLtoRH state has been calculated. The full reversal has only taken place in the middle of the nanopillar and most of the mag- netization does not switch fully since the shape and structureof our nanopillar device result in a strong magnetostatic cou-pling between the ferromagnetic layers. Although the partialswitching results in /H9004R/H20849I/H20850being smaller, the difference in resistance is still considerable and sufficient for device appli- cations. In summary, magnetostatic coupled pseudospin valve nanopillars are fabricated by a 3D focused ion beam lithog-raphy technique. The nanopillar devices demonstrate a rela-tively low switching current density in a small applied field.The micromagnetic simulation of MR shows good agreementwith our experimental results and explains that the orienta-tion of magnetization is critical for the reduction in criticalcurrent. Preparing an inhomogeneous magnetization in anearly switching state enables a very low switching currentrequired for current induced magnetization switching. This work was supported by the U.K. Engineering and Physical Sciences Research Council under the Spin@RTconsortium, and M. C. Wu would like to acknowledge thefinancial support of the Ministry of Education of Taiwan. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850. 2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850. 3J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 4A. Fert, V . Cross, J. M. George, J. Grollier, H. Jaffrès, A. Hamzic, A. Vaurès, G. Faini, J. Ben Youssef, and H. Le Gall, J. Magn. Magn. Mater. 272, 1706 /H208492004 /H20850. 5J. Akerman, Science 308, 508 /H208492005 /H20850. 6C. Chappert, A. Fert, and F. N. Van Dau, Nature Mater. 6, 813 /H208492007 /H20850. 7J. Z. Sun, Phys. Rev. B 62, 570 /H208492000 /H20850. 8K. Yagami, A. A. Tulapurkar, A. Fukushima, and Y . Suzuki, Appl. Phys. Lett. 85, 5634 /H208492004 /H20850. 9G. D. Fuchs, I. N. Krivorotov, P. M. Braganca, N. C. Emley, A. G. F. Garcia, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 86, 152509 /H208492005 /H20850. 10K. J. Lee, T. H. Y . Nguyen, and K. H. Shin, J. Magn. Magn. Mater. 304, 102 /H208492006 /H20850. 11Y . Jiang, S. Abe, T. Ochiai, T. Nozaki, A. Hirohata, N. Tezuka, and K. Inomata, Phys. Rev. Lett. 92, 167204 /H208492004 /H20850. 12F. J. Albert, N. C. Emley, E. B. Myers, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 89, 226802 /H208492002 /H20850. 13O. Ozatay, N. C. Emley, P. M. Braganca, A. G. F. Garcia, G. D. Fuchs, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, Appl. Phys. Lett. 88, 202502 /H208492006 /H20850. 14S. Mangin, Y . Henry, D. Ravelosona, J. A. Katine, and E. E. Fullerton, Appl. Phys. Lett. 94, 012502 /H208492009 /H20850. 15H. Meng and J. P. Wang, Appl. Phys. Lett. 89, 152509 /H208492006 /H20850. 16Y . Zhang, Z. Zhang, Y . Liu, B. Ma, and Q. Y . Jin, Appl. Phys. Lett. 90, 112504 /H208492007 /H20850. 17C. Bell, G. Burnell, D. J. Kang, R. H. Hadfield, M. J. Kappers, and M. G. Blamire, Nanotechnology 14, 630 /H208492003 /H20850. 18C. W. Leung, C. Bell, G. Burnell, and M. G. Blamire, Nanotechnology 15, 786 /H208492004 /H20850. 19M. C. Wu, A. Aziz, D. Morecroft, M. G. Blamire, M. C. Hickey, M. Ali, G. Burnell, and B. J. Hickey, Appl. Phys. Lett. 92, 142501 /H208492008 /H20850. 20M. C. Wu, A. Aziz, J. D. S. Witt, M. C. Hickey, M. Ali, C. H. Marrows, B. J. Hickey, and M. G. Blamire, Nanotechnology 19, 485305 /H208492008 /H20850. 21http://math.nist.gov/oommf, Version 1.2.0.4 of the software was used. 22B. Viala, G. Visentin, and P. Gaud, IEEE Trans. Magn. 40, 1996 /H208492004 /H20850. 23A. A. Tulapurkar, T. Devolder, K. Yagami, P. Crozat, C. Chappert, A. Fukushima, and Y . Suzuki, Appl. Phys. Lett. 85, 5358 /H208492004 /H20850. 24Y . Acremann, J. P. Strachan, V . Chembrolu, S. D. Andrews, T. Tyliszczak, J. A. Katine, M. J. Carey, B. M. Clemens, H. C. Siegmann, and J. Stöhr,Phys. Rev. Lett. 96, 217202 /H208492006 /H20850. FIG. 3. /H20849Color online /H20850Micromagnetic simulations. /H20849a/H20850At/H1100230 mT and 1.5 mA. The magnetization of the free layer oscillates in the C-state, flower state, and reversal C-state. /H20849b/H20850At/H1100270 mT and 1.5 mA for AP-to-P switch- ing. /H20849c/H20850At/H1100270 mT and 0.5 mA for P-to-AP switching. /H20849d/H20850The initial magnetization of the free layer obtained from the AP or P configuration inthe minor loop.122511-3 Wu et al. Appl. Phys. Lett. 94, 122511 /H208492009 /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: 138.251.14.35 On: Sun, 21 Dec 2014 01:08:59

1.1508163.pdf

Detection of coherent and incoherent spin dynamics during the magnetic switching process using vector-resolved nonlinear magneto-optics T. J. Silva, P. Kabos, and M. R. Pufall Citation: Applied Physics Letters 81, 2205 (2002); doi: 10.1063/1.1508163 View online: http://dx.doi.org/10.1063/1.1508163 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/81/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Experimental determination of the inhomogeneous contribution to linewidth in Permalloy films using a time- resolved magneto-optic Kerr effect microprobe J. Appl. Phys. 102, 053910 (2007); 10.1063/1.2772563 Two-dimensional patterns of spin-wave radiation by rectangular spin-valve elements J. Appl. Phys. 97, 10A717 (2005); 10.1063/1.1855014 Large-angle magnetization dynamics investigated by vector-resolved magnetization-induced optical second- harmonic generation J. Appl. Phys. 96, 6023 (2004); 10.1063/1.1811783 Magnetization dynamics in NiFe thin films induced by short in-plane magnetic field pulses J. Appl. Phys. 89, 7648 (2001); 10.1063/1.1359462 Subnanosecond magnetization dynamics measured by the second-harmonic magneto-optic Kerr effect Appl. Phys. Lett. 74, 3386 (1999); 10.1063/1.123353 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: 134.176.129.147 On: Sat, 13 Dec 2014 09:16:08Detection of coherent and incoherent spin dynamics during the magnetic switching process using vector-resolved nonlinear magneto-optics T. J. Silva,a)P. Kabos, and M. R. Pufall National Institute of Standards and Technology, Boulder, Colorado 80305 ~Received 27 March 2002; accepted for publication 29 July 2002 ! It is usually assumed that magnetic switching proceeds via coherent rotation under conditions of high symmetry. There is no a priorireason to expect an inhomogeneous response when a uniform magnetic torque is applied to a homogeneous ferromagnet. We test this assumption using vector-and time-resolved nonlinear magneto-optic measurements on a continuous Ni–Fe film. Whilecoherent dynamics are observed when the magnetization Mis initially oriented along the easy axis ~the preferred axis of Min the absence of external fields !, we find evidence for inhomogeneous spin dynamics when Mis initially oriented perpendicular to the easy axis, which suggests the generation of incoherent spin waves during the magnetic reorientation process.The inhomogeneity is sufficientto reduce the spatially averaged magnetic moment within the measured area by almost 50%.@DOI: 10.1063/1.1508163 # We describe a measurement technique that relies upon a nonlinear form of the magneto-optic Kerr effect 1,2to detect the time-resolved spin dynamics at the surface of a thin fer-romagnetic film. Using this technique, we have successfullyimaged the dynamics of the surface magnetization vectorwhen subject to a fast magnetic field pulse generated with a microwave waveguide. A particular strength of the vector-resolved measurement is the ability to verify the homogene-ity of the spin dynamics during the switching process. Sur-prisingly, we find that the degree of homogeneity is a strongfunction of the initial magnetic orientation in a thin metallicfilm, which suggests that a nonlinear coupling between themagnetization and spin-wave modes could be a source ofdifficulty in high-speed switching and the operation of futuremagnetoelectronic devices. Among the numerous methods that exist for the mea- surement of magnetic switching speeds, one of the most use-ful has been stroboscopic magneto-optical imaging, facili-tated by the increased availability of ultrafast lasers. 3,4The second-harmonic magneto-optic Kerr effect ~SHMOKE !has been used to measure precessional switching in thin films ofthe alloy Permalloy (Ni 81Fe19).1SHMOKE is sensitive to the rotation of the magnetization vector Min Permalloy in response to a slowly swept magnetic field.2We combine these two capacities of SHMOKE to determine the motion ofMin response to a magnetic field pulse.Afull description of the vector calibration procedure is provided in Ref. 4. Permalloy is an ideal material for these measurements because of its uniaxial anisotropy and low switching fields.The uniaxial anisotropy keeps Maligned along an in-plane easy axis in the absence of an applied field. Application of adc magnetic field in a direction orthogonal to the easy axis,or ‘‘hard’’ axis, rotates the magnetization in a continuousfashion into the field direction. By calibrating the SHMOKEsignal in response to a slowly swept magnetic field and fit-ting the data to conventional models of uniaxial anisotropy, 5we determine the appropriate fitting parameters to provide a one-to-one correspondence between the ellipsometric state ofthe second-harmonic light and the magnetization compo-nents parallel ( M x) and perpendicular ( My) to the plane-of- incidence. The polarization angle is measured with a photoelastic modulator and lock-in amplifier.Aphoton counter is used tomeasure the second harmonic intensity, with typical yields of10 3photons per second. Integration times of 3–4 h are re- quired to achieve signal-to-noise ratios for mxandmyof .100 and ;25, respectively, when acquiring a time trace with 100 ps time steps ove ra5n stime interval. To determine the dynamic response, the sample ~a 50- nm-thick film of Ni 81Fe19sputter deposited on a sapphire substrate 100 mm thick !is subjected to fast magnetic field pulses, generated by current pulses propagating in an under-lying 450 mm wide coplanar waveguide that is separated from the magnetic film by the sapphire substrate. The 5 mm laser spot is centered relative to the waveguide width, insur-ing a high degree of field uniformity at the measured region.Using the Karlqvist equation 6for the field produced by a finite-width current sheet, we calculate that the field pulsemagnitude at the sample should vary by less than 1 A/m(10 22Oe) over a 25 mm span near the middle of the wave- guide. Data are obtained with optical sampling methods. Thetime resolution is limited to 50 ps by the electronic jitter ofthe pulse generation system, with a commensurate 3 dBbandwidth of 7 GHz. A complete vector description of the magnetization re- quires measurement of the component perpendicular to thefilm plane, or M z. However, the precessional motion is highly elliptical for the films under study, with an ellipticityratioM z/My;0.02. Therefore, we assume Mz;0 for the vectorial analysis of the data. An important feature of such vectorial measurements is the ability to determine the coherence of the magnetic re-sponse. If all the spins that contribute to the magnetic mo-ment within the measured spot do not precess in phase witheach other, there must be a net reduction in the average a!Electronic mail: silva@boulder.nist.govAPPLIED PHYSICS LETTERS VOLUME 81, NUMBER 12 16 SEPTEMBER 2002 2205 0003-6951/2002/81(12)/2205/3/$19.00 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: 134.176.129.147 On: Sat, 13 Dec 2014 09:16:08moment.7It is a fundamental assumption in the derivation of the Landau–Lifshitz–Gilbert ~LLG!torque equation that M is uniform over the volume to be modeled.8Coherence at the experimental length scale is crucial to the validity of usingthe LLG equations to fit dynamics in ferromagnetic systems.Presumably, LLG could be used to simulate the dynamics ata finer length scale than our measured spot through the use ofmicromagnetic modeling methods. 9However, such an indi- rect extraction of dynamical parameters such as the dampingconstant is no longer trivial and is subject to the details of thenumerical simulation. 10It is therefore imperative that one ascertain whether Mresponds uniformly in a given experi- mental geometry before one can directly extract the LLGdamping parameter from the data using a single-domainmodel. We first measured the magnetic response of our sample to a slowly swept field applied along the hard axis. The rela- tive magnitude of the magnetization m5M/M sremained constant at its saturation value ~within error bars of 67%! during the field sweep, as expected for coherent rotation ofthe magnetization under adiabatic conditions. The dynamicresponse for this sample, in this particular field geometry, issimilarly uniform, as shown in Fig. 1. Here, the sample issubjected to a field pulse of 1200A/m ~15 Oe !along the hard axis.The field pulse has an onset time of 150 ps and durationof 2 ns.Adc field of 80A/m ~1O e!is applied along the easy axis to provide a preferred direction for M. By doing so, M should return to the same initial direction following termina-tion of the field pulse. The magnetic response is spatiallyuniform over the volume of the illuminated spot ~except for a slight dip at t51.2 ns), even though the magnetization is undergoing large-angle precessional motion. In this case, fit-ting of the data with LLG is reasonably justified and permitsan unambiguous extraction of the phenomenological LLGdamping parameter a50.016, in agreement with previously measured values.3 An alternative measurement geometry can be achieved by rotating the sample by 90°, so that the magnetic field fromthe waveguide is oriented along the film’s easy axis. The dcbias field is then applied along the hard axis to rotate themagnetization toward the hard axis. The magnetization isaligned parallel to the hard axis if the bias field is equal to ~or greater than !the anisotropy field of H k5320 A/m ~4O e !. We applied a hard-axis bias field Hbof at least Hk. This prevents any possibility that the switching process may pro-ceed via a process of nucleation and growth of domains, anotoriously irreproducible mechanism for magnetizationreversal. 11We then apply a field pulse ~of the same magni- tude used to obtain the data in Fig. 1 !along the easy axis with the resulting response shown in Fig. 2. The magnetiza-tion response is highly incoherent: the magnitude of the mag-netization is no longer constant during the switching process.Instead,mdips to almost 50% of saturation within 1 ns after the onset of the field pulse. Similar behavior was observedwhen the measurement was repeated at other locations overthe waveguide center conductor. We therefore conclude thata single-domain LLG model is not appropriate to character-ize the average dynamics of the measured volume in thisconfiguration. The turbulent magnetic state that reduces the measured FIG. 1. Time-resolved SHMOKE data for Permalloy with a dc bias field Hb580 A/m ~1O e!applied along the anisotropy axis and an orthogonal step pulse of 1.2 kA/m ~15 Oe !. The voltage wave form of the microwave pulse used to excite the sample is shown as an inset in ~a!. Both the time traceof ~a!thein-planeangle uand~b!themagnitude m5M/Msareshown. u50 is the direction of the applied bias field. The field geometry is shown as an inset in ~b!. The data for uare fitted to the classical LLG equation of magnetic motion with reasonable results. FIG. 2. Time-resolved SHMOKE data for Permalloy with a dc bias fieldH b5320A/m ~4O e!applied orthogonal to the anisotropy axis ~transverse bias!and a step pulse of 1.2 kA/m ~15 Oe !along the anisotropy axis. The step pulse duration is 2 ns.Atrace of the step wave form is superimposed onthe data as the dotted curve in ~a!. The sample is identical to that used to obtain the data in Fig. 2. Both the time trace of ~a!the in-plane angle uand ~b!the magnitude m5M/Msare plotted with solid dots. u50 is defined as the initial orientation of Min the direction of the applied bias field. Addi- tional magnitude data are shown for dc bias fields of 480 A/m ~open squares !and 640A/m ~open circles !. The data for mshow a pronounced dip that recovers on a time scale of a few nanoseconds. The depth of the dip isextracted by exponential fitting to the data. The data for uare fitted to the classical LLG equation of magnetic motion with poor results.2206 Appl. Phys. Lett., Vol. 81, No. 12, 16 September 2002 Silva, Kabos, and Pufall 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: 134.176.129.147 On: Sat, 13 Dec 2014 09:16:08moment is the result of spin–spin relaxation processes, which give rise to spin-wave modes.The spin waves in ques-tion must have wave numbers in excess of 10 4cm21to re- duce the spatially averaged value of mwithin the measured spot. Because the applied field pulse is highly uniform overthe volume of the measured spot and the sample is preparedin a coherent initial state, the magnetic response must exhibituniform precession immediately after the application of thepulse. With increasing time, however, the uniform modebreaks down into nonuniform excitations, characterized bywavelengths shorter than the illuminated spot. This results ina reduced measurement of m. In turn, these nonuniform modes decay via a variety of possible mechanisms. The pos-sible mechanisms for spin-wave decay includemagnon-phonon 12and magnon-electron13scattering. The re- covery of mto unity in Fig. 3 occurs with a longitudinal relaxation time of T151.1 ns. This is the time scale required for the eventual decay of the induced spin waves. Marginal stability is one possible explanation for the ob- served reduction in m. When biased along the hard axis with Hb.Hk, the net effective field that stiffens the individual spins is Hb2Hk.14WhenHb;Hk, the magnetization is highly susceptible to any sample imperfections that perturbthe nominally uniform state. However, it is possible to stiffenthe spin system to an arbitrary degree when H b.Hk. To test this hypothesis, we applied bias fields as large as 880 A/m~11 Oe !with little quantitative change in the temporal behav- ior ofm.The data for H b5480 and 640A/m ~6a n d8O e !are shown in Fig. 2. The insensitivity of our results to the exactvalue of H b~even for a bias field of twice the anisotropy strength !eliminates marginal stability as a plausible expla- nation for the observed incoherent response. It is the orien-tation ofMthat is important for the instigation of incoherent spin dynamics. Nonlinear mechanisms strongly affect ferromagnetic spin–spin relaxation. 15,16We varied the amplitude of the field pulse to determine whether nonlinear effects play a rolein our measurements. The maximum reduction in mis plot- ted as a function of the pulse amplitude in Fig. 3 for H b5320 A/m ~4O e!. Indeed, the dependence on field ampli- tude is highly nonlinear, implying that nonlinear spinwavegeneration is the reason for the observed reduction of m. Micromagnetic simulations for the case of negligible intrin-sic damping have predicted such nonlinear effects whenlarge-angle magnetic reorientation is induced with an appliedfield pulse. 17 Nonuniformities in the pulse field near the edges of the waveguide could spawn spin waves that propagate towardthe region where the measurements are made. For magneto-static surface wave modes 18in a 50-nm-thick film and k 5104cm21, the calculated group velocity is ng51.4 3104m/s. The requisite time for such a spin wave to propa- gate from the edge of the waveguide to the measured spot istherefore 16 ns, making it unlikely that waves generated atthe waveguide edge would contribute to the observed reduc-tion ofm. In conclusion, we find that the large-angle dynamic re- orientation of magnetization at precessional time scales canoccur coherently or incoherently, depending on the initialorientation of the magnetization relative to the anisotropyaxis.While the fitting of time-resolved magnetodynamic datawith LLG is a common means of analysis for the extractionof relevant material parameters, 1,3our vector-resolved results clearly show that the actual magnetization dynamics can dif-fer significantly from the coherent response presumed byLLG. When the dynamics are coherent, fitting of the datawith LLG is valid. Even when the dynamics are not coher-ent, we find that the magnitude of Mis still amenable to analysis. This permits the determination of the longitudinalrelaxation time T 1that describes the decay of incoherent spin wave modes within the sample. 1T. M. Crawford, T. J. Silva, C. W. Teplin, and C. T. Rogers, Appl. Phys. Lett.74, 3386 ~1999!. 2P. Kabos, A. B. Kos, and T. J. Silva, J. Appl. Phys. 87, 5980 ~2000!. 3W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 ~1997!. 4M. Bauer, R. Lopusnik, J. Fassbender, and B. Hillebrands, Appl. Phys. Lett.76, 2758 ~2000!. 5E. C. Stoner and E. P. Wohlfarth, Philos. Trans. R. Soc. London, Ser. A 240, 599 ~1948!. 6O. Karlqvist, Trans. Roy. Inst. Technol. Stockholm, 1,8 6~1954!. 7M. Sparks, Ferromagnetic Relaxation Theory ,~McGraw-Hill, San Fran- cisco, 1965 !, pp. 24–29. 8T. L. Gilbert and J. M. Kelley, Proc. Conf. Magnetism and Magnetic Materials, Pittsburgh, 1955, A.I.E.E. Publ., 1955, p. 153. 9J. Miltat, G.Albuquerque, andA. Thiaville, in Spin Dynamics in Confined Magnetic Structures I , edited by B. Hillebrands and K. Ounadjela ~Springer, New York, 2002 !, pp. 19–24. 10G. M. Sandler, H. N. Bertram, T. J. Silva, and T. M. Crawford, J. Appl. Phys.85, 5080 ~1999!. 11S. Kaka and S. Russek, J. Appl. Phys. 87,6 3 9 1 ~2000!. 12H. B. Callen, J. Phys. Chem. Solids 4, 256 ~1958!. 13V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769 ~1972!. 14D. O. Smith, J. Appl. Phys. 29, 264 ~1958!. 15H. Suhl, J. Phys. Chem. Solids 1,2 0 9 ~1957!. 16E. Schloemann, J. Appl. Phys. 33, 2822 ~1962!. 17V. L. Safonov, and H. N. Bertram, J. Appl. Phys. 85, 5072 ~1999!. 18Daniel D. Stancil, Theory of Magnetostatic Waves ~Springer New York, 1993!,p .1 1 3 . FIG. 3. Maximum reduction in m5M/Msas a function of pulse amplitude. For the vertical axis, mminis the minimum value of mduring the dynamic response to the field pulse.2207 Appl. Phys. Lett., Vol. 81, No. 12, 16 September 2002 Silva, Kabos, and Pufall 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: 134.176.129.147 On: Sat, 13 Dec 2014 09:16:08

1.1911431.pdf

Received 28 May 1968 12.10, 12.3; 5.21 Instrumentation for 80-Hz Seismic Communications K. IKRATH, W. A. SCHNEIDER, R. F. JOHNSON, AND K. J. MURPHY U.S. Army Electronics Command, Fort Monmouth, New Jersey 07703 A continuous-wave single-frequency seismic carrier for transmitting information by amplitude, phase, and frequency modulation has proven useful and practical in situations where electromagnetic means of com- munication are not feasible. An 80-Hz, 20-kg seismic transducer using 10 W maximum power was utilized for transmission through low-velocity media; an 80-Hz, 300-kg transducer using 200 W maximum power was utilized for transmission through high-velocity media. For reception, an 80-Hz resonant seismic trans- ducer equipped with a frequency trimmer was used. Efficient signal insertion into soil was achieved by employing an artificial mechanical-elastic transmission line as an impedance transformer. A mathematical model for a slotted steel tube, which operates as an impedance transformer, is presented with a mathematical description of the impedance transformation process. Seismic-coupling efficiency was derived by comparing an actual system with a theoretically ideal transducer-medium system. Experimental and theoretical results based on data recorded during tests on various seismic media are analyzed in terms of the mathematical model employed. Results verify the validity and soundness of design and operation of a seismic communi- cations system. INTRODUCTION OR several decades, active seismic systems have been employed in the geophysical exploration of the earth's crust and in locating and delineating geological structures that indicate the accumulation of petroleum, gas, and ores. In such geophysical work, seismic signals are usually generated as single pulses by detonation of explosives at or near the surface of the earth. This technique is restrictive for exploratory probing and its use is precluded in a communications system where many bits of information must be trans- mitted. Specifically, the destructive effects on the seismic medium, the complex and wide-spectrum character of impact-generated signals, and the dis- persive character of seismic media confine one to either a very complicated transient analysis of received signals or the inefficient rejection of all information other than arrival times. In contrast, a continuous-wave single-frequency seismic carrier for transmitting infor- mation over a small frequency band by amplitude, phase, or frequency modulation is practical and meaningful. To exploit the capability of using seismic acoustic signal communications in situations where electro- magnetic means are not feasible, the U.S. Army Electronics Command has designed, constructed, and evaluated an experimental seismic-acoustic communica- tions system for transmitting signals through various earth media. For transmission, this system employs a 20-kg transducer for use through low-velocity seismic media such as sand or other soft soils, and a larger 300~kg unit for use through high-velocity media such as hard rock. For reception, the system employs 80-Hz resonant seismic transducers as receivers (see Figs. 1 and 2). For this purpose, one of the small receiver transducers was equipped with mechanical frequency trimmers and high-impedance drive coils. This type receiver-transducer can be used on either soft- or hard-earth media. This seismic acoustic communications system has been used successfully in a series of experiments in various terrains and media, both on the surface and underground. I. DISCUSSION A. Selection of an Operating Frequency Mechanical vibrators are used to test the static and dynamic properties and bearing capacity of soils to be used as construction sites for large buildings and dams. At resonance frequencies, which are dependent upon the elastic moduli of the soil, such mechanical vibrators permit the efficient excitation of seismic waves. In principle, one could change the mass of the vibrator to compensate for variations in ground The Journal of the Acoustical Society of America 621 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.120.242.61 On: Fri, 28 Nov 2014 12:34:02IKRATH ET AL. ............. .............. ............. . .... : ...... ........... FIG. 1. Ten-W transmitter for soft media and receiver for hard and soft media. conditions and thereby obtain the same resonance frequency for different soil conditions. However, this method of tuning by changing vibrator masses is neither technically feasible nor operationally acceptable for communications, particularly when some degree of mobility is required because of the wide range of variations in the elastic moduli of seismic media. The problem of employing continuous seismic waves (CSW) for communications is further complicated by practical limitations of the usable frequency. Ultrasonic techniques are successfully used to study small samples of solids in the laboratory, but attenuation is too great for efficient use in the field. However, at low frequencies, difficulty in achieving high radiation efficiency with seismic transducers of reasonable dimensions is en- countered. Further limitations are imposed by the need to avoid regions where wind-generated noises at low frequencies or other natural microseismic disturbances interfere seriously with signal transmissions. One must also choose frequencies with minimum audible coupling to the air and avoid 50- and 60-Hz frequencies, which are used for power-line transmission. Thus, when selecting an operating frequency, a compromise between conflicting natural constraints and operational requirements must be made. A satisfactory choice of frequency for seismic communications was found to be 80 Hz. B. Signal Insertion Efficient seismic signal insertion required matching the input impedance (mechanical impedance of the interface area of the seismic transducer and the ground) to the output impedance of the signal source (electrical output impedance of conventional loudspeaker drive amplifiers for the range 2 to 50 2). The analogous impedance-matching problem for radio communications can be solved by employing a resonant antenna and a transmission-line section as an impedance transformer. Application of electromechanical analogies to the design of seismic transducers led to construction of the seismic transmitters and receivers (transducers) used in the experiments. These transducers (Figs. 1 and 2) contain a slotted steel tube, which serves as an artificial mechanical transmission-line section and impedance transformer. The slotted steel tube connects the drive coil of a conventional electrodynamic speaker system with a steel piston held in contact with the seismic medium by gravitational force or an equivalent mechanical spring force. The steel piston and magnet of the electrodynamic drive system are rigidly connected by a solid steel rod inside the slotted steel tube. The functioning of these seismic transducers is described by the equivalent mechanical circuit diagram in Fig. 3. Dimensions and weights of the components for both the small and large seismic transducers are listed in Table I. The mechanical impedance, which loads the electrodynamic drive system of the transducers, is given in terms of the parameters defined by the equiv- alent circuit diagram of Fig. 3, as follows - YoR-k j tanw/wo R.=jwm.-kZo. , (1) 1 + YoRlj tanw/wo where Y0= 1/Zo is the characteristic admittance and R1, the load impedance. For all practical operating conditions of the seismic transducers, tanw/wo /> 1, T^]t.. I. Design data for 80-Hz seismic transducers. Carbon-Steel Slotted Tube Small Large transducer transducer Young's modulus (kilograms/centimeters ') 2)< 106 Outer diameter (centimeters) 5.08 Mean diameter (centimeters) 4.76 Inner diameter (centimeters) 4.45 Slot width (centimeters) 0.08 Beam width (centimeters) 0.28 Slot length (centimeters) 5.00 Wall thickness (centimeters) 0.32 Slots/section on circumference 3 Slot sections along length 50 Static stiffness (newtons per meter) 105 Mass (kilograms) Slotted tube (active part) Drive coil and coil mount Ground piston Drive system (Indox V ceramic magnet) Coil: Tums No. 20 copper wire Mean diameter (centimeters) Piston diameter (centimeters) 2)< 106 21.83 20.76 19.69 0.32 0.68 10.48 1.07 6 32 2X 10 ø 0.640 14.500 0.160 3.090 18.935 285.100 168 700 4.55 7.40 30.5 60.33 K. Ikrath, W. A. Schneider, and R. F. Johnson, "Active Seismic Systems for Communications and Surveillance," Tech. Rep. ECOM-2695, AD No. 632082 (Apr. 1966), pp. 12-13. 622 Volume 45 Number 3 1969 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.120.242.61 On: Fri, 28 Nov 2014 12:34:02INSTRUMENTATION FOR SEISMIC COMMUNICATIONS and Zm e so that the following approximation of Eq. ! becomes valid' f0 Zo Zo 2 R.-' jwm2q J tanw/wo R' (2) One may note that the third term in Eq. 2 corresponds to the impedance inversion of R by a quarter-wave transmission-line section with a characteristic imped- ance Z0. Reactive contributions from the piston mass jwM, are small compared to those from the mechanical impedance of the piston-to-medium interface. The latter impedance is referred to as "the coupling impedance to the seismic medium," which under theoretically ideal conditions would be equivalent to the seismic radiation impedance. The third term in Eq. 2 then amounts to .conversion of normally very small coupling admittances into mechanical impedances, the reactive parts of which are tuned by the reactances presented by the first two terms of Eq. 2. The resultant purely resistive mechan- Fro. 2. Two-hundred-W transmitter for hard media (XTD-H). 2 m2 SLOTTED 1 TUBE i Zo ,o i I j MECHANICAL PORTION OF THE. TRANSDUCER Gg fd DRIVE FORCE Z -- MECHANICAL- ELECTRICAL SOURCE IMPEDANCE me m 2 COIL MASS Z o; o) o - CHARACTERISTIC IMPEDANCE AND FREQUENCY OF THE ELASTIC TRANSMISSION LINE (SLOTTED STEEL TUBE) i M p -- MASS OF THE PISTON << IwYg I Yg MECHANICAL ADMITTANCE OF THE SOIL-PISTON Cg COMPLIANCE G RADIATION CONDUCTANCE g FIC. 3. Equivalent mechanical circuit diagram of a transducer. ical load Ra= ZoaGa for the electrodynamic drive system is converted into an electromotional resistance, which appears at electrical input terminals of the seismic transducer. The characteristic impedance Z0 and frequency w0 of the slotted-steel-tube transmission line are chosen so that the electrical input impedances of the transmitter transducers cover the range from 4 to 30 and the resonance frequencies cover the range from 78 to 83 Hz. The exact values of the electrical input impedances and resonance frequencies are dependent upon the mechanical properties of the seismic medium and reflect damping by friction and possibly inelastic deformation of the media, in addition to radiation damping. C. Design of the Slotted Steel Tube Basically, the circumferential slots in walls of a steel tube from an aggregate of coupled cantilever beams, whose compliances and masses are analogous to lumped capacitances and inductances in the electrical four-pole networks used as artificial electrical transmission lines. Since the diameter of the steel tube is large compared to the wall thickness, flexural deflection-versus-force characteristics of cantilever beams constitute the com- pliances of the slot elements. The compliance (deflec- tion/force) for a slot element is given by a C=la/16bhaE , (3) where E is Young's modulus of the steel material; b, the wall thickness of the tube (beam width); h, the axial separation of slots (beam height); and l, the ' K. Ikrath and W. Schneider, "The Realization of Active Seismic Systems and Their Practical Applications," USAELRDL Tech. Rep. 2446, AD No. 601427 (Apr. 1964), pp. 7-10. The Journal of the Acoustical Society of America 623 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.120.242.61 On: Fri, 28 Nov 2014 12:34:02IKRATH ET AL. length of slot element [beam length as measured along the mean circumference (D,r) of the tube, i.e., rr ß 71' (4) s where n is the number of slots along the circumference referred to as a "slot section." The compliance of a slot section (C,) consisting of n slot elements is C--Cs/n (5) and the number of slot sections (ns,) along the axial length of the tube yields the total compliance n.. n.. (wDm) a Co=n..C..=--.C. .... (6) ß n, (n,) 4 16bhE The inverse of Co is the stiffness S0. The value of n is given by the height of the slots (h) and the heights of the beams (h) in relation to length (L) of the slotted part of the tube' L=(n,,+l).(h+h,)+h. (7) To check the accuracy of these design formulas for the slotted steel tube, static and dynamic load versus deflection curves were plotted from measurements made on a number of different samples. Discrepancies between calculated and statically measured compliance values were found to be less than 5%. Dynamic compliance values, derived from measurements of the first and third quarter-wave resonance frequency and the weight of slotted-steel-tube samples, were within 10-20% of values calculated from the static compliances. Thus, for design purposes, the characteristic mechan- ical impedance and frequency of slotted-steel-tube transmission lines are given by and Z0= (mo/Co) « (8) w0= 1/(moCo) , (9) where m0 is the mass and Co the static compliance of the active portion of the slotted steel tube. Numerical values for design parameters of the slotted steel tubes used in transducers for soft and hard media are listed in Table I, together with data for the various transducer components. For the soft media transducer, Z0=250 kg/sec, and w0=400 rad/sec. For the hard media transducer, Z0=5400 kg/sec, and w0=370 rad/sec. D. Equivalent Ideal Transducer Medium System The field and radiation impedances of mechanical radiators on the free surface of a semi-infinite solid were derived by Miller and Pursey. .4 However, seismic media have limited elastic ranges, are mostly heterogen- eous, and exhibit various degrees of gross anisotropy. These conditions create a problem with regard to distinguishing the coupling efficiency from the radiation efficiency of seismic transducers in the field. As pointed out in Sec. I-B, the relative coupling efficiency of a transducer to a seismic medium can be deduced from variations of the electromotional impedance of the transducer as a function of ground conditions. However, for evaluation of the radiation efficiency, measurements of transducer-induced vibrations must be made at various intervals on the surface or underground to distinguish radiation losses from power losses in the wave-excitation regionß Friction between piston and medium and inelastic behavior of the medium in regions of high stress beneath the edges of the piston are the major causes of power losses. Results of measurements from beyond the wave- excitation region must be reduced to values equal to actual radiated power or related parameters such as the radiation impedance of the transducer, as opposed to the mechanical coupling impedance. For this purpose, the concept of specifying the performance of seismic transducers in terms of parameter values of equivalent ideal transducer-medium systems is practical and meaningful. In applying the concept of equivalence to seismic transducers, one must take into account the seismic propagation medium and its elastic properties. There- fore, we must introduce an ideal equivalent transducer medium system to specify the radiation capability of seismic transducers. Thus, the ideal transducer medium system is defined by parameter values of the theoretical CSW vertical surface displacement amplitude-versus- distance curve that best fits the experimental CSW am- plitude-versus-distance curve corresponding to the area beyond the wave-excitation region. In addition, the pressure wavelength and shear wavelength values de- rived from the slopes of experimental CSW phase-versus- distance curves are used for numerical verification. Sets of normalized theoretical CSW vertical surface displacement amplitude-versus-distance curves Wnorm= F(; 0; n), (10) where Wnorm is the normalized theoretical CSW vertical surface displacement amplitude as a function of' (!) distance, r, normalized in terms of pressure wave length by = (2r/X)r=kr; (2) source dimensions corresponding to the radius, a G. F. Miller and H. Pursey, "The Field and Radiation Impedance of Mechanical Radiators on the Free Surface of a Semi-Infinite Isotropic Solid," Proc. Roy. Soc. London A223, 521-541 (1954). 4 G. F. Miller and H. Pursey, "On the Partition of Energy Between Elastic Waves in a Semi-Infinite Solid," Proc. Roy. Soc. London A233, 55-69 (6 Dec. 1955). 624 Volume 45 Number 3 1969 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.120.242.61 On: Fri, 28 Nov 2014 12:34:02INSTRUMENTATION FOR SEISMIC COMMUNICATIONS ro, of a vibrating vertical surface indentation normalized in terms of pressure wave by /0 = (2,r/XOro=kro; (3) n the ratio of pressure wavelength X to shear wavelength X2; were calculated by computer evaluation of the following integrafl' jo sin (r/o) F (; 0; n) = r- J0 (r)' (- 1) . X . [2-- (n/2)-r? ß (r?--l)l(-n) (11) The actual vertical CSW surface displacement ampli- tudes, w, are related to Wnorm by Eqs. 10 and 11 via w=--fn2/4Xl.e+it.F(; 0; n), (2) where f is the driving force acting on the transducer piston and u is the shear modulus. Computer evaluation of the integral Wnorm in Eq. 11 was carried out by application of contour integration in the complex plane. The contour consisted of a closed curve composed of two circular sections of radii a and b, respectively, which were connected by straight radial line sections at angles 0=0 (i.e., along the positive real r axes) and 0=5/110 rad, respectively. For n=X/X=V2, a and b were selected as 1.5 and 1.7; for n=2, a and b were selected as 2.1 and 2.2. Therefore, the pole of the integrand that occurs on the real axes fails between the values for a and b; whereas poles within the complex plane remain outside of the contour of integration. For n=V2, the values of the poles on the real axes were determined with (2.61803398) ,, and for n-2, with (4.59979192), by setting the denominator of the integrand of F in Eq. 11 equal to zero. The respective residue contribu- tions to the integral from these poles were evaluated using branch cuts straight downward from the poles and applying complex methods modified to account for the different signs of the square root expressions in the integrand on the left- and right-hand sides of the branch cuts. Along the real-axes section of the contour, contribu- tions to the integral were obtained by successive application of Simpson's rule with increasing spacing of sample points. Similarly, contributions from the circular sections of the contour and of the radial section inside the complex plane were obtained by a modified Simpsoh's rule for the complex plane as applied to F(z)dz, 5 Ref. 1, pp. 39-41. 20 I01 5 2 I .2 0.1 .05 .02 0.01 .005 .002 0'0010 2 4 6 8 I0 12 14 16 18 20 22 24 26 28 :50 /o '---" r/ro Fro. 4. Normalized vertical ground surface displacement, W, versus distance, r, in multiples of the piston radius, r0, for different values of the piston circumference to compression wavelength ratio, and for a 2:1 ratio of compression to shear wavelength. for r= a and r= b constant, with 0<0( 5/110 rad, and for with 0 = 5/110 rad= const, respectively. The accuracy of results decreases with increasing values of , for which the imaginary part of the integrand becomes large compared to contributions to the integral from the contour integration. Hence, differing from the usual asymptotic approximations for these type integrals that are valid at very large distances from the source, the described method of numerical computer evaluation of the integral in Eq. 11 yields the greatest degree of accuracy for the displacement- versus-distance function in the experimentally signifi- cant regions. Samples of curves for absolute values of I Wnom[ = JF(; 0; n) J and 0=0.1, 0.5, 1.0, respectively, as a function of the normalized distance /0, are given in Fig. 4. The characteristic CSW maxima and minima (showing constructive and destructive interferences, respectively, between several CSW propagation modes at certain distances from the transducer) are most significant for identification and selection of the theoretical curve that best fits the experimental curve under consideration. These distances must be in the elastic range beyond the wave excitation region but less than the depth of the medium. The wavelengths of these interfering propagation modes are shown by the slopes of the experimental CSW phase-distance curves of Figs. 5 and 6. The CSW phase-distance curve in Fig. 5 was measured on a 12-m thick layer of Tertiary Age Cohansey sand in the Lebanon State Forest, The Journal of the Acoustical Society of America 625 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.120.242.61 On: Fri, 28 Nov 2014 12:34:02IKRATH ET AL. 1800: 1600: 32OO 1400-- 1200 I000 - 8oo- 600- T oo- 3OOO 2oo 200 0[/,,,i .... i , , I,,,, ..... ! ........ , I 4 27m L- I0 20 30 40 "HERS FRO" XTD Fro. 5. Phase versus distance, transmitter-receiver transducer (XTD-RTD) of 80-Hz CSW signal on surface exposure of Cohansey sand (Tertiary Age), Lebanon State Forest, New Jersey, 30 Dec. 1964. New Jersey. The CSW phase-distance curve in Fig. 6 was measured on the surface of 259-m thick glacial ice over rock. 6 The same equipment and experimental setup as used at the Lebanon, New Jersey site were employed. The shear wavelength X2 was derived from slopes of the CSW phase-versus-distance curve within the dominant shear zone near the transmitter. The pressure wavelength 2,1 was derived from the slopes of the CSW phase-versus-distance curves within the dominant pressure transition region further away. The derived values are given in Table II. Note that for both sand and ice media, n-' 2 for Xl/X2. In Figs. 7 and 8, experimental CSW amplitude- distance curves and matching theoretical curves for the Lebanon sand and the glacial ice are shown. The theoretical curves are identical to the curves in Fig. 4 for 0=0.5 and 0.1, respectively. These parameters define the respective ideal transducer medium system for the Cohansey sand and the glacial ice. These 0 values were selected on the basis of the best proportional fit with the experimental curves when aligned with the most significant minima and the corresponding change of scale between normalized and absolute distances. In addition to restrictions due to the depth of the medium, the range of validity for the experimental- theoretical curve-fitting procedures is restricted as previously discussed, by the increasing inaccuracy of the theoretical curves for large values of /0 (due to the method of integration by Simpson's rule) on one hand, and by restrictions due to the distance relative to the depth of the medium. For the applicable media, i.e., the Cohansey sand at Lebanon, New Jersey and the glacial ice layer, distances cannot exceed the depths of 12 and 259 m, respectively. To verify further the validity of the equivalent ideal 6 R. F. Johnson and W. H. Fischer, "Seismic Propagation Studies in the Arctic Region Using Novel Type of Electromechan- ical Transducer," Tech. Rep. ECOM-2713, AD No. 636726 (July 1966). 260O 2400 2200 2000 1800 1600 1400 I00 I000 800 __ X 360o.23 6m ,, I _ / ./' '"7'_ __ / /71 80m .. 860 X-II3m I . - 27 rn - 1 I I I I I I I I I I I I I I I I I I0 20 50 40 50 60 70 80 90 I00 I10 120 I I I 160 170 I DISTANCE I EEE5 Fro. 6. Phase versus distance of 80-Hz CSW signal on glacial ice, 23 June 1965 (XTD power 3.8 V X0.65 A). transducer-medium system and to identify the propaga- tion modes contributing to the various minima in the CSW distance curves, phase relations between the modes are obtained from the following series-expansion representation for the vertical displacements' Ao B0 w .... eikørnL--' ei[kør+(r14)l'J V ß ' ' (k)« (k)l A1 'Jv ' ' ' ' .... e i [ k x r-- ( r14 ) ] -- ' ' ' (k2r) 'q-'ei(/2r-')- ..... e i[/2r-(r/2)] , (13) where the subscript 0 denotes Rayleigh surface-wave terms; the subscript 1, pressure-wave terms; and the subscript 2, shear-wave terms. The J i and By represent amplitudes of the corresponding modes normalized relative to the shear-wave propagation factor k= 2r/X. For example, A0 is the amplitude factor of the funda- mental Rayleigh surface wave mode, which decays in proportion to the inverse square root of the distance r; similarly, B0 is the normalized amplitude factor of the second higher-order Rayleigh surface wave mode. Note that the fundamental pressure mode, A z, and shear mode, A, decay in proportion to the inverse of the square of the distance from the source. Destructive interference between any two modes (i j) occurs at TABLE II. Shear and pressure wavelengths. Shear wave X2 Pressure wave (in meters) (in meters) For sand 3.05 6.1 For ice 11,3 23.6 626 Volume 45 Number 3 1969 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.120.242.61 On: Fri, 28 Nov 2014 12:34:02INSTRUMENTATION FOR SEISMIC COMMUNICATIONS TABLE III. Distances of minima from transmitter. Destructive Distance (meters) interfering Calculated Experimental Normalized modes ( ) r. r /0 value Cohansey sand (A A 2) 5.25 5.5 3.5 (AiB2)s 9.8 9.6 7 (A2B0)i 5 5.5 3.5 Glacial ice (AIA) 18.7 18 13 (A 1B2)a 35 33 19 (A 1B0)o 23.7 25 17 (AiB0)5 49.5 50 .... (A2Bo) 18.7 18 13 Beyond scale in Fig. 4. distances r., as given by r.= = [kg-- kl I (Xj/Xg)-- 11 2 In Eq. 14, a denotes odd numbers and Aij differences among phases ,r/4, ,r/2, ,r of the mode terms in Eq. 12. Required numerical values of the Rayleigh surface- wave propagation factors k0 and wavelengths X0 are obtained from the Rayleigh equation as a function of the parameter n. The Rayleigh equation is obtained by setting the denominator expression in the complex integrand of Eq. 11 equal to zero. Corresponding residual contributions to the integral in Eq. 11 yield the surface-wave terms in the series expansion for w of Eq. 13. The surface wavelength for n= 2 is equal to X0-' X2/1.075. Employing Eqs. 12 and 13 and numerical values for the surface pressure and shear wavelengths as derived from the CSW phase-distance curves for the Cohansey sand medium (Fig. 5)--(X0= 2.8 m, X1=6 m, and X2= 3 m)--and for the glacial ice medium (Fig. 6)--(X0= 10.5 m, X1=23.6 m, and X,11.3 m)--one obtains the distances (see Table III) of CSW amplitude minima from the transmitter. Notice the relatively close agree- ment and proportionality of calculated, experimental, and normalized values of the distances of CSW minima. Thus, the parameter values of the equivalent ideal transducer media systems n=2, 0=0.5 for the Cohansey sand case and n=2, 0=0.1 for the glacial ice case are verified, and the various mode terms associated with CSW minima identified. E. Coupling and Radiation Efficiency of the Transducer Evaluation of the effective radiation impedance or admittance of the transducer proceeds from the parameter values n and 0 of the equivalent ideal transducer-medium system in conjunction with complex values for F(; 0;n) in the range 0x</0x< 1. This xox'x\ THEORETICAL x ,f', //0:05 ,00 'i 10 PERI 10 20 0 Fro. 7. Amplitude versus distance XTD-RTD of 80-Hz CSW signal on surface exposure of Cohansey sand (Tertiary Age), Lebanon State Forest, New Jersey, 30 Dec. 19. range corresponds to radii of indentations in the equivalent ideal transducer-medium systems, which are referred to as "equivalent coupling radius of the transducer" and denoted by R0. The respective 0=0.5 and 0=0.1 yield equivalent coupling radii for the Cohansey sand 1.36 m x< R0 < 1.45 m, (15a) and for the glacial ice 1.38 m x< Ro( 1.44 m. (15b) Thus, despite the significantly different properties of the two media, except for n- 2, the equivalent coupling IOOO ß IOO E I / I/tINEUe . :,0 /,,,.v\., I NOIS[ LIBEL ' PREAPLIFI[R I J NOISE LEVEL t [ I i t I ,, ! 01 10 I00 I000 DISTANCE IN METERS m. 8. Amplitude versus distance of 80- CSW sgn] on gbcb] ce, 23 June 965 (D power 3.8 V 0.65 A). The Journal of the Acoustical Society of America 627 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.120.242.61 On: Fri, 28 Nov 2014 12:34:02IKRATH ET AL. radius of the transducer has essentially the same value. However, with regard to evaluation of the radiation efficiency of the seismic transducer on the sand and ice medium, respectively, one must note the difference between the wavelength-dependent values of ro as determined from the parameter o=(2r/X).ro. For Cohansey sand, the value is ro-0.47 m; for ice, it is ro-0.38 m. Since o, and thus ro, control the radiation admittance via F(; o; n) of Eq. 11, the dimension ro is called the "equivalent radiation source radius." The equivalent radiation admittance is defined as the ratio of the vertical displacement velocity jcow to driving force f in the region of wave excitation, 0 /o< 1, i.e., within the equivalent coupling radius. The equiv- alent radiation admittance derived from Eq. 11 is Yg=--jcow/f----jcon2/4Xt.F(; o; n), (16) where the shear modulus t may be expressed by = ,0622, p being the density, and c2, the phase velocity of the shear mode as determined by frequency (80 Hz) and shear wavelength X as derived from the experimental CSW phase-distance curves. Values for F(; 0.5; 2) and F(, 0.1; 2) (calculated by computer) vary only slightly in the respective wave- excitation regions; for example F(0; 0.5; 2)= -- 1.4291492+j0.5373937, F (0.45; 0.5; 2)= -- 1.2912628+ j0.4849856, and similarly F(0; 0.1; 2)- -- 10.4822597+j0.5835594, (18) F(0.09; 0.1; 2)= -- 10.2735616+j0.5811558. The representative intermediate value of F for Cohansey sand is F(0.25; 0.5; 2) '--- 1.32+j0.52, (19) and for glacial ice F(0.05; 0.5; 2) '--- 10.3+j0.58. (20) The approximate values for the equivalent radiation admittance in secondsXkilograms - are obtained by substituting Eqs. 19 and 20 into Eq. 16, using the following values' co=500 rad/sec; sand density is 2500 kg/m3; and ice density is 1000 kg/m 3. Thus, for the Cohansey sand Yg= 106X [-0.30+ j0.76 (21) and for the glacial ice Y= 10-7X [-0.15+j2.66. (22) In both cases, the reciprocal values of Y, i.e., the equivalent radiation impedances, are much larger than the inertance jcoMv=104 kg/sec of the transducer piston. Hence, the transformation law as given by Eq. 2 is valid, and at a resonance frequency of about 80 Hz, corresponding to the vanishing of the imaginary terms in Eq. 2, one obtains simply' R=ZoG, (23) where R represents the equivalent mechanical radiation damping resistance for the electrodynamic drive system. This resistance is due to transformation of the real part of the equivalent radiation impedance Ga by the slotted steel tube with a characteristic mechanical impedance Zo. Using the product of the drive-coil wire length L and magnetic flux density B of the electromagnetic drive system (0.7 V.sec/m), we obtain the equivalent elec- tromotional radiation damping resistance By comparison with actually measured electrical imput resistances of the transducer in relation to mechanically open- and short-circuit load conditions simulated by thick hard concrete foundations and soft hog-hair mats, one finds that the equivalent electromotional radiation resistance contributes only a small part to the over-all shunt-type damping resist- ance. For example, the input resistance of the transducer on thick concrete is approximately 8 2 at a resonance frequency of 78.6 Hz. On compliant hog-hair mats, the resistance is also about 8 2, but at a resonance frequency of 80 Hz. The measured frequency shift agrees closely with the calculated value of 1.27 Hz for open and short-circuit loading of the transducer using Eq. 2. On sandy soils, such as that at the Lebanon site, the input resistance is as low as 4 2. Reduction of the input resistance and associated widening of the frequency response curve amounts, in effect, to additional damping by a shunt resistance of 8 2. However, the equivalent electro- motional radiation damping in the Cohansey case is about 25 2. Thus, whereas the over-all coupling efficiency is about 50%, the radiation efficiency is only 24%. On hard glacial ice, the radiation efficiency is even less, roughly 1%, as seen by comparing the values of the equivalent radiation admittances (see Eqs. 21 and 22). F. Communications Range Numerical considerations carried out above are valid for transducer drive power levels of less than 3 W. Evidence of decreasing radiation efficiency with in- creasing drive power was obtained from measurements in the distant elastic region. Increasing the transmitter power above 3 W results in a less than proportional increase in signal levels received at a distance. For this reason, several transducers were used in an array and operated at drive power levels that did not 628 Volume 45 Number 3 1969 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.120.242.61 On: Fri, 28 Nov 2014 12:34:02INSTRUMENTATION FOR SEISMIC COMMUNICATIONS exceed the critical value of 3 W. For example, on the sandy soils at Fort Monmouth, New Jersey, a range of 600 m was achieved with a total of 12-W nominal 80-Hz drive power for a phase controlled linear array of four soft-media transducers and a single receiver. 7 This range may be further increased or the array drive power may be correspondingly decreased for the same distance by using coherent detection techniques. For an experiment in a noisy area at Fort Monmouth, the primary transmitter drive power was reduced from 6 W to 10 mW by application of coherent detection using a lock-in amplifier for detection of the receiver trans- ducer output. The different elastic and inelastic properties of the various seismic media and the performance character- istics of seismic transducers show clearly the need for at least two or possibly three transducer types using slotted steel tubes of different characteristic mechanical impedances and drive-power capacities. The large 200-W seismic transducers satisfy this need for hard media. These transducers are primarily intended for com- municating to deep underground sites embedded in hard rock. Surface ranges of 1-2 km were achieved at High Point State Park using one of the 200-W trans- ducers on outcrops of rock. a The depth range attained in experiments in the Sterling Hill Mine of the New Jersey Zinc Company, Ogdensburg, New Jersey, using 60 W power, was over 500 m. 9 Further, using about 3 W, flexural wave signals were transmitted across the ice cover of Lake George, New York and Greenwood Lake, New Jersey for over 2 miles? Transmission at Lake Champlain, New York from shore through earth, and from the surface of the ice to conventional hydrophone 7 K. Ikrath, W. A. Schneider, and R. F. Johnson, "Transmitter Arrays in Active Seismic Systems for Communications and Surveillance," Tech. Rep. ECOM-2730, AD No. 636301 (June 1966), p. 2. 8 R. F. Johnson, K. J. Murphy, W. A. Schneider, and K. Ikrath, "Continuous Seismic Wave Communications Experiments at 80-Hz in Diverse Terrains," Res. Develop. Tech. Rep. ECOM- 2921, AD No. 832794 (Jan. 1968), pp. 13-27. 9 K. Ikrath, R. F. Johnson, K. J. Murphy, and W. Schneider, "Seismic Communications to Underground Sites--Sterling Hill Mine Experiments," Res. Develop. Tech. Rep. ECOM-2919, AD No. 828962 (Dec. 1967), p. 12. 0 W. Kennebeck, "Excitation of Flexural Waves in Lake Ice with 80-Hz Resonant Seismic Transducers," Tech. Rep. ECOM- 2735, AD No. 639945 (July 1966). receivers under water, were successful for a range of more than 600 m with a 40-dB signal-to-noise ratio. n II. CONCLUSIONS Close correlation was found between the theoretical model of an 80-Hz seismic communications system and experimental data derived from tests on various media. The theoretical model isolates the radiation efficiency from the coupling efficiency of the transducer, and thus accounts for radiated power and power lost by friction and inelastic phenomena in the wave excitation process. With low power and simple receivers, relatively large communication ranges have been achieved between locations on the surface, and from surface locations to underground sites in a mine, from land to underwater, and similarly between locations on the ice cover of lakes, and from ice to underwater, indicating the soundness of the principles of design and operation of such a seismic communications systems. ACKNOWLEDGMENTS Many of our colleagues at the U.S. Army Electronics Command have actively contributed to the realization of the seismic communication system described. Special recognition is extended to Dr. S. Benedict Levin for scientific guidance, Lt. Gilbert Singer for devising the computer programs for the theoretical calculations, and Granville LeMeune and Horace Whichello (deceased) for fabrication and construction of seismic transducers. A list of relevant and related publications are cited as Refs. 12-15. 11 K. Ikrath, R. F. Johnson, W. Kennebeck, K. J. Murphy, R. Ridgeway, and L. Stascavage, "Communication and Target Detection through Ice by Means of Seismic Acoustic Signals," Res. Develop. Rep. ECOM-2900, AD No. 826187 (Nov. 1967), p. 22. 1 K. Ikrath, "Communications Via Seismic Waves Employing 80-Hz Resonant Seismic Transducers," IEEE Trans. Commun. Tech., COM-16, 439 (June 1968). 13 R. F. Johnson, "Photoelastic Modeling Techniques for Seismic Wave Propagation," Tech. Rep. ECOM-2769, AD No. 644590 (Oct. 1966). 4 K. J. Murphy and K. Ikrath, "Analysis of a 'Seismic Fence' for Intrusion Detection (Experimental and Theoretical Results)," Res. Develop. Tech. Rep. ECOM-2848, AD No. 658654 (May 1967). 5 F. Gilbert et al., "Seismic Propagation Studies for Seismic Communications," Final Rep. AF Contract 30-(602)-2113 for Rome Air Development Center (20 Oct. 1961), Texas Instruments, Inc., Geosciences Div. 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The Nature of Physics: A Physicist's Views On the History and Philosophy Of His Science Robert Bruce Lindsay Erwin Hiebert , Citation: Physics Today 23, 12, 49 (1970); doi: 10.1063/1.3021870 View online: http://dx.doi.org/10.1063/1.3021870 View Table of Contents: http://physicstoday.scitation.org/toc/pto/23/12 Published by the American Institute of Physics- "Iondrian was acquainted finds 3? in terms of enersywhich operates m a barere space-time con- i all trace of per- sonality or of 'secondary qualities' revealed in human perception, have been bamshed/ And as an example of scientific mystique: "[The Schrii- dinger equation] dealt with waves of a special kind referred to as psi-waves." As what must be considered a colossal oversight, no mention is made of Salva- dor Dali's magnificent and ingenious painting of Christ on the cross in which the cross was deliberately and know- ingly painted as a projection of a four- dimensional hypercube. The author is against the atomic bomb and claims that a number of artists "have responded to the horror of the atomic bomb." So anti-atomic bomb is the author that he attacks scien- tists working on the bomb as scientists who "placed their services as technical experts at the disposal of the political governments of their countries," and he further ostracizes them from "the community of natural scientists." I would strongly suggest that the author spend a little time in studying and un- derstanding the vast scientific, techno- logical and political aspects of the atomic bomb along with the histories of such men as J. Robert Oppenheimer. A book on art should not be an openforum for a political speech against the atomic bomb and 20th-century tech- nology as being the cause of Man's present dismal state of existence. tists throughout history have been de- picting Man's dismal state of decadence and lack of morality. A quote that I consider a particular gem is: "Looking at paintings, or experienc- ing other kinds of art, especially kinds which are not too immediately trans- parent but which demand some at- tempt by the spectator to enter into the experience of the artist's crea- tive process, is one of the best ways for a scientist to loosen the joints of his psyche, to 'roll the bones' of his ideas, and give himself a chance to dredge up from the obscure internal depths something, which will prob- ably have not the slightest obvious connection with the work of art he has been contemplating—but which may be fresh enough to be worth- while." The final inescapable conclusion about the book is that it is disorganized and confusingly written and expresses a strange mystical conception of both sci- ence and painting. The book, how- ever, if not taken too seriously, is great fun to read and to ponder. A. MICHAEL NOLL Bell Telephone Laboratories The Nature of Physics: A Physicist's Views On the History and Philosophy Of His Science By Robert Bruce Lindsay 212 pp. Brown U. P., Providence, R. I., 1969. $7.50 This volume contains the mature reflec- tions of a well known physicist who, is his own way, has come to grips with the philosophical and historical aspects of physics that have been his professional concern for about 40 years. The essays here presented are mostly new but lean heavily upon articles published since 1928. R. Bruce Lindsay approaches his in- vestigation of the nature of physics by examining the method that physicists have employed to study human experi- ence. Essentially this involves the de- scription and creation of experience and the enlargement of these two aspects of experience by understanding (that is, by the construction of theory). In reflecting on these matters it is crucial to recognize, he emphasizes, that experimentation is not merely con- trolled s inse perception. It also pro- duces experiences new to man. "Ex periment on may be described as t creatior experience." What is not clarify dsfactorily(for example on pagethat the terminal or theory stage method of physics is not soneatly to be set apart historically from either scientific description or experi- mentation. Rather, as we learn more about "the nature of physics" from our historical studies (by posing philosophi- cally significant questions) we discover, that the interaction of the imaginative, intuitive, logical and operational com- ponents of theoretical and experimental physics are significantly linked together in one way or another. An examination of the key to under- standing (theorizing about) physics is given in the main part, which deals with the structure of physical theory and the special philosophical problems encountered in physics. It is fair to say that Lindsay's position relative to theory appraisal is a compro- mise between the positivistic criterion of success and the absolutely unfettered and free invention of constructs and postulates. Accordingly, he accepts the somewhat tentative and arbitrary char- acter of physical theorizing as the price the physicist has to pay for the freedom to create, imaginatively, those con- structs and postulates that, in his bones, he feels to be essentially correct. Yet he knows they must be presented un- dogmatically to his colleagues for criti- cal examination. To sum it up: "Thephysicist who invents a new theory will use all the weapons in his arsenal to jus- tify its plausibility . . . [and he will demand] the right to continue to ex- ploit a theory if he feels confident that it is an ingenious idea and if it helps him understand a certain domain of ex- perience. In the last analysis we come back to faith in the value judgments of clever and imaginative people." Who is clever, and what is imagina- tive? These are questions that belong to the psychology of the scientific be- havior of creative individuals—con- cerning which our author suggests that we know, as yet, next to nothing. I be- lieve that he is right. But we can not afford to restrict our attention, as phil- osophers or historians, to the testimony of scientists who offer an account of the circumstances surrounding their own moments of creativity. Mostly they do not exist; and when they do, the evi- dence can be risky. I see no convincing reason for not examining such questions with the help of insights provided by the psychological, physiological and so- ciological study of group behavior. Not unless scientists are some kind of special breed. Not unless science is a discipline with a method exclusive to it- self. I seriously doubt that an analysis that limits itself exclusively to the logi- cal structure of physical theories can, in fact, tell us very much about what a physicist does when he claims to be doing physics. In this work Lindsay is very hard on rigid operationalism because it seems to rely on ultimately valid and verified ex- planations of physical experience. Such a naive realism disregards the les- sons of the history of physics and ig- nores "the frightful speed with which new physical experience is being creat- ed on a grand scale." Our author is even harder on Eddington's idealistic epistemology than on the kind of Bridg- man operationalism that leads to con- firmed realism. He seeks refuge, then, as a philosopher of physics, on the side of Poincare's conventionalism. For he argues that, even if we were to be per- suaded that all human knowledge about experience is created by the mind, "it seems reasonable to suppose that as the creation of experience multiplies, the methods the mind uses to cope with it will change." That is, we shall never quite know by performing mental gym- nastics alone or by carrying out experi- ments alone which theoretical con- structs in physics will be the most satis- factory. We must simply, through the creation of experience and with flexibil- ity of thought, settle for those con- structs that we deem most convenient. Lindsay's excursions into the history of physics are noteworthy mainly as an index of the kind of history of science that preoccupies a physicist or a physi- cist turned philosopher. He holds, cor- PHYSICS TODAY / DECEMBER 197049the BIOSPHERE g the MOON and SCIENCE in the 21stCENTURY T/ze mos£ provocative and inclusive approach to the environmental crisis to date— ENVIRONMENT, POWER, AND SOCIETY By HOWARD T. ODUM, University of Florida, Gainesville Environment, Power, and Society offers a macroscopic view of man and his part in the delicate balance of forces that sustains life in the bi- osphere. To analyze man-made and natural forces, the author discusses I the common denominator that uni- fies these seemingly disparate en- ergetic processes—power. Natural energy such as solar radiation, the hydrological cycling of the oceans, or the cyclic exchange of carbon, hydrogen, oxygen and nitrogen can be easily diagramed in terms of power. But, the author shows that the social, political and economic affairs of man also have quantitative power ratings and the important issues of man's existence and survival are as fully regulated by the laws of energetics as those of the simple phys- ical and chemical world. 1970 336 pages $9.95 ADVANCES IN ENVIRONMENTAL SCIENCES AND TECHNOLOGY Volume 1 Edited by JAMES N. PITTS, JR., University of California, River- side and ROBERT L. METCALF, University of Illinois "There should be no limit to the amount of words required to review this book. Most elaborate in its scope, arrangement, and coverage by nine outstanding research workers in their respec- tive fields... Pitts and Metcalf have chosen well for their contrib- utors and subjects in this volume on environmental science." —Choice At a time when the world is becoming increasingly urbanized and the standards of public health are rapidly deteriorating, it is vitally important to study the quality of the environment and the science and technology of its conservation. This first volume in a particularly relevant new series examines some of the en- vironmental problems our technology has created. The nine previously unpublished chapters range from an extensive intro- ductory overview of environmental sciences to such diverse topics as the Federal role in pollution abatement, aeroallergens and public health, and more technically orientated considera- tions of individual pollutants. 1969 356 pages $15.95THE LUNAR ROCKS By BRIAN MASON and WILLIAM G. MELSON, both of the Smith- sonian Institution "The lunar rocks represent a unique scientific adventure and an intellectual challenge of the first magnitude. From the data to be extracted from them, we expect to reconstruct the story of the origin and evolution of the Moon, which will also provide significant information about the early history of the Earth and the entire solar system."—from the Preface The first book to describe the samples collected by the Apollo missions, this volume examines the lunar rocks in terms of mineralogy, chemistry and structure. The authors present a sum- mary of what had been learned about the Moon prior to the first manned landings and a general account of the planning and execution of the Apollo landings. Special features include 40 photos of the Apollo 12 mission and rare blow-ups of the lunar samples. 1970 192 pages $8.95 SCIENCE AND TECHNOLOGY IN THE WORLD OF THE FUTURE Edited by ARTHUR B. BRONWELL, University of Connecticut "This book deals in futures—the fu- ture of science and engineering in certain broad fields that are under- going rapid change and that are destined to profoundly influence the world of the future. It is written by eminent scientists and engi- neers, as well as by leaders in government who have been deeply thoughtful of the forces shaping tomorrow's world."— from the Preface Topics are as diverse as they are provocative and treat rapidly evolving fields that will alter our way of life and our structures of knowledge in the next quarter century. Here are just a few— • Exploring the Mysteries of the Planets and the Cosmos—Albert Arking and Robert Jastrow • Architecture as Ultra Invisible Reality—R. Buckminster Fuller • The Ocean—A Scientific and Technical Challenge-Warns fi. Stewart, Jr. • An Elementary (?) Guide to Elementary (?) Nuclear Particles- Arthur H. Rosenfeld and Judith G. Goldhaber Additional articles discuss the unlocking of the atomic nucleus, hypersonic air transports traveling at 7,000 miles per hour, an instantaneous world-wide communication network, computers and energy sources. 1970 416 pages $11.95 50PHYSICS TODAY / DECEMBER 1970and other topicsor interestfrom Wiley-Inter science MATHEMATIC THEORETICAL PH^MATHEMATICAL AND THEORETICAL PHYSICS In 2 Volumes By EGILA. HYLLERAAS Foreword by Joseph O. Hirschfelder "The first edition of Mathematical and Theoretical Physics was published in Norwegian ...and was regarded as the 'Bible' by a whole generation of Norwegian physicists ...This is truly a great book written by one of the greatest physicists of the twentieth century. Nowhere else will one find all of theo- retical physics presented in such a simple, elegant, rigorous fashion!"—from the Foreword Extensively revised and up-dated, the English adaptation of Mathematical and Theoretical Physics appears in two volumes. The bulk of the author's endeavor encompasses five major themes: Mathematical Preparations for Theoretical Physics; Classical Mechanics; Thermodynamics; Kinetic Theory of Gases, and Statistical Mechanics; Electricity and Magnetism; and Atomic Theory. Volume 1 512 pages $15.00 Volume 2 528 pages $15.00 DIRECT NUCLEAR REACTION THEORIES By NORMAN AUSTERN, University of Pittsburgh A volume in the Monographs and Texts in Physics and Astronomy series, edited by R. E. Marshak A guide to modern research in direct nuclear reaction theories, this book evaluates and classifies published research reports and explores the background and practical development of this field. Direct reactions are discussed as a special case of nuclear structure theory. Students preparing for work in theoretical or experimental nuclear physics, as well as active researchers in this field will benefit from this systematic approach to the vast amount of published research articles. rnNTENTS: Wave Packets. Direct Reactions. Some Useful For- mulas B^ic Dl Theories. Applications of the DW Method. More ' oupled Channels. Wavefunction Models, High Energy. .:, pic Applications of Stripping. Polarizations and Angu- c' iations. DW Exchange and Recoil Effects. Unified Theo- in Theories. Index. *.,„„.. ig70 448 pages $19.95THE APPLICATIONS OF HOLOGRAPHY By HENRY JOHN CAULFIELD, Sperry Rand Research Center and SUN LU, Texas Instruments Inc. A volume in the Wiley Series in Pure and Applied Optics Advisory Editor, Stanley S. Ballard "It was our intention to write a book not for our fellow holo- jraphers, but for the many technical people who could use holography if they knew what it can do and how it can be done. Therefore, we have tried to make the book a self-contained introduction to holography and its applications." —from the Preface Written for nonholographers who wish to use it in their own field of interest, and who require a thorough understanding of its uses, this is the most complete account available of the uses to which holography can be put. 1970 160 pages $9.95 ADVANCES IN PLASMA PHYSICS Volume 3 Edited by ALBERT SIMON, University of Rochester and WILLIAM THOMPSON, University of California, San Diego "A much needed new series presenting critical, evaluative, and authoritative reviews by leading researchers in this rapidly ex- panding field."—New Technical Books Covering current research in plasma physics, this volume in- cludes three articles by specialists in their respective fields. The lead article contains a number of unusual mathematical ap- proaches to well-known plasma problems as well as an analysis of more recent problems involving solution of the Vlasov equa- tion in finite geometries. The second article presents a novel perturbation expansion of the electron Boltzmann equation to obtain the usual transport coefficients in a weakly ionized gas. The third contribution is a discussion of the method for treating nonlinear phenomena in plasma by the use of quantum-mechan- ical calculations. 1969 249 pages $14.95 SYMMETRY PRINCIPLES AND ATOMIC SPECTROSCOPY By BRIAN G. WYBOURNE, University of Canterbury, Christ- church, New Zealand This volume illustrates the application of the theory of compact groups to problems in atomic spectroscopy. It is the first book written for physicists approaching the subject via the theory of Schur Functions and the calculus of plethysm. Numerous ex- amples for calculating the dimensions of group representations, branching rules, manipulation of Young tableaux and the opera- tion of plethysm prepare the reader for independent applications of the theory of compact groups to any relevant field of physics and chemistry. A 200-page tabulation of the properties of com- pact groups is also included. 1970 313 pages $17.50 FOURIER METHODS IN CRYSTALLOGRAPHY By G. N. RAMACHANDRAN, Indian Institute of Science and R. SRINIVASAN, University of Madras A volume in the Wiley Monographs in Crystallography series, edited by M. J. Buerger A treatise on theoretical methods for transforming the data of x-ray diffraction by a periodic crystal into the electron density distribution in the crystal, this monograph is particularly con- cerned with procedures for deriving the structure via Fourier methods applied directly to x-ray intensities. This volume is essentially an account of the authors' procedures for deriving a structure, together with a discussion of related topics and ex- tensions of these procedures. 1970 259 pages $15.95 uiiteyWILEY-INTERSCIENCE a division of JOHN WILEY & SONS, Inc. 605 Third Avenue, New York, N.Y. 10016 In Canada: 22 Worcester Road, Rexdale, Ontario PHYSICS TODAY / DECEMBER 1970 51Celestial Mechanics Volume 1: Dynamical Principles and Transformation Theory by Yusuke Hagihara The launching of space vehicles has given rise to a broadened interest in the problems of celestial mechanics and the availability of computers has made practical the solution of some of the more numerically unwieldy of the problems. These circumstances only further enhance the importance of the ap- pearance of Celestial Mechanics, to be pub- lished in five volumes. This treatise is by far the most extensive of its kind and it rigor- ously develops the full mathematical theory. The emphasis is on the results obtained during the past hundred years, although the classical mechanics from the time of Laplace is also reviewed. This first volume is divided into six chapters The first develops the principles of analytical dynamics. The second chapter deals with quasi-periodic motions. The third is devoted to particular solutions of the three-body and many-body problems. Euler's and Lagrange's types of particular solutions are obtained for the n-body problem in a general manner, and the nature of the motion is fully analyzed. The fourth chapter takes up Lie's theory of continuous groups of transformations, with application to the n-body problem. In Chapter 5, the differential equations of the n-body problem are reduced by using the known integrals. The last chapter concerns Burns's and Poincare's theorems. This work was prepared editorially at NASA's Goddard Space Flight Center. $25.00 The Conceptual Foundations of Contemporary Relativity Theory by John Cowperthwaite Graves The central conceptual idea of the contem- porary theory of general relativity — or geo- metrodynamics — is the identification of matter with the structure of space-time. No entities foreign to space-time, like masses, charges, or independent fields are needed, and physics thus becomes identical with the geometry of space-time. This idea implies a philosophical description of the universe that is monistic and organic, characterized by an all-encompassing interdependence of events. Moreover it is an idea with deep roots in the history of philosophy. For these reasons, the author of this important study strives to clarify these philosophical and historical issues before proceeding to the details of the physical theory of geometrodynamics. Graves develops a general philosophical framework of "scientific realism," asserting that scientific theories have ontological im- port in determining the real structure of the world, and in particular that models play a central role as ontological hypotheses. He introduces two factors, the empiricist error and the logicist error that he feels have con- tributed to past misinterpretations of the nature of reality that were based on general relativity theory. $15.00 The MIT Press Massachusetts Institute of Technology Cambridge, Massachusetts 02142rectly in my opinion, that if the history of physics is to be of value it must be related inextricably to its philosophy. The reverse, that is, the relevance of the history of physics for the study of philo- sophical questions, is more controver- sial—at least where the latter becomes mostly an exercise in logical analysis. As an historian of science I feel strongly that Lindsay's historical discus- sions would have profited measurably in relation to his philosophical analysis if he had developed his brief but per- ceptive comments on the history of acoustics (electroacoustics, piezoelec- tricity, magnetostriction, vacuum-tube oscillators and amplifiers and transduc- ers and audition) rather than devote so much attention to Gilbert's magnetism, Gassendi's atomism and Galileo's analy- sis of free fall. He writes: "As the history of science in the last half century shows, there is no more reason to suppose that man will ever run out of questions about acoustics than there is reason to believe that he will run out of questions about the nucleus and its theoretical parti- cles." What a magnificent monograph in the philosophy and history of physics could be built around that theme. Let us hope that more physicists will under- take such analyses. Philosophers of science and historians of science would be (or should be) grateful for all such efforts. Somehow, science, its philoso- phy, and its history must hang together, and not separately. ERWIM HIEBERT Professor of History of Science Harvard University Analytic Functions and Distributions in Physics and Engineering By Bernard W. Roos 521pp. Wiley, New York, 1969. $19.95 This 500-page book is designed for theoretical physicists (and engineers) in their role as applied mathematicians of a traditional cast. The emphasis is on obtaining explicit analytic answers to tough problems—mainly in the do- main of partial-differential and integro- differential equations. In particular two chapters-a third of the book—are de- voted to solving the Boltzmann equa- tion as it arises in neutron-transport theory and plasma physics. The first part expounds the mathe- matical background needed to tackle such problems with today's sophisti- cated techniques. Five chapters: "Analytic Functions;" "Fourier Trans- forms, Casuality and Dispersion Rela- tions;" "The Wiener-Hopf Technique;""Sectionally Analytic Functions" and "Distributions" take the reader in gentle stages from Cauchy's theorem in com- plex-variable theory (essentially the only "prerequisite"). He arrives at a higher level, where he can derive, following H. A. Lauwerier's 1956 prize winning paper, an explicit formula for the electrostatic potential field caused by a point charge on the axis of a semi-infinite, hollow, conducting cyl- inder at fixed potential. The reader will be shown the Riemann-Lebesgue lemma for Fourier transforms and learn that the Dirac delta function and its derivations are "tempered distribu- tions of bounded support." He will become familiar with the Plemelj boundary-value formulas. In the en- suing applications the reader will ex- amine the famous phenomena of Landau damping in a collisionless plasma and will study the important normal-mode techniques introduced by N. G. van Kampen in 1955 and de- veloped by K. M. Case. Among the things the reader will not learn are, first, at the elementary level, the uses of the complex plane in discussing discrete linear systems, such as electric circuits and servomechanisms. Thus Nyquist's criterion for the sta- bilty of an amplified or servosystem appears in the book only on page 444 and following, where it is said to be "closely related to, but not identical with, the criteria of Penrose" (for the stability of a Vlasov plasma). At the other end of the scale we find that the author, Bernard Roos, remains quite firmly anchored to functions of one com- plex variable. The student who has heard that analytic functions and dis- tributions of many complex variables are important in modern field theory will look in vain for the definition of a "tube," for a statement of Schwartz's "nuclear" theorem, for multilinear func- tionals and for an explanation of the famous "Edge of the Wedge" theorem. (He will, however, find an introduction to these topics in the book PCT, Spin and Statistics and All That by R. F. Streater and A. S. Wightman, Benja- min, 1964.) Between these extremes there is, nevertheless, a wealth of mathematical material developed in the last two or three decades, which is of value to the professional physicist and which de- serves an introduction that will make it more accessible to him. As an applied mathematician with many years of varied experience at John Jay Hopkins Laboratory at Gulf General Atomic in San Diego, Ross is certainly competent to undertake this task. How well does he succeed? The answer is, perhaps, partly a matter of taste. The author's intention, to quote the blurb, is to "avoid abstract mathematics and details of proofs but concentrate on meaning 52 PHYSICS TODAY / DECEMBER 1970

1.5130282.pdf

AIP Conference Proceedings 2162 , 020072 (2019); https://doi.org/10.1063/1.5130282 2162 , 020072 © 2019 Author(s).Comparative study of spin waves using exchange and demagnetization field in bicomponent and air-gap stripes magnonic waveguides for transmission applications Cite as: AIP Conference Proceedings 2162 , 020072 (2019); https://doi.org/10.1063/1.5130282 Published Online: 29 October 2019 T. Vivek , S. Mounika , H. Bhoomeeswaran , and P. Sabareesan ARTICLES YOU MAY BE INTERESTED IN Tunability of output frequency using Spin Hall Angle in Spin Hall Spin Torque Nano Oscillator AIP Conference Proceedings 2162 , 020086 (2019); https://doi.org/10.1063/1.5130296 Magnetization switching in heterogeneous spin valve device with the aid of bi-quadratic coupling as well as orange peel coupling AIP Conference Proceedings 2162 , 020169 (2019); https://doi.org/10.1063/1.5130379 Influence of nanocellulose addition on the film properties of the bionanocomposite edible films prepared from maize, rice, wheat, and potato starches AIP Conference Proceedings 2162 , 020073 (2019); https://doi.org/10.1063/1.5130283Comparative Study of Spin Waves using Exchange and Demagnetization Field in Bicomponent and Air-gap Stripes Magnonic Waveguides for Transmission Applications T. Vivek1, S. Mounika2, H. Bhoomeeswaran1and P. Sabareesan1,a) 1Centre for Nonlinear Science and Engineering (CeNSE), School of Electrical and Electronics Engineering, SASTRA Deemed to be University, Thanjavur - 613 401, India. 2School of Electrical and Electronics Engineering, SASTRA Deemed to be University,Thanjavur - 613 401, India. a)Corresponding author: sendtosabari@gmail.com Abstract. Spin wave propagation in air-gap stripes and bicomponent magnonic waveguides are studied through the micromagnetic simulation. The magnonic waveguides composed of MandNstripes based on di erent values of saturation magnetization. In air- gap stripes magnonic waveguides, the air-gaps in N(i.e., Ms=0) are sandwiched between the Mof magnetic materials where the magnetostatic coupling take place due to the inhomogeneous field. In bicomponent magnonic waveguides(BMW), the exchange interaction is dominant due to the presence of two magnetic materials ( MandNstripes). The exchange interaction plays a major role in BMW existing at the interfaces of di erent magnetic materials which enhance the SWs propagation. Air - gap stripes based magnonic waveguides exhibit the transmission bands in range of 3-5 GHz and bicomponent exhibits 14-39 GHz, respectively. This work paves the way to develop the narrow and wide transmission bands based on magnonic waveguide which is suitable for signal transmission and processing devices. INTRODUCTION In recent years, to understand the spin waves (SWs) in magnonic waveguides (MWs) have been great attention in the microwave telecommunication industry, because SWs have a shorter wavelength than that of electromagnetic waves with the same frequency in microwave regime which paves the way to make the nano based devices. Magnonics is a budding field where the SW propagation in ferromagnetic materials [1] are studied. SWs represent the collective excitation of individual spins which able to carry information signal [2]. The information carried SWs are controlled using di erent geometry, various shapes and sizes of antidots /dots and external magnetic field [3]. MWs based on controllable SWs have alluring potential in microwave applications such as filters, logic gates and transducers [4]. Magnonic antidot waveguides (MAWs) have been considered the ideal candidates for making filtering devices in mi- crowave communication devices [5]. However, some researches have shown a keen interest in bicomponent magnonic waveguides (BMW) for making the signal transmission and processing devices [6]. Z K Wang et. al [7] fabricated a synthetic nanostructure magnonic crystals (MCs) composed of two di erent magnetic materials such as permalloy (Py) and cobalt (Co) are arranged in a periodic array. This work confessed the band gap tunability is achieved by varying the external bias field which could find many applications in the information carried SWs devices based on that magnonic crystal. Z K Wang et. al [8] calculated the first band gap and center frequency in periodic arrangement with the alternating stripes of Py and Co with respective widths through experimentally and theoretically. The width of respective stripes variation in BMW drastically modified the band structures which would be suitable to make the ecient transmission and processing of microwave signals on the nanoscale. In [9], SW propagation in BMW has been investigated where the MW structure is composed of square lattice array of circular iron (Fe) dots in yttrium iron garnet (YIG) matrix and the formation of band gaps is interpreted by varying the filling fraction, the lattice constant and applied a magnetic field. In [10], the SW propagation in the periodic variation of saturation magnetization MSon This is an example of first authornote. yThis is an example of second authornote. Proceedings of the International Conference on Advanced Materials AIP Conf. Proc. 2162, 020072-1–020072-5; https://doi.org/10.1063/1.5130282 Published by AIP Publishing. 978-0-7354-1907-0/$30.00020072-1a Py waveguide is studied through numerical simulation. The band structures arising in Py waveguide due to super- position between the lower and higher width modes. Further the depth of the band gap can be controlled based on the level of MSon waveguide which could be applicable for the spin-wave mode selective filter. As aforementioned studies have provoked us to proposed the MWs based on di erent MSwhich is composed of alternating MandNstripes. The narrow and wide transmission bands can be obtained due to change in demagnetiza- tion and the exchange field are the reason for the disturb and enhance SW propagation in MWs for di erent MS. The present work can be done through numerical simulation by Object-Oriented Micro-Magnetic Framework (OOMMF). The paper is organized as follows: A schematic representation of air- gap and bicomponent magnonic waveguides are given in section 2. The results and discussion of the dispersion curves, the static profile of demagnetization and exchange field of samples are discussed in section 3. Finally, the summary of work is discussed in section 4. SAMPLE DESCRIPTION AND MICROMAGNETIC STUDY FIGURE 1. (a) Schematic view of a magnonic waveguide with airstripes and (b) bicomponent magnonic waveguide with length =2400 nm, width =24 nm and thickness =3 nm. The M and N indicate the widths of their respective stripes which dominate the role of periodicity P=M+N The pictorial representation of 1-D magnonic waveguides (MWs) is composed of MandNstripes along with the length l=2400 nm, width w=24 nm and thickness t=3 nm as displayed in Figure 1. The periodicity P= M+Nare given by the sum of MandNthat mentioned the width of the respective stripes which is indicated by red vertical dashed lines (refer Fig.(1)). Based on those above studied in the previous section, the MWs are categorized into two types based on di erent MS: i) Air-gaps magnonic waveguide (AMW) and (ii) Bicomponent magnonic waveguide (BMW). In AMW the MSvalue is to be considered as zero in Nstripes medley with magnetic material stripe Mand in BMW two di erent MSvalues are used in both MandNstripes. For brevity, the samples are referred to as [12 Fe4Airgap ] and [12 Fe4Ni] respectively. Understanding of SW propagation in AMW and BMW through micromagnetic simulations are lead to the development of the next generation microwave signal devices. OOMMF is a micromagnetic simulation tool which gives a clear view of spin dynamics in MWs and it’s solve the Landau- Lifshiftz-Gilbert (LLG) equation (1). The LLG address the spin motion of individual magnetic moments which is given: @~M @t= (~M~He f f)+ Ms0BBBB@~M@~M @t1CCCCA (1) Here denotes the gyromagnetic ratio, Msis the saturation magnetization, is the Gilbert damping factor, and ~He f fis the total e ective field given by the sum of the external magnetic field ~Hext, the exchange field ~Hexc, the demagnetization field ~Hdemag and the magneto-crystalline anisotropy ~Hanis. The material parameters are taken during the simulation as follows: saturation magnetization Ms=17105A=m (Fe) and 4.8105A=m(Ni); exchange constant Aex=211012J=m(Fe) and 8.61012J=m(Ni); magnetocrystalline anisotropy constant K=48103J=m3(Fe) and 5.7103J=m3(Ni) respectively. OOMMF divides the samples into rectangular cuboids with a cell size of 3 33nm3and performed into two ways: (i) Static and (ii) Dynamics. In static, the steady magnetization state of the sample is achieved by subjecting the MAWs to an external bias 0H0=1.01 kOe along x direction under a Gilbert damping factor =0.5 . Once all the 020072-2magnetization attains static equilibrium state, the SWs are excited (dynamics) by an out-of-plane (z direction) sinc pulse (to excite many more SW modes) which is expressed in the mathematical form as given below. Hz=H0 sin2 fc(tt0) 2fc(tt0)!  sink c(xx0) kc(xx0)! (2) Where, x0=l 2=1.2m, fc=250 GHz and 0H0=1.01 kOe. The time evolution of magnetization for each cell is recorded at a regular time interval of 1 ps. The post processing method which helps us to obtain the dispersion curves upto 2ndBrillouian Zone (BZ) using fast Fourier transform (FFT) in MATLAB where the data obtained from OOMMF. Further, the spatial static magnetization distribution of demagnetization and exchange field is obtained from OOMMF. The results obtained from the simulation will be discussed in the forthcoming section. RESULTS AND DISCUSSION i) Air-gaps magnonic waveguide (AMW) FIGURE 2. (a) The dispersion characteristic reveals the forbidden and allowed band regions for [12 Fe4Airgap] and their corre- sponding (b) Static profiles of the demagnetization field and (c) exchange field is shown The dispersion characteristics of SWs are calculated upto 2ndBZ where diminished by vertical dashed lines, reveal the allowed and forbidden bands as shown in Fig .2(a). In [12 Fe4Airgap], there is an intrinsic forbidden band gap are observed below 50 GHz due to width confinement of nano-strips. The two forbidden band gaps (I: 33.69 and II: 83.22 GHz) are observed as shown in Fig .2(a). The opening of (I) at ( K=0) due to the SW Bragg scattering, which (I: 33.69 GHz) splits two narrow transmission bands (i: 54.2 - 57.62 (3.42 GHz) and ii: 91.06 - 93.75 (2.69 GHz)). The additional narrow transmission bands (iii : 176.5 - 181.4 (4.9 GHz)) is occurred due to the opening of (II: 83.22 GHz) band gap at ( K==a) results from the interaction between the initial and higher excited SW modes. The opening of (I: 33.69 and II: 83.22 GHz) in AMW is due to inhomogeneous static demagnetization field eect of the presence of air-gaps in Nstripes. In the absence of SW dynamics, the demagnetization field exists due to internal 020072-3energy (magnetostatic energy) which is given by ~Hdmag(~r)=5(~r) (~r)=Z !5: !m(~r0) 4j~r~r0j~dr0 (3) where~r0denotes the position vector. In the static case, the demagnetization field exists due to the geometry of sample and its field distribution is in-and-around the boundary of stripes which is shown in a demagnetization field plot in Fig .2(b) and Fig .3(b). The magnetostatic coupling arises because of inhomogeneous demagnetization field created by Nstripes is ob- served in Mstripes as a stair-case like structure which helps to enhance the SW propagation throughout the entire AMW. But in exchange plot, no stair-case like structure is observed in Mstripes (See Fig .2(c)). Hence, the demag- netization field predominated over the exchange field in AMW where the magnetostatic coupling takes places. The magnetostatic coupling SW based MWs are suitable for designing the narrow transmission band in microwave engi- neering. To achieve the wide transmission bands we are incorporated magnetic material in Nstripe instead of the air gap and its results will be discussed in the next subsection. ii) Bicomponent magnonic waveguide (BMWS) FIGURE 3. (a) The dispersion characteristic reveals the forbidden and allowed band regions for [12 Fe4Ni] and their corresponding (b) Static profiles of the demagnetization field and (c) exchange field is shown The exchange- dominated SWs are investigated by incorporating the dierent magnetic properties ( Ms) instead of air-gaps ( Nstripes) in MWs. The dispersion curves show the wide allowed bands for [12 Fe4Ni] in Fig .3(a). The intrinsic forbidden bands are observed below 60 GHz due to the width confinement. In [12 Fe4Ni], the three band gap widths (I: 18.07, II: 16.08 and III: 20.07 GHz) are observed (See Fig. 3(a)). The direct band gap (I) in [12 Fe4Ni] are appeared at ( K=n=a) due to Bragg scatters of SWs are formed the wide transmission bands (i : 61.28 - 75.2 (13.92 GHz)), (ii : 93.51 - 130.4 (36.89 GHz)) as seen in Fig. 3(a). The opening of (II: 16.08 GHz) at K=0 due to the periodic modulation of dierent Msproperties of materials along the SW direction. The II creates the (iii: 147.5 - 162.4 (14.9 GHz)) transmission bands (See Fig. 3(a)). The (III - 20.07 GHz) band gap opening at high frequencies range 020072-4results from the interaction between the initial propagating SW modes and reflected SW modes. Hence additional transmission bands (iv: 185.9 - 199.6 (13.7 GHz)) appeared in Fig .3(a). The band gap arising in BMW because of existing the exchange coupling at the interfaces between the two dier- entMsof constituent materials. This exchange coupling enhances the dynamical dipole coupling between neighboring stripes leads to SW propagation with low energy loss throughout the entire waveguides. The visualization of the ex- change field is presented in Fig .3(c) where stair-case like structure is observed in Mstripes which extended upto another Mstripes via Nstripes whereas no stair-case structure observed in demagnetization field (See Fig .3(b)). From Fig .3(b) and (c), it is observed that the exchange field is predominate than the demagnetization field in BMW which enhance the SW propagation with low energy loss in the entire waveguide. Hence, the exchange dominated SWs based MWs are promising candidates to design the wide band, transmitting of signal in nano scale devices. CONCLUSION In this work, we confessed that the dispersion relation is scrutinized that in air gap magnonic waveguide is highly suitable for narrow transmission bands having range 3-5 GHz. Owing to the inhomogeneous field are created by the presence of air-gaps ( Nstripes) in AMWs where the magnetostatic coupling is predominant. In BMW exhibit the transmission bands in the range of 14-39 GHz, the exchange coupling dominates as a result of incorporating another magnetic material in Nstripes which enhance the SW propagation throughout the waveguide with low energy loss. Hence, the constructed MWs is extended for transmitting the signal in the microwave regime which works eciently in narrow as well as a wide range. ACKNOWLEDGMENTS The work was supported for Vivek T by SASTRA Deemed University, Thanjavur, India, for providing the Teaching Assistance Fellowship and for Sabareesan P by DST, New Delhi, India, for providing financial assistantship through the Fast Track Fellowship under Grant SR/FTP/PS-061/2013. REFERENCES [1] P.A. Kolodin, B. Hillebrands, J. Magn. Magn. Mater, 161, 199, (1996). [2] A.V . Chumak, V .I. Vasyuchka, A.A. Serga, and B. Hillebrands, Nat. Phys. 11, 453, (2015). [3] C. Bayera, M.P. Kostylev, B. Hillebrands, Appl. Phys. Lett, 88, 112504, (2006). [4] K.S. Lee, S.K. Kim, J. Appl. Phys, 104, 053909, (2008). [5] D. Kumar, P. Sabareesan, W. Wang, H.Fangohr, A.Barman, J. Appl. Phys, 114, 023910, (2013). [6] G. Gubbiotti, S. Tacchi, G. Carlotti, N. Singh, S. Goolaup, A. O. Adeyeye, and M. Kostylev, Appl. Phys. Lett, 90, 092503 (2007). [7] Z. K. Wang, V . L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Jain and A. O. Adeyeye, Appl. Phys. Lett, 94, 083112, (2009). [8] Zhi Kui Wang, Vanessa Li Zhang, Hock Siah Lim, Ser Choon Ng, Meng Hau Kuok, Shikha Jain and Adekunle Olusola Adeyeye, ACS Nano, 4, 643-648, (2010). [9] F. S. Ma, H. S. Lim, Z. K. Wang, S. N. Piramanayagam, S. C. Ng and M. H. Kuok, Appl. Phys. Lett, 98, 153107 (2011). [10] Florin Ciubotaru, Andrii V . Chumak, Bjorn Obry, Alexander A. Serga and Burkard Hillebrands, Phys. Rev. B,88, 134406, (2013). 020072-5

1.4919882.pdf

Frequency tunable surface magneto elastic waves J. Janušonis, C. L. Chang, P. H. M. van Loosdrecht, and R. I. Tobey Citation: Applied Physics Letters 106, 181601 (2015); doi: 10.1063/1.4919882 View online: http://dx.doi.org/10.1063/1.4919882 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/106/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simulations of acoustic waves bandgaps in a surface of silicon with a periodic hole structure in a thin nickel film AIP Advances 4, 077138 (2014); 10.1063/1.4892076 Multilayer magnetostrictive structure based surface acoustic wave devices Appl. Phys. Lett. 104, 114101 (2014); 10.1063/1.4868530 Transient grating measurement of surface acoustic waves in thin metal films with extreme ultraviolet radiation Appl. Phys. Lett. 89, 091108 (2006); 10.1063/1.2336591 Acoustic microscope based on magneto-elastic wave phase conjugator Appl. Phys. Lett. 76, 3133 (2000); 10.1063/1.126547 Acoustic magnetic resonance in absorption and dispersion of surface elastic waves in multilayers Low Temp. Phys. 25, 148 (1999); 10.1063/1.593718 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: 134.129.120.3 On: Mon, 25 May 2015 22:24:42Frequency tunable surface magneto elastic waves J.Janu /C20sonis,1,a)C. L. Chang,1P. H . M . van Loosdrecht,2and R. I. Tobey1,b) 1Zernike Institute for Advanced Materials, University of Groningen, Groningen 9747AG, The Netherlands 2Department of Physics, University of Cologne, Cologne 50937, Germany (Received 11 March 2015; accepted 26 April 2015; published online 5 May 2015) We use the transient grating technique to generate narrow-band, widely tunable, in-plane surface mag- netoelastic waves in a nickel film. We monitor bot h the structural deformation of the acoustic wave and the accompanying magnetic precession and witness their intimate coupling in the time domain.Strikingly, when an in plane magnetic field is applied p arallel to the acoustic pr opagation direction, we witness its resonant coupling to the ferromagnetic resonance. VC2015 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4919882 ] In recent years, the influence of elastic waves on material magnetization has seen renewed interest. While the ground work for the subject matter is well established,1,2new technol- ogies and techniques have recently demonstrated how elastic-ity can be implemented as a spatially and temporallycontrollable mechanism by which to efficiently alter magnetic properties. These studies fall into two broad categories, those utilizing transducer based surface acoustic wave generation 3–9 and those implementing ultrafast optical excitation of longitu- dinal acoustic wave packets.10–14In the former scenario, sur- face acoustic waves (SAW) are generated by acoustic transducers, in most cases interdigitated metal electrodes,4,6,7,9 fabricated onto piezoelectric substrates. The coupling between elastic and magnetic degrees of freedom is understood to be a resonant process but is measured indirectly, for example, byacoustic power attenuation, or by the inverse spin Hall effectin a contacted metal. This method is constrained by the strict requirements of piezoelectric substrate material and achieves frequencies up to a few GHz in the higher harmonics of thetransducer fundamental frequency. The optical methods rely on generation of acoustic pulses via a thermoelastic process, 15–17leading to broadband longitudinally polarized acoustic wavepackets. These elasticpulses propagate into the depth of the material and appear on the “backside” where they interact with the magnetization of a thin magnetic material. The acoustic pulse length generatedin this geometry is largely dictated by the laser absorptiondepth in the material and thus, several picoseconds in dura- tion, leading to an impulsive interaction between acoustic and magnetic degrees of freedom. While the opticalapproach appears cumbersome and less amenable to techno-logical implementation, it is notable in that strains generated can exceed those of transducer methods by several orders of magnitude and approach levels of 1%, thus providing a man-ner in which to study dynamical behaviour in the limit of large strains (and their gradients) as well as approaching nonlinear regimes for the acoustic pulse propagation. Here, we bridge the divide between these approaches by optically generating narrowband, frequency tunable, surface acoustic waves, which resonantly excite a magnetizationprecession in a ferromagnetic film. We measure both the elastic distortion and its magnetization response independ-ently, thus allowing us to “visualize” the resonant couplingbetween the strain wave and magnetization. Strikingly, whenan external magnetic field is applied, we witness the excita- tion of the propagating ferromagnetic resonance, the fre- quency of which is tunable with grating periodicity andapplied field. Our method relaxes the substrate requirementsof the transducer based schemes and provides broader fre-quency tunability, while maintaining the planar geometrynot yet demonstrated using optical techniques. The resultantexcitations propagate at velocities approaching 6000 m/s anddistances of several tens of microns. To generate the elastic response, we implement the tran- sient grating technique, 16,18,19in which two spatially over- lapping short pulses of light generate a sinusoidal intensitygrating on the sample, and launch counterpropagatingin-plane Rayleigh surface acoustic waves. The response issubsequently probed in two ways: (1) through normal inci- dence Faraday rotation, which is sensitive to the magnetic component of the magnetoelastic excitation, and (2) throughphase matched transient diffraction which is sensitive to theelastic deformation. A schematic of our experimental geometry is shown in Figure 1(a). A 40 nm thick polycrystalline nickel film depos- ited on an MgO (001) substrate is placed at the intersection oftwo, linearly polarized, k pump¼400 nm pulsed laser beams (blue). The crossing angle between the beams, h,c a nb e adjusted, resulting in a sinusoidal intensity grating of variablewavelength, K. For the measurements described, the grating periodicities are varied between 1.4 lma n d8 lm while the excitation beam diameter remains constant at 350 lm, thus ensuring several tens to 100 s of grating periods excite the sample. In the Faraday configuration, we monitor the evolu-tion of the sample response in transmission with a second line-arly polarized laser pulse ( k probe¼800 nm, red in Figure 1(a)), whose complex polarization, HðtÞ¼hðtÞþi/C15ðtÞ, is measured to provide information about the magnetic component of themagnetoelastic wave. A magnetic field can be applied parallelto the k-vector, in this case, both oriented along the MgO(110) direction. Figure 1(b) shows a representative time-resolved Faraday rotation curve for excitations with spatial periodicity a)Electronic mail: j.janusonis@rug.nl b)Electronic mail: r.i.tobey@rug.nl 0003-6951/2015/106(18)/181601/4/$30.00 VC2015 AIP Publishing LLC 106, 181601-1APPLIED PHYSICS LETTERS 106, 181601 (2015) 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: 134.129.120.3 On: Mon, 25 May 2015 22:24:42ofK¼1:4lm. The temporal response is dominated by a sin- gle oscillatory component which persists for the entire mea- surement window. In Fig. 1(c), this Faraday response is shown at three representative cases of applied in-plane mag-netic field, where a suitable background has been subtracted. While all traces exhibit the same oscillatory frequency, it is their attenuation that is clearly altered by the applied field,most notably the intermediate case where the signal is nearly completely supressed. Additionally, the frequency and am- plitude of oscillation are also modified by choice of gratingperiodicity. In Figure 1(d), we show the effect of changing the grating period from K¼1:4lm, to 2.4 lm, to infinity, affected by pumping with either one of the excitation beams. As noted previously, the transient grating technique is most often used to excite surface acoustic waves in thin films, and therefore, we would like to reconcile the measuredfrequencies in the Faraday measurement to those measured in the transient grating geometry (details can be found in reviews on the topic 16). A representative curve is shown inFigure 2(a). The frequency response of both Faraday and transient diffraction data are analyzed in the same manner.We measure the temporal evolution of the respective signalsas a function of excitation grating wavelength and plot theoscillation frequency versus k¼2p=K, thereby arriving at a dispersion relation. The extracted frequency is the result of a single oscillator fit to the temporal evolution which includes both the possibility for ingrowth and decay time constants(measured at zero applied field), examples of which areshown in 1c for the Faraday response and 2a for the diffra-tion response. The respective dispersion relations for thetwo measurements are shown in Figure 2(b), which show near-perfect correspondence. This allows us to make a defin-itive assignment of the acoustic response as a RayleighSurface Acoustic Wave (Rayleigh SAW), whose velocity in the long wavelength limit is dictated by the Rayleigh veloc- ity of the substrate material 20and calculated to be21vRayleigh ¼4919 m =s along the (110) direction of MgO, similar to the measured value vRayleigh ¼5480620 m =s. We now revisit the effect of applying an in-plane mag- netic field to the Faraday signal and plot the Fourier trans-form of the temporal response as a function of applied field,as shown in Figure 3(b). We can identify in this plot the responses displayed in Figure 1(c), namely, the near com- plete suppression of Faraday amplitude is evident at /C25160 G, and the subsequent reemergence of amplitude at yet higherfield strengths. The acoustic frequency for this particular gra- ting periodicity, K¼1:4lm, is/C254GHz and indicated by the horizontal rectangle. Also included in the figure is the fre-quency response of optically excited uniform spin preces-sion, alternately called the Kittel mode or the ferromagneticresonance (FMR). The overlay is a result of independentmeasurements on our films (data not shown) following thework of van Kampen. 22It is evident that the reemergence of oscillation amplitude occurs precisely at the intersection ofthe acoustic wave frequency with that of the FMR, thus sug- gesting a coupling between the two phenomena, or more FIG. 1. (a) Two spatially overlapping pulses generate a sinusoidal excitation pattern, known to generate narrowband surface acoustic waves. The mag- netic response of the material is probed by normal incidence Faraday rota- tion. An in-plane magnetic field can be applied along the acoustic propagation direction. (b) The Faraday signal is dominated by a single fre- quency that persists for the entire sampling window. (c) The oscillation amplitude and damping is modified by an in-plane magnetic field, showingcomplete suppression at 160 G and recovery at 410 G. A single oscillator fit, which in growth and decay, is overlayed onto the 410 G data. (d) The oscil- lation frequency scales linearly with the excitation k-vector, and is absent when the sample is excited by a single pump pulse, in effect mimicing K!1 . All curves in (b) and (c) have had background removal and are ver- tically displaced. FIG. 2. (a) Representative transient diffraction signal for K¼2:4lm, show- ing large amplitude, single frequency oscillation with overlayed single fre- quency fit. (b) The extracted frequencies for both Faraday and transient diffraction are plotted versus wavevector and show near perfect correspon-dence. The velocity extracted from a linear fit is 5480 620 m/s, within 10% of the expected Rayleigh velocity.181601-2 Janu /C20sonis et al. Appl. Phys. Lett. 106, 181601 (2015) 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: 134.129.120.3 On: Mon, 25 May 2015 22:24:42precisely, the ability of the acoustic excitation to resonantly excite a precession of magnetization. Resonant coupling is further supported by plotting the Fourier amplitude and phase from the Faraday measurement,taken by vertically averaging the data enclosed in the rectan- gular region around the acoustic frequency as shown in Figure 3(a). Notably, the lineshape on resonance closely resembles a Lorentzian function (Lorentzian fit included forthe amplitude), while the extracted phase of oscillation expe- riences a pphase shift as the resonance is traversed, the response expected from a harmonic oscillator driven at afixed frequency. Therefore, we can understand that the exci-tation is a combined elastic deformation and magnetic pre- cession, where each component is measured individually by the either Faraday rotation or transient diffraction. When themagnetic field is appropriately tuned, the elastic wave reso-nantly couples to the ferromagnetic resonance, and results in an in-plane propagating magneto elastic wave. We come to a phenomenological understanding of these effects by considering the manner in which elastic and mag- netic degrees of freedom are coupled via the magnetoelastic contribution to the free energy. For a cubic polycrystallinematerial, this takes the form 23 Fme¼b½/C15xxm2 xþ/C15yym2 yþ/C15zzm2 z/C138 þ2b½/C15xymxmyþ/C15xzmxmzþ/C15yzmymz/C138; where bis a material dependent magnetoelastic coupling constant, miis the magnetic component along the idirection, and /C15ijare the components of the (time dependent) strain ten- sor. In the framework of the Landau-Lifshitz-Gilbert (LLG) equation, @M!=@t/M!/C2Hef f/C131/C131!, the gradient of the free energy with respect to magnetization acts as an effectivefield H ef f/C131/C131!¼ð /C0 @F=@mx;/C0@F=@my;/C0@F=@mzÞ, which ini- tiates precessional motion of the magnetization.6,12,23 Within this framework, we can explain the range of observed responses, in particular, the elastically drivenresonant coupling to the ferromagnetic precession at large fields as well as the low field, low frequency response shown in Figure 3(b). First, in the regime of large field par- a l l e lt ok( H!kk!kx!) and strong enough to saturate the magnetization along the in-plain direction, the term respon-sible for inducing the magnetization precession is that ofthe out-of-plane shear component /C15 xz, a strain that is inher- ent in the Rayleigh SAW, and one that has previously beenidentified for its efficient modulation of a magneticresponse. 24The propagating acoustic wave, via this shear strain, acts as a driving for ce by locally modulating the effective field direction at the frequency vg=K, dependent on the group velocity of the elastic wave and grating wave- length. The system is then br ought into ferromagnetic reso- nance when the elastically-mediated driving frequencymatches that of the field tuned FMR, resulting in theincreased oscillation amplitude of the Faraday response. The low field response can be understood by incorporat- ing into our description the possibility of a generalized mag-netic domain, whose orientation is determined solely by acompetition between shape anisotropy and magnetocrystal-line anistropy. XRD measurements indicate that a portion of the nickel film is (111) oriented, which is the easy magnet- ization axis for fcc nickel films and suggests an out of planecomponent to the magnetization orientation. In performing theabove calculation, one again arrives at a net torque providedby the shear strain on all domains. Importantly, this contribu-tion vanishes at /C2575 G, comparable to the coercive field required to orient all domains along the field direction, but notyet in resonance with the force applied by the passing acousticwave. The low frequency response apparent below 100 G weattribute to domain dynamics based solely on the fact that itdisappears at fields comparable to the coercive field. Finally, we note that the magnetic sensitivity in the Faraday measurements arises from a spatially modulatedmagnetization due to a sinusoidal temperature profile. Thestrain fields of two counterpropagating elastic waves act on FIG. 3. The main panel (b) shows the frequency response of the Faraday signal when an in plane field is applied. The acoustic frequency, denoted by the horizontal box, exhibits a nontrivialresponse. The reemergence of oscillation amplitude at /C25410Gcoincides with the frequency of the ferromagnetic reso- nance (red line). In (a), we show the oscillation amplitude and phase which exhibit a classical driven resonator response, including Lorentzian lineshapeandpphase shift across the resonance. The response at low fields is likely the result of domain reorientation.181601-3 Janu /C20sonis et al. Appl. Phys. Lett. 106, 181601 (2015) 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: 134.129.120.3 On: Mon, 25 May 2015 22:24:42local magnetic moments causing precession. Since our tech- nique is sensitive to the average value of magnetization, net Faraday rotation would be zero if all local magnetic moments were of equal amplitude. Our sensitivity comes,therefore, from propagation effects, whereby the one orienta- tion of the strain acts on the cold sample (the excitation troughs) while the opposite orientation acts on a hot, reducedmagnetization region (the crests). The importance of this result rests upon the fact that optically generated strains can exceed those generated by transducers by several orders of magnitude. Here, we expect to routinely generate strains in excess of 0.1% 25,26and in certain cases an order of magnitude larger.27Our demonstra- tion of pure in-plane magnetoelastic waves rivals pure spin wave propagation28as a means of transporting spin informa- tion over macroscopic lengthscales and could be imple- mented in a range of phononic devices29where a coupling between elastic and magnetic degrees of freedom can beused to store or transfer information. In summary, we have demonstrated a simple experimen- tal geometry with which to generate broadly tunable surfacemagnetoelastic waves. We showed a one-to-one correspon- dence between the acoustic and magnetic response, and dem- onstrated that under appropriate conditions of applied field,the strain excitation resonantly couples to a propagating, fre- quency tunable, ferromagnetic resonance. 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Reis, M. DeCamp, P. Bucksbaum, R. Clarke, E. Dufresne, M. Hertlein, R. Merlin, R. Falcone, H. Kapteyn, M. Murnane, J. Larsson, T. Missalla, and J. Wark, Phys. Rev. Lett. 86, 3072 (2001). 27V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, Nat. Commun. 4, 1468 2013. 28H. Yu, R. Huber, T. Schwarze, F. Brandl, T. Rapp, P. Berberich, G. Duerr, and D. Grundler, Appl. Phys. Lett. 100, 262412 (2012). 29P. H. Otsuka, K. Nanri, O. Matsuda, M. Tomoda, D. M. Profunser, I. A. Veres, S. Danworaphong, A. Khelif, S. Benchabane, V. Laude, and O. B.Wright, Sci. Rep. 3, 3351 (2013).181601-4 Janu /C20sonis et al. Appl. Phys. Lett. 106, 181601 (2015) 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: 134.129.120.3 On: Mon, 25 May 2015 22:24:42

1.1428804.pdf

High field ferromagnetic resonance measurements of the anisotropy field of longitudinal recording thin-film media C. J. Oates, F. Y. Ogrin, S. L. Lee, P. C. Riedi, G. M. Smith et al. Citation: J. Appl. Phys. 91, 1417 (2002); doi: 10.1063/1.1428804 View online: http://dx.doi.org/10.1063/1.1428804 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v91/i3 Published by the American Institute of Physics. Related Articles Ferromagnetic resonance and magnetooptic study of submicron epitaxial Fe(001) stripes J. Appl. Phys. 111, 123917 (2012) Damping effect on resonance bounds relationship of nanostructured ferromagnets and composites J. Appl. Phys. 111, 113912 (2012) Beyond ferromagnetic resonance: The inertial regime of the magnetization Appl. Phys. Lett. 100, 192407 (2012) Towards precise measurement of oscillatory domain wall by ferromagnetic Josephson junction Appl. Phys. Lett. 100, 152402 (2012) Theory of ferromagnetic wires resonating in the proximity of a ground plane: Application to artificial impedance surfaces J. Appl. Phys. 111, 064911 (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 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsHigh field ferromagnetic resonance measurements of the anisotropy field of longitudinal recording thin-film media C. J. Oates, F. Y. Ogrin,a)S. L. Lee, P. C. Riedi, and G. M. Smith School of Physics and Astronomy, University of St. Andrews, KY16 9SS, United Kingdom T. Thomsonb) Seagate Technology, 47010 Kato Road, Fremont, California 94538 ~Received 25 June 2001; accepted for publication 27 October 2001 ! The average value of the magnetocrystalline anisotropy field, Hk, is an important parameter for the characterization of magnetic recording media but is difficult to measure accurately due in part to theeffect of interactions between the grains. In order to evaluate H kwe have studied two model CoCrPtTa magnetic films using a number of complementary techniques: high field ferromagneticresonance ~FMR !~35.0–45.0kOe !,lowfield ~,20kOe !vectorvibrating-samplemagnetometryand torque magnetometry. The FMR measurements were performed at a number of discrete frequenciesin the range 75–93 GHz using a new quasi-optical spectrometer developed at the University of St.Andrews. The values of H kderived by FMR ~10.8 kOe !are approximately 10% greater than those obtained from conventional magnetometry ~9.6 kOe !. This difference is believed to be due to the presence of intergranular exchange coupling which reduces the measured value of anisotropy whenthe applied field is not sufficiently large to completely align the magnetic moments. © 2002 American Institute of Physics. @DOI: 10.1063/1.1428804 # I. INTRODUCTION Anisotropy and saturation magnetization are the two fundamental properties that determine the static characteris-tics of ferromagnetic materials. 1Magnetic anisotropy deter- mines the difficulty of changing the state of the atomic mag-netic moments for a given set of experimental conditionswhile the saturation magnetization measures the effect ofalignment of the atomic magnetic moments. In addition thedynamic properties of ferromagnetic materials during rever-sal are characterized by the Gilbert damping constant, a.2 Interest in ahas recently increased particularly in materials used for data storage such as CoCr-based thin films as datarates, and hence the time available to switch the magnetiza-tion, decrease to below 1 ns. 3,4Despite the fundamental na- ture of these quantities it remains an on-going experimentalchallenge to determine their values accurately, particularly when materials are in the form of thin films. In this study wereport comprehensive work aimed at determining the anisot-ropy and the Gilbert damping constant by high field ferro-magnetic resonance ~FMR !. We have chosen two model CoCrPtTa thin films of in- terest as media for data storage with different magnetic re-cording properties. The two samples were sputtered onsuper-smooth glass-ceramic substrates using a standard com-mercial dc magnetron sputtering system.The magnetic layersof both media were sputtered from the same CoCrPtTa alloytarget with the underlayers consisting of 50 nm thick CrMnfor mediaAand 40 nm/10 nm thick NiAl/CrMn for media B.Previous work on these samples shows that they have sub- stantially different recording properties; with mediaAhavinghigher noise than media B which was attributed primarily tothe differences in grain size and grain size distribution. 5,6 This paper examines the relative merits of the techniques used to determine these fundamental magnetic properties ofthin-film media and correlates the differences found in thesemodel samples to differences in their recording performance. II. EXPERIMENT The FMR measurements were performed at a number of discrete frequencies in the range 75–93 GHz ~35.0–45.0 kOe!using a new quasi-optical spectrometer developed at the University of St.Andrews.7As a preliminary experiment FMR and low field ~,20 kOe !torque magnetometry meth- ods were applied to a polycrystalline Co film. Good agree-ment was found between the two values derived for the ef-fective anisotropy which is discussed later in section four. Inthe FMR experiments the field is applied normal to the filmplane and is sufficiently large to decouple the interactionsbetweenthegrains.Inprinciplealltheparametersrequiredtocharacterize the material, Lande ´g-factor, saturation magne- tizationM s, anisotropy field Hkand the Gilbert damping factor a, may be obtained from the field for resonance and the linewidth of the FMR signal as a function of frequency.In practice, it was found that a more accurate data set can beobtained when M swas measured using a vibrating-sample magnetometer ~VSM !. Magnetization measurements were taken using a ADE/ Digital Measurement Systems model 10 vector VSM, with amaximum applied field of 20 kOe. The temperature was heldconstant at 20°C during all measurements. The saturationmagnetization was determined from hysteresis loops using a!Currently at the University of Exeter, School of Physics, University of Exeter, Stocker Road, ExeterEX44QL, UK. b!Currently at IBM, IBM—Almaden Research Center, 650 Harry Road, San Jose, CA95120.JOURNAL OF APPLIED PHYSICS VOLUME 91, NUMBER 3 1 FEBRUARY 2002 1417 0021-8979/2002/91(3)/1417/6/$19.00 © 2002 American Institute of Physics Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsthe instrument in conventional mode with the sample plane aligned parallel to the applied field.ANi foil standard of thesame diameter as the sample was used to calibrate the instru-ment. Corrections for the substrate and sample rod were at-tempted using two techniques. The first involved fitting alinear function to the high field region of the hysteresis loopsand using the slope as a correction factor. The second tech-nique consisted of removing the magnetic film from the sub-strate and remeasuring to give a point-by-point subtraction.The two techniques yielded extremely similar results and thelinear function method was adopted as this minimized theerror on individual data points. Anisotropy was measured using our in-house torque magnetometer. In this technique, the film is rotated from 0 to90 deg relative to the field direction. Details of this techniqueare further described in Ref. 8. @Table II ~Sec. IV !provides values to the saturation magnetization and anisotropy thatwere derived from the above techniques. # III. FMR MODEL OF LONGITUDINAL RECORDING MEDIA A. Free energy density equation In order to simulate the resonant conditions we applied a model in which the thin-film media consists of a large num-ber of crystallites ~grains !, each containing an easy uniaxial anisotropy axis. The orientation of the uniaxial axes is iso-tropic and confined to the film plane ~see Fig. 1 !.I ti sa s - sumed that the interaction between the grains is negligibleand that the properties of each crystallite in a high magneticfield is described by a single domain model. With these con-ditions the free energy per unit volume for each grain iswritten in the following form: U52M sH11 2NMs21Ksin2f, ~1! whereMsis the saturation magnetization vector, K 5MsHk/2 is the magnetocrystalline anisotropy constant, Hk is the anisotropy field and fis the angle between the grain’s magnetization and crystallographic axis. N5Nxax21Nyay21Nzaz2is the effective demagnetizing factor with directional cosines aiwhich define the orientation of the magnetization vector with respect to the coordinate axes. In the thin filmlimit the demagnetizing factor Nis simplified assuming that N x5Nz50 andNy54p. Minimizing Eq. ~1!will determine the magnetization’s static orientation which is used to simu-late FMR and torque data. B. Equation of magnetic motion To calculate the FMR response the equation of motion of the magnetization vector is used:9 v5g Mssinu0~EuuEww2Euw2!1/2, ~2! where vis the resonance frequency, gis the gyromagnetic ratio and Euu,EwwandEuwrepresent the second partial derivatives of the free energy taken at the magnetization vec-tor’s equilibrium position ( u0,w0) Euw5S]2E ]u]wD u5u0,w5w0, Euu5S]2E ]u]uD u5u0,w5w0, ~3! Eww5S]2E ]w]wD u5u0,w5w0. From Eqs. ~1!,~2!and~3!the resonance frequency is easily related to the parameters g,Hk,4pMs. In our configuration ~Fig. 1 !this relation is expressed as v5@~~4pMs2sin2w01MsHk!cos2u0 1HMssinu0cos~wH2w0!! 3~4pMs2sin2u0cos2w01HMssinu0cos~wH 2w0!!2~2pMs2sin2u0sin2w0 2HMscosu0sin~wH2w0!!2#1/2g Mssinu0. ~4! If the applied field His sufficiently high and directed normal to the film surface ~i.e.,wH590°!, the equilibrium position of the magnetization vector will therefore align with the ap-plied field: u0590°, w0590°. Thus for the perpendicular configuration in the high field limit, the resonant condition,@Eq.~4!#is simplified to Sv gD2 5~Hr24pMs2Hk!~Hr24pMs!, ~5! whereHris the resonant field. C. Solution to the FMR condition In principle it is possible to determine the three param- etersg,Hkand 4 pMsusing a minimum of three frequencies. However, in practice, due to the error in determining theresonant field it is difficult to determine 4 pMsandHksepa- rately but only the sum of the two variables, (4 pMs1Hk). FIG. 1. Orientation of the magnetization M, applied field Hand easy axes of the magnetic grains with respect to the film plane.1418 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Oateset al. Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsAt high frequencies, the function form of the equation is such that a small variation in Hrlead to a significant change in the individual values of Hkand 4 pMsas shown by the size of the intersection area in Fig. 2. Even for a fixed valueofgand a small error in H r~’0.5%!an error as high as 50% inHkis observed. This error can be reduced by applying frequencies with the maximum possible separation. Theangle between the solution lines that correspond to the twofrequencies increases resulting in a reduction to the area ofintersection ~see Fig. 3 !. Consequently, the spread in possible solutions of H kand 4 pMsdecreases.As an example, Table I provides results of the numerical simulation for two differentfrequencies with gand dHrfixed to 2.15 and 100 Oe, re- spectively. Applying frequencies over a broader range will reduce the error in the sample’s saturation magnetization and anisot-ropy significantly. However, even at a maximum possibleseparation ~for our spectrometer Df’200GHz !the best er- ror inH kstill remains not less than 61.1 kOe ( dHr ’100Oe) which is similar to the uncertainty commonly re- ported for other techniques used to measure anisotropy. Inorder to reduce errors further one should either try to im-prove the precision of the resonance field H rand/or employ other techniques that will measure one of the above param-eters separately. In this work we follow both directions. Wecarefully analyze the resonance line shape by fitting it to ananalytically derived distribution from which the resonancefield can be extracted as a parameter. We also use vibratingsample magnetometry to measure precisely the value of themedia’s saturation magnetization M swhich is used as a fixed parameter in Eq. ~5!.IV. RESULTS AND DISCUSSION It has been shown previously that FMR can provide use- ful information on anisotropy in thin films.10In many cases the analysis of anisotropy is built upon the angular depen-dence of the resonance field. 11For high field spectrometers based on superconducting coils it is not always easy to ar-range experiments with an angular variation of the appliedfield. This is due to space restrictions or mechanical com-plexities of the rotational mechanisms. An alternative is toset up an experiment where the applied field angle is fixedand the excitation frequency is varied. This is the approachwe adopt in high field, multi-frequency FMR measurementswhere the field was always normal to the plane of the film.As a preliminary example, multi-frequency measurements ona 30 nm polycrystalline Co film were performed to find theeffective anisotropy H eff. The cobalt film was grown using molecular beam epitaxy on a silicon substrate. The depositedcobalt layer consisted mainly of a polycrystalline hcp phase,with some fcc phase and stacking faults as determined bynuclear magnetic resonance. 12Given that the direction of the easy axes is mainly out-of-plane for the Co film, the reso-nance equation in this case is different to that for the mediasamples and given by the following relation: v g5Hr2Heff, ~6! FIG. 2. Graphical representation of solution to Eq. ~5!for two frequencies: 75 GHz and 93 GHz. The resonance fields were determined from the ex-perimental measurements on sample 15Aand equal to 35.85 kOe and 41.75 kOe, respectively. The resonance field error dHrwas estimated from fitting the resonance line shape into Eq. ~7!. Withgfixed to 2.15, it was found that for both samples dHr’100Oe ~which corresponds to the thickness of the lines at 75 and 93 GHz !. FIG. 3. Intersection of two solutions for 75 GHz and 270 GHz as simulated using Eq. ~5!. For both samples gis fixed to 2.15 and dHris taken as 100 Oe. TABLE I. Intersection of possible solutions for two different frequency ranges ~see Figs. 2, 3 !as simulated using Eq. ~5!. Parameters of ganddHr are fixed to 2.15 and 100 Oe, respectively. f1 f2 g dHr DHk D4pMs 75 GHz 93 GHz 2.15 100 Oe 8.4 kOe 5.0 kG 75 GHz 270 GHz 2.15 100 Oe 2.2 kOe 1.3 kG1419 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Oateset al. Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionswhereHeff54pMs2Hk. Equation ~6!was derived from Eqs. ~1!,~2!and~3!assuming that the orientation of the easy c axes and the applied field are perpendicular to the film plane. Figure 4 illustrates the resonance field dependence of thefrequency measured for the Co film. The experimental pointswere fitted with Eq. ~6!. The extracted values of gandH eff were 2.156 ~5!and 17.15 ~20!kOe, respectively. This was in reasonable agreement with previous measurement of Heffon the same film by torque magnetometry which was 16.2 ~7! kOe.13If it is assumed that the saturation magnetization for this film is the same as for bulk Co, the crystalline anisotropyfieldH kextracted in this way is of the order of 0.5 kOe. Consequently, the magnetocrystalline anisotropy constant Kv is deduced to be (3.15 61.45) 3105ergs/cc. This value is much less than that of bulk cobalt, which is typically (324)310 6ergs/cc.14,1and, most likely, is a result of the mix- ture in the crystalline phases and the polycrysalline nature ofthe sample. 13,15,16However, the emphasis here is on the analysis of the perpendicular effective field Heff, which is measured independently of the saturation magnetization and,therefore, is very useful for comparison with the equivalentmeasurements with other magnetization techniques, includ-ing torque magnetometry. It should be noted that in the configuration of the Co film the magnetocrystalline anisotropy has the same symmetry asthe demagnetizing field. This makes it impossible to measureH kseparately without contribution from the demagnetizing fields. Nevertheless, the effective field is measured with goodprecision from a relatively narrow range of frequencies with-out involving magnetometry.Asimulation of Eq. ~6!with the cobalt’s effective field measured as a function of the g-factor for two different frequencies: 75 and 93 GHz is shown inFig. 5. For thin film longitudinal media with easy axes in- planeof the film, unlike the cobalt film, it is theoretically possible to separate the magnetocrystalline anisotropy fromthe demagnetizing field @compare Eqs. ~5!and~6!#. How- ever, in practice, due to the high sensitivity to the resonancefield error dHrthe spread in possible solutions to Hkand 4pMsis very high and hence additional magnetization mea- surements are required. The FMR measurements on the media samples were car- ried out in the range of frequencies between 75 GHz and 93GHz. In contrast to previously published results by Igarashiet al. 17we were able to obtain very well resolved resonance lines which allowed us to determine the resonance fields witha precision of better than 60.25%. Figure 6 illustrates an example of a first order derivative absorption line shape of FIG. 4. Resonance frequency as a function of the resonance field for an hcp Co film. Solid line—fit using Eq. ~6!. FIG. 5. Graphical solution to Eq. ~6!for a polycrystalline Co film. Heff 517.15(20) kOe, g52.156(5). The resonance frequencies, 75 GHz and 93 GHz correspond to the resonance fields of 42.04 kOe and 47.95 kOe, re- spectively. The error in the estimate of the resonance field: dHr580Oe. FIG. 6. FMR spectrum measured on high noise media at 84 GHz. Circles— experimental data. Solid line—fit using Eq. ~7!Hr538750Oe 650Oe.1420 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Oateset al. Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionshigh noise ~sample 15A !media at 84 GHz. The resonance field is extracted from a fit to a first derivative Lorentzianline shape 18which is in the form y5aSHr2H DHrD19b23bSSHr2H DHrDD2 F31SHr2H DHrD2G2 , ~7! whereyis the FMR response, HandHrare the applied and resonance fields, DHris the resonance half peak-to-peak linewidth and aandbare the amplitudes of absorption and dispersion signals, respectively. The last two parameters are,in fact, very important for the determination of H r.I ti sa common feature of FMR spectrometers that the output signaloften contains a mixture of the absorption and dispersionphases.This is in contrast to the ideal situation when only theabsorption signal should be detected. Thus, by adding thepresence of a dispersion signal we can always fit the outputsignal correctly and determine the resonance field with highprecision. Figure 7 shows the variation of the excitation frequency ~ v/2p!as a function of resonance field Hrmeasured for the high noise ~15A!sample. The solid line is the result of a fit using Eq. ~5!with the value of the saturation magnetization given by the VSM measurement. The value of the anisotropyfieldH kproduced by the fit is given in Table II. It should benoted that the precision of the calculated value of Hkis di- rectly dependent on the precision of Ms. In the present case the uncertainty in Msarises as a combination of errors: the measured total magnetic moment m, magnetic layer thickness dand the sample area S~e.g., for sample 15A: m52096131026emu,d527.560.5nm, S519.6 60.2mm2.! It is interesting to note that similarly to the Co film, the values of the anisotropy field Hkmeasured by torque mag- netometry were close, but slightly lower than those measuredby FMR. This discrepancy in H kis believed to be not just a lack of experimental precision but a consequence of the in-trinsic characteristics of polycrystalline materials. In FMRthe external field is of the order of 40 kOe which is severaltimes larger than that required to saturate the media. As aresult only uniform precession of the magnetic moments isproduced. Any possible effects of coupling between thegrains of the material will not affect the resonant conditions.In contrast to FMR, torque measurements are typically car-ried out at lower fields ~up to 20 kOe !.Within this field range both the magnetostatic and exchange interaction can play asignificant role in the collective behavior of the magneticgrains which form the sample. This behavior is more com-plicated than that described by the simple model given here~1!and requires a more detailed micromagnetic approach. Although the interaction 5in these samples is quite weak it is sufficient to give a lower value of the anisotropy field mea-sured by low field ~,20 kOe !torque magnetometry. 8 V. DYNAMIC PROPERTIES FMR linewidth measurements provide details on the sample’s inhomogeneous broadening and relaxation con-stant. The relaxation constant is related to the switchingmechanism of the magnetization vector M s. In the present measurements the linewidth of the FMR signal was deter-mined in the same way as the resonant field: a fit to the lineshape using Eq. ~7!. The linewidth DH rin Eq. ~7!is defined as half of the field difference between the maximum andminimum of the first derivative absorption line shape. Thefrequency dependence of the peak-to-peak linewidth was de-rived from the following equation: 19 Dv5ag MsSEuu11 sin2u0EffD, ~8! where ais the Gilbert damping factor. An additional fre- quency independent term is included which contributes tothe inhomogeneous broadening. 20Thus the linewidth is ex- pressed as: DH~v!5DH~0!12av )g, ~9! where DH(v) and DH(0) is the frequency dependent and independent peak-to-peak linewidth. Figure 8 shows a fit tothe linewidth using Eq. ~9!andTable III shows the results for both media samples. The inhomogeneous broadening term in the low noise medium is twice that of the high noise medium. This mayresult from the following: The low noise media is comprised FIG. 7. Resonance frequency as a function of resonance field for high noise media ~sample 15A !. TABLE II. Parameters of the anisotropy field and g-factor as a result of the fit using equation 5. 4 pMsis measured separately by vector VSM. The Hk value by torque magnetometer is taken from Ref. 8. Sample nameHk~kOe! TorqueHk~kOe! FMR4pMs~kG! VSM g 15A~high noise media !9.6~4!10.8~2!4.9~1!2.150 ~5! 15B~low noise media !9.6~4!10.7~2!4.5~1!2.160 ~5!1421 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Oateset al. Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsof smaller grains ~11 nm !with better grain segregation than that of the high noise media where the mean grain size ~42 nm!is approximately four times greater. The segregation leads to better isolation of the magnetic moments and, there-fore, reduces exchange coupling between the grains. Conse-quently, the inhomogeneity of the medium is greater as aresult of a larger number of spins in different local magneticenvironments, nonetheless, the noise, associated with the ex-change coupling, is lower. This interpretation is also sup-ported by the values of the Gilbert damping factors. For thesample with larger grains ~high noise medium !the damping factor is greater. This indicates a faster relaxation time whichmay again be a result of stronger intergrain coupling in thissample. VI. CONCLUSIONS In this paper we have for the first time shown convinc- ingly that high field, multi-frequency ~75–93 GHz !FMR may be used to determine the static ( Hk) and dynamic prop- erties ~damping constant a!of realistic ~’3 Gbits/in2!longi- tudinal thin film recording media. While, in principle, Hk, Msand the Lande ´g-factor may all be extracted from the FMR data, in practice we find that fixing the value of Ms, which is easily obtained from magnetometry measurements,reduces the uncertainty in the values obtained for the otherparameters. The advantage of high applied fields is that neighboring grains are more decoupled, minimizing the ef-fect of interactions. Interactions often lead to a lower valueof magnetocrystalline anisotropy being reported when mea-suring materials such as CoCr based thin films ( H k ’6–9 kOe) using moderate applied fields from iron-cored electromagnets ,20 kOe. Two thin-film recording media chosen for this investiga- tionhavealreadybeenextensivelystudied.5,6Themediacon- sisted of the same composition of CoCrPtTa alloy sputteredonto different underlayer structures leading to very differentrecording properties. The anisotropy field measured by FMRwas 10.8 kOe for both media investigated which is 1.2 kOegreater than values of 9.6 kOe obtained from torque magne-tometry.The Lande ´g-factor derived was 2.15. Differences in the damping constant awere measured for the two media, with the high noise medium having a50.025 and the me- dium with a superior recording performance having a 50.016. Given the very different microstructures of these media it is possible to speculate that the anisotropy is morestrongly dependent on the alloy composition whereas thedamping factor and therefore the high frequency reversalproperties are more strongly governed by the microstructure.However, significant extra work will need to be completed inorder to validate such a speculation. ACKNOWLEDGMENT CJO gratefully acknowledges the EPSRC for providing funding to do this experiment. 1S. Chikazumi, Physics of Magnetism ~Wiley, New York, 1964 !. 2T. L. Gilbert, Phys. Rev. 100, 1243 ~1955!. 3A. F. Torabi, H. Zhou, and H. N. Bertram, J.Appl. Phys. 87, 5669 ~2000!. 4M. E. Schabes, H. Zhou, and H. N. Bertram, J. Appl. Phys. 87, 5666 ~2000!. 5E. T. Yen, S. Z. Wu, T. Thomson, R. Ristau, R. Ranjan, G. C. Rauch, C. Habermeier, and R. Sinclair, IEEE Trans. Magn. 35, 2730 ~1999!. 6L. Halloway and H. Laidler, IEEE Trans. Magn. 37, 1459 ~2001!. 7G. Smith, J. C. G. Lesurf, R. H. Mitchell, and P. C. Riedi, Rev. Sci. Instrum.69, 3924 ~1998!. 8F. Y. Ogrin, J. Magn. Magn. Mater. ~to be published !. 9J. Smit and H. G. Beljers, Philips Res. Rep. 10,1 1 3 ~1955!. 10J. F. Cochran, J. M. Rudd, M. From, B. Heinrich, W. Bennett, W. Schwarzacher, and W. F. Egelhoff, Phys. Rev. B 45, 4676 ~1992!. 11F. Y. Ogrin, ‘‘Ferromagnetic resonance studies of cobalt films and cobalt based multilayers produced by MOCVD,’’ Ph.D. thesis, University ofKeele, U.K. ~1996!. 12P. C. Riedi, T. Thomson, and G. J. Tomka, NMR of Thin Magnetic Films and Superlattices , Handbook of Magnetic Materials 12 ~1999!, edited by K. H. J. Buskow, ~Elsevier Science, B.V., 1999 !, Chap. 2. 13F. Y. Ogrin, S. L. Lee, and Y. F. Ogrin, J. Magn. Magn. Mater. 219,3 3 1 ~2000!. 14M. Prutton, Thin Ferromagnetic Films ~Butterworth, London, 1964 !. 15L.Albini, G. Carlotti, G. Gubbiotti, L. Pareti, G. Socino, and G. Turilli, J. Magn. Magn. Mater. 199,3 6 3 ~1999!. 16T. Hill, S. Stempel, T. Risse, M. Baumer, and H. J. Freund, J. Magn. Magn. Mater. 199, 354 ~1999!. 17M. Igarashi,T. Kambe, K.Yoshida,Y. Hosoe, andY. Sugita, J.Appl. Phys. 85, 4720 ~1999!. 18C. P. Poole, Electron Spin Resonance—A Comprehensive Treatise on Ex- perimental Techniques ~Wiley, New York, 1967 !. 19S.V.Vonsovskii, Ferromagnetic Resonance ~Pergamon, NewYork, 1966 !. 20C. Chappert, K. Le Dang, P. Beauvillain, H. Hardequint, and D. Renard, Phys. Rev. B 34, 3192 ~1986!. FIG. 8. Resonance linewidth for samples 15A ~circles !and 15B ~squares !as a function of frequency. Solid lines—result of fits using Eq. 9. TABLE III. Parameters of the inhomogeneous broadening DHrand the Gilbert damping factor aas a result of fitting to Eq. ~9!. Sample DHr(0)~Oe! a 15A~high noise ! 150684 0.0249 60.0025 15B~low noise ! 358669 0.0159 60.00211422 J. Appl. Phys., Vol. 91, No. 3, 1 February 2002 Oateset al. Downloaded 16 Jul 2012 to 128.95.104.109. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.4792496.pdf

Investigation of magnetic properties and microwave characteristics of obliquely sputtered NiFe/MnIr bilayers Nguyen N. Phuoc, Wee Tee Soh, Guozhi Chai, and C. K. Ong Citation: J. Appl. Phys. 113, 073902 (2013); doi: 10.1063/1.4792496 View online: http://dx.doi.org/10.1063/1.4792496 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v113/i7 Published by the American Institute of Physics. Related Articles Suppression of secondary electron yield by micro-porous array structure J. Appl. Phys. 113, 074904 (2013) Probing confined acoustic phonons in free standing small gold nanoparticles J. Appl. Phys. 113, 074303 (2013) Spatially periodic domain wall pinning potentials: Asymmetric pinning and dipolar biasing J. Appl. Phys. 113, 073906 (2013) Highly (001)-oriented thin continuous L10 FePt film by introducing an FeOx cap layer Appl. Phys. Lett. 102, 062420 (2013) Sn and Nb modified ultrafine Ti-based bulk alloys with high-strength and enhanced ductility Appl. Phys. Lett. 102, 061908 (2013) 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 24 Feb 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsInvestigation of magnetic properties and microwave characteristics of obliquely sputtered NiFe/MnIr bilayers Nguyen N. Phuoc,1,a)Wee Tee Soh,2Guozhi Chai,1and C. K. Ong2 1Temasek Laboratories, National University of Singapore, 5A Engineering Drive 2, Singapore 117411 2Center for Superconducting and Magnetic Materials, Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542 (Received 20 December 2012; accepted 1 February 2013; published online 15 February 2013) A comprehensive investigation of the magnetic properties and high frequency characteristics of NiFe/ MnIr bilayers with regards to oblique deposition angle was conducted in conjunction with an analysis based on the Landau-Lifshitz-Gilbert equation. It was found that exchange bias can be significantly enhanced with the variation of oblique deposition angle, which is interpreted in terms of the formationof inclined columnar structure of the films often observed in samples fabricated by this oblique deposition technique. Moreover, the uniaxial magnetic anisotropy field and the resonance frequency are increased with the increasing of oblique deposition angle. The variations of effective Gilbertdamping factor and the frequency linewidth with oblique deposition angle are also presented and discussed in details. VC2013 American Institute of Physics .[http://dx.doi.org/10.1063/1.4792496 ] I. INTRODUCTION Among various research themes in magnetism, exchange bias stands as one of the most interesting topics that have been received much attention from many research- ers.1–3This effect refers to the shift of the hysteresis loops in materials having ferromagnetic (FM)/antiferromagnetic (AF) interfaces when they undergo a field cooling procedure from a temperature higher than the N /C19eel point or when they are fabricated under an applied magnetic field.1–3Although extensively studied due to its extensive application in mag- netic recording industry4–7as well as potential applications in microwave devices,8–16the physical origin of this intrigu- ing phenomenon together with some associated effect is still in controversy and needs further research to elucidate. In theliterature, exchange bias has been investigated with respect to many influence factors such as thickness depend- ence, 1,8,11,12,14–16temperature dependence,1,6,16–18micro- structure dependence,1,3,18–20and fabrication condition dependence20,21of this effect. However, it appears that there is a lacking of research that focuses on the influence ofoblique deposition in exchange-biased thin films. 22–24In this work, we therefore aim to study the effect of oblique deposition on magnetic and microwave properties ofexchange-biased NiFe/MnIr bilayers. This study in the sense of fundamental research may assist us to have an insight of exchange bias. Besides, from application perspective, ourresearch work has a great implication for high frequency application based on magnetic thin films because both exchange bias 8–15and oblique deposition25–30are proven to be effective ways to tune the ferromagnetic resonance fre- quency. Hence, it may be interesting to investigate the possi- bility of combining these two ways together to push theresonance frequency to even higher range. With these objec- tives in mind, we carry out in the present research work, adetailed investigation to see how the oblique deposition angle affects both the static magnetic properties in thehysteresis loops and the dynamic magnetization in the per- meability spectra of the exchange-biased NiFe/MnIr bilayers and discuss the results in light of the analysis based on the Landau-Lifshitz-Gilbert (LLG) equation. II. EXPERIMENT Thin films of NiFe (125 nm)/MnIr (15 nm) bilayers were fabricated onto Si (100) substrates at room temperature using a radio-frequency (RF) magnetron sputter-deposition system with the base pressure of 7 /C210/C07Torr. The targets used in this fabrication are Ni 80Fe20and Mn 75Ir25alloy targets. A capping layer of SiO 2with the thickness of 10 nm was deposited on the top of the samples to protect them fromoxidation. The argon pressure was kept at 10 /C03Torr during the deposition process by introducing argon gas at the flow rate of 16 SCCM (SCCM denotes cubic centimeter per mi-nute at STP). The deposition setup is shown in Fig. 1, where the substrates were put at an oblique angle ranging from 0 /C14 to 45/C14. With this arrangement, the easy axis of the present films induced by oblique deposition is perpendicular to inci- dent plane. A magnetic field of about 200 Oe was applied during the deposition process along the easy axis induced byoblique deposition in order to assist the inducement of mag- netic anisotropy. The composition was determined by energy dispersive X-ray spectroscopy (EDS) and the thickness ofeach layer was controlled both by the deposition time and by keeping the deposition rate constant, which was verified by a thickness profile meter. For the structural properties of thefilms, an X-ray diffractometer using CuK aradiation was employed to characterize. A vibrating sample magnetometer (VSM) was used for the measurement of magnetizationcurves at room temperature. The permeability spectra over the frequency range from 0.05 GHz to 5 GHz were obtained by a shorted micro-strip transmission-line perturbationmethod 31using a fixture developed in our laboratory.a)Author to whom correspondence should be addressed. Electronic mail: tslnnp@nus.edu.sg. Tel.: 65-65162816. Fax: 65-67776126. 0021-8979/2013/113(7)/073902/6/$30.00 VC2013 American Institute of Physics 113, 073902-1JOURNAL OF APPLIED PHYSICS 113, 073902 (2013) Downloaded 24 Feb 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsIII. RESULTS AND DISCUSSION Figure 2presents X-ray diffraction patterns for a series of our NiFe/MnIr bilayer samples with oblique deposition angle changed from 0/C14to 45/C14. There are two prominent peaks observed which corresponds to the MnIr fcc (111)peak and NiFe bcc (110) peak. It can be seen that as the oblique deposition angle is increased, especially in the range from 30/C14to 45/C14, the NiFe bcc (110) peak is moved to the higher diffraction angle whereas the MnIr fcc (111) peak position is practically unchanged. This unambiguously indi- cates that there is a contraction in the lattice spacing of NiFebcc (110). The contraction of NiFe bcc (110) lattice spacing can be interpreted in terms of the formation of the columnar grains which are tilted due to the self-shadow effect 25–30,32– 34of the oblique deposition method. As the oblique angle is increased, the tilting of the columnar grains is increased leading to some distortion of the lattice of the films in such away that the lattice spacing is reduced substantially. 24,25 Shown in Fig. 3are several representative hysteresis loops of NiFe/MnIr bilayer films with different oblique depo-sition angles measured at room temperature along the easy axis and the hard axis. Here, the easy axis is the one in the same direction as the magnetic field applied during deposi-tion. The hard axis is the one that is perpendicular to the dep- osition magnetic field but it is still lying in the plane of the film. 12,13A detailed description for the definition of the easy axis and hard axis is provided in Fig. 1. It can be seen clearly in Fig. 3that all the easy axis loops are shifted to the nega- tive direction which is indicative of the presence of exchangebias coupling between the FM (NiFe) and AF (MnIr) layers. The hard axis magnetization curves show typical slanted loops implying that there is also an uniaxial magnetic anisot-ropy present in these films. More interesting, one can observe that the hard axis loops become more slanted as the oblique deposition angle is increased. This behavior suggests FIG. 1. Schemative view of the present oblique deposition system. FIG. 2. XRD profiles of NiFe-MnIr bilayered films grown at various oblique deposition angles. FIG. 3. Representative hysteresis loops measured along easy axis and hard axis at room temperature for different NiFe-MnIr films grown at various oblique deposition angles.073902-2 Phuoc et al. J. Appl. Phys. 113, 073902 (2013) Downloaded 24 Feb 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsthat the magnetic anisotropy is increased with increasing of the oblique deposition angle. More quantitative discussion on this behavior together with the contribution of each sepa-rated magnetic anisotropy will be given later. Now we focus on the measurement of dynamic magnetic characteristics of these films. The permeability spectra of thefilms measured at room temperature for some representative oblique angles are revealed in Fig. 4with both real and imag- inary parts. As demonstrated in Fig. 4(b), the peak in the imaginary permeability spectra is found to be shifted toward higher frequency range when the oblique deposition angle is increased. This behavior unambiguously indicates that theferromagnetic resonance frequency is increased with the increasing of oblique deposition angle. For a more quantita- tive analysis of these dynamic magnetic properties, the LLGequation 35as below is employed to fit the experimental dynamic permeability spectra, d~M dt¼/C0cð~M/C2~HÞþaef f M~M/C2d~M dt: (1) Here, M represents the magnetization of the films, H is the magnetic field, aeffis the dimensionless effective Gilbert damping coefficient ( aeffincludes intrinsic and several ex- trinsic sources of damping8,15,23), and cis the gyromagnetic ratio. By solving the LLG equation with the assumption of macrospin approximation and the presence of only in-plane uniaxial anisotropy in the films, one can obtain the expres-sion of the real and imaginary parts of permeability spectra as follows: 27,28 l0¼1þ4pMSc2ð4pMSþHKdynÞð1þaef f2Þ½x2 Rð1þa2 ef fÞ/C0x2/C138þð4pMSþ2HKdynÞðaef fxÞ2 ½x2 Rð1þaef f2Þ/C0x2/C1382þ½aef fxcð4pMSþ2HKdynÞ/C1382; (2) l00¼4pMef fcxa ef fc2ð4pMef fþHKdynÞ2ð1þa2 ef fÞþx2 ½x2 Rð1þa2 ef fÞ/C0x2/C1382þ½aef fxcð4pMef fþ2HKdynÞ/C1382: (3) Here, c,M S,HKdyn,a n d xR(xR¼2pfFMR) are the gyromag- netic ratio, the saturation magne tization, dynamic magnetic ani- sotropy, and ferromagnetic resona nce frequency, respectively. The saturation magnetization 4 pMScan be obtained from VSM measurement which is 10 kG while the dynamic magnetic ani- sotropy field H Kdynand the effective Gilbert damping aeffcan be considered as fitting parameters. Based on the above Eqs. (2)and(3), we can fit the experimental data very well with the theoretical curves, which are represented as the lines in Fig. 4. Figure 5provide a summary of the dependences of the exchange bias field H E, the coercivity measured along the FIG. 4. Representative permeability spectra of NiFe-MnIr films grown at various oblique deposition angles. (a) Real part l0. (b) Imaginary part l00. Lines are fitted from LLG equation. FIG. 5. (a) Variations of exchange bias field (H E) and coercivity (H C) along the easy axis as a function of oblique deposition angle. Lines are served as a guide to the eyes. (b) Variations of intrinsic uniaxial (H Kint), static (H Ksta), and dynamic (HKdyn) magnetic anisotropy fields as functions of oblique deposition angle.073902-3 Phuoc et al. J. Appl. Phys. 113, 073902 (2013) Downloaded 24 Feb 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionseasy axis (H C), the static magnetic anisotropy field H Ksta, the dynamic magnetic anisotropy field H Kdyn, and the intrinsic uniaxial magnetic anisotropy field H Kint. The exchange bias field H Eis defined as the shift of the center of the magnetiza- tion curve along the easy axis whereas the static magnetic anisotropy field H Kstais determined from the slope of the hard axis hysteresis loop.10,12,15,36,37It is noted that this static magnetic anisotropy field H Kstais the total effective static magnetic anisotropy field and is the sum of the unidir-ectional anisotropy field (which is exchange bias field H E) and the intrinsic uniaxial magnetic anisotropy field H Kint (HKsta¼HEþHKint).36,37In addition, one should note that the easy axis of H Kstais not changed with the increasing of oblique angle and it is in the same direction as the direction of the unidirectional anisotropy defined by the magnetic fieldapplied during deposition. Hence, we can obtain H Kintby subtraction of H Efrom H Ksta(HKint¼HKsta-H E).36,37The dynamic magnetic anisotropy field H Kdyn, on the other hand, is obtained from the LLG fitting procedure as mentioned above. As observed in Fig. 5(a), exchange bias field is gener- ally increased with oblique angle except a slight decrease inthe range from 16 /C14to 23/C14. This behavior may possibly be due to the above-mentioned self-shadow effect which results in a formation of tilted columnar structure leading to somedistortion of the lattice of the films. 25–30This distortion may assist in creating more pinning sites at the FM/AF interfaces which are important for emergence of the frozen AF spinsaccounting for exchange bias thus accounting for the enhancement of exchange bias with oblique angle as observed. 24Also presented in Fig. 5(a)is the variation of the coercivity H Cwith oblique deposition angle showing a slight increase of H Cwith oblique angle. The mechanism of H Cis rather complicated, which is dependent not only on the valueof the magnetic anisotropy but also on the reversal modes of the magnetization process. As the magnetic anisotropy is hugely changed while the coercivity is practicallyunchanged, one may tentatively draw a conclusion that the coercivity in this case is governed mostly by the reversal modes rather than by the magnitude of the magnetic anisot-ropy. We argue that it is also because the self-shadow effect that causes the intrinsic uniaxial anisotropy field H Kint increased with the increasing of the oblique deposition angle as seen in Fig. 5(b). This behavior was quite well established in the literature with many similar observations in various systems of single layered FM.25–30Due to the increasing of both exchange bias field H Eand the uniaxial anisotropy field HKintwith the oblique deposition angle, the total effective static magnetic anisotropy field H Kstais increased accord- ingly. However, it should be noted that this increment is mostly due to the contribution of H Kintexplaining why the trends of H Kstaand H Kintwith oblique deposition angle are quite similar. For example, when the oblique deposition angle is changed from 0/C14to 45/C14,HKintis increased from 6 Oe to a high value up to 110 Oe while H Eis only changed from 6 Oe to 9 Oe. The reason for these different behaviors lies in the fact that the self-shadow effect resulting in a formation of tilted columnar structure in oblique deposition governsH Kintand H Ein different ways. While this t ilted columnar structure containing elongated grains may lead to the emergenceof shape anisotropy and magnetocrystalline anisotropy causing a great enhancement of H K,23,25–30it may just only create more pinning sites at the FM/AF interfaces, thereby leading to amoderate increment in H E. Also as observed in Fig. 5(b),t h e difference between dynamic m agnetic anisotropy field H Kdyn obtained from LLG fitting and the static magnetic anisotropy field H Kstaestimated from M-H loops is negligible suggesting that there is practically no rotational anisotropy field present in these series of samples15,17,36,37 According to Kittel,38the ferromagnetic resonance fre- quency f FMR and the dynamic magnetic anisotropy field HKdyncan be related to each other through the following equation: fFMR¼c 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Hdyn KðHdyn Kþ4pMSÞq : (4) Since H Kdynis increased with the increasing of oblique depo- sition angle as discussed above and the saturation magnetiza- tion M Sis constant for all the samples, the ferromagnetic resonance frequency f FMRis increased with oblique deposi- tion angle accordingly as in Fig. 6(a). This characteristic has an important implication from the application perspective. By combining both exchange bias and oblique depositiontechnique, we can tune the resonance frequency from 0.9 GHz up to 3.2 GHz. The static permeability l S, however, is reduced when the oblique deposition angle is increased asobserved in Fig. 6(b). This reduction can be interpreted based on the following equation which reveals the relation- ship between the static permeability l Sand the effective static magnetic anisotropy H Ksta,11 FIG. 6. Dependences of (a) ferromagnetic resonance frequency (f FMR), (b) static permeability ( lS) with oblique deposition angle.073902-4 Phuoc et al. J. Appl. Phys. 113, 073902 (2013) Downloaded 24 Feb 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionslS¼1þ4pMS HKsta: (5) From the above formula, it can be easily deduced that as HKstais increased with oblique deposition angle as in Fig. 5(b) while M Sis constant, lSis expected to decrease when the oblique deposition angle increases. Figure 7summarizes the variations of the effective Gil- bert damping coefficient aeffand the frequency linewidth Df with oblique deposition angle. It is interesting to observe thatthe variation trend of a effwith oblique deposition angle as in Fig.7(a)is roughly inverse to the trend of the dependence of exchange bias H Ewith oblique angle as in Fig. 5(a). In par- ticular, when the oblique angle is increased from 0/C14to around 25/C14, exchange bias field H Eis reduced while the effective Gilbert damping coefficient aeffis increased. Then as the oblique angle increases from 25/C14to around 45/C14, the exchange bias H Eis increased whereas the effective Gilbert damping coefficient aeffis now decreased. In order to inter- pret these seemingly interesting contradicted trends, one should be reminded that the effective Gilbert damping factor measured in our case consists of two main contribution: oneis intrinsic and the other is extrinsic. 8,11,15,23While the intrinsic contribution solely comes from the material nature, the extrinsic one stems from various sources such asmagnetic anisotropy dispersion, material inhomogeneity, two-magnon scattering due to defects, grain size variation, surface roughness etc. 8,11,15,23Since the materials for these series of samples are the same, one may naturally expect that the intrinsic contribution may not change with the oblique deposition angle. Hence, the variation of aeffas in Fig. 7(a) is most likely due to the extrinsic contribution. Separatingthe originating sources of the extrinsic contribution is quite complicated and beyond the scope of the present paper. How- ever, in view of the contradicted variation trend of the effectiveGilbert damping coefficient a effin Fig. 7(a)and the exchange bias field H Ein Fig. 5(a), we may tentatively attribute the varia- tion of aeffto the magnetic anisotropy dispersion as one of the main contribution. As the role of exchange bias interaction between FM and AF spins is to keep the FM spins well aligned into a defined direction, when this exchange bias surface energyincreases (which manifests itself as the increasing of exchange bias field H E) the FM spins will be better aligned along the easy axis, leading to the decreasing of magnetic anisotropy dispersion.The decreasing of magnetic anisotropy dispersion then brings about the reduction of the extrinsic damping. This explains why a effis decreased when H Eis increased and vice versa. Appa- rently, other extrinsic contributions such as two-magnon scatter- ing, material inhomogeneity should not be ruled out and further studies where each contribution can be separated need to be per-formed to verify our argument. It is well known that the frequency linewidth Dfi s related to the effective Gilbert damping coefficient a eff through the following formula:15,17 Df¼caef fð4pMSþ2HKdynÞ 2p: (6) Since M Sis not changed with the variation of oblique deposi- tion angle and is also because 4 pMSis much larger than HKdyn, the change of Df is mostly due to the variation of aeff. This explains why the behavior of Df with oblique angle as in Fig.7(b)is quite similar to the variation of aeffin Fig. 7(a). IV. SUMMARYAND CONCLUSION To summarize, we have performed a detailed investiga- tion of the influence of oblique deposition angle on themagnetic and microwave properties of NiFe/MnIr exchange- biased bilayers. It was found that oblique deposition can tai- lor the magnitude of exchange bias significantly which issuggested to be due to the formation of tilted columnar struc- ture arising from a self-shadow effect in oblique deposited films. We also found that the uniaxial anisotropy field wassignificantly increased with oblique deposition angle due to this inclined columnar structure. As a result, the ferromag- netic resonance frequency is increased with oblique deposi-tion angle, which implies that exchange bias and oblique deposition technique can be combined together to push the resonance frequency to higher range. In our case, the reso-nance frequency can be pushed up to 3 GHz, which is quite promising for microwave applications. Our study also revealed that the effective Gilbert damping coefficient isstrongly dependent on the oblique deposition angle, which is ascribed mainly to the dispersion of magnetic anisotropy. ACKNOWLEDGMENTS The financial support from the Defence Research and Technology Office (DRTech) of Singapore is gratefully acknowledged. FIG. 7. Variations of (a) effective Gilbert damping factor ( aeff), and (b) fre- quency linewidth ( Df) on oblique deposition angle. Lines are served as a guide to the eyes.073902-5 Phuoc et al. J. Appl. Phys. 113, 073902 (2013) Downloaded 24 Feb 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions1J. Nogu /C19es and I. K. Schuller, J. Magn. Magn. Mater. 192, 203 (1999). 2R. L. Stamps, J. Phys. D: Appl. Phys. 33, R247 (2000). 3J. Nogu /C19es, J. Sort, V. Langlais, V. Skumryev, S. Suri ~nach, J. S. Mu ~noz, and M. D. Bar /C19o,Phys. Rep. 422, 65 (2005). 4B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D. R. Wilhoit, and D. Mauri, Phys. Rev. B 43, 1297 (1991). 5C. Chappert, A. Fert, and F. N. Van Dau, Nature Mater. 6, 813 (2007). 6N. N. Phuoc and T. Suzuki, J. Appl. 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Rev. 71, 270 (1947).073902-6 Phuoc et al. J. Appl. Phys. 113, 073902 (2013) Downloaded 24 Feb 2013 to 142.51.1.212. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.4991500.pdf

Temperature dependence of interlayer exchange interaction in La 0.7Sr0.3MnO3/ SrRuO3 heterostructure V. V. Demidov and G. A. Ovsyannikov Citation: Journal of Applied Physics 122, 013902 (2017); doi: 10.1063/1.4991500 View online: http://dx.doi.org/10.1063/1.4991500 View Table of Contents: http://aip.scitation.org/toc/jap/122/1 Published by the American Institute of Physics Articles you may be interested in Reduction of the magnetic dead layer and observation of tunneling magnetoresistance in La 0.67Sr0.33MnO 3- based heterostructures with a LaMnO 3 layer Applied Physics Letters 110, 212406 (2017); 10.1063/1.4984297 Size-induced changes of structural and ferromagnetic properties in La 1-xSrxMnO 3 nanoparticles Journal of Applied Physics 121, 214303 (2017); 10.1063/1.4984829 Magnetic properties, exchange bias, and memory effects in core-shell superparamagnetic nanoparticles of La0.67Sr0.33MnO 3 Journal of Applied Physics 121, 173902 (2017); 10.1063/1.4982893 Ultra-low magnetic damping of perovskite La 0.7Sr0.3MnO 3 thin films Applied Physics Letters 110, 112401 (2017); 10.1063/1.4978431 BaTiO 3/SrTiO 3 heterostructures for ferroelectric field effect transistors Applied Physics Letters 110, 232902 (2017); 10.1063/1.4985014 Announcement: Journal of Applied Physics eliminates publication fees effective 1 June 2017 Journal of Applied Physics 122, 010201 (2017); 10.1063/1.4989932Temperature dependence of interlayer exchange interaction in La 0.7Sr0.3MnO 3/SrRuO 3heterostructure V. V. Demidov and G. A. Ovsyannikov Kotel’nikov Institute of Radio Engineering and Electronics Russian Academy of Sciences, Mokhovaya 11-7, 125009 Moscow, Russia (Received 2 May 2017; accepted 21 June 2017; published online 7 July 2017) The parameters of the planar magnetic anisotropy and exchange interaction both in the epitaxial manganite/ruthenate (La 0.7Sr0.3MnO 3/SrRuO 3) heterostructure and in the manganite film (La 0.7Sr0.3MnO 3) have been studied using the ferromagnetic resonance (FMR) technique in the temperature range of 30–300 K. The temperature dependence of bilinear and biquadratic interlayer exchange interaction has been obtained from the angular dependences of the FMR spectra. It is shown that the interaction is determined by the short-wavelength oscillations of the interlayerexchange which are associated with the magnetic ordering of the interlayer between the ferromag- netic layers. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4991500 ] I. INTRODUCTION In this paper, we consider heterostructures from manga- nite films La 0.7Sr0.3MnO 3(LSMO) and SrRuO 3ruthenates (SRO) in which the features of oxide heterostructures aremost strongly manifested. Both materials, LSMO and SRO,have a perovskite crystal structure, which ensures epitaxialgrowth of film heterostructures. The Curie temperature ofthe LSMO film is in the range of 350–360 K. The magnetic anisotropy in addition to the biaxial inherent in the cubic structure of the LSMO film has an induced planar uniaxialanisotropy reaching hundreds of Oe at room temperature onthe (110) NdGaO 3substrate.1,2In turn, SRO films are ferro- magnetic metals with high conductivity and a Curie tempera-ture of about 150 K. A strong uniaxial magnetic anisotropyof/C2410 kOe (Ref. 3) makes it possible to use a SRO film in multilayer film structures as a layer with a rigidly fixed direction of magnetization in external magnetic fields up tofew kOe. In addition, the direction of the axis of easy magne-tization in the SRO film is determined by the strain thatarises during the epitaxial growth of the SRO film. Thisdirection can vary from 0 /C14to 90/C14with respect to the plane of the substrate,4depending on the substrate material. In partic- ular, epitaxial SRO films grown on NdGaO 3substrates have an easy axis direction along the normal to the substrateplane. Special attention should be paid to the fact that in thiscase the direction of the easy axis does not change withdecreasing temperature down to T ¼10 K. 4We note that in the case of the most widely used substrates from SrTiO 3 (STO), the easy axis is directed at the angle of 45/C14to the film plane near the Curie temperature and, almost linearly withtemperature, rotates by 15 /C14to the normal with decreasing temperature to T ¼4K .3,5Note that the easy axis changes its direction in LSMO films when using the STO substrate.6,7A distinguishing feature of SRO is the negative spin polariza-tion of free carriers. 3The negative sign of the magnetic polarization of SRO films was confirmed in experiments on hybrid heterostructures SRO/STO/Al8and magnetic tunnel junctions in trilayer films LSMO/STO/SRO9and LSMO/ LMO/SRO.10Here LMO is LaMnO 3.The bilayer LSMO/SRO structures belong to the so- called spin valves, which are intensively studied, because their use has led to the discovery of new interesting physicalphenomena, as well as the prospects of their application in various elements of spintronics. Recently, on a heterostructure consisting of a cuprate superconductor YBa 2Cu3O7-dand a spin valve, whose role was played by the two-layer structureof LSMO/SRO, a magnetic proximity effect was observed. 11 The gist of this effect lies in the fact that the magneticmoment penetrates into the superconducting part of the heter-ostructure from the composite ferromagnetic layer. 12,13 Recently, bilayer structures that consist of a layer with a strong perpendicular magnetic anisotropy in contact with a plane magnetic anisotropy have been investigated for their use as promising elements of storage cells in high-density magnetic random access memory (MRAM).14,15Much atten- tion is given to the trilayers F1/f/F2, in which F1 and F2 are ferromagnets with a Curie temperature of T C/C241000 K, and f is a spacer of a weak ferromagnet with a T Cclose to the room temperature. It is shown both theoretically and experimentally that rich spin-thermoelectric manipulations and as well as oscillations of the magnetoresistance in the frequency range from 0 to /C241 GHz can be expected on such structures.16–20 An important role in all of the above-mentioned effects is played by the interlayer exchange interactions that arise at the interface of neighboring ferromagnetic layers. It is known thatinterlayer interactions lead to a hysteresis shift of magnetic hys- teresis loops. 21This technique has been successfully applied for investigations of interlayer exchange17,19and, in particular, in the study of bilayer LSMO/SRO structures grown on STOsubstrates. 22,23However, in our opinion, a more informative method of investigating interlayer exchange is ferromagnetic resonance (FMR)14,19,24,25and especially when it is supple- mented with SQUID measurements.26 The purpose of this work is to obtain and analyze data on the magnitude and mechanisms of the formation of aninterlayer exchange interaction at the interface of manganiteLSMO and ruthenate SRO. Preliminarily, in a wide range of temperatures, the parameters of the planar magnetic anisot- ropy of manganite films were studied. 0021-8979/2017/122(1)/013902/6/$30.00 Published by AIP Publishing. 122, 013902-1JOURNAL OF APPLIED PHYSICS 122, 013902 (2017) II. SAMPLES AND MEASUREMENT PROCEDURES Epitaxial films of LSMO, SRO 10–100 nm thick and heterostructures from them, as well as with the LMO spacer, were deposited on the (110) plane of a single-crystal sub-strate of NdGaO 3(NGO) by laser ablation using a KrF laser emitting at wavelength k¼248 nm. The growth of films took place at the temperature of 600–800/C14C and an oxygen pres- sure of 0.2 mbar. The quality of LSMO films controlled by X-ray diffraction patterns and FMR spectra was quite high: the width of the LSMO reflex curve (002) did not exceed0.05 /C14, and the width of the FMR of LSMO films at room temperature was less than 50 Oe. The parameters of the mag- netic anisotropy were determined from the angular depend- ences of the FMR spectra, which were taken in a parallelorientation, when the external magnetic field was in the plane of the sample at all times, during its rotation around the normal. The FMR spectra were recorded using a standardBruker ER 200 magnetic resonance spectrometer, operating in the X-band ( x/2p¼9.6 GHz). To change the temperature, we used the Oxford ESR 900 cryogenic prefix. The method for determining the parameters of the mag- netic anisotropy consisted in processing the angular depend- ences of the resonant fields of the FMR spectra. The solutionof the linearized Landau–Lifshitz–Gilbert equation is usedfor the evolution of the magnetization Min an external con- stant magnetic field Hunder the action of the magnetic com- ponent of the radio-frequency field. This solution gives ananalytic connection between the external resonance field H 0 and the frequency x _under FMR conditions2,27 1þa2 ðÞx c/C18/C192 ¼/C18 4pM0þH0þHucos2uu þHc1þcos22uc 2/C19 H0þHucos 2uu ð þHccos 4ucÞ: (1) Here, ais the damping parameter, cis the gyromagnetic ratio, M0is the equilibrium magnetization, Hu¼2Ku/M0, Hc¼2Kc/M0, Ku is the uniaxial anisotropy constant, and Kc is the biaxial cubic anisotropy constant. As a result, the val- ues of Ku,Kc, and M0, as well as the angles between the easy axis of the uniaxial anisotropy and the external magnetic field uuand between the easy axis of the biaxial cubic anisotropy and the external magnetic field ucare determined from the angular dependence of the magnitude of the reso- nant magnetic field H0. Both easy axes lie in the plane of the substrate. III. EXPERIMENTAL RESULTS The temperature dependences of the uniaxial and biaxial magnetic anisotrop y for LSMO(50)/NGO, SRO(28)/ LSMO(40)/NGO, LSMO(40)/SRO (14)/NGO, and LSMO(30)/ LMO(13)/SRO(65)/NGO stru ctures are shown in Fig. 1. Here, the thickness of the layers in nm is indicated inside parentheses. It can be seen from this figure that all the film structures usedhave similar temperature depe ndences of the biaxial magnetic anisotropy [see Fig. 1(a)]. The result obtained is natural, sincethis anisotropy is due to the cubic symmetry of the crystal lat- tice of the LSMO substance itself, and therefore is weakly affected by the influence of neighboring layers. Earlier it wasshown that the temperature dependences of the biaxial mag- netic anisotropy of LSMO epitaxial films are practically inde- pendent of the substrates on which they were grown. 6 At the same time, the temperature changes in the values of the uniaxial magnetic anisotropy differ markedly both in magnitude and in the nature of the dependence. It was previ-ously found that LSMO epitaxial films grown on NGO have a maximum uniaxial magnetic anisotropy at room tempera- ture and exceeds the biaxial anisotropy by more than100 Oe. 28It can be seen from Fig. 1(b) that this excess remains the same order with decreasing temperature. In addi- tion, the uniaxial magnetic anisotropy in the single-layer LSMO/NGO structure predominates over the biaxial mag-netic anisotropy with a decrease in temperature, at least up to T¼30 K, which is not observed on other film structures. 6,7,29 In addition, the uniaxial magnetic anisotropy in the single- layer LSMO/NGO structure exceeds the biaxial magneticanisotropy as the temperature is lowered, at least to T ¼30 K, that is not observed in other film structures. 6,7,29The latter fact is important for the use of these structures in the devel-oped elements of microelectronics operating at low tempera-ture. In addition, it should be noted that our experiments demonstrated the invariability of the direction of the axis of easy magnetization of the uniaxial planar anisotropy in thestructure of LSMO/NGO with decreasing temperature. Let us turn to a more detailed consideration of the experi- mental data obtained on two-layer structures from LSMO andSRO. Let us consider the temperature dependences of the magnetizations for LSMO films in the LSMO/SRO/NGO and SRO/LSMO/NGO structures. The thickness of the LSMOlayer in both bilayer structures was 40 nm, and the SRO layer was 14 nm in the first case and 28 nm in the later. Figure 2 shows the values of the parameter M 0obtained from fitting the angular dependences of the FMR spectra from the LSMOlayer using formula (1)at different temperatures. It should beFIG. 1. Temperature dependences of the parameters of the planar magnetic anisotropy for the LSMO epitaxial film in the LSMO(50)/NGO (rhombus), SRO(28/LSMO(40)/NGO (circles), LSMO(40)/SRO(14)/NGO (squares),and LSMO(30)/LMO(13)/SRO(65)/NGO (triangles). The thickness of the layers in nm is indicated in parentheses. (a) The fields of cubic anisotropy. The solid line is the averaged temperature dependence of the cubic anisot- ropy field. (b) Uniaxial anisotropy fields.013902-2 V. V. Demidov and G. A. Ovsyannikov J. Appl. Phys. 122, 013902 (2017)noted that the parameter M0in formula (1)can be identified with the equilibrium magnetization only in the absence of exchange interaction with other magnetic layers, that is, at temperatures above the temperature of the ferromagnetic tran- sition TSROin the neighboring SRO layer. The experimental points in this area were used to fit curves calculated from thetheory of the Weiss molecular field, 30which describe the tem- perature behavior of the equilibrium magnetization (see the lines in Fig. 2). It can be seen from Fig. 2that the experimental points clearly diverge from the calculated curves for true magneti- zation M0obtained using the Weiss theory in the temperature range where the SRO layer is in the ferromagnetic state.Moreover, for LSMO/SRO/NGO, the points pass higher, while for SRO/LSMO/NGO, the points are lower than the corresponding calculated curve. One possible explanation is that the magnetization vec- tor of the LSMO layer can come out of the film plane. Indeed, the authors of Ref. 31have shown, using the exam- ple of studying FMR spectra on high-quality epitaxial ferro-magnetic films of iron of various thicknesses, that the magnetization lying in the plane of the substrate at large film thicknesses begins to leave the substrate plane with a decrease in the thickness of the film. Such an evolution of the magnetization is equivalent to a decrease in the demag- netization factor, which in turn leads to an apparent increase in the magnetization using expression (1). To test the possi- ble influence of this effect, we investigated the temperature dependence of the angle of deflection of the external mag- netic field from the substrate plane b, at which the value of the resonance field in the LSMO/SRO/NGO structure is min-imal. In this way, the direction of the easy axis for magneti- zation was determined. We note that the vector of the external magnetic field remained in the plane assigned by the normal and the direction of the axis easy for magnetization of a uniaxial plane anisotropy, determined by the standard procedure at room temperature. 2,27As can be seen in the inset of Fig. 2, the angle bremains practically zero in the temperature range for which the parameter M0for the LSMOlayer in Fig. 2deviates from the calculated curve describing the temperature dependence of the real magnetization. Another explanation is connected with the fact that relation (1)loses its validity under the conditions of coexis- tence of two magnetically ordered layers LSMO and SRO, since the interlayer exchange interaction should be taken into account in the free energy of the magnetic system. In this case, the total free energy per unit area will have theform 32 F¼/C0 d1M1;HðÞ /C0d2M2;HðÞ /C0Ku1d1 M2 1M1;nu1 ðÞ2 /C0Ku2d2 M2 2M2;nu2 ðÞ2þ2pM2 1zd1þM2 2zd2/C0/C1 þ1 2d1M1;^NcM1/C0/C1 /C0J1M1;M2 ðÞ M1M2/C0J2M1;M2 ðÞ2 M1M2 ðÞ2: (2) Here, indices 1 and 2 refer to the LSMO and SRO layers, respectively. It was assumed in writing (2)that due to suffi- ciently large layer thicknesses, the magnetization of each fer- romagnetic layer is uniform in the volume of the layer andthe magnetic moment of each such layer precesses as a whole. The first two terms in (2)describe the energy of the Zeeman interaction. The next pair describes the uniaxial magnetic anisotropy energy with the corresponding constants K u1,u2 and unit vectors nu1,u2. Then there is a term describing the demagnetization energy (the zaxis is directed along the normal to the film), and the contribution of the cubic mag-netic anisotropy characteristic of LSMO with the crystallo- graphic anisotropy tensor ^N c. The last two terms describe the bilinear and biquadratic exchange with the corresponding constants J1andJ2. We have introduced the biquadratic term because a magnetically ordered interlayer with a thickness of several atomic quantities33,34is formed during the contact of the ferromagnetic layers of LSMO and SRO, which pro-motes the appearance of a biquadratic interaction. 35In addi- tion, the orthogonal direction of the magnetizations of free single-layer films LSMO and SRO also requires accounting for the biquadratic exchange. At last, we note that the energy of the interlayer exchange is proportional to the area of thecontacting surfaces, and the Zeeman energies and anisotropy energies are proportional to the volume of the layers. The proportionality of the volume leads to the proportionality of the corresponding terms to the thicknesses of the ferromag- netic layers d 1,2. Before solving the Landau–Lifshitz–Gilbert equation, which determines the resonance ratio of the FMR, it is neces- sary to find the equilibrium directions of the magnetizations in the LSMO and SRO layers. To do this, we must find the minimum of the free energy (2)for spherical angles. Only then, using the angles found, one can solve a system of twocoupled Landau–Lifshitz–Gilbert equations in which the expressions for the effective field H effof each of the layers will have the following form: Heff j¼/C01 dj@F @Mj: (3)FIG. 2. Temperature dependence of the M0parameter of the LSMO epitaxial film in the LSMO/SRO/NGO structures (circles) and SRO/LSMO/NGO (rectangles) determined using formula (1). The lines show the calculated dependences of the magnetizations of the LSMO layer in these structures (see text). The inset shows the temperature dependence of the angle corre- sponding to the minimum resonant magnetic field when the external mag- netic field vector deviates from the substrate plane (see text).013902-3 V. V. Demidov and G. A. Ovsyannikov J. Appl. Phys. 122, 013902 (2017)Here we used the fact that the layers have a sufficient thick- ness at which the magnetization in each layer can be assumedto be homogeneous in volume, and therefore for each layerthe variational derivative can be replaced by a partial deriva-tive. The index jtakes the value 1 or 2. Next, it is necessary to solve the system of linearized Landau–Lifshitz–Gilbertequations for high-frequency magnetizations m j¼Mj–M0j (M0jis the equilibrium magnetization of the j-th layer), which are sought in the form mj(t)¼mj•exp (/C0ixt). Such a calcula- tion technique was used for the situation when the external constant magnetic field H was laying all the time in the plane of the substrate, the magnetic component of the microwavefield was directed along the normal to the plane of the sub-strate, and the rotation of the heterostructure was made aroundthis normal. As a result, a resonance ratio was obtained forthe LSMO layer ð1þa 2Þx c/C18/C192 ¼ðAþHJ1sinH/C0HJ2cos 2HÞ /C2ðBþHJ1sinHþHJ2sin2HÞ: (4) Here we have introduced the notations AandB, correspond- ing to the factors on the right-hand side of the resonance relation (1):A/C174pM01þH0þHu1cos2uuþ0:5Hc1ð1þ cos22ucÞand B/C17H0þHu1cos 2uuþHc1cos 4uc. Besides, HJ1¼J1=ðM1d1ÞandHJ2¼2J2=ðM1d1Þare effective fields acting on the magnetization of the LSMO layer due to bilin-ear and biquadratic exchange with the ferromagnetic layerSRO, and His the angle between the easy axis of magnetiza- tion for the SRO layer and the equilibrium magnetizationdirection M 02of this layer. Assuming that the equilibrium magnetization of the LSMO layer is directed along the exter-nal magnetic field H 0, and taking into account the negative polarization in the SRO layer, we find from the minimum ofthe free energy (2)over all spherical angles that the vector M 02rotates in the plane determined by the vectors nu2and H0, and expression for Hhas the form: sinH¼rHJ1/C0H0 Hu2/C04pM2/C0rHJ2: (5) Here r/C17ðM1d1Þ=ðM2d2ÞandHu2¼2Ku2/M02is the magni- tude of the magnetic anisotropy field in the SRO layer. In addition, it was taken into account that the easy axis of mag-netization for the SRO layer grown on the NGO substrate isdirected along the normal to it 4in the derivation of relations (4)and(5). Thus, we described the experimental angular dependen- ces of the FMR spectra of the LSMO layer on the LSMO/SRO/NGO heterostructure in the temperature region belowthe Curie temperature of the SRO layer using expression (4), taking into account the relation (5). An example of the angu- lar dependences of the experimental values of the resonancefield and the half-width of the FMR line obtained atT¼50 K is shown in Fig. 3. The insets in Fig. 3(a)show two experimental spectra for angles corresponding to the maxi-mum widths of the FMR. These spectra were described byLorentz lines with allowance for both circular components ofthe microwave field. It was assumed that the temperaturedependence of the magnetization of the LSMO layer corre- sponds to a calculated curve obtained from fitting the experi-mental data for high-temperature measurements (see the solid curve in Fig. 2), and the parameter H cvaried in accor- dance with the averaged values shown in Fig. 1(a). In addition, it can be seen from the experimental data shown in Figs. 3(a)and3(b) that the values of the resonance fields and the widths of the FMR lines are comparable witheach other. In this case, the broadening of the FMR line affects the value of the resonant field through the damping parameter a. Because of this, it was assumed that the angular dependence of the damping parameter ais proportional to the width of the FMR line when processing the experimental data. The results of this treatment are shown in solid lines in Fig. 3. Separately, attention should be paid to the values of the angle H[see Fig. 3(c)] obtained in the description of the experimental data by the relations (4)and(5). It is seen from Fig.3(c)that the account of the interlayer interaction deflects the magnetization of the SRO layer from the direction of easymagnetization by a considerab le value. Moreover, for some values of the angle u, the magnetization vector of the SRO layer is oriented almost para llel to the magnetization of the LSMO layer. The latter remark is especially important in thedevelopment of MRAM 14,15and other elements of spintronics. The main result of this work is the obtaining of tempera- ture dependences of the interlayer exchange interaction in the LSMO/SRO/NGO heterostructure. Processing of the angular dependences of the FMR spectra taken at differenttemperatures using relation (4)made it possible to obtain the temperature dependences of the parameters H u2,J1, and J2, which are shown in Fig. 4. Figure 4(a) demonstrates that the magnetic anisotropy of the SRO layer increases practically linearly with decreasing temperature, and its value corresponds to the values charac-teristic for this substance. 3Figure 4(b) shows that the inter- action between the ferromagnetic layers of LSMO and SROFIG. 3. (a) The angular dependences of the resonance field for the LSMO layer in the LSMO/SRO/NGO heterostructure (insets: experimental FMR spectra (noise curves) and fitting curves by Lorentzian (smooth curves), (b) the half-widths of the FMR spectrum, and (c) the angle between the easy axis of magnetization for the SRO layer and the equilibrium magnetization direc- tionM02of this layer H,o b t a i n e da tT ¼50 K. The circles and squares are experimental data and the lines and triangles are the fits using the relation (4).013902-4 V. V. Demidov and G. A. Ovsyannikov J. Appl. Phys. 122, 013902 (2017)in the LSMO/SRO/NGO structure has a ferromagnetic char- acter (J 1>0). Note that the conclusion about the ferromag- netic interaction between the LSMO and SRO layers in theLSMO/SRO/NGO structure could already be done from thedata for the M 0parameter shown in Fig. 2, which is expressed in their deviation upward from the magnetization curve. There are different theories that explain the existence of an interlayer exchange interaction. In our case, the tempera-ture dependence can be satisfactorily described using theappearance of interlayer exchange by conduction electrons ofthe interlayer according to the theory of RKKY (Ruderman–Kittel–Kasuya–Yoshida), which leads to the relation 36 J1TðÞ¼J10ðÞT=T0 sh T=T0ðÞ: (6) A fitting according to this formula is shown in Fig. 4(b) with a dashed line. There is quite a satisfactory coincidence.However, the value of the parameter T 0¼40 K obtained as a result of the fitting differs significantly from the expected value. Indeed, the characteristic temperature T0is of the order of/C22hvF=2pkBdin this formula, where vFis the Fermi velocity for electrons participating in the interlayer exchange, kBis the Boltzmann constant, and dis the thickness of the interlayer. In our case, vF/C24108cm/s,37,38which gives the value of T0 /C24103K for d/C2410/C07cm. The solid curve in Fig. 4(b) is obtained by the best fitting according to the formula J1ðTÞ¼J0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1/C0T=T0p (7) with the values J0¼(1.1460.09) erg/cm2and T0¼(154 67) K. The relation (7)occurs in the case of short- wavelength oscillations of the interlayer exchange associatedwith spin-density oscillations in a magnetically orderedinterlayer between neighboring ferromagnetic layers. 35,36 We have already noted above that when a ferromagnetic layer LSMO and SRO contact, a ferromagnetically orderedinterlayer with a thickness of several atomic quantities is formed. 33,34In this case, the parameter T0has the meaning of the ordering temperature of this layer, and its value is nat-urally close to the Curie temperature of the SRO layer.The temperature variation of the constant of the biqua- dratic interlayer interaction [see Fig. 4(c)] is satisfactorily described by a linear dependence J 2ðTÞ¼J0 2ð1/C0T=T0Þ (8) with the values J0 2¼(1.060.1) and T0¼(162613) K, which, like in the case of J1, is close to the Curie temperature of the SRO layer. Thus, it can be asserted with sufficient certainty that in our LSMO/SRO/NGO structure, the interlayer interaction has a ferromagnetic character, and the temperature depend- ences of the constants of this interaction indirectly confirmthe presence of a magnetically ordered layer arising in thecontact area of the SRO and LSMO layers. At the same time, the values of the parameter M 0at T<TSRO in the two-layer structure SRO/LSMO/NGO, obtained by processing the angular dependences of the FMRspectra according to the formula (1), lie below the true value of the equilibrium magnetization (see Fig. 2). This fact indi- cates the antiferromagnetic nature of the interlayer interac-tion. The same effect was observed in a similar bilayerstructure, grown on a substrate of STO, by other scientificgroups, 22,23which also connect the apparent decrease in the magnetization of the LSMO layer by the antiferromagnetic interaction of adjacent ferromagnetic layers. Unfortunately, we did not find in the literature data on the direction of theequilibrium magnetization in the SRO layer grown on theLSMO film, which did not allow us to make a correct numer-ical analysis of the experimental data for the SRO/LSMO/NGO system. In the future we hope to solve this problem. IV. CONCLUSION The uniaxial and cubic anisotropies of epitaxial manga- nite LSMO films grown on a substrate of neodymium gallateincrease with decreasing temperature, and the uniaxialanisotropy remains dominant in the temperature range of30–300 K, and the directions of the easy magnetization axesretain their orientation. The FMR resonance relation for theLSMO/SRO/NGO heterostructure, taking into account bothbilinear and biquadratic contributions of the exchange inter-action between two ferromagnetic layers, was obtained. The magnitude and nature of the interlayer interactions in the bilayer LSMO/SRO structure in the temperature range of40–150 K were determined from a comparison of the experi-mental data for FMR parameters with fitting curves obtainedfrom the Landau–Lifshitz–Gilbert equation. It is shown thatthis interaction is determined by the short-wavelength oscil-lations of the interlayer exchange, which are associated withthe magnetic ordering of the interlayer between the ferro-magnetic layers. The obtained results confirm the presenceof a magnetically ordered spacer that arises in the interface between the SRO and LSMO layers. ACKNOWLEDGMENTS The authors are grateful to V. A. Atsarkin, I. V. Borisenko, A. B. Drovosekov, and D. I. Kholin for usefuldiscussions. This work was supported by the programs of theFIG. 4. Temperature dependences of the magnetic anisotropy field in the SRO layer Hu2, bilinear J1and biquadratic J2exchange constants in the structure of LSMO/SRO/NGO. 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5.0042437.pdf

Phys. Plasmas 28, 050901 (2021); https://doi.org/10.1063/5.0042437 28, 050901 © 2021 Author(s).Lithium, a path to make fusion energy affordable Cite as: Phys. Plasmas 28, 050901 (2021); https://doi.org/10.1063/5.0042437 Submitted: 04 January 2021 . Accepted: 28 April 2021 . Published Online: 20 May 2021 A. de Castro , C. Moynihan , S. Stemmley , M. Szott , and D. N. Ruzic COLLECTIONS Paper published as part of the special topic on Papers from the 62nd Annual Meeting of the APS Division of Plasma Physics ARTICLES YOU MAY BE INTERESTED IN Transport in non-ideal, multi-species plasmas Physics of Plasmas 28, 050401 (2021); https://doi.org/10.1063/5.0048227 Turbulence in space plasmas: Who needs it? Physics of Plasmas 28, 032306 (2021); https://doi.org/10.1063/5.0041540 Forward scattering and filamentation of a spatially smoothed laser pulse in the hydrodynamic and kinetic frameworks Physics of Plasmas 28, 052701 (2021); https://doi.org/10.1063/5.0043931Lithium, a path to make fusion energy affordable Cite as: Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 Submitted: 4 January 2021 .Accepted: 28 April 2021 . Published Online: 20 May 2021 A.de Castro,a) C.Moynihan, S.Stemmley, M.Szott, and D. N. Ruzica),b) AFFILIATIONS University of Illinois at Urbana-Champaign, 104 S. Wright Street, Urbana, Illinois 61801, USA Note: This paper is part of the Special Collection: Papers from the 62nd Annual Meeting of the APS Division of Plasma Physics. Note: Paper VT2 1, Bull. Am. Phys. Soc. 65(2020). a)Authors to whom correspondence should be addressed: alfonsodcc11@gmail.com and druzic@illinois.edu b)Invited speaker. ABSTRACT In this tutorial article, we review the technological, physics, and economic basis for a magnetic fusion device utilizing a flowing liquid lithium divertor (molten metal velocity in the range of cm/s) and operating in a low-recycling plasma regime. When extrapolated to magnetic fusionreactor scale, the observed effects of a liquid lithium boundary on recycling reduction, confinement increase, and anomalous heat transport mitigation may offer a fundamentally distinct and promising alternative route to fusion energy production. In addition, this lithium-driven low recycling regime could accelerate fusion’s commercial viability since such a device would be smaller, dramatically decreasing plant andelectricity costs if all technological complexities are solved. First, the theoretical basis of the energy confinement and fusion performance aswell as the related possibilities of low recycling regimes driven by flowing lithium plasma-facing components are reviewed. Then the paperemphasizes the technological obstacles that need to be overcome for developing the necessary systems for such a flowing liquid lithium solu- tion at reactor scale and details how many of these have been overcome at laboratory and/or proof-of-concept scale. Finally, the current and planned scientific and engineering endeavors being performed at the University of Illinois at Urbana-Champaign regarding this alternativereactor option are discussed. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0042437 I. INTRODUCTION If humankind wishes to reduce its dependence on fossil fuels in order to lessen the effects on global warming, climate change, and bio- sphere degradation, the only alternative capable of high energy density,24/7 availability is nuclear power. For fusion energy to be part of thenuclear solution, it would need to follow the same trajectory as fission.This means that to merely provide 1% of the world energy demand,fusion would need to follow an exponential growth phase in which 10fusion power plants should be built by 2060 and a hundred by 2070. 1 For this to happen, fusion reactors need to be cheaper and smallerthan the ITER-like devices which are envisioned for DEMOnstration(DEMO) reactors. There must be a fusion energy device that is lesscomplex, easier, and faster to construct. 2Such an approach would lower the investment risk and construction period of the fusion reac-tor. Furthermore, it would also increase the innovation cycle as thetechnological and scientific knowledge, derived from the experimenta-tion/operation of such prototypes, would be acquired faster. Today, the conventional pathway to magnetic fusion derives from the 1990s state of the art and technology limitations. Exceptingthe replacement of carbon plasma-facing components (PFCs)(divertor plates with tungsten and first wall with beryllium), from the engineering and operational point of view the machine being built is very close to its first conceptual design. 3,4Consequently, the current ITER-like DEMO power plant scenarios consider a minimum size with major radius (R) in the range of 6–9 m.5This solution supposes building larger and more complex reactors than ITER. Consideringthe enormous cost ( /C2420/C210 9dollars) and the prolonged construc- tion time required for the device being constructed at Cadarache (France), a larger future prototype may face even more stringent eco- nomic and practical impediments. Electric power plants have a key metric—the cost of electricity (COE). For nuclear systems, a huge fraction of the COE is the capital cost (investment) for construction. Generally that capital cost is approximately proportional to machine volume ( /C24R3).6Therefore, the path to reduce the COE must be to reduce size while maintaining the same electrical output. Smaller size, high-fusion-power-density reac- tors will need better plasma performance and energy confinement.7In this article, we consider the effects of a possible low-recycling regime driven by a flowing lithium divertor configuration which might solve these issues and provide a path to lower COE for fusion. This would Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-1 Published under an exclusive license by AIP PublishingPhysics of Plasmas TUTORIAL scitation.org/journal/phpproduce burning plasma conditions at high gain factor (Q) in smaller size reactors and also utilize a plasma-facing component (PFC) solu- tion with less replacement and maintenance requirements which areessential for a higher power plant availability. While the theoretical basis of energy confinement and fusion performance of a low- recycling regime driven by flowing lithium PFCs is reviewed, the bulkof the paper emphasizes the technological challenges that have been or need to be overcome to develop the necessary systems for such a flow- ing liquid lithium (FLiLi) PFC configuration. The article is structured as follows. Section IIreviews the problem of energy confinement in magnetic fusion devices that determines the minimum size required for a reactor and how lithium-driven low-recycling regime may move the paradigm to a lower size reactor. Section IIIexplains the power exhaust problem and the positive effects of a lithium PFC solution that may relax the heat handling require-ments of the reactor. Section IVanalyzes the role of the proposed reactor configuration in the reduction of eventual electricity costs. Section Vexplores the required technologies to accomplish a low recy- cling scenario driven by a flowing liquid lithium PFC configuration, showing the works performed and planned at UIUC to solve the associated issues and provide an affordable reactor solution. Finally,Sec.VIoutlines the main conclusions related to this innovative path- way to commercial fusion and makes a case for utilizing JET in an attempt to surpass the breakeven conditions and if the low recycling theory astounding predictions are correct perhaps achieve Q factor performance similar to that planned for ITER, in the next years. II. PHYSICS CONSIDERATIONS OF THE FUSION PROBLEM A. A question of energy confinement and transport In the 1950s obtaining energy from nuclear fusion reactions on Earth in a controlled way was viewed as an extremely hard challenge that needed to overcome many formidable obstacles both from the physical basis and the required technology. 8,9The conditions for prac- tical energy generation from thermonuclear reactions10are frequently summarized in the fusion triple product that for the case of D-T plas- mas is expressed in terms of temperature (T), density (n), and energy confinement time ( sE) of the D-T ionic species, n/C1T/C1sE>2/C21021keV s m/C03 ðÞ : (1) For magnetic fusion, this implies values in the range of n /C241020m/C03, T/C2410–20 keV, and sEaround few seconds. The energy confinement time is the key parameter that represents a direct measure of the effec-tiveness of the plasma to be heated and isolated from energy dissipa- tion, being defined as 11 sE¼W Ph/C0dW dt; (2) where W is the kinetic energy stored in the plasma and P his the auxil- iary heating power injected into the plasma. Its direct extrapolation from present devices to future projected reactors is based on empiricalscaling laws. The parameter sensitively depends on the nature of the energy transport processes in the plasma that are intercoupled and related to intrinsic instabilities that underlay in the plasma properties,its geometrical characteristics as well as the interaction with the mag- netic and electric fields. Additionally, such plasma stability and therelated energetic fluxes are associated with the nonperfect confinement of the plasma and the unavoidable interaction between it and the con-taining device. 12 In plasmas with noncircular cross section, the Lawson triple product (L TP) is found to scale as5,13 LTP¼nTsE/e3H2/C1j7=2/C1B3/C1R2 q3; (3) where B is the magnetic field, R the major radius, q the safety factor, e the inverse aspect ratio (r/R, defined as the quotient between the minor and major radius of the toroidal device), and jthe elongation of the plasma loop. The H parameter is the so-called confinement fac-tor that measures the equivalence with the projected ITER basedenergy confinement empirical scaling ðs ELMy EÞ:14It is important to note that Eq. (3)shows an equivalent scaling of L TPwith H and size (R). Conditions with H >1 would represent improved energy confine- ment with respect to the empirical as sE¼H/C1sELMy E. Therefore, one could try to achieve similar triple product in smaller devices if the Hfactor may be increased in the same proportion in which the reactorsize was reduced. Increase plasma confinement has been pursued since the beginning of research in nuclear fusion. However, the achievement of confinement values well beyond the ITER scaling (H >1.5) is not an easy and/or straightforward question. As we will see later, the use ofa flowing lithium divertor scenario might enable the opportunity toachieve this high confinement route to magnetic fusion. For practical purposes, a Q fusgain factor (Q fus¼Pfus/Ph), which n e e d st ob eh i g h e rt h a nu n i t y( Q ¼1/C17breakeven) to extract net power, is also defined, easily indicating the thermonuclear efficiency of the reactor. In steady state, considering that in the D-T reaction approximately 20% of the energy is taken by the alpha particles, thereis a direct (and mathematically consistent and increasing) relationbetween Qf usand L TP,B ,T ,a n d sE,5,13 Qfus¼5A/C1LTP 5/C0A/C1LTP; (4) where A >0 is a constant. In a magnetic fusion reactor, the plasma needs to be heated and kept hot, isolating it from the reactor walls by using strong toroidal and poloidal magnetic fields, trying to maximize the triple product, and improving the energy confinement time and the plasma temperature (both parameters being globally coup led) until surpassing the breakeven conditions. Accomplishing such a milestone is an exceptionally difficulttask as was early inferred by the first experimental appr oaches that dem- onstrated an energy transport leve l greatly exceeding expectations. 15–17 First considerations assumed that the transport coefficients such as thermal diffusivity ( v) in a plasma could be directly derived from the results obtained by Chapman and Enskog for a diluted gas.18,19 Therefore, for plasma particles under a magnetic field B, with a density n and temperature T, presenting a collision frequency /C23which scales as/C23/C24n/(T3/2/C1m1/2)20and Larmor gyroradius q/C24T1/2/C1m1/2/(e/C1B), the rate of such classical heat diffusion is given by21 vc//C23/C1q2/C24n/C1m1=2 T1=2/C1e2/C1B2: (5) Usually, an associated inversely proportional energy confinement time (sE) to the heat transport coefficient is considered; thus in the classicalPhysics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-2 Published under an exclusive license by AIP Publishingtransport approximation, the energy confinement time is found to scale as sE/C24B2/C1T1=2/C1e2 n/C1m1=2: (6) Unfortunately, the real experimental values of confinement time were remarkably lower compared to those predicted by this classical model. Earlier the enhanced transport in a plasma had been observed byBohm 22when studying magnetic arcs for isotope separation. To explain the anomalous results, he proposed a less favorable scaling for the heat diffusion coefficient ( vB) in the plasma where particle gyrofre- quency X¼e/C1B/(m/C1c) substituted collision frequency in Eq. (5),r e s u l t - i n gi na ne n e r g yc o n fi n e m e n ts c a l i n ga s sE/v/C01 B/X/C1q2/C0/C1/C01/C24B T: (7) Such scaling is considered as a lower limit case for the confinement time values and the associated transport was defined by Taylor as “themaximum value which the transverse diffusion can ever attain.” 23 Bohm diffusion would imply the necessity of considerably moreintense magnetic fields and much larger devices to achieve fusionconditions. The results obtained in the 1950s were more in agreement with the Bohm conjecture, but in the 1960s, experiments in the tokamak T3 in the USSR showed an increase in plasma confinement, resulting inthe achievement of high plasma performance and keV range tempera- tures never registered before. 24Fortunately, in the last decades, toka- mak plasmas have been found to also follow generally lower diffusionrates that have originated the gyro-Bohm scaling 25,26for thermal diffu- sivity ( vGB) and derived energy confinement time. It is the considered for ITER and future reactor performance extrapolations being basedon the ratio ( q/r), v GB¼q r/C1vB; (8) sE/C241 vGB/C24r q/C1B T; (9) where qis the ion gyroradius and r is the minor radius of the device. Such prediction gives increased values for the energy con- finement time in a factor (r/ q/C291) compared to Bohm scaling. Consequently, larger reactors would be associated with higher con-finement times. Although not as ha rmful as Bohm-like transport, this scaling prediction is still anomalously larger when compared to classical expectations. In toroidal devices, these observed trans-port levels are assumed to be the consequence of considerablymore complex processes than simple, original phenomenological explanations of Bohm diffusion. First, toroidal geometry and curvature aspects entail a magnetic field gradient and effects in the particle motion producing gyrocentershift (neoclassical effects 26–28)a sw e l la st r a p p i n ga n db o u n c i n go f particles in banana orbit drifts27during their movement along field lines. Such effects provide additional contributions (enhancement) totransport coefficients. In the same way that classical transportsupposes an ideal, minimum rate of heat loss in a confined plasma,neoclassical theory specifies a lower limit for the transport rate in a device where magnetic confinement is approached by means of toroi- dal geometry. Neoclassical transport is divided into different regimes(Pfirsch–Schluter, plateau, and banana) depending on collisionality[ratio between collision frequency ( /C23) and the frequency of the par- ticles transiting around the torus, x T]. More importantly, the formati on of coherent, macroscopic turbulent structures within the confined plasma is linked to large-scale flows that increase transport and energy losses (by a factoreven beyond an order of magnitude when compared to neoclassicalpredictions). The creation of the t urbulent structures is strongly linked to the transport processes driven by gradients. 29They are considered as the free energy sources that trigger the nonlineargrowth in the amplitude of the microinstabilities. Among themany instabilities linked to different gradients, the ion-temperature gradient (ITG) mode is considered as the main candi-date for explaining the resultant, global turbulent transport, andtherefore the limiting factor to the energy confinement time andthe temperature profiles in the most advanced tokamaks and stella-rators. 30,31ITG mode is also expected to dominate the confinement time in ITER.32Eventually, turbulent transport affects the distribu- tion and gradients within the plasma, thus creating a feedbackscheme that couples the turbulence origin term and the subsequenteffects 33as shown in Fig. 1 . Consequently, the correct modification (inhibition) of gra- dients may be seen as a basic action to mitigate/inhibit turbulence.Reducing turbulence in fusion plasmas is a must and has been amajor area of investigation within controlled fusion research. As aresult, zonal flows (ZF), 34E/C2B shear flow and/or plasma rota- tion35,36have been proposed and used to reduce/ameliorate turbu- lence and increase plasma stabilization. However, none of theseactions rely on the inhibition/reduction of the turbulence source,i.e., the temperature gradients responsible for this turbulent trans-port in the first place. In Secs. II B–II E ,w ew i l ld e t a i lt h ee f f e c t so f a lithium boundary in hydrogen recycling and concomitant edgetemperature gradient reduction. FIG. 1. Feedback scheme for the turbu- lent transport processes driven by gra-dients in magnetically confined toroidalplasmas.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-3 Published under an exclusive license by AIP PublishingB. Influence of plasma material interactions on energy transport The physics of a confined plasma, its transport processes and ulti- mately the energy confinement are strongly influenced by the plasma-material boundary. In the larger machines that are the basis for ITERand DEMO, the temperature gradients are especially important in the plasma-edge region. In such devices, the more promising plasma sce- nario for a reactor is based on the so-called H mode confinementregime. 37It is characterized by the creation of a narrow pedestal in the radial direction with high temperature that abruptly decreases radially toward the plasma boundary, thus presenting a very pronounced tem- perature gradient.38At the same time, the density gradient is damped in such region, normally presenting a much weaker density gradientor even a flattened profile. Both edge temperature and density gradients are affected by the plasma material interactions with the surrounding materials. First due to the “recycling” influx of cold hydrogen atoms (thermalized to wall temperature) returning to the plasma. This source element immedi-ately cools the plasma edge down and at the same time increases its density. The recycling coefficient is defined as the ratio between the returning hydrogen particle flux coming from the plasma-facing sur-face over the total hydrogenic flux from the plasma that impacts suchsurface, R¼ Hydrogenic influx back to plasma from surface Hydrogenic flux to surface from plasma: (10) Such a parameter depends on the nature and chemistry of the chosen plasma-facing material. In ITER-like scenarios based on a high Zrefractory metal divertor (tungsten), this coefficient is generally quite high (0.95 or even close to unity when saturation of tungsten with hydrogenic atoms takes place in the range of few nm which is the pen-etration depth of corresponding ions). However, lithium behavior istotally opposite as it is capable of retaining large hydrogenic content that escapes from the confined plasma, lowering the recycling. Second, the plasma material interaction, originated by electron, ion, impurities, and neutron fluxes impinging the plasma facing com- ponents (PFCs) produces ejection of atoms, ions, and compoundsfrom the wall interfaces by different physical and chemical mecha-nisms (sputtering, evaporation, sublimation, codeposition, etc.). Such impurities reach the plasma edge and may penetrate in the confined plasma, thus contaminating it. Obviously, they also contribute to thecooling of the plasma as they will extract energy during their succes-sive ionizations and may be transported within the confined volume. The maximum fraction of impurities that plasma may contain without collapsing strongly depends on the atomic number of the impurity(Z), being orders of magnitude larger for low Z elements (lithium, beryllium, carbon, etc.) compared to high Z elements such as tungsten. 39,40 C. Effects of neutral recycling reduction by lithium in plasma confinement and performance Lithium absorbs incident hydrogen at a very high rate. Therefore, a fresh lithium surface greatly reduces recycling. The impressive effects of lithium in the plasma boundary of a fusion device were firstobserved in the TFTR tokamak, operating in limiter mode withoutdivertor (circular plasma cross section). It was found that pulsespreceded by lithium pellet injections showed a notable increase in the energy confinement time of the discharge 41in a keV temperature plasma edge.42Although several strategies were implemented in the machine trying to improve wall conditioning and thus lower recycling and increase confinement,43the improvements associated with lithium injection were the most successful. The resultant plasma confinement produced a combination of higher central densities and lower H asig- nals in the plasma edge. TFTR D-T operation also demonstrated that D-T fusion reac- tion yield could be aided by reducing the hydrogen recycling flux from surrounding walls and/or by improving energy confinement with the discovered lithium wall conditioning.44As demonstrated later, both actions were, indeed, intimately related.45Furthermore, the highest values of confinement time ever registered in the machine (factor of two larger than normal discharges with deute- rium) were observed when lithium was injected in the plasma edgein the liquid state, also multiplying the triple product by a factor of 4 46in discharges with peak density and plasma current in the core as well as higher and wider temperature profiles over the plasma cross section. Confinement improvements have been also seen in different rele- vant machines when the global content of neutrals in the plasma edge was reduced and/or controlled regardless of the method utilized to doit, for example, using efficient divertor pumping. 47In JET tokamak operating with carbon walls, the higher48uptake of hydrogen by the graphite surfaces was associated with lower recycling and better plasma performance when compared to the ITER-like wall (ILW) scenario49where the presence of tungsten negatively affected con- finement.50Nitrogen seeding was introduced to induce detachment and limit the tungsten influx acting as a palliative for the confine- ment problems. Nevertheless, restoration of confinement (H /C251) was also possible in JET-ILW using other strategies directly related to hydrogen recycling reduction by means of more efficient pump- ing. For example, when divertor strike point was moved to a region where neutral gas pumping was favored.51Additionally, improved H factor (both in core and pedestal) was observed at lower aver-aged collisionality and peaked core density in conditions where the role of neutral content in the scrape-off layer (SOL) was pointed out. 52In LHD stellarator, helium wall conditioning53and divertor cryopumping54reduced recycling and helped to control the edge density, increasing plasma temperature, and global confinement. Higher recycling conditions were correlated with poorer confine- ment in JT60 tokamak55and in ASDEX Upgrade, the presence of a higher density region in the plasma boundary was linked to a con- tribution from the neutral influx and shown to degrade the pedestal structure, pressure, and confinement.56 Hence, there is wide and clear scientific evidence from the most relevant worldwide machines that decreased hydrogen recycling and neutral content in the edge/SOL improves energy confinement and plasma performance. In the case of lithium, the triggering of these effects appeared to be caused by its chemical affinity by hydrogen iso- topes. This fact leads to the massive and efficient trapping of hydro- genic ions escaping from confined plasma and the suppression of the subsequent returning influx of cold neutrals to the plasma edge. Consequently, plasma edge cooling is directly avoided and an auto- matic increase in the edge temperature may be logically expected with outstanding implications in the physics of the confined plasma and itsPhysics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-4 Published under an exclusive license by AIP Publishingenergy and particle transport as it is directly linked to a decrease in the temperature gradients. D. Lithium-based reactors as possible low recycling, high-confinement route to more compact fusion The effect of hydrogen recycling reduction driven by a lithium absorptive boundary in reactor performance was first analyzed theo-retically by Krasheninnikov et al. 57They considered the absorption of the plasma flux in the edge by lithium walls in a zero-recycling regime.Then, they analyzed the subsequent global heat transport from the plasma core to the edge. The work showed that in this ideal approxi- mation, the main physics of the plasma transport processes is funda-mentally distinct. As recycling is reduced, the particle influx in theedge decreases such that there are no cold fuel particles that create thetemperature gradient. The subsequent low recycling plasma regimewas characterized by striking predictions. These assertions includedthe presence of a hot, low collisionality plasma edge with strong sup-pression of the temperature gradient from the core and the notableexpansion of the high-temperature plasma volume able to producefusion power. At the same time, the inhibition of the strong gradients in electron and ion temperatures produce a much more stable plasma where turbulence (crucial ITG instability) would be strongly sup-pressed. Finally, the presence of a hot and homogeneous plasma edgewould be also linked to the presence of a more even, finite current den-sity profile. This translates into a more stable plasma with a largerbeta, a parameter directly proportional to the triple product (and thusto be maximized in any fusion reactor). Beta ( b)i sd e fi n e da st h er a t i o between plasma pressure and magnetic pressure, scaling as b/C24 n/C1T B2: (11) Consequently, in such an operational regime, the fusion performance in ITER-like scenarios would be considerably enhanced. All these outstanding benefits may be possible due to the unique atomic and chemical characteristics of lithium. It is the lowest atomic number (Z ¼3) element capable of being used as a divertor plasma- facing material, thereby minimizing the effective charge in the plasma(Z eff). Its strong affinity for hydrogen isotopes, where lithium acts as electron donor, leads to the formation of Li-H chemical hydride ionicbonds. Additionally, it possesses a low melting point (180 /C14C), excel- lent heat handling capabilities, and a very low first ionization energy(5.39 eV). The total trapping of hydrogen isotopes in liquid lithiumhas been demonstrated in ion beam and reactor relevant flux linearplasma device experiments up to a temperature of 400 /C14Ci nt h el i q u i d lithium that was progressively converted into hydride.58,59Concerning the nature of this hydrogenic retention on liquid lithium, it is interest- ing to note that in lithium conditioned carbon walls covered by thinfilms, the presence of oxygen, associated with the lithium atoms, playsa major role within the retention process. 60However, the scenario pro- posed here (a flowing lithium divertor with liquid thickness in therange of mm supported in a pure metallic substrate and with no pres-ence of graphite) is quite different when compared to the researchwith mixed carbon-lithium walls. In any case, the unavoidable pres-ence of impurities on the liquid lithium flowing within the divertormay actually play a role in the hydrogenic uptake, but if the lithium is renovated (thus reducing the presence of impurities and passivationlayers) and carbon is not present at all, it seems logical to think that the dominant mechanism may be more in agreement with the hydrideformation results found by Baldwin and Doerner. 59 On the other hand, the diffusion of hydrogen isotopes in liquid lithium61and the renovation of the liquid surface induced in a flowing liquid PFC make the low recycling regime compatible with a continu-ous operation scenario. The 450 /C14C upper-temperature limit appears mandatory to reduce the lithium evaporative flux that may contributeto the accumulation of lithium in the core and concomitant dilution inthe confined plasma. 62It should be noted that if such dilution in the core exceeds a limit, the fusion power will strongly decrease, thus pre- cluding any possible benefit of the lithium usage in terms of fusion performance. In this respect, any possible measurement of the lithiumconcentration in the core would be very useful to continuously moni-tor such key parameter, although it is necessary to keep in mind thatthis real-time diagnosis will not be trivial. Consequently, the most con-servative and direct action to control the lithium influx will be to keepits temperature below 450 /C14C. Additionally, if such temperature limits are surpassed, the massive lithium vaporization (exponential withincreasing temperature) will cool the plasma edge down by means ofvapor shielding and/or radiation mechanisms. This will create a tem- perature gradient in that region. In this strongly evaporative scenario, the lithium boundary would act in the opposite manner when com-pared to the goal of reducing recycling and edge temperature gra-dients. Therefore, for a low recycling operation driven by hydrogenicabsorption on the liquid PFC, the temperature of the lithium boundarymust be maintained below the indicated massive evaporationthreshold. With respect to this expected flux of lithium going into the plasma by means of evaporation and sputtering, the low ionizationpotential of lithium is responsible for its high sputtered ion fraction(60%) experimentally determined under particle bombardment(Fig. 2 63). In the divertor boundary, these secondary sputtered ions will interact with the plasma edge structure and then may be screened and accelerated back to the PFC surface due to the sheath potential, thus being promptly redeposited and consequently not contributing tothe net erosion rate. Furthermore, even the fraction that is evaporated/sputtered in the form of neutral lithium atoms may be re-deposited ina hot plasma edge due to the very low 1st ionization energy of lithiumand the high-temperature plasma structure that will interact with suchneutral impurities. The usual criterion for prompt redeposition, basedon general impurity transport considerations, establishes that the phe-nomenon will take place when the ionization length in the plasmaboundary of the neutral species is smaller than its Larmor radius ( k Liþ <qLiþ). It may be expressed as vLi hr/C23i/C1ne<102ffiffiffiffiffiffiffiffiffiffi ffil/C1Tip Z/C1B; (12) where v Liis the velocity of the lithium atoms depleted from the surface (it may be just thermal if the atom is evaporated or on the order of thebinding energy if it is sputtered), hr/C23iis the average rate coefficient for the 1st ionization of lithium by electron impact (a parameter thatstrongly increases with temperature), lis the reduced mass of the lithium atom respect to the pro ton, Z the atomic number, and B the magnetic field (in Gauss units) on the plasma boundary. Hence, thevalues for both parameters basical ly depend on the characteristics of t h ep l a s m ae d g e( n e,Ti). In the practice, in a low temperature, detachedPhysics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-5 Published under an exclusive license by AIP Publishingplasma edge that characterizes high Z metal divertors, the criterion for prompt redeposition may be frequently not accomplished and impuri-ties can penetrate into the plasma, affect the core as well as being trans-ported within the plasma and/or migrate to another in-vessel surface.However, in configurations with a hotter edge that would result from the low recycling operation with lithium, a much higher temperatureclearly favors the prompt redepositi on criterion previously exposed. At reactor-relevant divertor scale, a very high prompt redeposi- tion fraction ( /C2195%) of the sputtered lithium has been predicted by the modeling works carried out by Brooks et al. 64Therefore, the prob- lems related to net lithium erosion and concomitant core dilutionwould be strongly reduced. Globally, this high fraction of lithiumprompt redeposition is considered to explain the extremely low con-tamination in the plasma core by lithium observed in every relevantfusion experiment conducted so far. At this respect, it must beremarked that even in the case of the previously mentioned high- performance discharges in TFTR with very hot (keV) plasma edge, the contribution of lithium impurities to the (eventually very low) Z eff value was found to be considerably small when compared to otherimpurities as carbon or oxygen that penetrated further in the plasma. Reducing recycling to zero means that all the effects of edge neu- trals in temperature gradients van ish. In an ideal approach, if such gra- dients are totally suppres sed, the plasma would behave under isothermal conditions. 65The conditions for this theoretical case are much closer to equilibrium when compared to conve ntional tokamak operation since it would have fewer sources of free energy (gradients). The plasma temper-ature would be higher in the edge, showing a flattened profile and thusdiminishing the effects of collisions in transport. Other interesting char-acteristics are the presence of an exp onentially decaying radial density profile (decreasing with poloidal flux), peaking in the center of theplasma but with a very low density close to the separatrix. Such plasmaedge structure is an absolutely opposite case compared to the conven-tional, high Z divertor ITER/DEM O approach and its envisioned high recycling regime where there is a strong temperature gradient close tothe separatrix (beyond the pedestal) and the edge density is high due to the effect of the high recycling as an edge particle source. In fact, in high recycling machines operating in H-mode, the presence of the plasma pedestal has been claimed to be driven by the spontaneous generation of an edge transport barrier (ETB) 66that con- siderably suppresses turbulence in the edge causing higher tempera-ture and energy confinement time of the plasma. Furthermore,different relevant machines have also shown the creation of an internaltransport barrier, 67placed closer to the plasma core in locations where the q factor had an integer value, resulting in a plasma with strong poloidal rotation68and regions with reduced temperature gradients69 that significantly improved plasma performance. In analogy, low recy-cling regimes would be essentially an extreme case of these H-mode conditions, where suppression of external plasma cooling in the edge would allow the extension of the energy content of the “pedestal”wider with higher temperature across a larger plasma volume. In a lithium low recycling regime, 70,71the plasma energy losses would be dominated by particle diffusion rather than conductive thermal diffusivity. Combined with full neutral beam injection (NBI) core fueling (no external gas puffing), the plasma would have only a source of hotparticles in the core and one efficient particle sink in the edge, i.e., the absorbing flowing lithium elements. As formulated in Ref. 72, consider- ing that particle flux involved from the core to the edge would depend on the NBI input, C core/C0edge¼CNBI¼PNBI=e/C1ENBIðÞ ; (13) being PNBIand ENBIthe power and energy of the NBI fueling system, e the elemental electron charge, and a corresponding flux for from the edge to the wall ( Cedge-wall) affected by recycling (R), Cedge/C0wall¼Ccore/C0edge=1/C0RðÞ ; (14) the power evacuated by plasma particles, considering a plasma with averaged temperature (T i,Te) would be FIG. 2. Results of liquid lithium sputtering obtained by Allain and Ruzic. [Graphs adapted with permission from J. P. Allain and D. N. Ruzic Phys. Rev. B 76, 205434 (2007). Copyright 2007 American Physical Society.] On left, it is shown the clear enhancement in sputtering with temperature for the case of Dþbombardment, regardless the energy of the projectiles. However, the total yields tend to clearly saturate beyond 500 eV incident energy with yields well below 1. On the right, the seconda ry ion sputtered fraction is represented, showing that around a 60% of the sputtering is in the form of Liþions in the range of 200–400/C14C for D, He and Li projectiles. As explained in the main text, this particularity in conjunction with the low first ionization energy of lithium has direct consequences in the penetration of such eroded impurities wit hin the confined plasma. At divertor relevant scale, the scenario will be dominated by a very high prompt redeposition fraction for the evaporated/sputtered lithium species, t he fact that will be essential to minimize the detrimental effects in the fusion performance potentially caused by its accumulation in the core and the related D-T dilution.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-6 Published under an exclusive license by AIP Publishing5=2/C1TeþTi ðÞ /C1Cedge/C0wall¼PNBIþPa/C0Prad; (15) with Pradand Pathe power terms given by radiation losses and alpha particle generation. Then, we can obtain the basic low recycling regime relation between plasma temperature, NBI heating, and recycling pro- posed by Zhakarov,71 1=2/C1TeþTi ðÞ ¼1=5/C1ENBI/C11/C0RðÞ /C11þPa/C0Prad ðÞ/C0/C1 =PNBI: (16) Equation (16) shows that if the particle source in the edge is mini- mized, then the average temperature value of ions and electrons, extended widely along the edge due to the reduced gradient, increases to hot values in the range of keV with lower recycling. Intuitively, this expression may be also seen as an immediate consequence of the high diminution of the cooling term in the edge caused by suppression of the cold recycled neutrals, thus creating a plasma that is effectively iso- lated from the wall boundary. The influx of neutrals may be also con- sidered as a source of the so-called plasma-beam instability.73Again, it is important to remember that the necessity of isolating the plasma forits efficient confinement and heating was a major requirement since the very early magnetic fusion ideas, 8,9,15being a clear objective of the plasma surface interaction (PSI) discipline. Because of the isolation from recycled neutrals, the edge temperature would be decoupled from the heat transport characteristics of the core, and the heat flux from the edge to the wall would be dominated by particle diffusion of ions rather than thermal conductivity. This is a substantial difference when compared to high recycling regimes, where heat conduction of ions is frequently affected by ITG modes and concomitant turbulence, showing associated heat losses that are greatly enhanced. Such heat losses originated by the nature of the plasma edge have motivated theuse of more intense heating power trying to increase the temperature o ft h ep l a s m aa n di m p r o v et h ep e r f o r m a n c eo ft h ed e v i c e .H o w e v e r , this “brute strength” strategy alone does not solve the original question as further heating of the plasma core accompanied by strong recycling in the edge may even worsen the temperature gradients and hence the turbulent transport and heat dissipation problem regardless the level of external power injected into the plasma. Moreover, considering a hot, low recycling plasma with low colli- sionality (banana regime), in a tokamak geometry, and approximating the perpendicular velocity of the ions as the ratio between Larmor radius and the ion–ion collision time, Zhakarov gives an expression 74 for the heat flux attributed to ion particle diffusion by integrating overthe toroidal volume, yielding a result that is independent of the high edge temperature. This heat flux would be directly proportional to the square of the edge density, whose profile would decrease exponentially with poloidal flux. 65Following this formulation, the heat flux to the walls would depend on the edge density but not on the (high) edgetemperature resulting from the low recycling regime. It means that a high edge temperature does not proportionally increase the power density to the divertor. Therefore, if the edge density can be externally adjusted (for example, by means of resonance magnetic perturbations as suggested in Ref. 74)up to a sufficiently low and controlled value, the heat flux reaching the divertor PFCs may be made compatible with the power density limits of the lithium boundary and consequently consistent with the previously comments related to the possibility of an excessive temperature rise and the related problems of massiveevaporation, edge radiative cooling, and/or core dilution.The postulated flattened temperature profiles along the plasma edge as a consequence of the changes in energy transport caused by low recycling have been recently observed in lithium Tokamak experi- ment (LTX) tokamak during the termination phase of Ohmic-heated discharges where the gas fueling was ceased in a machine operatingwith plasma boundaries massively covered by lithium. 75The energy confinement time (measured for electrons) was enhanced up to a fac- tor of 3 (200% enhancement) during the discharge when compared to Ohmic heating scaling law with global recycling coefficients decreasing up to 60%. Moreover, during the termination of the shot (where fuel-ing was suppressed and the temperature profile became flattened), the energy confinement time did not decrease with the global density decay. These observations were the first proof of principle confirma- tion of the low recycling regime’s basic predicted feature. However, the extrapolation of such results to reactor scale cannot be directly done without further research at more reactor-relevant scales (in terms of size, geometry, plasma stored energy as well as during longer time-scales). In this respect, LTX has been upgraded to allow NBI heating, thus planning to go deeper in the understanding of the low recycling regime and trying to demonstrate that the achieved novel regime may be supported by total core (NBI) fueling. 76 More recent works have simulated the hypothetical performance of a JET size tokamak operating with D-T fueling and liquid lithium walls. In this remarkable and striking research, it is theorized that a decrease in the global recycling up to R ¼0.5 would be translated into a spectacularly increased thermonucle ar performance, producing fusion power of 23–26 MW and a fusion gain factor up to 5–774close to the maximum Q objectives of ITER. This would occur despite ITER being double JET in size (R) and magnetic field. The continuous (not limitedin time due to passivation/saturatio n on static lithium surfaces) recy- cling reduction in JET by using flowing lithium on the divertor plate would increase fusion output by one order of magnitude compared to the previous D-T JET results (Q /C250.7) 77obtained with a high recycling plasma. Although such astonishi ng results obviously need a sound experimental validation, the possibility of attempting a more modest goal (i.e., only overpass breakeven c onditions increasing the Q factor in a5 0 % ,b e y o n dQ ¼1) by using a flowing lithium divertor to pursue a pronounced confinement gain within the low recycling conditions seems noteworthy. With the proper development of the incipient tech- nology that will be widely exposed in Sec. V,t h e s ee x p e r i m e n t sc o u l d be attempted in a device that is already constructed and operable. On the other hand, examples of the exceptional plasma perfor- mance theorized by Zhakarov are exactly what has been seen in LTX75 (and previously in its predecessor CDX-U, as we will comment later,where the energy confinement was enhanced in factor 3), the only machines where sufficiently low recycling for the achievement of theclaimed regime has been achieved, but also in the primal experience gained in D-T TFTR operation where L TPwas increased by a factor up to 64 with lithium wall conditioning.45Concerning this question, it should be recalled that the mentioned machines (TFTR, CDX-U, and LTX) were limiter devices and the direct extrapolations to a diverteddevice like JET may not be straightforward. Further experimentation in diverted devices with very low recycling conditions driven by flow- ing lithium components at a relevant scale appears essential to demon- strate technology integration as well as confirm the proof of principle to extend the low recycling framework that LTX provided in nondi- verted plasmas to divertor-relevant devices.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-7 Published under an exclusive license by AIP PublishingAny device operating with a high temperature, low recycling regime would also have a higher boperation and a natural and strong enhancement of fusion power as the burning zone of the confined plasma would be increased to a much larger plasma volume, thus opening the path to more compact, smaller, easier and faster to con-struct, and less expensive fusion reactors, where learning, innovation, and feedback time lapses may be strongly reduced as well. The increase in the edge temperature and beta would also entail a higher conductiv- ity of the plasma 78in the boundary that would have beneficial effects related to stability, as already theorized in 1963 for high temperature,collisionless plasmas. 79Furthermore, tearing modes80are mainly caused by high resistivity values in the plasma edge. The eventual effects of such instabilities are linked to plasma disruptions that need to be virtually suppressed in a reactor for economically feasible opera- tion. Disruptions cause dramatic damage of the inner walls of the device. Associated repair and replacement of the PFCs will negatively affect the operational cost and the availability of the reactor.Stabilization of the MHD activity in the edge has been already observed in low recycling experiments with lithium. 81Additionally, ELMs suppression has been achieved with lithium as well because of decreased recycling and the pedestal profile stabilization.82,83 Going back into the original theoretical studies and challenges of the physics of confined plasmas, it is obvi ous that the first key question in the fusion problem was to minimize the energy dissipation to benefit triple product and fusion performance. Low r ecycling particularly achieved by fuel-absorbing lithium walls may ach ieve much better fusion conditions. If the input of cold neutrals to the edge is avoided, the free energy source associated with the created temperat ure gradient is damped. As postulated in the 1960s, in a plasma scenario where the temperature gradient induced transport is vanished, the d iminution of energy losses and heat flux in the divertor may be automatically expected.84Additionally, if tem- perature gradients are significant ly reduced and, at the same time, they are accompanied by exponentially deca ying density gradients, the condi- tions would be the opposite compared to those that drive high thermaldiffusion and intrinsic unstable modes (relative ion T gradients exceeding corresponding density gradients). Moving deeper into the theory, Galeev and Sagdeev 85found a universal, intrinsic instability that is present in any plasma with a finite value for the thermal diffusivity. However, they also showed that ifthermal conductivity is removed from the analysis (i.e., temperature gradient vanishes and thermal diffusion does not play any role in the energy flux), then this instability is damped, opening the door to a more stable high-temperature plasma scenario. This is exactly the basic premise of the isothermal tokamak theory and the recent results obtained by Zhakarov where the dependence of fusion gain obtained in a lithium walled JET tokamak would be small with respect to thevalues of the heat transport coefficients, even in cases where such val- ues are increased by two order of magnitude with respect to the neo- classical ones. A low recycling regime in the ideal case of an isothermal structure is also characterized by an automatic strong poloidal rota- tion 65that may be identified as a sign of natural stability, having the same effect compared to E /C2B shear flows present in the plasma edge for particular conditions on electron density profiles where turbulence and transport are reduced.86The natural, strong peak in density profile at the center will be a characteristic as well in the exposed lithium- driven regimes with core fueling. This density peaking is a factor that has been recently claimed as possible cause of the mitigation of ionheat transport and the concomitant observed improved confinement in W7-X stellarator with pellet fueling.87 Certainly, a low recycling device may be expected to present other microinstabilities, mainly driven by density gradients [although other ones such as trapped electron modes (TEM) may be expected as well]. Another universal instability is also associated with high temper-ature, collisionless plasmas where a density gradient is present. 79As the low recycling theory framework infers, the radial density profile will exhibit an exponential decay and then these density gradient insta-bilities will be automatically present. However, the stability conditions to control it (potentially shear flows and/or induced rotation) appear more benevolent when compared to lower temperature collisionalregimes so any necessary plasma stabilization seems easier to be driven in such lithium driven collisionless regime 88than in present main-line fusion devices with high recycling. E. Confinement improvements worldwide driven by lithium plasma boundary The use of lithium in fusion devices has been considerably extended worldwide as a powerful wall conditioning technique and/or conforming PFCs.89Utilizing lithium in both ways has basically shown to increase energy confinement in all cases. Russian teams started towork in concepts of liquid lithium limiters based on capillary porous systems (CPS) where liquid lithium was supported in a porous mesh 90 for liquid surface stabilization via capillary forces. Soon thereafter experiments were conducted in the Russian T-11 tokamak using a rail CPS limiter and showing the reduction in hydrogen recycling and the good capabilities and robustness of lithium CPS when handling nomi-nal and transient heat loads. 91,92Among many other tests, experiments with liquid lithium have included its utilization in conventional and s p h e r i c a lt o k a m a k sa sw e l la ss t e l l a r a t o r sb ym e a n so fc o a t i n g so rL iinjection, 81,82,93–95additional CPS limiters96,97(in TJ-II stellarator and FTU tokamak) and flowing ones,98,99showing different grades of ben- efit and improvement in the plasma performance mainly dependingon plasma-surface area covered by lithium and/or its amount present in the plasma boundary, factors directly linked to the recycling reduc- tion. Notable results in relevant machines include the achievement ofquiescent, novel edge localized mode (ELM) free mode operation in long duration discharges in Experimental Advanced Superconducting Tokamak (EAST) 100,101and the suppression and pacing of ELMs by lithium coatings and pellet injection in the NSTX-U plasma edge.81,82 Regarding flowing liquid metal solutions, slow-medium flow with both flat surface (FLiLi, Flowing Liquid Lithium Limiter)98and trenched [Liquid Metal Infused Trenches (LiMIT) concept that will bemore widely introduced in Sec. V]p l a t e sh a v eb e e nt e s t e di nt h eE A S T tokamak. Experiments with three different generation FLiLi plates have been conducted. The concept is based on the flow of a thin layer(/C241 mm thickness) of lithium over a rectangular plate that is inserted in the midplane of EAST for plasma exposure. The flow of lithium was provided by external electromagnetic (EM) pumping (velocities inthe range of cm/s) and is aided by gravity. 98,99The testing has shown beneficial effects in terms of confinement and reduced recycling, han- dling heat fluxes up to 3.5 MW/m2.102,103For example, the L-H transi- tion threshold was found to progressively decrease during the operation with the flowing limiter, enabling longer H modes and improving the energy content of the plasma. At the same time, hydro- gen recycling and impurity plasma content were reduced and notably,Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-8 Published under an exclusive license by AIP PublishingELM size and frequency were also progressively diminished. LiMIT concept incorporated a characteristic trenched surface that improvedthe wetting and global coverage of the plate substrate also offering [thermoelectric magnetohydrodynamic (TEMHD)] capabilities for self-flow of the liquid lithium along the PFC surface. LiMIT alsoincreased the heat handling capabilities clearly beyond 4.5 MW/m 2 (Ref. 104) appearing as a robust limiter that did not present any noticeable damage after the testing. Complete details about the LiMITconcept, its testing as PFC and the technological evolution of the pro- totypes are addressed throughout Sec. V. On the other hand, the strong effects of reduced recycling in energy confinement time were observed in the CDX-U tokamak. In experiments combining full solid coatings on the walls and liquid lith-ium limiters, they found that decreasing recycling (with R Hcoefficients around 50%–60%)105increased the energy confinement time up to values that exceed the ITER scaling by a factor of 3. These high con-finement conditions were driven by the suppression of the plasma edge cooling by thermalized neutral fuel atoms that characterize high recycling regimes, where the formation of strong temperature gra-dients and concomitant turbulent structures that enhance energytransport and losses take place. These CDX-U results showed that con- ditions for the low recycling plasma regime may be created by using lithium surfaces that actively absorb the particle fluxes in the edge,thus eliminating the cold particle source responsible for the thermal conduction driven by gradients and the associated poor confine- ment. 81Those results were later amplified by measuring a flat temper- ature profile in its successor LTX as commented in Subsection II D.75 III. POWER EXHAUST IN FUSION DEVICES AND RELATED SCENARIO WHEN USING A FLOWINGLIQUID LITHIUM BOUNDARY A. General considerations In fusion devices, the nominal power loading from plasma will be mostly concentrated on the divertor PFCs. They will need to survive enormous heat and particle loads, not only in steady state, but alsoduring transients [vertical displacement events (VDEs), disruptions, runaways, and ELMs 106] During operation, the unavoidable erosion of exposed materials must be limited to ensure a sufficient lifetime whenusing solid PFCs. For ITER, its tungsten divertor establishes a maxi- mum nominal power load around 10 MW m /C02.107For DEMO-like devices the longer pulsed operation and the associated higher neutronfluence will constrain the limits to a value of 5 MW m /C02.108Given a maximum allowed erosion for the tungsten divertor elements of 5 mm thickness in two years’ lifetime,109slow transients surpassing the limit will be only allowed for short times (10 s) for a limited number of cycles. Type I ELMs must be suppressed, and disruptions and/or verti- cal displacement events totally avoided.110High Z tungsten elements are not compatible with a hot plasma edge, so to accomplish this oper-ational scenario, the temperature in front of the divertor plates needs to be sufficiently low (5 eV) to stay below the sputtering threshold. For a tentative, high Z, high recycling DEMO reactor operating with Q fus¼10, where heat losses to the divertor will be dominated by strong gradients beyond the pedestal, this power to be exhausted willbe in the range of 500 MW. 109To limit the heat flux to the divertor, a high fraction of the power must be radiated. Radiated power exhaust fraction (f rad) needs to be around 90%–95% both in the divertor (detachment111,112) and core (higher Z seeding).113Obviously, this willdegrade the confinement in the core and needs to be carefully con- trolled so as not to surpass the L-H threshold and/or induce radiative collapse [such as Multifaceted Asymmetric Radiation From the Edge (MARFE) events114] and at the same time be sufficient to mitigate the heat flux to divertor elements until the 5 MW/m2established limit. This high radiation scenario and its tight safety margins are yet to be demonstrated and will need an extremely efficient control as any fail- ure will be notoriously harmful for the reactor operation and the integ- rity of the inner walls. The primary factor to evaluate the power finally reaching the sur- face of the divertor is the parallel heat flux across the separatrix (q sep) that follows this expression: qsep/C24Psep=R/C1kq/C1Bp=Bt/C0/C1; (17) where P sepis the total power across the separatrix, R is the major radius of the machine, B pand B tare the poloidal and toroidal magnetic fields, andkqis the exponential power decay length in the SOL. This parame- t e ri sc r u c i a lf o rt h ee s t i m a t i o no f the peak heat load deposited on the divertor target. Smaller values of kqwill cause higher heat fluxes and more challenging power exhaust handling in the divertor. Predictions or estimations of this parameter are essential to eval- uate the design of future fusion devices (i.e., ITER and DEMO). Multimachine investigations (Eich scaling) have demonstrated that kq scales inversely with poloidal magnetic field115and does not scale favorably with machine size (R). Since power at the separatrix scales faster than /C24R, and the power decay length does not follow the P sep/R scaling,108these results have a significant impact in future machines as P sepwill be signifi- cantly increased, so to achieve a constant heat flux, an increase in kq would be needed. ITER experts group assessed values of kq¼5m mt o be achieved in partial detachment conditions for an acceptable plasma performance with tolerable material damage and suitable power dissi- pation.107Unfortunately, that value is a factor 5 larger than direct extrapolations of the Eich scaling and consequently determining ahighly radiative operation scenario as previously commented. B. Liquid lithium and high temperature SOL scenario Theoretical work on the scaling of the divertor power decay width has been carried out by Goldston considering a heuristic-drift model 116in low puffing H-mode plasmas in which the divertor would significantly act as a particle sink. The model estimates the resultant SOL kqwidth as the product of the residence time of the ions in the SOL and their perpendicular averaged drift velocity, resulting in a scal- ingkq/C24qp/C1e,w h e r e qpis the ion poloidal gyroradius ( /C24Ti1/2/Bh)a n d ethe inverse aspect ratio, then the scaling may be assessed as kq/C24e/C1Ti1=2=Bh; (18) where B his the poloidal magnetic field. This expression poses an explicit relationship that analytically agrees with the Eich scaling115 with respect to the magnetic field dependence and it also favors config-urations where ion temperature is enhanced. In the lithium low recycling proposed configuration, the liquid surfaces would act as an outstanding particle sink resulting in an edge temperature much higher compared to high recycling ITER-like devi- ces. The ITER-like SOL will be highly collisional with maximum tem- peratures on the order of 100 eV beyond the pedestal, which needs toPhysics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-9 Published under an exclusive license by AIP Publishingdiminish to <5 eV in front of the divertor targets. In a lithium regime, the SOL conditions would be almost collisionless with a high fraction of trapped particles.117Consequently, the particles reaching the diver- tor plates by pitch angle scattering, in reactors with a very hot(/C2410 keV) temperature and conversely tiny density in the SOL, would have k qthat would also scale with Larmor poloidal ion gyroradius.118 Such differences increase the power width by a factor offfiffiffiffiffiffiffiffiffiffi ffi Thot SOL Tcold SOLq .T h a t value may be as high as an order of magnitudeffiffiffiffiffiffiffiffiffiffiffiffi /C24keV 5/C010 eVq/C18/C19 .T h e resultant hot edge-SOL structure, where ion temperature gradients and associated turbulent transport would be strongly reduced, would have had one order of magnitude lower divertor heat loads, relaxing the power exhaust handling requirements of the divertor solution.Therefore, a low-recycling lithium-divertor machine could be smallerwith a higher power density and have no worse of a power load than that imagined for ITER and DEMO. It appears plausible to optimize such benefits in order to compensate the power exhaust increasederived from the smaller inner area of an envisioned more compactreactor or caused by potential higher magnetic field. 119–121Likewise, it is also worth mentioning that the liquid lithium solution might also be combined with advanced divertor magnetic configurations that offerthe possibility of spreading the heat fluxes over larger divertor areas asdouble null, X, Super-X, Snowflake, or long-legged ideas 122–125as well as with, otherwise, potentially required seeding scenarios.62 These implications on future fusion reactors and associated divertor designs should be of prime importance since the heat exhaustproblem in the divertor continues to be one of the most importantchallenges. Rather than attempting to solve the problem by cooling down the plasma that surrounds the divertor elements to induce detachment and assure the survival of the PFCs, a lithium walled reac-tor would suppose a completely different approach that would attackthe problem from the root. In a completely different physics scenario, the heat flux from the core is reduced as turbulence decreases and con- finement of energy improves. Therefore, the heat loss rate, driven byparticle flux, entering the SOL is damped as a result of the combina-tion of trapping of particles in the region, the very low density values, the reduced temperature gradient, and an exhaust mechanism of pitch angle scattering for the majority of particles. C. Additional power handling benefits The utilization of flowing liquid lithium surfaces as PFCs has addi- tional advantages since the heat load extraction will not be limited to con-duction as in the case of conventional solid PFCs. On the contrary, liquid lithium streams can dissipate power b y means of convection, evaporation and that lithium vapor can shield the divertor through radiation. Enhancement of the heat handling capabilities by convection has been observed in flowing liquid lithium experiments when exposed toreactor-relevant plasmas. 99Massive evaporation at sufficiently high temperatures may produce the creation of a vapor cloud that may be screened in the SOL due to the prompt redeposition of lithium in theplasma boundary. 64Cyclic vapor shielding in lithium PFCs has been observed in Magnum PSI in cold divertor-like plasmas when the tem- perature of the lithium surface increased up to 700/C14C.126If this vapor cloud is sufficiently thick, the heat dissipation may be important espe-cially when combined with lithium radiation produced by the ioniza-tion/excitation of these vaporized atoms. Figure 3 shows such lithiumradiation in the plasma boundary (color red for Li I transition and green for Li II) of EAST tokamak during the previously mentionedexperiments with LiMIT plate. The theoretical considerations 127–130have shown that a signifi- cant power dissipation is possible by means of noncoronal radiation of lithium in plasmas where high evaporative fluxes are present, requiring temperatures clearly beyond 600/C14C on the liquid metal surface. At such temperature, however, no net retention of hydrogen particlesmay be expected on the flowing lithium divertor, thus totally preclud-ing the achievement of lower recycling, but also the lithium contami-nation and accumulation in plasma and/or solid first wall may compromise the reactor operation due to the commented core dilution but also due to fuel codeposition problems on the first wall that, addi-tionally, might threaten the tritium inventory limits inside the reactor(see more details about this specific drawback issue in Subsection VE). For such reasons, this vapor shielding scenario will not be compati-ble with a low recycling divertor configuration able to continuously flow and absorb the hydrogenic plasma exhaust flux. Nonetheless, these phenomena may be extremely helpful in a reactor, when the lith-ium elements face a possible off-normal transient event. Immediate,strong evaporation that will follow the intense localized heating of thelithium surface will be a natural defense mechanism that will protectthe in-vessel elements. The volumetric dissipation of the transient heat will evolve as a radiative collapse rather than as a strong disruptive epi- sode concentrated on the solid wall that will produce irreversible dam-age on solid PFCs. 131In this sense, the exceptional characteristics of lithium to protect substrates have been observed in experimentsexposing lithium PFCs to extremely high, reactor-relevant transient FIG. 3. Lithium radiation (showed in color) within the plasma boundary observed during the LiMIT operation in EAST tokamak. The LiMIT limiter plate is the rectan- gle at the end of the arrow.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-10 Published under an exclusive license by AIP Publishingheat loads at the level of what may be expected in the more cata- strophic transient events.132,133 Finally, it is interesting to mention the lithium vapor box concept proposed by Goldston et al.134as perhaps an extreme case of the lith- ium radiative divertor scenario. In this concept, a liquid lithium pool would be statically confined in a differentially pumped divertor box presenting different sections operating at distinct temperatures.Operating at the proper temperatures it would prevent from excessive lithium vaporization to the plasma by means of lithium condensation on cold ( /C24300–400 /C14C) baffles and, at the same time, would be able to detach the plasma from the box strike points operating at hotter tem- perature ( /C24600/C14C). Recently, the authors have determined that a lith- ium evaporative temperature of 580/C14C would be sufficient to volumetrically dissipate the divertor power exhaust reaching the SOL (/C24200 MW is estimated in this study as conservative value) of a rele- vant fusion power plant operating with this detached divertor configu-ration driven by lithium vaporization and radiation. This conceptual study was carried out developing a power balance and a detachment model in the lithium vapor box. 135The proposed technology is now under development in order to be tested at laboratory and linear plasma device scale. However, the reader should note that the idea is conceptually incompatible with the flowing lithium divertor scenario,as the liquid lithium would be intrinsically static and operating clearly beyond the temperature limits for proper hydrogenic absorption by lithium and associated conversion to hydride addressed in Ref. 59. Consequently, in this concept, the claimed (and outstanding) benefits in recycling and concomitant edge cooling suppression, energy con- finement, and plasma performance enhancement of a low recyclingboundary driven by a flowing divertor configuration may not be cer- tainly expected as the vapor box concept aversely aims to induce the cooling (detachment) of the divertor plasma edge. IV. ECONOMIC CONSIDERATIONS OF A FLOWING LIQUID LITHIUM DIVERTOR SOLUTION Several studies have approached the economics of fusion energy, focusing on the dependence of the electricity cost (COE) with physical and technological aspects. 136,137Such estimations can be averaged and evaluated in terms of COE percentage: direct cost (investment) for the construction of the reactor and associated installations ( /C2460%), cost for the replacement of divertor and blanket elements (20%–30%) andsum of fuel, operation, maintenance, and decommissioning costs (10%–20%). The direct investment cost is clearly increased in larger size machines, being considered directly proportional to machine vol-ume (/C24R 3).10At this respect, it is worth mentioning that the projected size for ITER and DEMO is considerably larger than the actual size of fission reactors. A light-water-reactor (LWR) core can actually fitinside the ITER center stack. LWRs are a mature technology where a $5B capital cost can make a 1000 GWe power plant in 5 years which will last for 40 years of near continuous operation. If the world turnsto a nuclear option to replace fossil fuels, fusion must be on the same order of costs, although it is also necessary to keep in mind that the much higher external cost related to the nuclear fission wastes will alsoaffect to the comparative economic feasibility of both technologies. In a lithium-based low recycling regime, burning conditions and high fusion Q gain could be achieved in smaller reactors compared to the case of high recycling DEMO scenarios. Just considering the pre-dictions 74for JET tokamak operation where it might produce a similarfusion performance compared to the expected in ITER-like Q factor could be obtained in a machine with half the size and approximatelyhalf the magnetic field value, resulting in a construction cost that w o u l db eo n l ya ne i g h t hw h e nc o m p a r e dt oad o u b l e ds i z em a c h i n e .A s the capital cost of reactor construction and its depreciation is around60% of COE 137,138the associated reduction in a factor 7/8 might imply an outstanding decrease in COE around 50%. This assertion will be valid if we consider fusion devices with the same magnetic field whose construction cost would approximately scale with R3as inferred in Ref. 6. For devices with increasing magnetic field, however, the effect of such parameters should be taken into account as the higher field will increase the associated cost of the m agnets and other related subsys- tems. In this respect, perhaps a direct cost scaling with R3/C1B2(expres- sion proportional to the stored magnetic energy) may be more realisticfor the case of higher magnetic field reactors. It is also important to note that the reactor size and concomitant plasma volume will also affect the fusion power that may be extracted from the reactor. The dependence of the fusion power with engineer- ing (external values of toroidal field, major radius, and inverse aspectratio) and internal performance parameters such as safety factor q andnormalized beta may be written as 121 Pfus/C24b2 N/C1B4/C1R3/C1/C154 q2: (19) On the other hand, as inferred in Ref. 139, the maximum reactor per- formance given a minimum device size will be reached by operating at highest possible values of Greenwald density fraction, normalized beta, and also H factor. Both beta and H are expected to be clearly increasedin a lithium low recycling regime, 42,57,71hence the influence of the lower size in the fusion power might be compensated through such effects. Additionally, Costley et al.139scanned the fusion performance parameters (triple product and Q fus) depending on size for fixed values of P fus¼200 MW, Q /C245 and usual Greenwald density, aspect ratio, and beta values (being all these values ITER range). They showed that operating at high confinement (H ¼1.5), ITER range Q factors might be achieved in smaller devices around R ¼3 m (JET size) with a higher magnetic field in the range of 8 T, being 5.3 T the toroidal on axis mag-netic field of ITER. This magnetic field value is smaller compared to other scenarios where engineering and stability advanced, high field tokamaks are considered. 121 More interestingly, if confinement might be increased up to H values of 1.9 being accompanied by high elongation spherical shape,the magnetic field requirements would be clearly relaxed until very plausible levels comparable to JET metrics. Such confinement increase is clearly within the enhancement seen in machines operating withwalls massively coated by lithium 75,105than even increased the H fac- tor threefold. In this way, the doors would be open to improve gener- ated power (compensating in this way the effect of a lower machine size) by moderately increasing the magnetic field [P fus/C24B4as Eq. (16) shows] up to ITER values that would not compromise the main actualconcerns about plasma stability and would not worsen the power exhaust scenario where k qis expected to scale inversely with poloidal B. In this aspect, a liquid lithium solution would ameliorate such kq constraints giving an opportunity to increase the magnetic field with- out compromising the power exhaust requirements. Another important economic/technological challenge for fusion reactors is related to the extreme heat/particle exhaust conditions andPhysics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-11 Published under an exclusive license by AIP Publishingthe very energetic generated neutron fluence and their effects in the PFCs and structural materials. Radiation damage on breeders and pro- gressive deterioration/destruction of solid PFCs will determine the global machine duty cycle and their possibilities to be economically competitive. Consequently, the use of advanced materials with longerlifetime will be clearly necessary to reduce the cost of fusion electricity up to affordable levels. 140The approach based on a flowing liquid lith- ium divertor can increase the lifetime of such components, as the liq- uid surface exposed to plasma will be immune to permanent deterioration and destruction. If liquid loop technologies141for injec- tion, pumping, tritium recovery as well as cleaning and recirculation of the liquid PFC are developed, the liquid lithium component can be continuously replaced, purified, and recycled back inside the reactor in real-time. This advanced configuration would be especially important in the case of the divertor material, whose replacement frequency is established at two years even considering the most optimistic scenar- ios.112,113However, it is important to bear in mind that the technologi- cal readiness level (TRL) of the mentioned systems currently corresponds to prove of concept. Future research will need to raise the technologies until higher TRLs consistent with more realistic reactor scenarios when compared to potential power plant applications. Additionally, the effects of these incipient technologies on the operat- ing costs may be expected, although the arguments presented in this chapter suggest that the benefits in reactor size and direct constructioncost would be more important. Conceptual studies have suggested a minimum availability of 75% for a fusion power plant to be competitive. 142Furthermore, engi- neering analysis envisions as baseline scenario a two-year lifetime for divertor and four-year replacement of the breeder blanket elements,being this replacement coincident with the second one of the diver- tor. 143The remote handling replacement strategy entails six months period to replace just the blankets being equal compared to the time to replace the blankets and divertor because the divertor elements must be removed first in any case. It also considers a shutdown time forreplacing only the divertor of four months, also adding a one-month cooling period before each replacement and one month of condition- ing and pumping before reactor restart. The resultant approach will consist of a total, periodic cycle of 48 operational months in a total period of 62 months that would fulfill the availability demands with a global value of 77.8%. A self-replenishing liquid lithium PFC reactor solution may entail a considerable reduction in the cost and shutdown time associ- ated with divertor replacement, also increasing the duty cycle and the availability of the power plant. Considering as conservative assump- tion that the lifetime of the divertor may be increased up to the level of the blanket one (thus passing from two years to four years), the main-tenance cycle would contain only one reactor shutdown of six months after 4 years operation. In Fig. 4 , a radial chart comparing both opera- tional/maintenance scenarios for conventional DEMO and flowing liquid lithium option is presented. It shows that availability of the plant would be increased up to 85.7% for an equivalent operational time(48 months) that would be completed in a total period of only 56 months, result that implies an 8% equivalent reduction in COE. The dwell time (pumping, conditioning, etc.) is reduced a 50% (from 4 months to 2 months), the shutdown time for replacements is reduced a 40% (from 10 months to 6 months), and the number of divertor replacements would be a half, so the associated cost would be reducedin the same factor. The cost of divertor and blanket replacement is approximately assessed around 23% 137and 30%138of the COE, con- sidering as conservative estimation that the specific contribution ofdivertor replacement may be a half of the blanket one, the 50% reduc-tion in divertor replacement would be directly translated in an addi-tional 3.8%–5% COE reduction. Globally, a high confinement smaller reactor solution with a liq- uid lithium divertor, in which power generation might be similar com-pared to ITER size machines by means of beta and H improvement aswell as a possibly conservatively larger magnetic field, would be associ-ated with reductions in COE up to 50% for the smaller size and lowerconstruction cost, 5% for the lower divertor replacement requirements, and 8% for the related higher availability of the plant resulting from a less time-consuming replacement and maintenance. In this respect, itmust be specified that the considerations made herein are not basedon existing experimental data (as flowing lithium divertors have notoperated and flowing lithium limiters have only been used a fewtimes). On the contrary, they are based on conservative estimationstaking into account the characteristic self-healing nature of the liquidPFCs that theoretically opens the possibility of enlarging the PFC life-time. For such flowing liquid metal (LM) solutions, the incipient tech- nologies (widely presented in Sec. V)n e e dt ob ed e m o n s t r a t e da t proper TRL in order to show that they are capable of providing alarger operational time and also to prove that they do not introduceadditional problems that might limit the reactor duty cycle. Another possible technical benefit of this lithium configuration might be related to the extraordinary impurity gettering and pumpingcapabilities of lithium. Pumping of impurities (mainly residual vac-uum and/or seeded species) is carried out in the subdivertor region oftokamaks by means of cryogenic units that saturate quickly and whoseregeneration may compromise the reactor duty cycle. The technicalneeds of cryopump regeneration (possibly affected by ammonia FIG. 4. Radial chart presenting the tentative two different operational and mainte- nance scenarios in a 4-year operation cycle (48 months). With a self-healing, self-replenishing liquid lithium divertor, contingency aspects (dwell and shutdown time) could be considerably reduced (factors 50% and 40%), dividing by two the PFC replacement (and related costs), finally increasing plant availability from 77.8% to85.7% and giving an extra security margin for the minimum fulfillment of 75% thatis considered to make fusion energy minimally competitive.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-12 Published under an exclusive license by AIP Publishinggeneration), where the necessary massive outgassing would obligate to stop the plant operation, is already threatening the much shorter duty cycle of ITER if nitrogen is used as divertor radiator144and similar constraints might be expected in future reactor prototypes operating in complex seeding and/or strong cryopumping requirements. Any reduction in the cryopumping needs would be translated in relaxation in the regeneration needs and/or in the number of necessary units with immediate benefits in the reactor availability and also makes the pumping system simpler with cheaper and less time-consuming main- tenance. Regarding this question, the direct comparison and extrapola- tion of the Li divertor impurity pumping capabilities respect to the usual cryopumping systems in a reactor are intrinsically difficult, being a question that seems to not be addressed yet in the specialized litera-ture. In principle, one could guess that lithium gettering will help in the approached rector scenario but it is also true that the presence of the liquid lithium on the divertor surfaces may be problematic for the cryopump placement/installation and therefore might reduce the effi- ciency of the cryopumping system (loss of vacuum conductance, direct obstruction of the pumping channels by lithium…). Therefore, in order to assess extra cryopumping needs due to the lithium presence, but also to guarantee that the flowing lithium configuration is consis- tent with the cryogenic system, it is important to note that engineer- ing/design efforts toward an optimization/integration of both divertor subsystems will be necessary to eventually develop a proper impurity pumping scenario compatible with a flowing lithium divertor. Finally, when fully operative, an important requisite for high availability of fusion power plants will be an almost total absence of disruptions. 110In this respect, lithium wall configurations may signifi- cantly lower the risk and probability of disruptions. The enhanced flat- tened temperature profiles over the plasma volume would entail a decrease in the turbulent transport and associated plasma instabilities such as resistive tearing modes that drive transient collapse. Even if a disruption happens the liquid material will protect the structural com- ponents of the reactor, thus preventing destructive damages that, oth- erwise would need to be replaced, thus dramatically effecting reactor availability and the COE. V. TECHNOLOGICAL CHALLENGES OF FLOWING LIQUID LITHIUM PFCs W h i l et h i sp a p e rh a sm a d eac a s ef o rl i t h i u mP F C s ,i th a sn o ty e t examined their shortcomings or how to actually make it all work. This section will examine this subject, highlighting the work at Illinois. A. Self-flowing liquid metal PFC concepts and power handling At UIUC, flowing liquid metal PFCs have been developed by employing thermoelectric magnetohydrodynamic (TEMHD) effects to drive self-generated movement on liquid lithium infused in small trenches where a stabilized flow can be used to mitigate high heat fluxes and provide a clean lithium surface to the plasma. This phe- nomenon was first observed by Jaworski and co-workers.145Because of such effects, the thermoelectric current created between two dissim- ilar metals when subjected to a thermal gradient (being provided by the plasma heat flux and the internal cooling of the element) can pro- duce a driving Lorentz force when held in a transverse magnetic field.146Following this approach, in 2011 UIUC researchers designed and constructed the first LiMIT (liquid-metal infused trenches) PFC.The complete theoretical analysis of the TEMHD self-flow concept is detailed in Ref. 147. The first LiMIT design was based on narrow trenches (width /C24mm) infused in a liquid lithium bath (depth /C245–10 mm). This con- figuration was tested at laboratory scale in MW/m2range electron beam exposure experiments, demonstrating stable flow under mag-netic fields in different (vertical, horizontal, and obliquely oriented)configurations, 148,149with flow velocities in the range of 5–15 cm/s regardless of the geometrical orientation of the plate. Figure 5 shows both 3D and cross-sectional schematics of this first LiMIT design aswell as results from calculations of the thermoelectric currents and thespecific force induced in the liquid lithium bulk. Successful testing ofthe LiMIT system has been carried out in the HT-7 tokamak, 150the Magnum PSI linear plasma device151and very recently in the EAST superconducting tokamak104,152where a LiMIT plate was coupled to the preexistent full liquid lithium loop of FLiLi antecessors. In Magnum PSI, the tests showed that the circulating liquid lith- ium was capable of safely handling heat loads of 3 MWm/C02during timescales of 5 s151with velocity measurements (up to 70 cm/s) that were consistent with the predicted TEMHD based models. Testing inHT-7 (2012) was the first full-scale test of the system at high toroidalmagnetic fields (1.6 T). The observed flow velocity /C244c m / s m a t c h e d the theoretical prediction within the error bars. 153Higher fields slowed the flow velocity, but sufficient speed remained to ensure a clean absorptive surface. Even with only a partial fill on the lithium plate (that anyway covered a very small area of the in-vessel area), the globalplasma confinement properties were shown to improve with a 10%increase in confinement time. 150 The experiment in EAST aimed to investigate the lithium effect on the plasma and to explore if the TEMHD concept and thetrenched surface coupled to a loop may help in an actual fusion deviceregarding key issues as heat handling and wettability. The LiMIT plate(Fig. 6 ) was manufactured from molybdenum [titanium-zirconium- molybdenum alloy (TZM)] with dimensions of 320 mm /C2300 mm /C220 mm, with 0.5 mm depth trenches that were orientated vertically and perpendicular to the toroidal magnetic field. The plate was cou-pled to a full lithium loop system including an EM pump, a distributorplaced on the top of the plate, a lithium collector on the base and anexternal reservoir with an injection system for the lithium. Forty shots were done to test the LiMIT plate against different plasma heating power, distance from the separatrix, cooling pressuresin the plate for TEMHD, and operating temperature. Figure 3 shows the plate inserted on the midplane of EAST during plasma operation. Analysis of the data is still ongoing but has demonstrated heat han-dling of 4.5 MW/m 2(being the analysis of high power NBI shots, with expected much larger handled heat flux, still pending), improvedplasma performance, and was not damaged by the plasma. 104,152This LiMIT plate has many similarities to the flowing liquid lithium (FLiLi) system which uses gravity to produce a slow thin flow.98 Improvements to accomplish a power handling even higher are pre- sented in Subsection VC. B. Liquid lithium surface stability and dryout The first fundamental problem derived from using a flowing liquid lithium PFC is the stability of its surface under intense heatloads, magnetic fields, MHD activity, and induced j /C2BL o r e n t z forces. In such a scenario, Rayleigh–Taylor instability (RTI) andPhysics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-13 Published under an exclusive license by AIP PublishingKelvin–Helmholtz instability (KHI) on the liquid PFC can be trig- gered. These instabilities can lead to ejection of liquid metal droplets from the melt unless stabilized by surface tension.154,155Generally, RTI is gravity driven events produced when the interface between alighter and a heavy fluid is perturbed, thus producing bubbling on thelow-density fluid and spikes on the high-density one as a consequence of the conversion of potential energy into kinetic energy. 156On the other hand, KHI instabilities occur when two inviscid fluids are in rel-ative and irrotational motion characterized by a discontinuity on thedensity and tangential velocity profile at the interface. Such disconti-nuity generates a shear flow and induces vorticity on the boundary layer that becomes unstable after the instability grows creating vortex/ spiral structures that eventually eject. The process entails a conversionof kinetic energy (taken from the mean flow) into potential energythat is translated in a relative movement (up and down) of the heavyand light fluids, respectively. 157To combat those phenomena, capillary porous systems, being explored mostly in Russia and Europe, utilize the strong surface tension effects of the porous substrate-liquid metal interface to avoid such instabilities. However, to operate continuouslyin a low recycling regime a macroscopically flowing system to providea fresh, clean absorptive boundary will be needed. By varying trench dimension and exposing a LiMIT system to pulsed plasmas a stability criterion was developed 155which explained previous results in DIII-D and NSTX.154,158It also showed that thin enough trenches ( /C241m mi n width) will not eject droplets. Depending on the average velocity of the liquid metal stream, flowing schemes can be classified in slow-medium flow (the only flow- ing PFC technology tested to date, with velocity in the range of few totens of cm/s) and fast flow solutions being proposed as future configu-rations (velocity in the range of m/s with much thicker, /C24few cm, FIG. 5. 3D and sectional schematics of liquid-metal infused trenches (LiMIT) first concept explaining how the temperature gradient produces a thermocurren t which in-turn cre- ates flow due to a j/C2Bforce. Upper right shows current density and specific force calculations for typical operating conditions. FIG. 6. The LiMIT plate used in the EAST winter campaign of 2019/2020. The front face trenches (left) are vertical and perpendicular to the magnetic field while the Langmuir probesare circled in red. Right: back side of the platewith the cooling lines, heater elements, and thermocouples.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-14 Published under an exclusive license by AIP Publishingliquid metal layers159). Higher velocity flows have a much higher heat handling capability as the fast speed would also increase the contribu-tion of convection in the dissipation, with the liquid lithium streamacting as a coolant itself. However, such velocity and the nature of therelated flow also interplay on the surface stability, so ejection concerns are much more important as speed increases. Consequently, for future divertor concepts considering fast, annular flow of liquid lithium,along the divertor, much more advanced external methods/configura-tions would need to be developed to assure a stable fast-flowing solu-tion in a high magnetic field divertor environment. 160–162In this respect, a recently published paper has addressed a new interestingconcept called “divertorlet” 163that pretends to reduce the velocity of the liquid metal at the free surface by means of multichannel designthat will minimize this flow length. Besides, the fast flow will be main-tained in the vertical direction as opposed to the one that faces theplasma, thus trying to combine the high heat flux capabilities of fastflow schemes with a more stable and conservative liquid metal surfacefacing the plasma. For such comparative, lower complexity reasons, the slow- medium flow has been the more developed to date. Two different vari- eties of plates (flat plate FLiLi and trenched LiMIT) have been tested as flowing liquid lithium limiters in EAST tokamaks in experimentswhere the PFCs were integrated in a EM pumping scheme. For bothplate designs, the system was able to pump lithium and continuouslyrefill the liquid surface facing the plasma within the tokamak at theplasma edge. The slow-medium flow solutions provided a stable flow-ing surface able to operate in relevant tokamak environments,although a number of technological constraints (related to wetting andmaterial compatibility, topics approached in Subsections VD and VG) were found and need to be improved and solved for a longer duty cycle, real reactor solution. 103,152 A second concern related to surface stability is the perturbation of the thickness of the lithium layer, leading to the underlying plate orstructures being exposed to the plasma when bombarded by a nonho-mogeneous, highly localized heat flux. This phenomenon called dry-out 164and produces a depression of the flowing liquid lithium surface where the heat load is concentrated. The resultant thermal gradient on the lithium surface interplays with magnetic field generating localacceleration that depress the liquid interface, exposing the solid surfacebeneath and producing a local increase in the level of the liquid in thedownstream lithium region. Figure 7 shows a sketch of the surface dryout phenomenon.C. New LiMIT geometries that ameliorate dryout, improve surface stability, and power handling The lessons learned from the results of LiMIT testing in the vari- ety of devices exposed in previous subchapters revealed some techno- logical concerns about its feasibility. Main difficulties were related to dryout and nonfull coverage with lithium of the PFC, aspects bothtechnologically key. In order to ameliorate these issues, the dryout ofliquid lithium layers on the components must be mitigated for improving the stability of the liquid metal surface, also assuring a homogeneous coverage of the substrate. These enhancements havebeen shown essential in order to advance configurations able to betested at longer timescales and in more relevant devices. Consequently, during the last few years, Center for Plasma Material Interactions (CPMI) at Illinois has developed different experimentaland modeling works as well as manufacturing techniques in order toexplore potential novel geometrical designs and engineering for the LiMIT elements that can minimize these problems and potentially handle more relevant reactor power loadings ( /C2110 MW/m 2)w i t h o u t deterioration of the elem ents. Computational modeling is being imple- mented for analyzing the TEMHD flow in the proposed elements by using COMSOL Multiphysics in order to predict experimental behavior and select the most advantageous plate configurations for further experi-mentation and designing. The exper iments carried out were centered in the exposure of different designs of LiMIT plates to MW/m 2range heat fluxes provided by an electron beam i n the SLIDE (solid–liquid divertor experiment) facility widely described elsewhere.165 The first prototype consisted of a two-dimensional (2D) surface with different rectangular posts and spaces placed along the plate (see Fig. 8 ). Three different size configurations were tested within this geometry: 1 /C21m m2posts with 2 mm separation, 2 /C22m m2posts with 2 mm separation, and 2 /C22m m2posts with 4 mm separation. The other solution proposes the utilization of a modular 3D hybrid CPS-LiMIT system with partial use of the capillary effect to decrease dryout and flow depression and thus improve the liquid lithium sur-face stability. The advanced 3D porous “ordered foam” configurationswere prepared by using laser methods to 3D print stainless steel struc- tures ( Fig. 8 ). The experimentation was carried out with three different geometrical sizing configurations (see figure caption for details). Regarding the 2D postgeometry, the measured velocities were in the range of 2–8 cm/s in good agreement with the modeling works. Crosstalk flow and swirling along perpendicular direction to the main flow was predicted by simulations and observed in the test ( Fig. 9 ). FIG. 7. Schematic of dryout caused by localized plasma flux in a plate with a flowing liquid lithium layer. [Adapted and reproduced with permission from M. Szott and D. N. Ruzic, Fusion Eng. Des. 154, 111152 (2020). Copyright 2020 Elsevier B.V.164]Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-15 Published under an exclusive license by AIP PublishingIt resulted in an amelioration of the dryout problem that, however, was not totally suppressed.165,166 The overall performance of the 3D ordered foam LiMIT plates was excellent in terms of improved wetting and coverage and also regarding surface stability and dryout mitigation. Additionally, the heat handling capabilities of the solution were improved respect topreviously tested geometries. Figure 10 shows a COMSOL simulation of the experiment show- ing the velocity profile of the lithium flow along the LiMIT module consisting of a 1 /C21m m 23D modular structure with separation of 3m mg a p s . Full coating with effective wetting was achieved on the texturized surface. Rapid swirling of the liquid was monitored showing bulk flowthroughout the structure. Enhanced surface stability beyond the stan- dard LiMIT design was provided by excellent capillary action that, otherwise, was compatible with TEMHD drive. Power exhaust han-dling showed an increase in the known LiMIT-style PFC operating window by 127%, being enhanced from 3 to 6.8 MW/m 2of impinging heat flux. At the same time, the robustness of the element againstdamage or liquid ejection/dryout was excellent. No dryout or ejectionwas observed. 165 Future testing of this PFC concept will occur as a flowing liquid lithium divertor plate for the ST40 tokamak,167in a project with Tokamak Energy Ltd. that will pair the lithium low recycling benefits with the high field, spherical torus approach. The LiMIT element willbe integrated into a loop containing a complete set of accessoryelements that are being designed as well: a load lock section for lith- ium, EM pumps, a lithium reservoir, pumping lines, and distributionand collection units placed within the plate, as well as sensors, flow meters, and feedback systems for a safe, continuous operation. Figures 11and12show sketches of the different designed elements. 168 D. Wetting, spreading and distribution of liquid lithium In flowing liquid metal PFCs, proper wetting on the underlying solid support structure and homogeneous distribution for flow needs to be achieved at temperatures compatible with the material vapor pressure and potential substrate corrosion limitations. As liquid metalspose high values of surface tension this adequate wetting behavior isnot trivial. For lithium low recycling, the upper operational tempera-ture limit is set at 450 /C14C approximately. Additionally, the wetting con- trol will be selective depending on the surface location. Regions ofplasma exposure must always be covered by liquid metal; however, other surfaces should not wet to prevent wicking of the liquid metal away from the desired zones. As seen in experiments performed onHT-7 and EAST tokamaks with both flowing liquid lithium limiters(FLiLi and LiMIT) 102–104,150,152absence of total lithium coverage on the substrate was reported and resulted in non-optimal operation ofthe PFC. During the last few years, CPMI at Illinois has studied the static wetting properties of liquid lithium on fusion-relevant substrates and lithium compounds. 169,170Minimum temperatures about 300–350/C14C were required to wet with contact angle below critical value (defined as90 /C14between liquid droplet and substrate) stainless steel, molybdenum, tungsten, tantalum, and TZM and lithium was found to wet its oxide,nitride, and carbonate compounds at lower temperatures, a result thatimplies that the unavoidable passivation of the lithium layers will notworsen the wetting. During flowing operation, however, spreading and homogeneity in the lithium stream may demand a smaller contact angle of the liquid possibly driven by higher temperatures, althoughexperimentation also demonstrated that flowing liquid lithium tem-perature may be reduced once the metal wets. Additionally, mirror fin-ishing (by means of fine grade, surface polishing) of surfaces hasshown that lithium may wet a surface as soon as it melts. Other suc- cessful methods to improve wetting are glow discharge conditioning of the substrates and the evaporation on these substrates of a thin lithiumfilm. 169Local and efficient heating of the liquid lithium surface in con- tact with the filled plate by an electron gun seems also interesting toimprove the wetting as has been already showed in tin filled CPS. 171 Preventing lithium flow to undesired locations in the machine can be FIG. 8. Left: Example of 2D post geometry consisting in 2 /C22m m2rectangular posts with 4 mm2separation used in the experi- ments and modeling works. Right: Thethree different 3D porous LiMIT geometriestested in SLiDE fabricated by 3D printing technology: (a) design with 1 /C21m m 2 structure with 2 mm separation, (b) the 1/C21m m2structure with 3 mm separation, and (c) the 0.5 /C20.5 mm2structure with 3 mm separation. FIG. 9. Crosstalk flow (shown in color dotted lines) in the 2D LiMIT postgeometry.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-16 Published under an exclusive license by AIP Publishingachieved through local surface modification. By controlling the rough- ness of substrate materials, the temperature at which liquids will wetthose materials can be adjusted. At Illinois, a laser structuring process was developed that induces microstructure and nanostructure forma- tion on the surfaces of stainless steel and molybdenum. On both sub-strates, structuring of the surface was observed to produce ahydrophobic effect that increased the wetting temperature by 80 /C14C.172 Following these laboratory scale lessons, LiMIT trenched geome- try operation in EAST showed good wetting behavior that also assured surface stability, having a global wetting pattern favored by the textur-ized surfaced when compared to FLiLi. Nevertheless, the total, homo-geneous distribution of the lithium layer along the plate surfaceremained challenging. About 87% of the plate 104,152seemed to have had lithium entering and flowing along the trenches, improving the global wettability respect to the flat FLiLi plate (around 70%). Someareas remained absent of lithium coverage, probably due to deficientlithium spreading from the distributor. This finding seemed to be orig-inated by the clogging of a part of the distribution holes due to the pos-sible formation of solid lithium impurities due to passivation. Consequently, to improve the lithium spreading and thus enabling the real full coverage of the PFC with lithium, the design of the distributorwill need to be improved in order to avoid such problems. Such activi-ties are being developed at CPMI to design, fabricate and test differentdistributor designs containing texturized surfaces aimed to enhance the spreading of lithium over all the plate channels by using differenttwo-dimensional geometries as rectangular, rhomboidal, or cylindrical posts. E. Pumping of liquid lithium/lithium hydride (Li-LiH) and hydrogenic extraction As the liquid metal is a conductive fluid, the j /C2B Lorentz force can be used to induce the movement by providing a current in the liq- uid bulk that would also interact with the magnetic field of the fusion device and/or other possible external fields. Through this electromag- netic (EM) pumping scheme, the magnetic drag of the flow on the liq- uid lithium divertor elements is found to scale as Dp MHD/C24r/C1v/C1L/C1B2; (20) where ris the conductivity of the liquid metal, v its average velocity on the element, L is its equivalent (in a hydraulic sense) length, and B the toroidal magnetic field. The total power to produce this flow will be strongly reduced in slow-medium flow PFC solutions when com- pared to fast flow ones as the velocity is reduced by two orders of mag- nitude ( /C24cm/s vs /C24m/s average velocity ranges). As the formulation of the general MHD pumping scheme shows, the absolutely dominant component of the total MHD drag will be the contribution of the pip- ing system rather than the plates/distributor or reservoir as the involved velocity will be clearly larger.98Simple extrapolations of this formulation considering a tentative DEMO-like prototype with major radius of 5 m, minor radius of 3 m, total equivalent length of the pip- ing system (that dominates the total MHD drag) of 200 m (pretty large and conservative value) with a total lithium mass flow of 0.5 kg/s (value that surpasses the necessary rate expressed later in this chapter for continuous hydrogenic extraction in the power plant prototype) flowing through pipes with 5 mm radius and 1 mm wall thickness gives a power requirement (usually calculated multiplying the MHD drag by the total lithium volumetric flow) in the range of 150–200 kW, which is clearly an insignificant fraction of the envisioned GW range power output.110,141Additionally, lithium-driven low-recycling regimes and its confinement enhancement will minimize the require- ments in magnetic field of the fusion device for a given performance. Hence, use of liquid lithium will pose an advantage in terms of MHD FIG. 10. Distribution of lithium velocities on the 3D modular structure obtained withCOMSOL. Values were in the range of2–5 cm/s with extensive cross flow and distribution in both perpendicular direc- tions of the surface. FIG. 11. Preliminary designs of a LiMIT plate (with an approximated plasma exposed area of 500 cm2) and accessory elements for testing in ST 40 tokamak. The installation of the PFC solution, which will cover approximately 1/16 of thelower divertor outer tiles area, is programmed to start in late 2021.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-17 Published under an exclusive license by AIP Publishingdrag and magnetic pumping requirements respect to any other liquid metal option such as tin, tin-lithium, or galinstan alloys where lowrecycling would not be feasible. More important than the energy required to pump the lithium is the amount of deuterium/tritium (D/T) trapped in the lithium and how one can extract it. The composition and nature of the lithium stream to be pumped will be progressively and unavoidably changed.The most important compound that will be formed is lithium hydride 173that precipitates as solid salt in the liquid bulk when solubil- ity limit (1% molar ratio approximately at 400/C14C174)is overpassed. Solubility of other lithium compounds as oxides, hydroxide, or nitride in liquid lithium is very low as well, then they will also contribute to the formation of a slurry stream with different physical properties thatwill affect to its circulation as a fluid. Those impurities pose a much lower value (orders of magnitude) of electrical conductivity, 175ap r o p - erty that is essential and needs to be high to induce efficient and effec-tive EM pumping. Second, the solid impurity particles will settle to the bottom of the PFCs/pipes/elements of the loop, thus producing possi- ble clogging/blocking problems that will affect the overall flow capabil-ities. Furthermore, beyond a given threshold, the system might be incapable of pumping such solid impurities at high concentration, thus needing extra, auxiliary systems for solid filtration and separation.Therefore, experiments have been performed on Li/LiH mixtures to ensure rapid extraction while avoiding LiH buildup. Tritium self-sufficiency is desire di nf u s i o np i l o tp l a n t sd u et ot h e extremely limited external inventories of this isotope worldwide. From a commercial standpoint, this isot ope is only produced in Canada Deuterium Uranium (CANDU) fission r eactors at approximate rates of 130 g/year 176(although some production exis ts related to defense appli- cations), requiring a complicated se paration process to extract it from heavy water. Only two facilities wo rldwide are operative as potential suppliers. For such reasons, tritium is also very expensive (with prices even beyond 100 000 $/gram).177Therefore, any real fusion reactor will need to breed its own tritium fuel by using a lithium-containing com- pound that will be transmuted into tritium when interacting with the fusion neutrons in the reactor breeding blanket units. Tritium is radioactive by means of negative bemission with a lifetime of 12.32 years, so its storage in the reactor must be limited andit needs to be fully used and minimally lost in the components andauxiliary systems. The administrative limits for the in-vessel tritium accumulation at ITER are determined as 700 g of total moveableinventory. 178The tritium breeding rate in the blanket modules is also affected by a low supply margin that limits the total tritium that can be lost from the fuel cycle to approximately 0.1% of the fueled tritium.179 For reactors based on the traditional ITER-DEMO approach (big size,high recycling solid walls, with fusion reactions happening only in thevery plasma core), the maximum burning efficiency in the reactor will be around 1%–1.5%. 180The remaining unused tritium will need to be removed from the gas exhaust system and in-vessel components where hydrogen isotopes may be unavoidably retained by means of different mechanisms (implantation, bubble trapping, codeposition, bulk diffu-sion). To carry out such crucial actions, external tritium processing plants are being designed with a duty cycle that needs to be carefully synchronized with the tritium inventory requirements in PFCs and pumping system. On the other hand, the higher performance of the low-recycling regime is expected to increase the burning effi- ciency 57,71,74in the reactor as temperature and triple product may be extended over much larger plasma volumes. Such more efficient burn-ing will directly relax the recovery requirements of any tritium proc- essing plant. However, in this sense, the most extraordinary possibility of a reactor solution based on flowing liquid lithium PFCs is that effi- cient trapping of hydrogen isotopes in flowing lithium components can uniquely offer a possibility for its control, mobilization, continuousrecovery, and reinjection into the plasma in real time, also adding the advantage of maintaining the lithium plasma facing surface fresh and clean from impurities and thus achieving a stationary operation throughout a full liquid lithium loop as proposed by Ono et al. 141 Within this scheme, the extraction of hydrogen isotopes and impurities absorbed by the floating lithium layer would be carried out continuously by means of lithium/lithium hydride distillation,175a technology whose development is underway at CPMI with the rest of mentioned liquid loop technologies. The concept envisions a station-ary loop with mass flow of liquid lithium in the range of hundreds of g/s circulating at moderate velocities (cm/s) as capable to be sufficient to absorb the hydrogenic flux from plasma, compensate evaporation/ erosion, and conform a steady state liquid lithium absorptive bound- ary in the divertor. To achieve such solution, development of suitabletechnologies at reactor relevant scale needs to be accomplished to FIG. 12. External elements /C3of the loop being designed for the testing of the LiMIT divertor module in ST40 tokamak./C3EM pump, flow meters, valves, and lithiumpumping lines to the machine are notshown for simplicity.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-18 Published under an exclusive license by AIP Publishingenable the critical demonstration of tritium real time recovery and inventory limit control in a lithium PFC configuration, showing that the massive absorption of tritium in flowing liquid lithium compo- nents is not a major drawback, and on the contrary, provides the sin- gular possibility to be used to recuperate and refuel the tritium in real time. It is important to remember that tritium recovery and concomi- tant fuel self-sufficiency are conditions that any future reactor will need to accomplish regardless of its plasma-facing material choice. Such continuous, real time recovery possibility does not exist for con- figurations based on solid traditional materials such as tungsten. Even considering the small short-term hydrogenic retention of tungsten, the long-term diffusion and permeation of H-isotopes will produce the much larger long-term retention (even up to 10% atomic ratio) of radioactive fuel176also implying a more difficult outgassing from the solid walls at much higher temperatures. Real concerns about the effi-ciency of such thermal recovery processes in considered solid materials have been found with investigations showing a difficult time (up to 1 month treatments) and energy-consuming procedure (with desorp- tion needing temperature rises beyond 1000 /C14C for significant release) for the recovery of tritium from solid divertor PFC and codepo- sits.181–183Additionally, it will be impossible to outgas, handle and refuel this in-vessel inventory during reactor operation as the trapped gaseous fuel cannot be mobilized out of the vessel and consequently will be released directly into the confined plasma, dramatically affect- ing its performance. The implementation of the auxiliary tritium recovery technolo- gies from lithium will not require a major additional requirement as a tritium gas processing plant will be needed anyway to process the gas- eous tritium removed from the pumping system and PFCs. Likewise, the temperature requirements for the total outgassing of tritium from lithium are less important (lithium hydride totally decompose below 700/C14C and hydrogen release peaks at lower temperatures) when com- pared to the removal from solid materials, and the liquid loop may directly transform the tritium inventory into a moveable one, prone to be processed faster and more efficiently. However, to approach theproposed reactor scenario in an integrated way, it is necessary to consider that the evaporation/erosion and the coupled migration of lithium out of the flowing divertor may be a possible source of uncon- trolled Li-fuel codeposition in remote hidden or plasma-shadowed metallic, solid first wall regions. Such issue might jeopardize the tri- tium inventory and safe operation of the reactor in the similar way compared to the current concerns with solid materials, but also aggra- vated by the high undesired hydrogenic retention in such regions out of the lithium loop. Concerning this issue, however, experiments in both linear plasma and gas exposure 184,185have inferred that the oper- ation with a hot tungsten first-wall (T /C21400/C14C) would form a very thin (sub-micron size) lithium film where lithium hydride is unstable and thus the long-term fuel retention may be similar when compared to pure tungsten. Consequently, if most of the particle flux is concen- trated and absorbed in the flowing liquid lithium divertor, the influ-ence of this remote tritium codeposition would be very minimal and would not significantly aggravate the problem with respect to a pure W walled reactor. After the exposure to the plasma, the flowing lithium stream will contain a mix of Li and LiH/LiD/LiT as well as different impurities unavoidably originated by gettering/passivation processes (Licompounds as oxides, carbonate, nitride, etc.) and/or corrosion. Beyond a solubility limit (determined by temperature 174)of the hydride in the liquid, the formation and precipitation of solid hydride take place with the separation of two phases: the alpha ( a) phase where liquid lithium is in equilibrium with a minor temperature-dependentfraction of dissolved hydrogen and the beta ( b) phase where stoichio- metric hydride is present. Within this system, dilution or transition of hydrogen from btoaphase, stability of hydride and hydrogenic desorption from both phases depends on thermodynamic equilibrium determined by Sievert’s law, 173whose main parameters are external hydrogen pressure and volumetric temperature, with the thermody- namic decomposition of pure hydride taking place at a temperature around 690/C14C. As commented, the fuel recuperation is envisioned to be carried out by using a thermal method for the distillation of lithium–lithiumhydride mixtures. 175First investigations regarding hydrogen outgas- sing from this kind of mixtures showed that the hydrogen recovery rate strongly depends on the hydrogen fraction and Li-LiH proportion of the mixtures,186indicating that higher LiH fractions determine a larger desorption flux of hydrogen. Consequently, the main idea is to treat LiH enriched streams (previously separated from liquid lithium by centrifugal, filters and/or cold trap technologies) that can be heatedup to 700 /C14C in order to recuperate hydrogen at a rate that may com- pensate the tritium losses expected in future reactors. For such pur- pose, the first generation of Li/LiH distillation column was developed, assembled, and operated at UIUC.187The distillation apparatus basi- cally consists of a base bucket where the Li-LiH mixture is heated by using an inductive system up to 700/C14C and two modular condensa- tion stages (at temperature of 350 and 320/C14C) for pure Li recovery. The outgassed hydrogen escapes from the top of the column through a small sniffer tube that directs it to an analysis chamber. The tempera- ture and hydrogen evolution with time is recorded with thermocou- ples and differentially pumped mass spectrometry (residual gas analyzer, RGA). Absolute calibration for hydrogen gas correlates theobtained RGA signals to total recovery rates of hydrogen. A cross- sectional sketch of the column and its internal elements after testing, showing almost total lithium removal in the base bucket and Li pres- ence mostly concentrated on the first condensation stage, can be visu- alized in Fig. 13 . InFig. 14 , the evolution of desorbed hydrogen flux in the column depending on temperature 188is shown where three different regions are visible (marked in roman numbers), characterized by a different hydrogen outgassing rate and associated temperature. The first one appears at lower temperature and probably corresponds to hydrogen outgassing associated with impurity depletion (mainly residual water). The second one shows a peaked hydrogen depletion rate (approxi-mately 4.3 /C210 22H2molecules/m2s) at a temperature close to 700/C14C. It is the highest desorption rate that probably combines hydro- gen desorption from alpha phase as a result of b!atransitions but also including outgassing from pure hydride regions of the mixture. The operation of the column at this point would need only a distilla-tion area of 0.35 m 2to compensate the fuel losses previously inferred in an ITER-like lithium walled reactor (around 3 /C21022atoms/s [571]). The experimental values (size of the heated bucket section of 62 cm2and necessary heating power of 2640 W) allow to calculate the necessary power density that may be used to extrapolate the necessary power to heat up the previously obtained, reactor-scale distillationPhysics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-19 Published under an exclusive license by AIP Publishingarea, giving a value around 150–200 kW. The third region shows a more constant outgassing rate that may be mainly produced due to decomposition of pure hydride after the total depletion of hydrogen from the alpha phase. It corresponds to a hydrogen outgassing rate of1.72/C210 22H2molecules/m2s. The system operating at this pointwould need an active area around 0.87 m2and power heating around 500 kW. This heating power for the Li-LiH distillation will unavoid- ably impact the economy of the plant with an additional cost. However, the operation scenarios here inferred would suppose (in theworst case) an affordable energetic cost that will be only a 1% fraction FIG. 13. Cross-sectional sketch showing the column and their modular elements and pictures showing the aspect of theinternal elements after Li/LiH distillationexperimentation. FIG. 14. Time evolution of hydrogen out- gassing from Li/LiH registered during a test performed in the distillation column.Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-20 Published under an exclusive license by AIP Publishingof the auxiliary heating power of the plasma (500 kW power for the Li-LiH distillation within a ITER-like reactor, a device that will use 50 MW of heating power for 500 MW of total generated fusionpower). The incorporation of the distillation technology into a full liquid lithium loop at pilot plant scale will require the coupling of this tech- nology with centrifugation and/or filtration capable to concentrate ahydride-rich slurry that will maximize the tritium recovery. In this way, this LiH enriched stream can be diverted from the loop toward the distillation column, and a fraction rich in liquid lithium may bepurified in yttrium filters and/or surface cold traps before returning tothe flowing liquid lithium divertor. For such purpose, the incorpora- tion and coupling of the distillation technology with innovative ideas that may take advantage of the magnetic field of the reactor to inducecentrifugal separation 189may be contemplated. A schematic of the processes, considered for a 3 GW-thermal power plant where tritium recovery needs to balance the fueling requirement (1% burn), is showninFig. 15 . Based on such scheme proposed by Ono et al. , using the mea- sured recovery rates and incorporating mass flow global balances, extrapolations of the total tritium inventory within the loop schemehave been performed. Complete details of such calculations will be published in a specific and separate article being prepared. Generally, the mass flow formulation considers centrifugation to preconcentrateLiT up to 33% and 0.5 g/s tritium fueling rate, resulting in 3.5 m 2of distillation area and 12 MW heating power that would be needed for the separation process. Additionally, assuming a machine with R ¼5, divertor area of 150 m2, total lithium flow rate /C240.4 kg/s circulating with 2 mm thickness, and lithium average velocity of 1 cm/s (based on a 3 m inner radius geometry), the total lithium content of the loop willbe around 55 L with a tritium inventory that will eventually depend on the efficiencies of the centrifugal and distillation separation processes. Scans performed varying such parameters show that it would notexceed 1.2 kg at plausible efficiency values around 60%. Such quantityis within the order of magnitude of the ITER tritium inventory and considerably small when compared to the inventories expected to be necessary for the D-T operation start-up in conventional DEMO sce-narios. In them, the low tritium burnup (1%) expected within a high Zsolid wall-3 GW power reactor will determine a derived scenario dom- inated by tritium fueling/exhaustion that would require higher tritium inventories (up to 21 kg) to assure a continuous operation as process-ing and reserve time in the range of 6 h are expected. 181,190To amelio- rate such constraints, research priorities in the direction of improving the burnup fraction and tritium processing technology have been claimed.191,192In this sense, lithium low recycling ideas envision a burning efficiency much higher, a consequence of the higher temper- ate and enhanced confinement over a more extended confined volume and larger beta,43,57,70thus more efficiently using the fuel and mini- mizing the related tritium requirements to start-up the reactor.Continuous recovery with distillation is a faster and more efficient option that can greatly decrease processing and reserve time as well as the derived start-up inventory. These are all arguments about how theliquid lithium configuration can also help in these nontrivial questionsthat are common regardless of the considered fusion reactor configuration. F. Helium ash exhaust Continuous and efficient exhaustion of helium (He) out of the reactor will be necessary to maintain steady state conditions in the FIG. 15. Integration of the distillation column technology into a stationary loop necessary for a flowing liquid lithium configured fusion reactor envisioned to produce 3 GW-thermal power with 1% tritium burnup fraction. [Figure adapted with permission from Ono et al. , Nucl. Fusion 57, 116056 (2017). Copyright 2017 International Atomic Energy Agency.141]Physics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-21 Published under an exclusive license by AIP Publishingplasma core in any proposed prototype. Projected pumping require- ments appear important at reactor while helium atoms are hard topump by cryopanels due to their low evaporation point and chemical inert nature. However, the gettering of helium by the wall materials of the reactor would unavoidably help in this task, being the role of this codeposition claimed as crucial for any power plant scenario. 193 Trapping of helium in tungsten is limited in time due to material satu- ration at very low retention levels below 1020atoms/m2at tempera- tures up to 700/C14C.194Higher temperatures moderately increase the retention but aggravate the formation of fuzz tendrils and bubbles thatwould be irreversible and considerably damage the tungsten tiles. While a possible toroidal pumping duct idea 71to externally pump the helium ashes in a flowing lithium configuration has been proposed; onthe other hand, conceptual studies postulated that flowing liquid metal components may be also advantageous for inducing continuous helium trapping on lithium driven by strong liquid convection. 195 Experimentally, even in slow-flowing systems, He has been shown tobe retained in liquid lithium, potentially at impurity boundaries. Thisreduction in the helium recycling was observed at Illinois in its flowing lithium retention experiment (FLiRE) facility with liquid lithium streams. 196–198It has also been shown in tests performed by Hirooka et al. , employing moving liquid lithium coatings.199Although the proper scaling of such benefits at relevant tokamak scenario needs to be demonstrated, any contribution in helium pumping by the flowing liquid lithium solution will not be limited in time. When integrated and accommodated with the suitable pumping systems in an opti-mized divertor, the configuration might help in relaxing the external pumping requirements necessary for helium removal at steady state operation. G. Material compatibility, temperature limitations, and safety Liquid lithium is known to corrode many commonly used metals such as copper or aluminum and leaches chromium from steel. 200–202 This situation may be aggravated in flowing configurations due to the induced erosion/abrasion on the solid surfaces. Therefore, these corro- sion considerations are important when approached the proposed reactor scenario in an integrated way. Materials such as tungsten, molybdenum, and 316 stainless steel have shown good or fair compati-bility with liquid lithium at laboratory time scales, thus possibly offer- ing plausible options as PFC substrates. Nonetheless, the utilization of such options for the structural materials might be inappropriate con-sidering neutron activation, cost, or other engineering issues. To study the compatibility with low activation structural candidates and to con- firm the good perspectives with the PFC substrates at longer, reactor-relevant time scales, specific experimental efforts to determine unknown lithium corrosion patterns in unexplored materials and alloys are necessary. In a collaboration with General Fusion company, CPMI has recently developed a novel dynamic lithium corrosion testbed where eight samples of four different materials (privately andconfidentially determined by the company) are immersed and spun in liquid lithium during timescales of 100 h per run. After the testing, the mechanical properties and the chemical changes induced on the sam-ple surfaces are studied by tensile stress testing and surface characteri- zation techniques. First results have been satisfactory in the question of detecting changes in the material surfaces associated with lithiumdeposition and in their mechanical properties, showing the usefulnessof the facility in discerning what material options seem incompatible with lithium and which candidates may be initially considered. However, the 100 h laboratory timescale is not representative when compared to the contact time between the molten metal and the structural materials that are envisioned for a fusion reactor. Theduration of contact of reactor materials with liquid metals is clearly longer by a few orders of magnitude. Therefore, for the encourag- ing materials screened, much longer timescale experimental valida- tions would be the next step within the selection of structural candidates in order to validate the potential substrates that may becompatible with lithium within the temporal window in which an eventual reactor will operate. For a low-recycling PFC solution to work, the lithium must retain the deuterium and tritium except during off-normal events. This limits the exit temperature of the flowing lithium to the 400–450 /C14C range. Therefore, if the lithium is introduced at 200/C14C, the average tempera- ture of the divertor structure would be around 300/C14C. Thermodynamic power extraction efficiency through a boiling water cycle (T-Cold ¼100/C14C) when T-Hot is only 300/C14Ci sa tm o s t3 5 % , and likely lower. This does not preclude efficient electrical generation since 80% of the fusion energy is carried by neutrons, and the blanket can operate at much higher temperatures than the divertor plate sys-tem. The inside of a fusion device is a vacuum, so some degree of ther- mal isolation between the divertor and the blanket is feasible. After all, any fusion device would have to be superconducting, and therefore keep the magnetic field coils at cryogenic temperatures. If it is possible to maintain a temperature gradient between the blanket and the coils, it is possible to maintain a much smaller temperature gradient between the blanket and the divertor. To prevent risks associated with water, cooling systems will prob- ably need to be gas based. Although He cooling capabilities are nor- mally considered below water cooling systems, advanced He coolingconcepts with very high heat transfer rates 203,204are being developed. The exploration and optimization of these innovative technologies appear paramount to open the possibility of achieving divertor rele- vant power exhaust capabilities in the range of the ITER divertor needs (/C2110 MW/m2) using He cooling instead of more conventional liquid water systems. Of course, these technologies will need to demonstrate the projected performance in experimental tests up to the relevanttechnological readiness level (TRL) corresponding to potential reactor installations. In this sense, in a collaboration with Micro Cooling, Inc., advanced prototypes of such cooling systems will be tested, coupled to a LiMIT plate. The configuration basis is the use of texturized plates with trench dimensions and spacing in the submillimeter range (/C24100lm) and innovative manufacturing techniques to configure an actively cooled heat sink that will maximize the contact area of thecoolant (helium) flowing internally through microchannels, thus improving the extraction of the heat flux deposited on the lithium sur- face. Modeling and simulation works have pointed to potential heat handling capabilities at the level of 20 MW/m 2without raising the lith- ium temperature beyond 400/C14C. Remaining below this temperature limit on the flowing liquid lithium components is essential to achieve a low recycling regime, assuring a stationary absorptive boundary also compatible with the limited evaporation and concomitant edge cooling and core dilution. The planned experimentation at the laboratory scale is focused on determining if the proposed geometry is compatible with proper wetting and flow pattern and will experimentally determine thePhysics of Plasmas TUTORIAL scitation.org/journal/php Phys. Plasmas 28, 050901 (2021); doi: 10.1063/5.0042437 28, 050901-22 Published under an exclusive license by AIP Publishingreal heat handling capabilities of the design, thus checking if the power exhaust performance predicted by the simulations (ITER divertor rele- vant) can be achieved. In terms of safety, molten sodium has been used commercially to cool fission power plants. The key is that the molten metal is not underhigh pressure. Every slow to medium flow system attempting to oper-ate in the low-recycling regime has a free surface of lithium exposed tovacuum. Pressure relief is therefore automatic and the maximum pres- sures in the lithium pumping systems will need to be low. Finally, related to diagnostics and other in-vessel components of the reactor, itis necessary to take in mind that even local and temporal temperatureexcursions on the liquid lithium surfaces might produce strong evapo-ration of liquid Li that might directly affect those elements. This mayresult especially important for the case of windows, copper gaskets, brazing joints, and so on. Clever design/engineering of their emplace- ment, as well as proper utilization/replacement strategies, will beessential to assure their long-term compatibility within the lithiumdivertor environment. VI. CONCLUSIONS Lithium’s ability to reduce recycling, increase energy confine- ment, and mitigate anomalous heat transport theoretically opens a dif-ferent and promising alternative route to fusion energy that mightaccelerate the development of commercially viable fusion powerthrough more affordable, smaller size, yet higher-power-density devi- ces. Using this approach, electricity costs might be greatly reduced if all the technological constraints can be resolved. The scientific argu-ments, observed physics, and phenomenology of the lithium low-recycling framework in this narrative offer the possibility of favoringthe thermonuclear efficiency of the reactor, also helping in technologi-cal hurdles related to plasma surface interaction and combating many of the physical/technological challenges that any fusion reactor will face. Lithium as PFC has shown to dramatically improve plasma stabi-lization and energy confinement, ameliorating the problems of turbu-lence and instabilities. Therefore, the utilization of a flowing liquid lithium divertor is a technological scenario that may open the pathway to a less expensive approach to fusion reactors. More compact, smaller reactors will need significant beta and confinement improvements greatly beyond theITER scaling and require the finding of operable, high confinement,high-performance scenarios. In this sense, the flowing lithium divertorapproach is an incipient technology that attempts to enhance thefusion performance by means of a continuous and notable reduction of neutral recycling on the divertor plasma boundary, being an advanced plasma performance scenario that needs to be corroboratedat reactor-relevant scale. With this developing technology operating,the continuous low recycling operation may be envisioned and, if thetheoretical regime is corroborated, the fusion reactor approach wouldbe based on smaller machine size, increasing energy confinementtime, higher edge plasma temperature, burning fraction, and volume as well as enhanced fusion power and Q gain factor. Additionally, a high-temperature edge configuration driven by the low recycling maydiminish the power exhaust handling constraints and the liquid natureof the divertor elements may help in increasing the PFC lifetime andthe power plant availability, utterly decreasing the PFC replacementcosts. It even poses a method to recover tritium in real time and relax its inventory start-up requirements. All these technical considerationsappear fundamental to finally establish the feasibility of a future fusion power plant that may be economically competitive. These benefits offlowing lithium can be combined with higher magnetic field configu- rations and perhaps spherical geometries (high elongation, low aspect ratio) and/or advanced divertor magnetic structures/configurationssuch as double null, super X, or snowflake. To accomplish this approach, new, emergent, and developing technologies need to be demonstrated and developed to the maximum technological readiness level corresponding to a continuous reactor operation. Cutting-edge technological prototypes have been success-fully tested at laboratory and/or mid-size tokamak scale showing potential and encouraging solutions to most objections claimed against lithium utilization. Work at the University of Illinois at Urbana-Champaign is pioneering and leading endeavors with national, inter- national, and commercial partners to demonstrate and meet all the critical major engineering/technological challenges at higher TRL andmove to the reactor scale. This paper has reviewed the work on manyof these aspects and shows that flowing liquid-lithium technology is ready for the next step. One possible next step could be to experimentally explore the low-recycling theory, trying to investigate the accessibility to suchregimes and the enhanced performance at the largest existent relevanttokamak (JET), by means of a LiMIT or FLiLi style flowing lithium divertor PFC solution. Such kinds of divertor plates are currently being designed to be tested on the ST-40 tokamak divertor within a full lith-ium loop configuration. To add such a device to JET as a final lithium campaign for the machine follows the same technological and scien- tific roadmap of previous magnetic devices: the HT-7 and TFTR toka-maks. These final runs achieved their highest levels of performance. The scientific arguments and the accumulated research experience exposed in this review provide the necessary motivation to aggressivelyadvance the engineering/technology necessary to enable flowinglithium-enhanced magnetic fusion devices. ACKNOWLEDGMENTS The authors want to gratefully recognize the funding and/or collaboration opportunities being conducted through contracts with U.S. Department of Energy: Contract Nos. ALPS-DEFG02–99ER54515and DE-SC0020685, Subcontract No. 20DOE001–1 (SBIR program in collaboration with Micro Cooling Concepts, Inc., California, USA), and U.S. Department of Energy Contra ct No. DE-AC02–09CH11466, with the Institute of Plasma Physics-Chinese Academy of Sciences (ASIPP).Funding has also been provided by Tok amak Energy Ltd. (Oxfordshire, UK) and General Fusion (Canada). DATA AVAILABILITY The data that support the findings of this study are available from Tokamak Energy Ltd., General Fusion, and Micro Cooling, Inc. Restrictions apply to the availability of these data, which were used under license for this study. Data are available from the authors uponreasonable request and with the permission of Tokamak Energy Ltd., General Fusion, and Micro Cooling, Inc. REFERENCES 1N. J. 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1.3460639.pdf

Validation of the transition state theory with Langevin-dynamics simulations J. Schratzberger, J. Lee, M. Fuger, J. Fidler, G. Fiedler et al. Citation: J. Appl. Phys. 108, 033915 (2010); doi: 10.1063/1.3460639 View online: http://dx.doi.org/10.1063/1.3460639 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v108/i3 Published by the American Institute of Physics. Related Articles Magnetic anisotropy in ordered textured Co nanowires Appl. Phys. Lett. 100, 252405 (2012) Extraction of dielectric and magnetic properties of carbonyl iron powder composites at high frequencies J. Appl. Phys. 111, 114104 (2012) L10 structure formation in slow-cooled Fe-Au nanoclusters Appl. Phys. Lett. 100, 211911 (2012) Fast switching of a ground state of a reconfigurable array of magnetic nano-dots Appl. Phys. Lett. 100, 192412 (2012) Spatial control of magnetic anisotropy for current induced domain wall injection in perpendicularly magnetized CoFeB|MgO nanostructures Appl. Phys. Lett. 100, 192411 (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 21 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsValidation of the transition state theory with Langevin-dynamics simulations J. Schratzberger,1J. Lee,1M. Fuger,1J. Fidler,1G. Fiedler,1T. Schrefl,2and D. Suess1,a/H20850 1Department of Solid State Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria 2University of Applied Science, Matthias Corvinus-Stra /H9252e 15, 3100 St. Poelten, Austria /H20849Received 10 February 2010; accepted 8 June 2010; published online 11 August 2010 /H20850 Finite-element Langevin-dynamics simulations are performed in order to extract the attempt frequency of small magnetic particles as a function of an applied perpendicular field. The obtainedvalues of the attempt frequency are in excellent agreement with the analytical results of /H20851Kalmykov, J. Appl. Phys. 96, 1138 /H208492004 /H20850/H20852. It is shown that an external field that is applied perpendicularly to the easy axis with a strength of just about 1% of the anisotropy field is strong enough that theframework of the transition state theory /H20849TST /H20850for broken symmetries can be applied. It is concluded that for most realistic structures, the attempt frequency can be numerically calculated by brokensymmetry—TST formulism. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3460639 /H20852 I. INTRODUCTION Magnetic grains in the nanoscale regime are basic con- stituents of various magnetic systems, ranging from mag-netic recording media to biological applications. These smallgrains can be approximated as single-domain particles. Forthe case of uniaxial anisotropy and zero field, the systemexhibits two equivalent ground states of opposite magnetiza-tion, which are separated by an energy barrier /H9004E. According to the Néel–Brown model, the mean time /H9270spent in one of the states obeys an Arrhenius law /H9270=/H92700e/H9004E/kBT, /H208491/H20850 where /H92700−1=f0is the attempt frequency. A detailed knowledge of the average lifetime of a nano- magnet is not only important for predictions of the thermalstability but also for calculating the coercive field at finitetemperatures. 1,2 Recently, various simulation techniques have been suc- cessfully applied to calculate the energy barrier numericallyfor magnetic structures. 3,4However, in order to obtain a full picture for the thermal stability, a detailed knowledge of theattempt frequency is required. Based on the work ofKramers, 5Langer,6and Langer and Turski,7the calculation of the switching rate amounts to the evaluation of the totalprobability current of a stationary nonequilibrium distribu-tion through a surface near the saddle point, which can bedescribed using the Fokker–Planck equation. Kramers de-rived the escape rate of point Brownian particles with sepa-rable and additive Hamiltonians from a potential well for the/H20849i/H20850intermediate to high damping /H20849IHD /H20850limit and /H20849ii/H20850the very low damping /H20849VLD /H20850limit. For all damping regimes, it was assumed that the energy barrier was much larger than thethermal energy. Kramers developed an ingenious method oftreating these two damping limits but mentioned in his paperthat he could not find a general method to obtain a formulathat holds for any damping regime. Much later Mel’nikovand Meshkov 8obtained an escape rate formula valid for all damping regimes what has come to be known as the Kramersturnover problem. The paper is structured as follows: In Sec. II, we will review the results of the calculation of the attempt frequencyfor various damping limits and for different symmetries ofthe potential for a single-domain particle. In Sec. III, we willreview Langer’s approach for the transition state theory/H20849TST /H20850and apply the general multidimensional formula to derive the attempt frequency for a single-domain particle,when the external field is applied perpendicularly to theuniaxial easy axis. In Sec. IV, the analytical results are com-pared with finite-element Langevin dynamics simulations. Asummary and outlook are given in Sec. V. II. ATTEMPT FREQUENCY OF SINGLE-DOMAIN PARTICLE A. Axially symmetric potential In the case of an external field applied, h/H20648, exactly par- allel to the easy axis Brown9derived, by extending the Kramers method to spins where the Hamiltonians are in gen-eral nonseparable, the attempt frequency f 0/H20849h/H20648/H20850=/H9251/H9253 1+/H92512/H20881HK3 2KBT/H9266/H208491−h/H20648/H20850/H208491−h/H206482/H20850. /H208492/H20850 Here, it is important to note that the attempt frequency de- pends on the volume V of the particle and the temperature T.Due to the symmetry of the magnetic system, Eq. /H208492/H20850holds for all values of the damping constant /H20849see Ref. 10for de- tails /H20850. B. Broken symmetry If the symmetry of the system is broken, the completely degenerate class of saddle points transforms to one saddlepoint, and the escape rate for spins now exhibits the sameKramers damping regimes as exists for point particles. The-oretical predictions of the attempt frequency for a nonaxially a/H20850Electronic mail: suess@magnet.atp.tuwien.ac.at.JOURNAL OF APPLIED PHYSICS 108, 033915 /H208492010 /H20850 0021-8979/2010/108 /H208493/H20850/033915/7/$30.00 © 2010 American Institute of Physics 108, 033915-1 Downloaded 21 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionssymmetric potential were first derived for two limiting cases: /H20849i/H20850the IHD limit11,12and /H20849ii/H20850the VLD limit.13Coffey et al.14,15also derived in the IHD limit an explicit equation for the attempt frequency when the symmetry was broken by anexternal field at an angle /H9274to the easy axis. Good agreement of these formulas with experimental results of single-domainparticles was obtained. 15A detailed derivation of the IHD formula for magnetic systems is given in Ref. 16. Coffey et al.10and Déjardin et al.17have also shown that the Mel’nikov–Meshkov formalism can be extended to spins inorder to estimate the relaxation time for nonaxial single-domain particles for all values of the damping constant. Fol-lowing Déjardin et al. 17and Kalmykov,18the attempt fre- quency of a single-domain particle subjected to a magneticfield applied at an angle /H9274to the easy axis for all values of the damping constant can be written as f0,general /H20849h/H20850=f0/H20849h/H20850A/H20849/H9251S1/H20850A/H20849/H9251S2/H20850 A/H20849/H9251Si+/H9251Si/H20850, /H208493/H20850 where /H9251Siis the energy loss per cycle in well iand f0/H20849h/H20850is the attempt frequency in the IHD. A/H20849/H9251Si/H20850can be calculated by using the depopulation factor A/H20849/H9251Si/H20850= exp/H208751 /H9266/H20885 0/H11009In/H208491 − exp /H20850/H20851−/H9251Si/H20849x2+1 /4/H20850/H20852 x2+1 /4dx/H20876. /H208494/H20850 For a single-domain particle, the action Siin the well Ican be calculated by evaluating the integralSi=/H9252/H20886 V/H20849/H9258,/H9278/H20850=V0/H20877/H208491 − cos2/H9258/H20850/H11509 /H11509cos/H9258V/H20849/H9258,/H9278/H20850d/H9278 −1 1 − cos2/H9258/H11509 /H11509/H9278V/H20849/H9258,/H9278/H20850d/H20849cos/H9258/H20850/H20878. /H208495/H20850 The integral is evaluated along a path P 1/H20849the escape contour on one side of the saddle point /H20850and also the escape contour along the other side of the saddle point /H20849P2/H20850. The energy along each path is constant and equals the saddle point en- ergy. The path P 1is shown in Fig. 1by the red curve. For the case of a perpendicularly applied field, Kalmykov et al. ap- proximated the action Siby the following series: Si=1 6K1V kBT/H20881h/H208751−13 6h+11 8h2−3 16h3¯/H20876. /H208496/H20850 The asymptotic solutions for the VLD and IHD limits have been compared with the universal formula Eq. /H208493/H20850in detail by Kalmykov.18In Ref. 18, the validity of the universal for- mula in the entire damping regime is shown by comparing itwith the exact solution obtained by continued fraction meth-ods. C. Crossover between broken symmetry and axial symmetric potentials The nonaxially symmetric asymptotes will not reduce to the axially symmetric ones without adjustment, such ascrossover formulas, which bridge the two asymptotes. Thisfact is stressed in the studies of diluted magnetic samples,where, due to the random anisotropy for some particles, theexternal field is applied almost parallel to the easy axis, FIG. 1. /H20849Color online /H20850Minima and saddle point of the energy of a single-domain particle for two different values of the external field h. The path P1shows the path along the action S1, evaluated in Eq. /H208495/H20850.033915-2 Schratzberger et al. J. Appl. Phys. 108, 033915 /H208492010 /H20850 Downloaded 21 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionswhere for other particles, the symmetry is broken. In Ref. 19, it is stated that these experiments cannot be explained byusing the formulas for the broken symmetry, such as in Ref.14. The crossover problem was solved by Garanin et al. , 20 who developed crossover formulas between the axially- symmetric asymptotes and the asymptotes for the brokensymmetry for various damping constants. The full math-ematical details of the various crossover formulas can befound in Ref. 10. III. TST FOR MULTIDIMENSIONAL SYSTEMS The TST represents a powerful method of prediction of the rate of activated processes. Initially, in order to includefluctuations and disspipation in the TST Kramers consideredin his famous paper a Brownian particle moving along the xaxis, taking into account frictional and random forces im-posed by the heat bath. 5Later, Langer6generalized the Kramers approach to multidimensional systems, applying itto the nucleation of a multicomponent system. Although theTST was originally used for the calculation of the reactionrates for molecules, it has been shown that any system thatevolves from a well-defined initial state to a final state can betreated within this framework. Following the approach of Langer, 6the determination of the attempt frequency amounts to calculating the total prob-ability current of a stationary nonequilibrium distributionthrough a surface near the saddle point. Let us start with asystem, which is described by a set of variables /H9257i,i =1,..., K, which describes the Kdegrees of freedom of the system. In order to be able to calculate the attempt frequency,one has to know the following properties: /H20849i/H20850 the state /H9257i,minat the minimum and the state /H9257i,spat the saddle point, /H20849ii/H20850the free energy E/H20849/H92571,...,/H9257k/H20850at the minimum and at the saddle point, as well as the curvature of the free energy at the minimum and the saddle point, /H20849iii/H20850the equation of motion, so that the dynamics of the system can be described close to the saddle point. Knowledge of the above properties allows one to calcu- late the calculation of the attempt frequency, which can bewritten as f 0=/H9261+ 2/H9266/H90240, /H208497/H20850 where /H9261+denotes for the dynamical prefactor and /H90240is the ratio of the well and saddle angular frequencies. A. Calculation of /H9261+ For the calculation of /H9261+, one needs to know the mag- netic configuration at the saddle point. The prefactor /H9261+is obtained by solving the noiseless linearized equation of mo-tion for a configuration close to the saddle point. For a sys-tem with Nspins, the noiseless equation of motion is given by the Landau–Lifshitz–Gilbert /H20849LLG /H20850equation as/H11509Mi /H11509t=−/H9253M/H11003Heff,i+/H20873/H9251 M0/H20874Mi/H11003/H11509Mi /H11509t, /H208498/H20850 where /H9253=2.210 /H11003105/H20849sA /m/H20850−1is the gyromagnetic ratio, Midenotes the magnetization vector for the spin iin Carte- sian coordinates, and Videscribes the corresponding volume of the spin i. In order to transform the system to a coordinate system that describes the K=2Ndegrees of freedom, we re- write the LLG equation in spherical coordinates with con-stant radius. Furthermore, we substitute H eff,i/H11015 −1 //H92620Vi/H20849/H11509E//H11509Mi/H20850. We get for the LLG equations fk/H20849/H92571, ...,/H92572N/H20850=/H20898/H11509/H9258i /H11509t /H11509/H9278i /H11509t/H20899 =−/H9253 JsVi/H208491+/H92512/H208501 sin/H9258i/H20898/H9251sin/H9258i/H11509E /H11509/H9258i+/H11509E /H11509/H9278i −/H11509E /H11509/H9258i+/H9251 sin/H9258/H11509E /H11509/H9278i/H20899, /H208499/H20850 where /H92572i−1=/H9258i /H92572i=/H9278iandf2i−1/H20849/H92571, ...,/H92572N/H20850=/H11509/H9258i /H11509t f2i/H20849/H92571, ...,/H92572N/H20850=/H11509/H9278i /H11509t/H2084910/H20850 and E=E/H20849/H92581,/H92781, ...,/H9258N,/H9278N/H20850=E/H20849/H92571, ...,/H92572N/H20850. /H2084911/H20850 Since we need to know only the magnetization dynamics close to the saddle point, we can linearize the LLG Eq. /H208499/H20850 around the saddlepoint, /H9257¯k. We get /H20898/H11509/H92571 /H11509t ] /H11509/H92572N−2 /H11509t/H20899/H11015/H20898f1/H20849/H9257¯1, ...,/H9257¯2N−2/H20850 ] f2N−2/H20849/H9257¯1, ...,/H9257¯2N−2/H20850/H20899 +/H20904/H20898/H11509f1 /H11509/H92571¯/H11509f1 /H11509/H92572N−2 ]/GS] /H11509f2N−2 /H11509/H92571¯/H11509f2N−2 /H11509/H92572N−2/H20899/H20904 SP /H11003/H20898/H92571−/H9257¯1 ] /H92572N−2−/H9257¯2N−2/H20899, /H2084912/H20850 where fk=fk/H20849/H92571,...,/H92572N−2/H20850and /H20849/H9257k−/H9257¯k=/H9263k/H20850. At the saddle point of the energy, fk/H20849/H9257¯1,...,/H9257¯2N−2/H20850vanishes. Hence, we rewrite the equation above and get033915-3 Schratzberger et al. J. Appl. Phys. 108, 033915 /H208492010 /H20850 Downloaded 21 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions/H20898/H11509/H92711 /H11509t ] /H11509/H92712N−2 /H11509t/H20899/H11015Hdyn/H20898/H92711 ] /H92712N−2/H20899, /H2084913/H20850 where Hdyn=/H20904/H20898/H11509f1 /H11509/H92571¯/H11509f1 /H11509/H92572N−2 ]/GS] /H11509f2N−2 /H11509/H92571¯/H11509f2N−2 /H11509/H92572N−2/H20899/H20904 SP. /H2084914/H20850 The solution of the dynamics close to the saddle point leads to the ansatz /H9263/H6109=/H9263/H61090e/H9261t, /H2084915/H20850 which describes the exponential change in the magnetic con- figuration close to the saddle point. By inserting the ansatz/H2084915/H20850into /H2084913/H20850,w eg e t /H9261 /H9263/H61090=Hdyn/H9263/H61090. /H2084916/H20850 Due to the construction of the matrix Hdynthe eigenvalue problem according to Eq. /H2084916/H20850has only one positive eigen- value. This positive eigenvalue is /H9261+of Eq. /H208497/H20850. Hence, the problem of calculating /H9261+, which is required to evaluate Eq. /H208497/H20850reduces to the determination of the positive eigenvalue of the matrix Hdyn. B. Calculation of Ω0 /H90240is obtained by evaluating the curvature of the free energy Eat the saddle point and at the minimum. The cur- vature is obtained by calculating the second derivative of theenergy. H stat=/H20898/H11509h1 /H11509/H92571¯/H11509h1 /H11509/H92572N−2 ]/GS] /H11509h2N−2 /H11509/H92571¯/H11509h2N−2 /H11509/H92572N−2/H20899 =/H20898/H115092E /H11509/H92571/H11509/H92571¯/H115092E /H11509/H92572N−2/H11509/H92571 ]/GS ] /H115092E /H11509/H92571/H11509/H92572N−2¯/H115092E /H11509/H92572N−2/H11509/H92572N−2/H20899, /H2084917/H20850 where hk/H20849/H92571,...,/H92572N−2/H20850=/H20849/H11509E//H11509/H9257k/H20850. The ratio /H90240in Eq. /H208497/H20850is obtained by calculating the determinant of the matrix Hstatas /H90240=/H20881det /H20849Hstat/H20850/H20841min − det /H20849Hstat/H20850/H20841sp/H2084918/H20850 From Eqs. /H2084916/H20850and /H2084918/H20850, the attempt frequency can be cal- culated using Eq. /H208497/H20850. We emphasize that these formulas are valid only in the IHD. In order to generalize them to anyvalue of the damping constant, the formalism resulting in Eq. /H208493/H20850can be applied. C. Example: single-domain particle with external field perpendicular to easy axis In this section, we will apply the previous results in or- der to calculate the attempt frequency of a single-domainparticle, where the external field is applied perpendicularly tothe uniaxial easy axis. In Ref. 21, a similar derivation is performed under the assumption of a hard axis anisotropyperpendicular to the easy axis. The total energy of a magneticparticle can be described by the theory of micromagnetics as E= /H20885/H20853A/H20849/H11612u/H208502−KE/H20849eˆeasy·u/H208502−J·Hext−J·Hdemag /H20854dV, /H2084919/H20850 where JS/H20851T/H20852is the magnetization polarization, Ke/H20851J/m3/H20852is the crystalline anisotropy constant, and A/H20851J/m/H20852is the ex- change constant. For a magnetic particle whose dimensions are smaller than the domain wall width, we assume that themagnetization remains homogeneous within the particle.Hence, we will neglect the first term in Eq. /H2084919/H20850.I fw ea s - sume a spherical particle, we can also neglect the last term inEq. /H2084919/H20850. In the following, we assume that eˆ easypoints in the x-direction and the external field Hextpoints in the y-direction. Introducing polar coordinates /H20849ux=sin/H9277cos/H9272, uy=sin/H9277sin/H9272, and uz=sin/H9277/H20850, we get, E=−VK E/H20849sin2/H9277cos2/H9272+2h/H11036sin/H9277sin/H9272/H20850, /H2084920/H20850 with h/H11036=HyJs 2KE. /H2084921/H20850 The two minima of the energy are situated at /H9258min,1=/H9266/2, /H9278min,1=arcsin /H20849h/H20850and/H9258min,2=/H9266/2,/H9278min,1=/H9266−arcsin /H20849h/H20850. The saddle point configuration is /H9258sp=/H9278sp=/H9266/2. The contour plot of the energy for two different values of the external field h is shown in Fig. 1. Substituting Eq. /H2084920/H20850into Eq. /H2084917/H20850, we get at the mini- mum Hstat,min =/H208752h2VK e+2 /H208491−h2/H20850VK e 0 0 2/H208491−h2/H20850VK e/H20876, /H2084922/H20850 and at the maximum Hstat,sp=/H208752hVK e 0 0 /H20849h−1 /H208502VK e/H20876 /H2084923/H20850 Using Eq. /H2084922/H20850and Eq. /H2084923/H20850for the evaluation of Eq. /H2084918/H20850, we get for the statical prefactor /H90240=/H208811+1 /h. Substituting Eq. /H2084920/H20850into Eq. /H208499/H20850and Eq. /H2084923/H20850,w ec a n calculate the dynamical prefactor, which leads together withthe static prefactor to the attempt frequency in the IHD:033915-4 Schratzberger et al. J. Appl. Phys. 108, 033915 /H208492010 /H20850 Downloaded 21 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsf0=1 2/H9266/H9261+/H90240=1 2/H9266/H20875/H92531Ke Js/H208491+/H92512/H20850/H20849/H9251/H208491−2 h/H20850 +/H20881/H92512−4h/H20849h−1 /H20850/H20850/H20876/H208811+1 h, /H2084924/H20850 which agrees with the results obtained by Coffey et al. ,14 which was derived as a special case of the Langer theory. IV. COMPARING TST WITH LANGEVIN-DYNAMICS SIMULATIONS As mentioned in Sec. II, the formulas for the attempt frequency for the broken symmetry and the axial symmetriccase do not converge to each other when the external fieldh→0 without additional adjustment. 20In order to judge the applicability of the formula for the broken symmetry /H20851Eq. /H2084924/H20850/H20852and the formula for the axial symmetric case /H20851Eq. /H208492/H20850/H20852, we perform micromagnetic simulation using Langevin-dynamics, where we adjust the broken symmetry by applyingperpendicularly applied fields h=H ext/Haniwith different am- plitudes. In real magnetic structures, it seems reasonable thatdue to imperfections, in most cases the symmetry is brokento a certain degree. A detailed discussion about the require-ments for the application of the formulas for broken symme-try, in order to explain experimental results, can be found inRef. 19. Recently, various groups have claimed good agreement between Langevin-dynamics simulations and analytical ex-pressions for the attempt frequency for various dampinglimits. 22–24Usov and Grebenshchikov24compared the relax- ation time of a single-domain particle with a nonaxially sym-metric double-well potential with Langevin-dynamics simu-lations and found good agreement of the magnetizationrelaxation process. Vouille et al. 22compared Langevin- dynamics simulations with the formulas of Ref. 15and found favorable agreement with Coffey’s formulae. Suh et al.23 have shown that there is an excellent agreement between the attempt frequency obtained from Langevin-dynamics simu-lations and the theoretical formulas of the attempt frequency/H20851Eqs. /H208492/H20850and /H208493/H20850/H20852for nanomagnets with thin-film geometry. In Refs. 22–24, the Langevin-dynamics simulations were performed in the macrospin approximation, where the nano-magnet was represented by one magnetization vector only. Inthe following Langevin-dynamics simulations, the magnet issubdivided into several finite elements in order to resolvemagnetization inhomogeneities. The mesh size dependenceis minimized by using a scaling approach. 25 To study the average lifetime of a single particle, a series of simulations are performed, where the particle’s magneti-zation along the easy axis is measured as a function of time.Thus, the mean life time /H9270of Eq. /H208491/H20850becomes directly ac- cessible. This type of measurement is called telegraph noisemeasurement because of the expected stochastic fluctuationbetween the two states of lowest energy /H20849see Fig. 2/H20850.A s /H9270 increases exponentially with decreasing temperature, it is very unlikely that an escape process will be observed at lowtemperatures. However, applying a constant field perpen-dicularly to the easy axis reduces the height of the energybarrier. When the energy barrier is sufficiently small, enoughswitching events will occur in order to obtain /H9270with a rea- sonable accuracy. In order to extract /H92700from/H9270, we perform various simulations. The magnetic properties are compiled inthe caption of Fig. 2. In order to be able to calculate the average lifetime of a particle, it is essential to define theswitching event properly. In the following simulations, wedefined a particle to be switched when both of the followingcriteria apply: /H20849i/H20850the z-component of the magnetization /H20849component parallel to easy axis /H20850has to overcome a thresh- old of m z/ms=/H110060.87 and /H20849ii/H20850the magnetization has to per- form at least one full precessional cycle around the minimumof the switched state. 26In order to estimate the attempt fre- quency if /H9270is measured, Eq. /H208491/H20850is used. In order to be able to obtain accurate fits, we use only /H92700=1 /f0as a fit param- eter. The energy barrier E0is obtained from the simulation using the nudged elastic band method.3 In the first simulation, we studied the influence of the discretization size on the attempt frequency. Figure 3shows FIG. 2. /H20849Color online /H20850The time evolution of the magnetization at finite temperature is shown. The input parameters are as follows: h=0.2, cube length a=1.0 nm, T=18 K, JS=0.5 T, KE=3/H11003106J/m3, and /H9251=0.05. The number of finite elements to discretize the cube is 13. FIG. 3. /H20849Color online /H20850Results of a cube with for different finite element mesh sizes; the other parameters are the same as in Fig. 2./H20849a/H20850mesh size =1.0 nm and /H20849b/H20850mesh size=0.5 nm.033915-5 Schratzberger et al. J. Appl. Phys. 108, 033915 /H208492010 /H20850 Downloaded 21 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsa weak dependence of f0on the mesh size. In all of the following simulations, a mesh size of 1 nm was assumed,which leads to 13 finite elements. Figure 4shows the attempt frequency as a function of temperature for different values of the applied perpendicularfield. According to Eq. /H2084924/H20850, the attempt frequency does not depend on temperature in the IHD. However, the Langevinsimulations clearly show an increase in f 0as function of 1/T. The results of the Langevin simulations are clearly supportedby extending the analytical simulation to the general equa-tion, which is valid for all values of the damping constant. Inthe comparison, we evaluate f 0with Eq. /H208493/H20850and use the approximation of Eq. /H208496/H20850. The dependence of the attempt frequency as a function of the external field is shown in detail in Fig. 5. Again, the Langevin simulations are compared with analytical results.The analytical formula for the IHD for the broken symme-tries Eq. /H2084924/H20850clearly overestimates the attempt frequency. In the limit when the external field approaches zero, the attemptfrequency obtained by Langevin dynamics simulations con-verge to a value, which is also smaller than the prediction ofEq. /H208492/H20850. Again, the general equation Eq. /H208493/H20850for the attempt frequency, which is valid for all values of the damping con-stant, describes the simulation results very well. Althoughthe analytical formula Eq. /H2084924/H20850diverges for h→0, which one must remember is an artifact 10,20of the steepest descentsapproximation used to evaluate the various integrals, the general equation for the attempt frequency /H20851Eq. /H208493/H20850/H20852seems to converge to a finite value. The comparison between Eq. /H208493/H20850 and the results of f0obtained by Langevin dynamics simula- tions are summarized by the contour plot of Fig. 6. Although sharp contours require a large amount of simulation points,the overall agreement can be seen well. V. CONCLUSION AND OUTLOOK Langevin-dynamics simulations of the single-domain particle confirm the validity of Eq. /H208493/H20850. The broken symmetry cases seem to be relevant for all realistic samples studied inthis paper. As a consequence, the formalism summarized inEqs. /H208497/H20850–/H2084918/H20850seems to be a useful framework to estimate the thermal stability for large magnetic structures, including ar-bitrary shaped geometries and inhomogeneous magnetizationconfigurations. This will allow us to estimate the long-termthermal stability of magnetic structures that cannot be ac-cessed by Langevin-dynamics simulations. The attempt fre-quency in the IHD limit can be calculated numerically if: /H20849i/H20850 the second derivative of the energy at the saddle point and atthe minimum is known and /H20849ii/H20850the magnetization dynamics around the saddle point can be expressed as shown in Eq./H2084912/H20850. All of these properties are usually accessible in micro- magnetic simulations, which will allow for the calculation ofthe attempt frequency in the IHD limit. Starting from thisapproximation, correction to the general formula, which isvalid for all values of the damping constant, can be tacklednumerically. A promising route will be to investigate the en-ergy loss per cycle in both wells next to the saddle point. ACKNOWLEDGMENTS The financial support of the FWF Projects No. P20306, No. F4112-N13, and the support of the European ProjectTERAMAGSTOR /H20849Grant No. FP7-ITC-2007-2-224001 /H20850are acknowledged. 1R. W. Chantrell, N. Walmsley, J. Gore, and M. Maylin, Phys. Rev. B 63, 024410 /H208492000 /H20850. 2M. P. Sharrock, J. Appl. Phys. 76,6 4 1 3 /H208491994 /H20850. 3R. Dittrich, T. Schrefl, D. Suess, W. Scholz, H. Forster, and J. Fidler, J. Magn. Magn. Mater. 250,1 2 /H208492002 /H20850. 4E. Paz, F. Garcia-Sanchez, and O. Chubykalo-Fesenko, Physica B 403, 330 /H208492008 /H20850. FIG. 4. /H20849Color online /H20850Attempt frequency as a function of the temperature. The magnetic parameters are the same as in Fig. 2. The dotted lines are analytical results according to Eq. /H208493/H20850, using the approximation of Eq. /H208496/H20850. FIG. 5. /H20849Color online /H20850Attempt frequency as a function of the perpendicular external field strength. The magnetic parameters are the same as in Fig. 2. /H20849IHD /H20850is the analytical results in the intermediate to high damping limit according to Eq. /H2084924/H20850/H20849Ref.9/H20850. is the analytical result for the symmetric case according to Eq. /H208492/H20850. The two dotted lines at the bottom /H20849blue and red /H20850are analytical results, valid for all damping values according to Eq. /H208493/H20850using the approximation of Eq. /H208496/H20850 FIG. 6. /H20849Color online /H20850Contour plot of the attempt frequency as a function of temperature and external field h./H20849left /H20850f0according to Eq. /H2084924/H20850./H20849right /H20850f0 obtained by Langevin-dynamics simulations. The magnetic parameters are the same as in Fig. 2.033915-6 Schratzberger et al. J. Appl. Phys. 108, 033915 /H208492010 /H20850 Downloaded 21 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions5H. A. Kramers, Physica /H20849Utrecht /H208507,2 8 4 /H208491940 /H20850. 6J. S. Langer, Ann. Phys. /H20849N.Y. /H2085041, 108 /H208491967 /H20850. 7J. S. Langer and L. A. Turski, Phys. Rev. A 8, 3230 /H208491973 /H20850. 8V. I. Mel’nikov and S. V. Meshkov, J. Chem. Phys. 85, 1018 /H208491986 /H20850. 9W. F. Brown, Phys. Rev. 130, 1677 /H208491963 /H20850. 10W. T. Coffey, D. A. Garanin, and D. McCarthy, Adv. Chem. Phys. 117, 483 /H208492001 /H20850. 11D. A. Smith and F. A. de Rozario, J. Magn. Magn. Mater. 3, 219 /H208491976 /H20850. 12W. F. Brown, IEEE Trans. Magn. 15, 1196 /H208491979 /H20850. 13I. Klik and L. Gunther, J. Stat. Phys. 60, 473 /H208491990 /H20850. 14W. T. Coffey, D. S. F. Crothers, J. L. Dormann, L. J. Geoghegan, and E. C. Kennedy, Phys. Rev. B 58, 3249 /H208491998 /H20850. 15W. T. Coffey, D. S. F. Crothers, J. L. Dormann, Y. P. Kalmykov, E. C. Kennedy, and W. Wernsdorfer, Phys. Rev. Lett. 80, 5655 /H208491998 /H20850. 16L. J. Geoghegan, W. T. Coffey, and B. Mulligan, Adv. Chem. Phys. 100, 475 /H208491997 /H20850.17P. M. Déjardin, D. S. F. Crothers, W. T. Coffey, and D. J. McCarthy, Phys. Rev. E 63, 021102 /H208492001 /H20850. 18Y. P. Kalmykov, J. Appl. Phys. 96, 1138 /H208492004 /H20850. 19H. Kachkachi, W. T. Coffey, D. S. F. Crothers, A. Ezzir, E. C. Kennedy, M. Noguès, and E. Tronc, J. Phys.: Condens. Matter 3077 ,1 2 /H208492000 /H20850. 20D. A. Garanin, E. C. Kennedy, D. S. F. Crothers, and W. T. Coffey, Phys. Rev. E 60, 6499 /H208491999 /H20850. 21H. B. Braun, J. Appl. Phys. 76,1 5 /H208491994 /H20850. 22C. Vouille, A. Thiaville, and J. Miltat, J. Magn. Magn. Mater. 272–276 , E1237 /H208492004 /H20850. 23H.-J. Suh, C. Heo, C.-Y. You, W. Kim, T.-D. Lee, and K.-J. Lee, Phys. Rev. B 78, 064430 /H208492008 /H20850. 24N. A. Usov and Y. B. Grebenshchikov, J. Appl. Phys. 105, 043904 /H208492009 /H20850. 25M. Kirschner, T. Schrefl, F. Dorfbauer, G. Hrkac, D. Suess, and J. Fidler, J. Appl. Phys. 97, 10E301 /H208492005 /H20850. 26S. Wang and P. B. Visscher, J. Appl. Phys. 99, 08G106 /H208492006 /H20850.033915-7 Schratzberger et al. J. Appl. Phys. 108, 033915 /H208492010 /H20850 Downloaded 21 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.4803065.pdf

Stepwise behavior of the core trajectory in magnetic vortex dynamics under an alternating-current magnetic field Je-Ho Shim, Hong-Guang Piao, Sang Hyuk Lee, Suhk Kun Oh, Seong-Cho Yu et al. Citation: J. Appl. Phys. 113, 173904 (2013); doi: 10.1063/1.4803065 View online: http://dx.doi.org/10.1063/1.4803065 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v113/i17 Published by the American Institute of Physics. 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 06 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsStepwise behavior of the core trajectory in magnetic vortex dynamics under an alternating-current magnetic field Je-Ho Shim,1Hong-Guang Piao,2Sang Hyuk Lee,1Suhk Kun Oh,1Seong-Cho Yu,1 Seung Kee Han,1Dong Eon Kim,3,4and Dong-Hyun Kim1,a) 1Department of Physics, Chungbuk National University, Cheongju 361-763, South Korea 2Laboratory of Advanced Materials, Department of Materials Science and Engineering, Tsinghua University, Beijing, China 3Department of Physics and Center for Attosecond Science and Technology, POSTECH, Pohang 790-784, South Korea 4Max Planck Center for Attosecond Science, Pohang, 790-784, South Korea (Received 5 February 2013; accepted 11 April 2013; published online 1 May 2013) We predict that the radial distance of a magnetic vortex core from the disk center shows a stepwise behavior during initial excited motion under an alternating-current magnetic field by means of micromagnetic simulations. The stepwise behavior is clearly observed around the resonance frequency and depends on the amplitud e and frequency of the external magnetic field. It has been found that the stepwise behavior originates from the relative phase difference between the gyrovector and the radial distance of the vortex core. VC2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4803065 ] I. INTRODUCTION Recently, magnetic vortex structures have attracted much interest due to possible applications of various spintronic devi- ces based on magnetic vortex structures.1For instance, micro- wave generation from magnetic vortex core motion excited bya spin transfer torque is considered to be a promising technique for radio frequency generation. 2,3Another possible application is a magnetic memory scheme based on a magnetic vortexstructure, 4as it has been reported that vortex core switching can be controlled by tuning alternating-current (AC) field fre- quency. Several works have been devoted to understanding thecontrolling mechanism of the vortex core during the core’s gyrotropic motion by a pulsed magnetic field, 5a horizontal AC external magnetic field, or an AC spin current.6–10 To fully understand vortex core dynamics, not only the core switching mechanism under an AC magnetic field or AC spin current but also the transi ent and steady-state motion of the vortex core under AC excita tion should be well understood. The understanding of core dynamics under AC excitation becomes more essential consideri ng possible spintronic appli- cations based on magnetic vortex structures since the devices include fast AC operation for pr actical applications. Very recently, it has been reported that vortex core dynamicsbecomes complex and nonlinear around the resonance condi- tion under an AC magnetic field, 11w h i c hi sa ni n t r i g u i n gr e s u l t considering the simple Thiele eq uation describing excited vor- tex core motion.12The nonlinearity of the core motion has been known to be explainable by the simple dynamic correc- tion of a gyrovector and a dampin gt e n s o ri nt h eT h i e l ee q u a - tion. Nonlinear dynamics of the core motion driven by a spin- polarized current has been als o explored with an analytical model,13predicting that the nonlinearity mainly arises from magnetostatic and Zeeman energies. Although the simpledynamic correction of the gyrovector depending on a vortex core position and thus, on the magnetostatic and Zeeman ener- gies as well, was valid in repr oducing the nonlinear and com- plex vortex core dynamics, little is known of the detailedmechanism of the gyrovector modification for the core under an AC field or current. In this work, we report that there exists a strong correlation between the gyrovector and core radial dis-tance from the disk center, whic h is predicted to exhibit step- wise behavior of the AC-forced vortex core motion. II. MICROMAGNETIC SIMULATION AND ANALYTICAL MODEL We have carried out micromagnetic simulations to investigate the magnetic vortex dynamics of a ferromag- netic disk under various AC magnetic fields based on the Landau-Lifshitz-Gilbert equation.14The Gilbert damping constant of the Landau-Lifshitz-Gilbert equation was varied from 0.01 to 0.1, and the simulation cell size was 2/C22/C25n m3. In the simulation, the material parameters of Permalloy were considered with an exchange stiffness coef- ficient of 13 /C210/C012J/m and a saturation magnetization M s of 8.6 /C2105A/m. The disk thickness was 5 nm, and the disk radius was varied from 150 to 500 nm. A sinusoidal AC field was applied along a certain direction on the disk plane, and the field direction was defined to be the x-axis. The am-plitude of the AC external magnetic field was varied from 0.5 to 2.5 mT. The AC frequency was varied as well, from 50 to 250 MHz around the resonance frequency of each diskwith a different radius. The resonance frequency was deter- mined by analyzing the steady-state motion of the vortex core. The Thiele’s equation 12is similar to an equation describing damped harmonic oscillation, /C0~G/C2~X0/C0D$/C1~X0þk~X¼uð^z/C2~HÞ; (1) a)donghyun@cbnu.ac.kr 0021-8979/2013/113(17)/173904/5/$30.00 VC2013 AIP Publishing LLC 113, 173904-1JOURNAL OF APPLIED PHYSICS 113, 173904 (2013) Downloaded 06 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionswhere the derivative is with respect to the time, ~Gis a gyro- vector, D$is a damping tensor, kis a stiffness coefficient, ~H is an external magnetic field, and uis a constant depending on sample geometry.15The gyrovector depends on the geo- metrical spin configuration of the disk as follows:12 ~G¼/C0ðMs jcjsinhðrh/C2r/ÞdV; (2) where Msis the saturation magnetization, cis the gyromag- netic ratio, his the polar angle of local spin from the disk axis (z-axis), and /is the azimuth angle of local spin around the disk axis. In most cases, the gyrovector is approximated to beconstant, which has been found to be invalid for vortex core motion under an AC field. 12It has been known that a gyrovec- tor simply modified by considering z-component magnetiza-tion distribution on the disk works well in describing vortex core dynamics under an AC field, 12where the modified gyro- vector ( D~G) is approximated to be proportional to the distance of the vortex core ( RVC) from the disk center, as in Eq. (3). D~G¼/C01:6/C210/C03RVCðkg=m/C2sÞ: (3) III. RESULTS AND DISCUSSION In Fig. 1, we have plotted initial transient core trajecto- ries with variation of the disk radius (R) from 150 to 500 nm.An AC field was applied with a fixed amplitude of 1 mT and with a different frequency (f) corresponding to a resonance frequency for each case of disk radius. R and f are denotedin this figure. The core trajectory was normalized by disk ra- dius. It can be easily seen that the relative maximum kicking distance of the core from the disk center becomes larger fora larger disk, which is expected from the fact that the vortex structure becomes more rigid for a smaller disk. It is interesting to note that there exists stepwise behav- ior in the core trajectory, in which a circular motion (solid) with a temporarily constant radius is followed by a motion with an increasing radius (open), which is again followed bya circular motion (solid) with a temporarily constant radius and so on. The trend remains clear until the magnetic core reaches the maximum radial distance but is still observableafterward. This stepwise behavior is found for all cases in the detailed gyrotropic core motion of Fig. 1. To investigate further, we have analyzed representative results of D~Gand R VCfor the case of Fig. 1with the same simulation parame- ters. In Fig. 2, it is clear that there exists a strong correlation between RVCandD~Gin all cases, which supports our approximation of simple proportionality between RVCand D~Gwith Eq. (3). Note that detectable periodic oscillations are observed both for RVCandD~Gin all cases. Since an external field is applied along the x-axis in the present simu- lation, the equilibrium position is shifted along the y-axis. FIG. 1. Initial vortex core motion for disks with different radius (R) from 150to 500 nm for an AC field amplitude of 1 mT at each resonance frequency (f). Open circle represents R VCwhen the ra- dius increases while a closed circle is represents when the radius decreases.173904-2 Shim et al. J. Appl. Phys. 113, 173904 (2013) Downloaded 06 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsThe shift of the equilibrium position along the y-axis leads to an elliptic core trajectory. The elliptic core trajectory is observed throughout the entire timescale, even in the steady-state. An elliptical trajectory allows the core to pass themaximal radial distance two times when the core crosses the longer axis of the ellipse, whereas the core passes the mini- mal radial distance two times as well, leading to a doubledfrequency compared to the external AC frequency. The FIG. 2. D~GandRVCduring initial 30 ns period for the AC field frequency of res- onance and the AC field amplitude of 1 mT. FIG. 3. Initial vortex core motion fordisks with different damping parameter (a) for a disk with a radius of 250 nm. The AC field amplitude is 2 mT. Open circle represents R VCwhen the radius increases while a closed circle is repre- sents when the radius decreases.173904-3 Shim et al. J. Appl. Phys. 113, 173904 (2013) Downloaded 06 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsdoubled frequency in the initial phase and the steady-state phase is confirmed to be the same. The small oscillations of RVCandD~Gseem to be a little bit off from each other, but still retaining overall proportionality. We have also examined the effect of damping parame- ter change on stepwise behavior. As shown in Fig. 3,t h e core exhibits a clear stepwise trajectory for all cases of damping parameter ( a) ranging from 0.01 to 0.1, where the AC field amplitude has been fixed to be 2 mT and the trajec-tories are plotted until the core reaches the maximum radius. The kicking motion of the core seems to be significantly reduced, as expected, but there still exists a clear indicationof the stepwise trajectory even in the case of a¼0.1. Thus,the stepwise behavior of the initial core motion under an AC magnetic field, found in the present study, seems to be quite universal irrespective of damping parameters or diskradii. To have a detailed analysis of the observed stepwise behavior, fast Fourier transform (FFT) amplitude and phasewere determined. In Fig. 4, the FFT results of R VC(Fig. 4(a)) andD~G(Fig. 4(b)) are plotted for the case of a 2 mT and 120 MHz AC field. The Gilbert damping constant is 0.01, andthe disk radius is 250 nm. One of the main amplitude peaks of FFT at 10 MHz was found for all R VCandD~G, which can be ascribed to the envelope frequency ( xen).xenis the fre- quency of the amplitude envelop of the x (or y) position of FIG. 4. FFT amplitude (closed square) and phase (open square) of (a) RVCand (b) D~G. FIG. 5. D~GandRVCfor the initial 80 ns period, which is categorized into three regions. D~Gvs.RVCis plotted on the bottom for three regions of increasing (bottom left), decreasing (bottom center), and decaying region (bottom right).173904-4 Shim et al. J. Appl. Phys. 113, 173904 (2013) Downloaded 06 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsthe initial vortex core motion for a regime of transient motion before the dynamics becomes steady-state forced AC motion.12It is observed that another major amplitude peak at 240 MHz exists only for RVC, whereas no peak can be found at this frequency for D~G. The frequency originates from the doubling of the external AC field frequency (120 MHz), sincethe radius has squared sinusoidal x- and y-position terms and, thus, has a doubled frequency. It is interesting to note that the main two peaks of FFT are not related to any spin wavemode. It is well known that spin wave generation sensitively depends on the rate of external field increase. In the present case, an AC field with a frequency of less than 200 MHz isconsidered to be relatively slower than a pulse field with a rise time of sub-ns. The FFT analysis of Fig. 4also supports our consideration that the observed stepwise core behavior isnot related to spin wave behavior. The phases of R VCandD~Gcorresponding to the enve- lope frequency peak (10 MHz) are at about /C0180/C14and 0/C14, a ss h o w ni nF i g . 4. Thus, RVCis expected to follow a cos(xentþp) phase while D~Gto follow a cos( xent) phase. The relative phase difference between RVCandD~Glead to a linear relation, as in Eq. (3). On the other hand, the phases ofRVCandD~Gcorresponding to the doubled AC field fre- quency (240 MHz) are at about /C090/C14and 270/C14, leading to a relative phase delay of 2 p, which indicates that there is no effective phase difference. Thus, we can approximate that RVCandD~Gare composed of two terms, in which one fol- lows envelope frequency ( xen) and the other follows doubled AC field frequency ( xAC). The envelope frequency terms RVCandD~Ghave opposite phases of p, so that the ra- tio between RVCandD~Gis negative and oscillates around the constant value in Eq. (3). The AC field frequency terms ofRVCandD~Ghave the same phases, so that the ratio between RVCandD~Gis positive and constant. The superpo- sition of these two terms produces the observed stepwise behavior. Stepwise behavior of the vortex core dynamics under an AC magnetic field is prominent only in the initial transient region. In Fig. 5,D~G,RVC, and the ratio between the two (bottom three figures) are plotted together for the initial 80 ns. All the simulation parameters are kept the same as Fig. 4. It is interesting to note that stepwise behavior is observed not only for the initial increasing phase of D~Gand RVC, as shown in the bottom left figure, but also for the decreasing phase as in the bottom center figure. However,stepwise behavior becomes hard to detect later than 40 ns for smaller envelope amplitude, as in the bottom right figure, which is understood based on the two contribution terms fol-lowing x enandxAC, as previously discussed.IV. CONCLUSIONS In conclusion, we have observed the stepwise behavior of the vortex core radial distance and the gyrovector for vor- tex structure under an AC magnetic field, which originatesfrom the in-phase dynamics of D~GandR VCwith AC field frequency and out-of-phase dynamics with envelope fre- quency. We propose that vortex core radial distance can betuned to have a specific constant value under an AC external field by controlling AC field frequency and amplitude. ACKNOWLEDGMENTS This work was supported by the Korea Research Foundation (NRF) Grant (Nos. 2010-0004535 and 2010- 0021735) funded by the South Korean government (Ministryof Education, Science and Technology). This research was also supported in part by the Global Research Laboratory Program [Grant No 2009-00439], the Leading ForeignResearch Institute Recruitment Program [Grant No 2010- 00471], and the Max Planck POSTECH/KOREA Research Initiative Program [Grant No 2011-0031558] through theMEST’s NRF funding. 1M. Rahm, J. Stahl, and D. Weiss, Appl. Phys. Lett. 87, 182107 (2005). 2V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Nat. Phys. 3, 498 (2007). 3A. Dussaux, B. Georges, J. Grollier, V. Cros, A. V. Khvalkovskiy, A. Fukushima, M. Konoto, H. Kubota, K. Yakushiji, S. Yuasa, K. A. Zvezdin, K. Ando, and A. Fert, Nat. Commun. 1, 8 (2010). 4B. Pigeau, G. de Loubens, O. Klein, A. Riegler, F. Lochner, G. Schmidt, L. W. Molenkamp, V. S. Tiberkevich, and A. N. Slavin, Appl. Phys. Lett. 96, 132506 (2010). 5S.-B. Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran, J. St €ohr, and H. A. Padmore, Science 304, 420 (2004). 6K.-S. Lee, K. Y. Guslienko, J.-Y. Lee, and S.-K. Kim, Phys. Rev. B 76, 174410 (2007). 7R. Hertel, S. Gliga, M. F €ahnle, and C. M. Schneider, Phys. Rev. Lett. 98, 117201 (2007). 8M. Weigand, B. Van Waeyenberge, A. Vansteenkiste, M. Curcic, V. Sackmann, H. Stoll, T. Tyliszczak, K. Kaznatcheev, D. Bertwistle, G. Woltersdorf, C. H. Back, and G. Sch €utz, Phys. Rev. Lett. 102, 077201 (2009). 9M. Kammerer, M. Weigand, M. Curcic, M. Noske, M. Sproll, A.Vansteenkiste, B. Van Waeyenberge, H. Stoll, G. Woltersdorf, C. H. Back, and G. Sch €utz,Nat. Commun. 2, 279 (2011). 10K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono, Nature Mater. 6, 270 (2007). 11J.-H. Shim, H.-G. Piao, S.-H. Lee, S.-K. Oh, S.-C. Yu, S.-K. Han, and D.- H. Kim, Appl. Phys. Lett. 99, 142505 (2011). 12A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). 13A. Dussaux, A. V. Khvalkovskiy, P. Bortolotti, J. Grollier, V. Cros, and A. Fert, Phys. Rev. B 86, 014402 (2012). 14M. J. Donahue and D. G. Porter, OOMMF User’s Guide: http://math.nist.- gov/oommf (2002). 15K. Yu. Guslienko, Appl. Phys. Lett. 89, 022510 (2006).173904-5 Shim et al. J. Appl. Phys. 113, 173904 (2013) Downloaded 06 Jun 2013 to 193.1.100.108. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.4916112.pdf

Electric field tunability of microwave soft magnetic properties of Co 2FeAl Heusler alloy film Shandong Li,1,a)JieXu,1Qian Xue,1Honglei Du,1Qiang Li,1Caiyun Chen,1RuYang,1 Shiming Xie,1Ming Liu,2Tianxiang Nan,3Nian X. Sun,3and Weiquan Shao1 1College of Physics Science, Key Laboratory of Photonics Materials and Technology in Universities of Shandong, and Laboratory of Fiber Materials and Modern Textile, The Growing Base for State Key Laboratory, Qingdao University, Qingdao 266071, China 2Electronic Materials Research Laboratory, Key Laboratory of the Ministry of Education & International Center for Dielectric Research, Xi’an Jiaotong University, Xi’an 710049, China 3Electrical and Computer Engineering Department, Northeastern University, Boston, Massachusetts 02115, USA (Presented 4 November 2014; received 6 September 2014; accepted 15 November 2014; published online 24 March 2015) Co2FeAl Heusler alloy film with 100 nm in thickness was sputtered on (011)-cut lead zinc niobate-lead titanate (PZN-PT) single crystal slabs. It was revealed that this multiferroic laminateshows very large electric field (E-field) tunability of microwave soft magnetic properties. With the increase of electric field from 0 to 8 kV/cm on PZN-PT, the anisotropy field, H K, of the Co 2FeAl film along [100] direction of PZN-PT is dramatically enhanced from 65 to 570 Oe due to the strongmagnetoelectric (ME) coupling between ferromagnetic Co 2FeAl film and ferroelectric substrate. At the same time, the damping constant aof Co 2FeAl film dramatically decreases from 0.20 to 0.029. As a result, a significantly shift of self-biased ferromagnetic resonance frequency, fFMR, from 1.86 to 6.68 GHz with increment of 3.6 times was obtained. These features demonstrate that Co2FeAl/PZN-PT multiferroic laminate is promising in fabrication of E-field tunable microwave components. VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4916112 ] I. INTRODUCTION Recently, multiferroic composite materials have drawn increasing attention due to that the magnetoelectric (ME) coupling structures offer an E-field manipulation of magnetic properties (converse ME effect) or vice versa (direct MEeffect). 1–4The ME coupling gives rise to many novel multi- ferroic materials and devices. Of them, one important branch is electrostatically tunable ferromagnetic/ferroelectriccomposites, which led to many tunable devices such as a picotesla sensitivity magnetometer filters, 5resonators, phase shifters, etc.6–9Comparing with the conventional magnetic field tuned microwave magnetic devices, the electrostatically tunable microwave multiferroic devices exhibit some advan- tages, such as more energy efficient, compact, lightweight,and less noisy, etc. 10 The ME coupling strength in multiferroic composites is determined by many factors, such as piezoelectric/magnetoe-lastic parameters of the ferroic/magnetic phases, the inter- face interactions, the ME coupling mode, and the orientation of the magnetic and electric fields. The strain/stress-mediatedmultiferroic composites have effective energy transfer between electric and magnetic fields; however, at microwave frequencies, the strong ME coupling is difficult to beachieved due to the large loss tangents of ferroic/magnetic phases, leading to a very limited tunability in electrostati- cally tunable microwave multiferroic devices. The typicaltunable frequency range is less than 150 MHz, and the tuna- ble magnetic field less than 50 Oe. 8,11,12Layered multiferroic heterostructures with magnetic thin films give rise to greatopportunities for obtaining strong ME coupling at micro-wave frequencies due to improved interfaces, minimizedcharge leakage of ferroelectric materials, and low losstangents of magnetic thin films. 10,13Heusler alloys, such as NiMnSb, Co 2FeAl, Co 2MnSi, Co 2FeSi, exhibit low coerciv- ity H C, low damping constant a, and high anisotropy field, HK, due to their itinerant-electron characteristics, showing a potential high-frequency ferromagnetic properties.14–16 (011)-cut PZN-PT single-crystal slabs with 6% lead titanate have high anisotropic piezoelectric coefficients.17Therefore, in this study, Co 2FeAl Heusler alloy films as ferromagnetic films were deposited on (011)-cut single-crystal PZN-PTsubstrates to explore the ME effect in the layered ferromag-netic/ferroelectric multiferroic heterostructure. II. EXPERIMENTAL PROCEDURE The 100-nm Co 2FeAl Heusler alloy films were depos- ited on PZN-PT substrates at room temperature under 2.8 mTorr Ar atmosphere with a floating rate of 20 sccm,along with a RF power of 80 W for Co 2FeAl target. The (011)-cut single crystal PZN-PT substrates with a dimensionof 5 mm[100] /C25 mm[01-1] /C20.5 mm[011] have been pasted on the sample turntable with their [100] directionalong the radial (R) direction. The magnetic properties weremeasured by a vibrating sample magnetometer (VSM). Themicrowave frequency performances of the multiferroic a)Author to whom correspondence should be addressed. Electronic addresses: lishd@qdu.edu.cn and dylsd007@163.com. 0021-8979/2015/117(17)/17B722/4/$30.00 VC2015 AIP Publishing LLC 117, 17B722-1JOURNAL OF APPLIED PHYSICS 117, 17B722 (2015) composites were evaluated by use of a vector network ana- lyzer with co-planar waveguide fixture. III. RESULTS AND DISCUSSION Figure 1shows the E-field dependence of hysteresis loops for the Co 2FeAl/PZN-PT multiferroic laminates along [100] and [01-1] directions, respectively. As illustrated inFig. 1(a), the as-deposited Co 2FeAl film does not exhibit detectable magnetic anisotropy; however, when an externalE-field is applied on the [011] direction of PZN-PT substrate,a well-fined uniaxial magnetic anisotropy was formed due tothe ME coupling [see Fig. 1(b)]. The E-field not only enhan- ces the squareness ratio of film along the easy axis directionof [01-1] [see Fig. 1(c)] but also dramatically increases the magnetic anisotropy field, H K, along hard axis of [100]. As illustrated in Fig. 1(d), the E-field dependent magnetic anisotropy field, H K, dramatically increases from 65 Oe at 0 kV/cm to 570 Oe at 8 kV/cm along the HA [100] direction.A large ME coupling coefficient of 63.1 Oe cm/kV isobtained in Co 2FeAl/PZN-PT multiferroic laminates. The ferromagnetic resonance frequency, fFMR, of ferromagnetic films can be expressed by Kittle equation, fFMR¼c 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HK/C3HKþ4pMS ðÞp ; (1) where cis the gyromagnetic ratio (c 2p/C242.8 MHz/Oe), H Kis the anisotropic field in plane, and 4 pMSis the saturation magnetization of ferromagnetic films. From Eq. (1), it can be concluded that large enhancement of H Kmanipulated by E-field will give rise to a large upward shift of fFMR. In other words, the ferromagnetic resonance of Co 2FeAl films will be driven to a high frequency by E-field.Figure 2shows the E-field dependence of permeability for the Co 2FeAl/PZN-PT multiferroic laminates. As expected, the ferromagnetic resonance frequency was dra-matically shifted from 1.86 to 6.68 GHz, when the E-field increased from 0 to 8 kV/cm. The f FMR increases by 3.6 times due to the ME coupling via the interface between Co2FeAl film PZN-PT substrate. A large fFMRenhancement ratio of 602.5 MHz cm/kV was achieved. This E-field-induced fFMRshift can be explained by the strain/stress-mediated electrostatic-field-induced in-plane magnetic anisotropy field H eff. For the Co 2FeAl/PZN-PT multiferroic laminates, the Kittel equation [Eq. (1)] can be rewritten as fFMR¼c 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi HKþHeff ðÞ /C3 HKþHeffþ4pMS ðÞp ;(2) FIG. 1. The E-field dependence of hys- teresis loops for the Co 2FeAl/PZN-PT multiferroic laminates along [100] and [01-1] directions, respectively. (a) The as-deposited Co 2FeAl film at 0 kV/cm, showing isotropic loops along [100] and [01-1] directions of PZN-PT sub-strate; (b) at 8 kV/cm, showing a well- defined magnetic anisotropy induced by E-field; and (c) and (d) the E-field dependent loops along [01-1] and [100] directions, respectively. FIG. 2. The E-field dependence of permeability for the Co 2FeAl/PZN-PT multiferroic laminates.17B722-2 Li et al. J. Appl. Phys. 117, 17B722 (2015)where H effis the orthogonal in-plane compressive and tensile stress corporately induced internal effective magnetic field,which could be positive or negative, and in our case it can be expressed as H eff¼3kY MS1þ/C23ðÞd31/C0d32 ðÞ E; (3) where Y is the Young’s Modulus, /C23is the Poisson’s ratio, k is the magnetostriction constant, d 31¼/C03000 pC/N along [100] and d 32¼1100 pC/N along [01-1] are linear aniso- tropic piezoelectric coefficients of PZN-PT, and E is theapplied external E-field strength. H effis the effective mag- netic anisotropy field under various E-fields.18From Eqs. (3) and(2), it can be concluded that the E-field will give rise to an increase of H effvia ME coupling, and, therefore, drive the ferromagnetic resonance upwards shifting to a high frequency. The free-energy density, E total, in the studied system includes the Zeeman energy, E zeeman , the demagnetization energy ;Eshape, the stress energy, E stress, and the uniaxial ani- sotropy energy, E uni. It can be written as Etotal¼Ezeeman þEshapeþEstressþEuni: (4) For the studied system, the main interaction between the magnetic film and PZN-PT substrate occurs at the interface (i.e., in-plane), so the contribution of demagnetization energy can be ignored. In the case of a certain magnetic field,the variation of total free-energy density is dominated by thecompetition and transformation between E stressand E uni. The uniaxial magnetic anisotropy of magnetic film is induced byE-field tunable stress, and the uniaxial magnetic energy canbe manipulated by E-field via the magnetoelectric coupling.Therefore, the anisotropy field exhibits an E-field tunability,e.g., H Kincreases with the increase of E-field. Figure 3shows the relationship between magnetic ani- sotropy field, H K, and ferromagnetic resonance frequency, fFMR, as well as the damping constant a, at various E-fields. The Gilbert damping constant, a, represents magnetic loss of the Co 2FeAl magnetic film, which is deduced by fitting the permeability spectra, as shown in Fig. 2, using Landau- Lifshitz-Gilbert equation. As illustrated in Fig. 3(a), the fFMR and H Kincrease almost linearly with the E-field, and the E- field dependence of H KandfFMR exhibits a same trend at various E-fields. These facts demonstrated that the E-fieldinduced H Kdominates the ferromagnetic resonance fre- quency, as described in Eqs. (1)–(3). On the other hand, the damping constant, a, dramatically reduces from 0.20 to ca. 0.029 with the increase of E-field from 0 to 8 kV/cm, indicating that the magnetic loss of the multiferroic composites effectively decreases by the E-field.It is excited to note that the ME coupling in the studied mul-tiferroic laminates not only enhances the ferromagnetic reso-nance frequency but also reduces the magnetic loss atmicrowave frequencies. This merit of the multiferroic com-posites gives great opportunity in practical application. The intrinsic stress, in general, is randomly distributed in the magnetic films, which give rise to a high dampingconstant and decreasing resonant frequency. The investi- gated Co 2FeAl film at zero E-field exhibits bad ferromag- netic resonance characteristics due to the magnetic isotropy of as-deposited Co 2FeAl film. As illustrated in Figs. 1,2and 3(b), the Co 2FeAl film shows a higher coercivity of 65 Oe, lower fFMRof 1.86 GHz with broad resonance peak, and very large damping constant, a, of 0.20. The large and random distributed intrinsic stress is responsible for the bad micro-wave performance of the Co 2FeAl film at E ¼0 kV/cm. However, if the E-field was applied on the PZN-PT sub- strate, a biaxial anisotropy stress, compressive stress along [100] direction and tensile one along [01-1] direction of thePZN-PT substrate will be induced by E-field. 19According to magnetoelastic energy equation, E K¼/C03 2kSrcos2h, for a positive kS(as the case in this study for the Co 2FeAl films), a compressive stress will drive the magnetic moments to be aligned parallel to [01-1] direction of PZN-PT substrate. Similarly, the tensile stress also leads the magnetic momentsalong [01-1] direction. As a result, a uniaxial magnetic ani-sotropy with the magnetically easy axis along [01-1] direc-tion is formed due to the ME coupling effect. As reported inour previous work, 20,21if the biaxial stress induced uniaxial magnetic anisotropy is formed in the ferromagnetic films, the enhanced ferromagnetic resonance and improved micro- wave magnetic properties will be obtained due to the align-ment of magnetic moments. As illustrated in Figs. 1–3, with the increase of E-field, the coercivity and damping constantreduce, and the H KandfFMRdramatically increase. IV. CONCLUSIONS Strong magnetoelectric coupling effect, corresponding to a large magnetoelectric coefficient of 63.1 Oe cm/kV, wasobserved in Heusler alloy Co 2FeAl/PZN-PT multiferroic heterostructure, leading to continuously E-field tunablemicrowave frequency characteristics with f FMR upwards FIG. 3. The E-field dependence of (a) magnetic anisotropy field, H K, and fer- romagnetic resonance frequency, fFMR, and (b) Gilbert damping constant a.17B722-3 Li et al. J. Appl. Phys. 117, 17B722 (2015)shifting from 1.86 to 6.68 GHz, i.e., an increment of DfFMR¼4.82 GHz or increment ratio of DfFMR/fFMR¼256% in E-field range of 0–8 kV/cm. The magnetoelectric couplingbetween Co 2FeAl film and PZN-PT substrate not only enhances the ferromagnetic resonance frequency but alsoreduces the magnetic loss at microwave frequencies, whichgives great opportunity in practical application of the multi- ferroic composites. ACKNOWLEDGMENTS This work was financially supported by National Nature Science Foundation of China with Grant No. 11074040 and Key Project of Nature Science Foundation of ShandongProvince with Grant No. ZR2012FZ006. 1N. Spaldin and M. Fiebig, Science 309, 391 (2005). 2W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature 442, 759 (2006). 3J. F. Scott, Nat. Mater. 6, 256 (2007). 4C. A. F. Vaz, J. Hoffman, C. H. Anh, and R. Ramesh, Adv. Mater. 22, 2900 (2010).5J. Zhai, Z. Xing, S. Dong, J. Li, and D. Viehland, Appl. Phys. Lett. 88, 062510 (2006). 6G. Srinivasan, A. S. Tatarenko, and M. I. Bichurin, Electron. Lett. 41, 596 (2005). 7G. Srinivasan, I. V. Zavislyak, and A. S. Tatarenko, Appl. Phys. Lett. 89, 152508 (2006). 8Y. K. Fetisov and G. Srinivasan, Appl. Phys. Lett. 88, 143503 (2006). 9C. Pettiford et al. ,IEEE Trans. Magn. 43, 3343 (2007). 10M. Liu, J. Lou, S. D. Li, and N. X. Sun, Adv. Funct. Mater. 21, 2593 (2011). 11A. Ustinov, G. Srinivasan, and B. A. Kalinikos, Appl. Phys. Lett. 90, 031913 (2007). 12A. S. Tatarenko, V. Gheevarughese, and G. Srinivasan, Electron. Lett. 42, 540 (2006). 13M. Liu et al. ,Adv. Funct. Mater. 19, 1826 (2009). 14S. D. Li et al. ,J. Alloys Compd. 427, 15 (2007). 15M. Oogane et al. ,J. Appl. Phys. 101, 09J501 (2007). 16S. D. Li et al. ,J. Appl. Phys. 111, 07C705 (2012). 17L. C. Lim, K. K. Rajan, and J. Jin, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 54, 2474 (2007). 18M. Liu et al. ,Appl. Phys. Lett. 98, 222509 (2011). 19M. Liu et al. ,Adv. Mater. 25, 1435 (2013). 20S. D. Li et al. ,J. Appl. Phys. 115, 17C723 (2014). 21S. D. Li et al. ,J. Appl. Phys. 113, 17C727 (2013).17B722-4 Li et al. J. Appl. Phys. 117, 17B722 (2015)Journal of Applied Physics is copyrighted by AIP Publishing LLC (AIP). Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. For more information, see http://publishing.aip.org/authors/rights- and- permissions.

1.2839266.pdf

Ferromagnetic resonance linewidth reduction in Fe ∕ Au multilayers using ion beams C. Bilzer, T. Devolder, C. Chappert, O. Plantevin, A. K. Suszka, B. J. Hickey, A. Lamperti, B. K. Tanner, B. Mahrov, and S. O. Demokritov Citation: Journal of Applied Physics 103, 07B518 (2008); doi: 10.1063/1.2839266 View online: http://dx.doi.org/10.1063/1.2839266 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/103/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ferromagnetic resonance study of Co/Pd/Co/Ni multilayers with perpendicular anisotropy irradiated with helium ions J. Appl. Phys. 109, 033917 (2011); 10.1063/1.3544474 Determination of the saturation magnetization of ion irradiated Py/Ta samples using polar magneto-optical Kerr effect and ferromagnetic resonance Appl. Phys. Lett. 96, 022503 (2010); 10.1063/1.3291051 Broadband ferromagnetic resonance linewidth measurement of magnetic tunnel junction multilayers Appl. Phys. Lett. 94, 012506 (2009); 10.1063/1.3054642 Ion irradiation of Co/Pt multilayer films J. Appl. Phys. 93, 7226 (2003); 10.1063/1.1557311 Using ferromagnetic resonance to measure magnetic moments of ultrathin films (abstract) J. Appl. Phys. 81, 4475 (1997); 10.1063/1.364982 [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: 136.165.238.131 On: Fri, 19 Dec 2014 22:50:49Ferromagnetic resonance linewidth reduction in Fe/Au multilayers using ion beams C. Bilzer,1,a/H20850T . Devolder,1C. Chappert,1O. Plantevin,2A. K. Suszka,3B. J. Hickey,3 A. Lamperti,4B. K. T anner,4B. Mahrov,5and S. O. Demokritov5 1Institut d’Electronique Fondamentale, CNRS UMR 8622, Université Paris-Sud, 91405 Orsay, France 2CSNSM, CNRS/IN2P3, UMR 8609, Université Paris-Sud, 91405 Orsay, France 3E. C. Stoner Laboratory, School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 4Department of Physics, University of Durham, South Road, Durham, DH1 3LE, United Kingdom 5Institute for Applied Physics, University of Muenster, Corrensstr. 2-4, 48149 Muenster, Germany /H20849Presented on 9 November 2007; received 12 September 2007; accepted 5 December 2007; published online 15 February 2008 /H20850 In order to optimize their magnetic properties, Fe /Au multilayers were treated by pregrowth and postgrowth ion-beam bombardments. The ferromagnetic resonance linewidth was used as our mainfigure of merit. The pregrowth treatment of the MgO substrate using a 60 eV atomic oxygen beamresulted in a reduction of the inhomogeneous linewidth broadening in comparison with a samplegrown on an untreated substrate. This homogeneity increase is linked to the removal of substratecarbon contamination by the chemically active oxygen. It correlates with the reduced interfaceroughness. The postgrowth sample irradiation using 30 keV He +ions also reduces the inhomogeneous broadening in the linewidth. Fe and Au have a miscibility gap, but the demixing iskinetically quenched at room temperature. Ion collisions locally minimize the interface energy byproviding the energy necessary for localized demixing, resulting in a smoothing effect. Combined,the pregrowth and the postgrowth irradiations lead to the lowest observed linewidth. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2839266 /H20852 Engineering the magnetic properties of thin films and multilayers is imperative to reveal novel effects or optimizethe functioning of devices. In particular, in the latter, most often ultrathin layers are employed. Due to the reduced di-mensions, the interface properties, e.g., its roughness, 1gain considerable importance. While optimizations of the growthprocedure by, for example, using buffer or seed layers 2are possible, ion irradiation constitutes an alternative method,which can be used in addition. Two different treatments ofFe /Au multilayers are assessed in this article. A pregrowth ion-beam treatment of the MgO substrate using atomic oxy-gen: cleaning the substrate in this manner promises improvedgrowth on top. Postgrowth irradiation using He +ions: while ion irradiation is often associated with mixing, this is not thecase here. As Fe and Au are thermodynamically immiscible,we assume that the energy transferred by the ion collisionsshould make localized demixing at the interfaces possible.Finally, we analyze if both treatments can be combined toachieve an even more pronounced effect. Two Fe /Au multilayer samples were grown by molecu- lar beam epitaxy. Their nominal structure isMgO /H20849100 /H20850/Fe/H208493/H20850/Au/H2084919/H20850//H20851Fe/H208491/H20850/Au/H208490.9 /H20850/H20852 20. All thick- nesses are expressed in nanometers. The 1 /H110031c m2sized MgO substrates were mounted on a rotating molybdenumplate to improve homogeneity. Prior to growth, they wereannealed for 1 h at 500 °C removing water and hydrogen.The 3 nm Fe seed layer was deposited at the same tempera-ture. It leads to an improvement of the growth conditions forthe subsequent 19 nm Au buffer layer, 2,3which was depos- ited at 200 °C. A surface roughness of the Au layer on theorder of the interlayer spacing is expected. 4The actual Fe /Au multilayer was deposited at 70 °C to minimize inter- diffusion of the alternating layers. The growth pressure was6.9/H1100310 −9Torr. The sample A-non was grown on a MgO substrate, which was additionally pregrowth ion beamtreated. The procedure consists of an exposition for 2 min toa neutralized atomic oxygen beam with the nominal energyof 60 eV, providing an ion dose of 6 /H1100310 16ions /cm2. While the substrate was irradiated in vacuum /H2084910−4mbar /H20850,i tw a s kept in air until the growth in the molecular beam epitaxy system. It was shown by Auger spectroscopy that the treat- ment removes the carbon contamination, which is found onthe surface of MgO substrates. 3The second sample, B-non, was deposited on a substrate only subject to the describedannealing procedure. The samples were characterized using the vector net- work analyzer ferromagnetic resonance /H20849FMR /H20850: the thin films are placed on a coplanar waveguide, which creates asmall oscillating field perpendicular to a constant magneticfield applied by an electromagnet. A small angle precessionis excited, which is detected using the network analyzer. Adetailed description of the measurement procedure is foundelsewhere. 5Varying the frequency of the small excitation field, the measured data allow the calculation of the dynamicsusceptibility spectra of the sample. This is illustrated in Fig.1, plotting the imaginary part of the susceptibility of both samples, under an applied field of 45 mT. a/H20850Electronic mail: claus.bilzer@ief.u-psud.fr.JOURNAL OF APPLIED PHYSICS 103, 07B518 /H208492008 /H20850 0021-8979/2008/103 /H208497/H20850/07B518/3/$23.00 © 2008 American Institute of Physics 103 , 07B518-1 [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: 136.165.238.131 On: Fri, 19 Dec 2014 22:50:49From each spectrum, we evaluate two quantities: the FMR frequency fresand the full linewidth at half maximum /H9004f. The resonance frequencies almost coincide. They allow to determine the effective magnetization using the appropri-ate equation for the resonance condition. 6We find /H92620Meff =0.97 T for the A-non sample and 0.99 T for B-non, wherewe assumed a gfactor of g=2.1. These values are confirmed by conventional magnetometry. They are in agreement withexpected values for the effective magnetization of 1 nm thickFe layers, as the saturation magnetization is reduced by asurface anisotropy contribution. 7 The resonance linewidths differ considerably. To permit an interpretation, Fig. 2plots/H9004fas a function of the external applied field Happl. The linewidths clearly exceed the values observed on single Fe layers. This is attributed to the multilayer, wherethe measured linewidth is extracted from a superposition ofresonances of all individual layers, which are slightly shifteddue to different effective fields. Therefore, the extractedproperties can only be related to the ensemble of layers. Thelinewidth and the intrinsic Gilbert damping parameter /H9251are related as /H9004fint=/H9251/H9253/H92620 2/H9266/H20851Meff+2 /H20849Happl+Haniso /H20850/H20852, /H208491/H20850 where /H9253is the gyromagnetic ratio and Hanisothe anisotropy field. The expected curve for /H9004fintis included in Fig. 2/H20849dot- ted line /H20850. The measured curves deviate in shape from /H9004fint.This is related to the inhomogeneous broadening: this phe- nomenon is well known in field-swept FMR,8where it is observed as a finite zero-frequency field linewidth /H9004Hinhomo . In frequency-swept FMR, on the other hand, the extrinsicfrequency linewidth /H9004f inhomo is not constant with resonance frequency but decreases with its inverse. The analyticaldescription 9yields /H9004finhomo/H11008/H9004Hinhomo /fres. It adds in quadrature to the intrinsic linewidth,10 /H9004fmea2=/H9004fint2+/H9004finhomo2. /H208492/H20850 Equation /H208492/H20850allows to separate the extrinsic contribution in Fig.2:/H9251was determined to be 0.022. It is difficult to quan- titatively attribute /H9004finhomo to the spread in one magnetic property. However, it is reasonable to assume that a majorcontribution to the inhomogeneity is related to roughness, asthe individual magnetic layers in the sample consist of onlyfew atomic layers. A qualitative comparison of the extrinsiclinewidth contributions confirms a higher homogeneity in theA-non sample grown on the cleaned substrate than in theB-non sample. Atomic force microscopy images on the sur-face of the uppermost Au layer support the connection be-tween the lower interface roughness and the higher homoge-neity: the root mean square roughness was determined to be0.27 nm for the A-non sample and 0.44 nm for B-non. Theeffect of the substrate cleaning can thus still be observedafter a stack of 42 layers. Another strong indication for the better growth condi- tions on the cleaned substrate is the observation of a secondmagnetic resonance in the spectrum of the A-non sample /H20849see Fig. 1/H20850. It is caused by the lowermost 3 nm thick Fe seed layer. From its resonance frequency, we obtain /H92620Meff =1.5 T. We could not observe this resonance for B-non:grown on the rougher substrate, it most likely has a magneticdead layer, its signal too low for detection. An alternative means of influencing the layer quality is the postgrowth light-ion irradiation. 11The irradiation param- eters were determined from stopping and range of ions inmatter SRIM simulations:1230 keV He+ions lead to a nuclear stopping power of 1.28 eV /Å in Fe. This low value main- tains the crystallographic structure, typically displacingsingle atoms only a few interatomic distances. Though Feand Au are immiscible, demixing of the FeAu alloy at themultilayer interfaces is kinetically quenched at room tem-perature. The ion collisions provide the atom mobility toovercome this kinetic limit, yielding an effective interfacesmoothing effect. Due to the immiscibility, it is thermody-namically favorable to reduce the interfacial area and thecorresponding interfacial energy. 13The penetration depth of the incident ions is estimated to be 113 nm, i.e., they remainin the substrate. The two initial samples were diced into several pieces. They were irradiated with 30 keV He +ions at a low current of 0.25 /H9262A/cm2, avoiding sample heating. A-low and B-low are pieces of the respective nonsamples irradiated with a flu-ence of 1 /H1100310 15ion /cm2. A-high and B-high represent pieces of the initial samples irradiated with a fluence of 3/H1100310 15ion /cm2. Figure 3shows the linewidths of the samples B-non, B-low, and B-high at 45 mT applied field, centered around 0 FIG. 1. /H20849Color online /H20850Imaginary part of the complex susceptibility of the A-non and B-non samples for the same applied field of 45 mT. The line-width is broader for the B-non sample. The small second resonance is onlyvisible for the A-non sample. FIG. 2. /H20849Color online /H20850Resonance linewidth as a function of the applied magnetic field for the A-non and B-non samples. A fit including the extrinsiclinewidth contribution is shown for A-non. The intrinsic linewidth using thefitted damping parameter is plotted.07B518-2 Bilzer et al. J. Appl. Phys. 103 , 07B518 /H208492008 /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: 136.165.238.131 On: Fri, 19 Dec 2014 22:50:49for better comparison. The irradiation led to a slight increase in effective magnetization, yielding a higher resonance fre-quency for the same field. This increase is not understood. The linewidth decreases when irradiating with a higher fluence. Figure 4/H20849b/H20850plots the measured linewidths for the B samples as a function of applied field. The lower linewidthfor the irradiated samples is attributed to a lower inhomoge-neous broadening. In analogy to the preceding argument, thelower extrinsic linewidth can be related to a lower interfaceroughness. Indeed, the lowest inhomogeneous broadening is ob- served for the A-high sample in Fig. 4/H20849a/H20850. This sample was deposited on the pregrowth ion-beam cleaned MgO substrateand was postgrowth irradiated with the higher fluence. We conclude that the smoothing effect of both methods add up.The efficiency of the postgrowth treatment still has room forimprovement by further augmenting the fluence, as defectcreation is omitted due to the low ion energy. In summary, we have shown that it is possible to im- prove the interface roughness of Fe /Au multilayer samples by ion-beam irradiation. A pregrowth substrate treatment us-ing a dissociated 60 eV atomic oxygen ion beam removesthe carbon contamination on MgO substrates due to its highchemical activity. This allows us to decrease the interfaceroughness of all subsequently deposited layers. The proce-dure is even still effective after exposing the substrate to airfor a longer period. The postgrowth irradiation using 30 keVHe +ions also improved the interface smoothness. The low energy avoids defect creation. The ion collisions provide theatom mobility necessary to achieve localized demixing of theFeAu alloy at the interfaces. Increasing the irradiation flu-ence decreases the inhomogeneous linewidth broadening.The pregrowth and postgrowth treatments combined lead tothe lowest extrinsic broadening, i.e., the highest homogene-ity. C.B., A.K.S., A.L., and B.M. acknowledge the financial support provided through the European Community’s MarieCurie actions /H20849Research Training Networks /H20850under Contract No. MRTN-CT-2003-504462, ULTRASMOOTH. 1A. T. G. Pym, A. Lamperti, B. K. Tanner, T. Dimopoulos, M. Ruhrig, and J. Wecker, Appl. Phys. Lett. 88, 162505 /H208492006 /H20850. 2A. Cole, B. J. Hickey, T. P. A. Hase, J. D. R. Buchanan, and B. K. Tanner, J. Phys.: Condens. Matter 16, 1197 /H208492004 /H20850. 3M. Rickart, B. F. P. Roos, T. Mewes, J. Jorzick, S. O. Demokritov, and B. Hillebrands, Surf. Sci. 495,6 8 /H208492001 /H20850. 4M. M. J. Bischoff, T. Yamada, A. J. Quinn, R. G. P. van der Kraan, and H. van Kempen, Phys. Rev. Lett. 87, 246102 /H208492001 /H20850. 5C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P. P. Freitas, J. Appl. Phys. 101, 074505 /H208492007 /H20850. 6G. Counil, J.-V. Kim, T. Devolder, P. Crozat, C. Chappert, and A. Cebol- lada, J. Appl. Phys. 98, 023901 /H208492005 /H20850. 7S. Visnovsky, R. Lopusnik, M. Nyvlt, A. Das, R. Krishnan, M. Tessier, Z. Frait, P. Aitchison, and J. N. Chapman, J. Magn. Magn. Mater. 198–199 , 480 /H208491999 /H20850. 8M. Farle, Rep. Prog. Phys. 61, 755 /H208491998 /H20850. 9C. Scheck, L. Cheng, and W. E. Bailey, Appl. Phys. Lett. 88, 252510 /H208492006 /H20850. 10O. Mosendz, B. Kardasz, D. Schmool, and B. Heinrich, J. Magn. Magn. Mater. 300, 174 /H208492006 /H20850. 11T. Devolder, H. Bernas, D. Ravelosona, C. Chappert, S. Pizzini, J. Vogel, J. Ferre, J. P. Jamet, Y. Chen, and V. Mathet, Nucl. Instrum. Methods Phys.Res. B 175–177 , 375 /H208492001 /H20850. 12J. F. Ziegler, J. P. Biersack, and U. Littmark, The Stopping and Range of Ions in Matter /H20849Pergamon, New York, 1985 /H20850, Vol. 1. 13C. Rumbolz, W. Bolse, S. Kumar, R. Chauhan, D. Kabiraj, and D. Avasthi, Nucl. Instrum. Methods Phys. Res. B 245, 145 /H208492006 /H20850. FIG. 3. /H20849Color online /H20850The imaginary part of the complex susceptibility of the three B samples for the same applied field of 45 mT centered around 0/H20849shifted by their respective resonance frequency /H20850. The linewidth is smaller for a higher irradiation dose. FIG. 4. /H20849Color online /H20850Resonance linewidth as a function of the applied magnetic field for the three A samples /H20849a/H20850and for the three B samples /H20849b/H20850.A t lower fields, the linewidths are clearly larger for the B samples compared tothe A samples.07B518-3 Bilzer et al. J. Appl. Phys. 103 , 07B518 /H208492008 /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: 136.165.238.131 On: Fri, 19 Dec 2014 22:50:49

1.3639305.pdf

Analytic treatment of the precessional (ballistic) contribution to the conventional magnetic switching Ya. B. Bazaliy Citation: Journal of Applied Physics 110, 063920 (2011); doi: 10.1063/1.3639305 View online: http://dx.doi.org/10.1063/1.3639305 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/110/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin torque switching and scaling in synthetic antiferromagnet free layers with in-plane magnetization J. Appl. Phys. 111, 123914 (2012); 10.1063/1.4729776 Ballistic (precessional) contribution to the conventional magnetic switching Appl. Phys. Lett. 98, 142501 (2011); 10.1063/1.3570635 Minimal field requirement in precessional magnetization switching J. Appl. Phys. 99, 013903 (2006); 10.1063/1.2161421 Influence of surface anisotropy on the magnetization precessional switching in nanoparticles J. Appl. Phys. 97, 10J302 (2005); 10.1063/1.1845891 Computer simulations of magnetization switching of elongated magnetic particles with shape defects J. Appl. Phys. 93, 9865 (2003); 10.1063/1.1575494 [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27Analytic treatment of the precessional (ballistic) contribution to the conventional magnetic switching Y a. B. Bazaliya) Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA and Institute of Magnetism, National Academy of Science, Kyiv 03142, Ukraine (Received 16 June 2011; accepted 11 August 2011; published online 29 September 2011) We consider a uniformly magnetized particle (i.e., a macroscopic magnetic moment) with an easy axis anisotropy. The particle’s moment is switched from “up” to “down” direction by an external magnetic field applied parallel to the easy axis and continuously swept from a positive to anegative value. In addition, a small constant perpendicular bias field is present. It is shown that while the driving field switches the particle’s moment in a conventional damped way, the perpendicular field creates an admixture of the precessional (ballistic) switching that speeds upthe switching process. Precessional contribution produces a non-monotonic dependence of the switching time on the field sweep time with a minimum at a particular sweep time value. We derive analytic expressions for the optimal point, and for the entire dependence of the switching time onthe field sweep time. Our approximation is valid in a wide parameter range and can be used to engineer and optimize of the magnetic memory devices. VC2011 American Institute of Physics . [doi: 10.1063/1.3639305 ] I. INTRODUCTION Conventional magnetic switching by an externally applied magnetic field His based on the transition to the lowest energy state under the action of damping torques.Currently this type of damped switching constitutes the basis of magnetic recording in hard disk drives and other devices. The speed at which the moments of the magnetic bits can beswitched between the two easy directions has an obvious implication for the technology performance, setting the limit for the information writing rate. In general, magnetic switch-ing time s mdepends on the material parameters, sample size, and shape. Here we are interested in the dependence of smon the rate of change of the driving magnetic field. We will callthe time s hrequired to flip the external magnetic field a “field sweep time.” This time is of course finite in any real device used for magnetic recording. Clearly, very long field sweeptimes will make the magnetic switching very slow. One can argue then that, since s mwill decrease with decreasing sh, the best case scenario for switching is the instantaneous flipof the external field with s h¼0. However, it was found numerically1that in realistic conditions the function sm(sh)i s not monotonic and has a minimum at a particular value ofthe sweep time s /C3 h. The field sweep time corresponding to this minimum is optimal and any further decrease of shwill be counterproductive in terms of the technology perform-ance. An analytic expression for s m(sh) was announced in Ref. 2. That paper also explained the physical reason behind the existence of the switching time minimum. In the presentpaper we provide the detailed derivation of the approximate analytic expressions for the function s m(sh) and the optimal field sweep time s/C3 h. We then discuss the limits of their valid-ity and compare analytic approximations with the exact numeric results. Before proceeding to the derivations, we would like to place the phenomenon under investigation into context. Mag- netic switching and its speed are important technological pa- rameters and were studied by many authors. We willconcentrate on a uniformly magnetized single domain particle with easy axis anisotropy energy. The state of such particle is completely determined by its total magnetic moment M. While the magnetization distributions in real magnetic par- ticles are not always uniform, the deviations from uniformity are supposed to vanish as the size of the particles decreasesand the exchange interaction overpowers the demagnetization effects. Magnetic switching of a uniformly magnetized parti- cle under the action of a static applied field is described by theStoner-Wholfarth astroid 3in the H-space (Fig. 1,a x i s ^zis pointing along the easy axis). If the applied field is changed infinitely slowly, and the thermal fluctuations of the momentcan be neglected, magnetic switching starts when the trajec- tory of the driving field in the H-space crosses the astroid boundary. The switching process that follows takes a finitetime which depends on the crossing point. Since the field changes infinitely slowly, His constant during the switching. If the magnitude and direction of the applied field are allowed to change during the magnetic switching process, the number of switching scenarios becomes infinite. In the most general case one wants to optimize the function H(t)t o minimize the switching time. Different minimization schemes are discussed theoretically, 4–6but their experimen- tal realization is distant as they all require external fields thatchange exceedingly rapidly and have carefully controlled magnitude and direction. A large body of research is devoted to a restricted case of pulsed applied fields with fixed direction. The easiest pulse shape to produce experimentally is a field jump from a)Electronic mail: yar@physics.sc.edu. 0021-8979/2011/110(6)/063920/13/$30.00 VC2011 American Institute of Physics 110, 063920-1JOURNAL OF APPLIED PHYSICS 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27an initial value Hito a final value Hf. The jump can be in- stantaneous (step function) or have a certain rise time (smeared step function). After the jump the field stays at thefinal value H f. The equilibrium directions of the moment before and after the jump are described by the Stoner- Wohlfarth picture with H¼HiandH¼Hf, respectively. Before the field jump the moment resides in the minimum Miof the magnetic energy corresponding to the initial field. After the field jump the moment eventually reaches the equi-librium M f, corresponding to the minimum of the energy at the final external field. (We will consider situations where the energy has only one minimum.) The switching timedepends on both the initial and final directions of the moment or, equivalently, on both H iandHf.7(Switching by an infinitely slowly varying field discussed earlier in this sec-tion is equivalent to an infinitesimally small jump across the astroid boundary. In this case the switching time depends on one parameter, since H i¼Hf). Switching by a field jump leads to some unusual properties even for the idealized case of an instantaneous jump. For example, switching below the Stoner-Wholfarth limit becomes possible,7–10and is accom- panied by an interesting counter-intuitive phenomenon: The usual property of switching time to decrease with the increasing magnitude of Hf, observed for Hfoutside of the astroid, may reverse when Hfis inside the astroid.11 If instead of the field jumps one uses pulses with the field switched from zero to a fixed value for a finite time period,more options for switching time minimization become avail- able. In the absence of magnetic anisotropy the fastest 180 /C14flip of a moment is achieved by applying Hperpendicular to Mi. The field is turned on, the moment starts to precess around H. After a half of the precession period it reaches the direction Mf¼/C0Mi, at which time the field has to be switched off. This scheme constitutes the simplest example of “precessional” or “ballistic” switching. In the pr esence of anisotropy precessional switching becomes more complicated, but it is stillpossible.12–16The key ingredients of the precessional switching are extremely short pulses of pr ecise duration and sufficiently strong field magnitude. Ideally their rise and fall times shouldapproach zero, but this is very hard to achieve experimentally as the pulse duration should be of the order of picoseconds. Precessional switching was observed using the fields createdby pulses of synchrotron radiation 17or by a careful pulse shap- ing using femtosecond lasers,18but is not yet used in applica- tions. The speed of the precessional switching increases withthe field magnitude, but cannot be increased infinitely due to the breakdown of the ferromagnetic state of the sample. 19 Another widely discussed class of switching scenarios is the application of the high-frequency oscillating field, or an RF signal. When the frequency of the RF signal is close to the eigenfrequency of the magnetic moment, a magneticresonance occurs and a continuous precession state is estab- lished. The amplitude of the precession is growing as the power of the RF signal is increased. Sufficiently large sig-nal would in principle be able to increase the amplitude so much as to switch the magnetization from “up” to “down.” Experiments observed an RF-assisted switching 20happen- ing inside the Stoner-Wohlfarth astroid. In this case the RF signal helps the external field to drive the transition. To improve the RF-assisted switching the schemes with vari-able (“chirped”) frequency are designed theoretically. 21–26 Their goal is to keep the frequency equal to the instantane- ous, amplitude-dependent resonance frequency of themagnet. Various instabilities may prevent the purely RF switching by destroying the coherent single-domain state of a sample. However, they are supposed to be suppressed inthe magnetic particles with the sizes below 10 /C020nm. 11,24 The phenomenon considered in the present paper is dif- ferent from all of the above. We consider conventionaldamped switching by a field jump, but concentrate on the switching time s mdependence on the jump rise time sh.T h e function sm(sh) exhibits an unexpected minimum which was not discussed in the literature. As it will be shown in the Conclusions, the minimum of sm(sh) results from an admixture of a precessional (ballistic) switching to the con-ventional switching, an effect which can be named a ballistic- assisted switching. Normally, the ballistic contribution of a small perpendicular field is quenched by the large anisotropy,but here it is restored by the time-dependence of the switching field during the rise time of the jump. This type of ballistic contribution can be observed for infinitesimally small perpen-dicular fields, which distinguishes it is from the purely ballis- tic case where a finite field comparable to the anisotropy fields is required. Our analysis also shows that the phenomena ofballistic-assisted and RF-assisted switching are complimen- tary. Both can be treated on equal footing by considering the averaging of the perturbation field torque during the preces-sion cycle of the moment in the strong anisotropy field. II. MODEL Our treatment is based on the Landau-Lifshitz equation, and does not take into account thermal fluctuations. We con-sider a single domain magnetic bit with uniform magnetiza- tion, sometimes also called a “macrospin” in the literature. It FIG. 1. Top: Stoner-Wohlfarth astroid in the H-space and the trajectory of the magnetic field change (dashed line) during the rise time of the pulse with axial field sweep from þH0to/C0H0at a constant perpendicular bias H \. The lower panels (A) and (B) correspond to the initial and final directions of the field. They show the “up” and “down” minimum points Mu,M d, and maximum points Xof the magnetic energy.063920-2 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27is described by its total magnetic moment M¼M0n, where n is a unit vector. The magnet has an easy anisotropy axis directed along z, and its anisotropy energy is given by Ea¼/C0 ð 1=2ÞKn2 z. This anisotropy creates two equilibrium directions of the moment along þzand/C0z. The switching field H¼H(t)^zis directed along zas well. It favors the þzdirection for H>0 and /C0zforH<0. For large enough magnitudes, jHj>K=M0, there is no equilib- rium in the direction opposite to the field. When the external field is pointing along the easy axis direction, magnetic switching relies on the fluctuations near the equilibrium position. Without them a moment pointingexactly along þzwill not be switched by any amount of negative applied field. Instead it will remain at the point of unstable equilibrium, until an initial fluctuation occurs andthen grows with time. The switching time in this case strongly depends on the fluctuation magnitude dh, being in- finite for dh¼0. 27–31In order to model the required initial fluctuation we apply a small bias field H?¼H?^xperpen- dicular to the easy axis. This field is set to be constant in time. It creates a controlled deviation of the initial magnet-ization from the field direction and makes the problem well defined. The switching dynamics is described by the Landau-Lif- shitz-Gilbert (LLG) equation. In the case of a uniformly magnetized particle it is an ordinary differential equation describing the evolution of a single vector M(t), _M¼/C0c@E @M/C2M/C20/C21 þa M0½M/C2_M/C138; where cis the gyromagnetic ratio, E¼Ea/C0(HþH\)/C1Mis the magnetic energy, and ais the Gilbert damping constant. In terms of the unit vector n(t) the LLG reads _n¼/C0@e @n/C2n/C20/C21 þa½n/C2_n/C138 (1) with e¼cE/M 0. Using the spherical angles ( h,u), defined so that n¼fnx;ny;nzg¼f sinhcos/;sinhsin/;coshg, one gets a system of equations ð1þa2Þ_h¼/C01 sinh@e @//C0a@e @h; (2) ð1þa2Þ_/¼1 sinh@e @h/C0a sin2h@e @/: (3) Actual materials are characterized by a/C281 and we will always calculate up to the linear terms in a, e.g., 1 þa2/C251. In our case the energy has the form eðh;/Þ¼/C0x0 2n2 z/C0hðtÞnz/C0h?nx ¼/C0x0 2cos2h/C0hðtÞcosh/C0h?sinhcos/;(4) where x0¼cK/M 0,h¼cH,h?¼cH?. The smallness of the bias field is ensured by the condition h?/C28x0. The field sweep is assumed to be linear in time and given by the expressionshðtÞ¼þ h0;ðt<0Þ hðtÞ¼h01/C02t sh/C18/C19 ;ð0<t<shÞ hðtÞ¼/C0 h0;ðt>shÞ:(5) Field evolution in H-space and positions of the minima and maxima of the magnetic energy for the initial and finalfield orientations are shown in Fig. 1. We start with the positive field h¼þh 0>x0which guarantees that the mag- net is initially pointing close to the þzdirection. This state will be called an “up-equilibrium.” At the end of the reversal the moment reaches the “down-equilibrium,” corresponding toh¼/C0h0. The initial and final directions of the moment are determined from the conditions @e=@/¼0,@e=@h¼0 with h¼6h0. This gives u¼0 and an equation for h reading x0sinhcosh6h0sinh/C0h?cosh¼0: In the limit h?/C28x0one finds the following approximations for the values of the polar angles in the up- and down- equilibria: hu/C25h? x0þh0;hd/C25p/C0h? x0þh0: (6) As the field is swept from positive to negative values, the up- equilibrium disappears and the magnetic moment starts to move in a spiral fashion toward the down-equilibrium, approaching it exponentially. To define a finite switchingtime we have to introduce a provisional cut-off angle h sw and calculate the time it takes to reach hswduring the switching process. The remaining distance from hswtohd takes extra time, but this time interval does not depend on the field sweep time since in the regimes studied in our pa- persh<sm, and the remaining motion happens at a constant external field. We use the commonly adopted1,7,9,11value of hsw¼p/2. III. NUMERIC RESULTS: NON-MONOTONIC DEPENDENCE OF THE SWITCHING TIME The LLG equation can be easily solved numerically and the switching time dependence sm(sh) can be obtained. Figure 2shows the results of such modeling for a particular FIG. 2. Switching time as a function of field sweep time (time is measured in units of x0/C01). Parameter values are a¼0.01, h0¼3.5x0,h?¼0.001 x0. The thin solid line is the approximate expression derived in this paper.063920-3 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27parameter set (see figure caption). The minimum of smis clearly observed. We will show in the subsequent sections that the switch- ing time can be approximated by an expression sm¼3h0þx0 4h0shþln½h0=ph2 ?sh/C138 2aðh0/C0x0ÞþsRða;h0;x0Þ; where sRis independent of shand h?. As one can see in Fig. 2the correspondence between the actual (numeric) and approximate curves is quite good and reproduces the mini-mum of s m. A note on the numeric calculation is due here. It is more convenient to follow the time dependence of the total energye(t), than that of h(t). The reason for that is as follows. In the regime considered here the switching time and field sweep time satisfy s m>shand the switching threshold h¼p/2 is reached when the external field is already time-independent, h¼/C0h0. According to the LLG equation, at constant exter- nal field the time derivative of energy is strictly negative _eðtÞ¼/C0 a_n2<0; ande(t) is a strictly decreasing function of time. This prop- erty greatly simplifies the solution of the equation e(sm)¼esw, where eswis the new cut-off, introduced instead ofhsw. Note that in contrast to e(t) the time dependence of h(t) is non-monotonic, which can be easily understood by recalling that in the a/C281 limit the spiral motion of the moment approximately follows the equipotential lines. Dueto the presence of the bias field h \the latter differ from the h¼const lines, leading to the oscillations of h(t) in time. In other words, the fact that e(h,u) depends on both spherical angles makes the cut-offs hswandeswnot completely equiva- lent. Nevertheless, they serve the same purpose with the lat- ter being a more convenient choice. We adopt the cut-offvalue e sw¼e(p/2,p/2)¼0, closest to the original definition.1 IV. ANALYTIC APPROXIMATION FOR THE SWITCHING TIME The switching process consists of two stages. The first stage is the field sweep time interval, 0 <t<sh. The second stage is the motion in the constant field for the time interval sh<t<sm. Below we use two different approximations to find the magnetic dynamics in each stage. A. First stage The idea for the first stage approximation is to assume that the deviation of nfromþzis small. The rationale for that is provided by the following argument. The initial posi-tion of the moment is given by h u/C25h?=ðx0þh0Þ/C28 1, i.e., is very close to þz. We then assume that proximity to the þzholds throughout the first stage if the sweep time is not too long. The precise condition imposed by this assumption onshis not clear at this point but will be obtained after we do the calculations. According to the above, we linearize the LLG equation near the þzpoint. In the linearized equation the unknownsare the two projections nx;ny/C0/C1 of the unit vector. Both are small for nclose to þz. One gets a linear system _nx¼/C0aðx0þhÞnx/C0ðx0þhÞnyþah?; _ny¼¼ð x0þhÞnx/C0aðx0þhÞny/C0h?: Introducing a notation xðtÞ¼x0þhðtÞwe rewrite it as _nx _ny/C18/C19 ¼/C0axðtÞ/C0 xðtÞ xðtÞ/C0 axðtÞ/C12/C12/C12/C12/C12/C12/C12/C12n x ny/C18/C19 þah? /C0h?/C18/C19 :(7) The matrix on the right hand side can be diagonalized by changing variables to n¼nxþiny,g¼nx/C0iny. We get two decoupled equations _n¼ði/C0aÞ½xðtÞn/C0h?/C138; _g¼/C0 ð iþaÞ½xðtÞg/C0h?/C138; which turn out to be complex conjugates of each other. Con- sequently, we can solve either one of them. Denoting l¼i/C0a, we search for the solution of the first equation in the form nðtÞ¼AðtÞelÐt 0xðsÞds: For future notation we define a phase function uðtÞ ¼Ðt 0xðsÞds. The solution is found to be nðtÞ¼nð0ÞeluðtÞ/C0lh?eluðtÞðt 0e/C0luðuÞdu: Going back to ðnx;nyÞwe obtain a solution nx¼e/C0aunx0cosu/C0ny0sinu/C8 /C0h?ðS/C0aCÞ/C138cosu/C0ðCþaSÞ/C138sinu ½/C138 g ; ny¼e/C0aunx0sinuþny0cosu/C8 /C0h?ðS/C0aCÞ/C138sinuþðCþaSÞ/C138cosu ½/C138 g ;(8) where we have defined SðtÞ¼ðt 0eauðsÞsin/ðsÞds; CðtÞ¼ðt 0eauðsÞcos/ðsÞds:(9) In our case the initial conditions are given by nx0¼h? x0þh;ny0¼0; (10) thus both projections nxandnyare proportional to h?and we should be able to satisfy the assumption of small deviation from the origin for sufficiently small bias field. Below wewill calculate how small should h ?be to ensure small devia- tions in Stage I. For the linear field sweep (5)the phase /(t) is a quad- ratic function063920-4 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27u¼ðt 0x0þh01/C02s sh/C19/C18/C19/C18 ds¼ðx0þh0Þt/C0h0t2 sh: In this case the integrals (9)can be found exactly and expressed through the error function of complex argument (Appendix A). Here we will consider a useful approximation valid in a large region of parameters. The phase /(t) has one maximum on the interval [0, sh] at the point tm¼sh(x0þh0)/2h0 (recall that x0<h0). The presence of a maximum means that the integrals for C(sh) and S(sh) can be approximated by a steepest descent (stationary phase) method in the case of alarge change of /on the integration interval [0, s h]. This cer- tainly requires the inequality x0sh/C291 to hold but, as it turns out below, sometimes an even stronger condition isneeded. The steepest descent calculation is performed in Appendix Aand gives an approximation S¼e a/mffiffiffiffiffiffiffipsh h0r sin/m/C0p 4/C16/C17 þa 2cos/m/C0p 4/C16/C17 hi ; C¼ea/mffiffiffiffiffiffiffipsh h0r cos/m/C0p 4/C16/C17 /C0a 2sin/m/C0p 4/C16/C17 hi ; um/C17uðtmÞ¼ðx0þh0Þ2 4h0sh:(11) Substituting this into Eq. (8)using /(sh)¼x0shand per- forming the calculations we find nxðshÞ¼e/C0ax0shh? x0þh0cosx0t /C0eaDuh?ffiffiffiffiffiffiffipsh h0r ðsinDu/C0a 2cosDuÞ; nyðshÞ¼e/C0ax0shh? x0þh0sinx0t /C0eaDuh?ffiffiffiffiffiffiffipsh h0r ðcosDuþa 2sinDuÞ;(12) where we have defined Du¼uðtmÞ/C0uðshÞ¼ðx0/C0h0Þ2 4h0sh; Du¼Du/C0p 4:(13) We observe that in both formulae the first term on the right hand side is initially small and further decreases as a func- tion of sh, while the second term increases with sh. Therefore the condition for small deviations can be formulated as the smallness of the second term eaDuh?ffiffiffiffiffiffiffipsh h0r /C281: Explicitly separating the product x0sh, we can write the con- dition on the bias field h? x0/C28ffiffiffiffiffiffiffiffiffi h0 px0r ffiffiffiffiffiffiffiffiffiffi 1 x0shr exp/C0aðx0/C0h0Þ2 4h0x0ðx0shÞ"# ;(14)which will guarantee the validity of the small deviations assumption. Additional inequalities (A4) enabling the approximation (11) are listed in Appendix Aand have to be satisfied as well. We will return to their discussion in Sec. V. B. Second stage During Stage II the external magnetic field is constant, H¼/C0H0^zþH?^x. The action of the bias field H?can be viewed as a perturbation of the axially symmetric problem with H?¼0 and H¼/C0H0. For the unperturbed problem the switching time is a known27–31as a function of the angu- lar deviation hinof the angular deviation of the magnetiza- tion from the easy axis at the beginning of Stage II. To find the perturbation corrections to the axially sym- metric problem we employ the method of approximate dif- ferential equation for the total energy ein the limit of small Gilbert damping constant (see, e.g., Ref. 32). In the a!0 limit the motion of the moment M(t) can be viewed as a fast precession along the e¼const lines and a slow motion from one equipotential orbit to the next one nearby. Up to the lin-ear terms in athe change of energy upon one precession cycle around an orbit is given by De¼/C0aþ C@e @n/C12/C12/C12/C12/C12/C12/C12/C12dn¼/C0af1ðeÞ; (15) and the period of this cycle is T¼þ Cdn @e=@njj¼f2ðeÞ; (16) where the integrals are taken along the constant energy orbit CðeÞon the unit sphere. In this approximation the differential equation for e(t) reads32 de dt¼De T¼/C0af1ðeÞ f2ðeÞ¼/C0awðeÞ: (17) Its solution is given by t¼/C01 aðe2 e1de wðeÞ; and determines the time required to move from the orbit with energy e¼e1to the orbit with e¼e2. To find the integrals (15) and(16), we have to find the orbits CðeÞ. When h?is small, one can expect that the equi- potential orbits will be close to those in the unperturbed case with h?¼0. The latter are the circles of constant polar angle. Indeed, in the absence of the bias field e¼e0(h) and for any given energy the polar angle is given by an inverse function h¼h0ðeÞ. This statement is true most of the time, however important exceptions exist. As shown in Fig. 3, the orbits are indeed relatively close near the equator. But near the North pole Nof the sphere the orbits of the perturbed energy are small circles around the maximum point X, while the original orbits are small circles around N. They are not relatively close in the sense that the perturbation is larger than the orbit size.063920-5 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27One can remedy the situation by switching to a coordi- nate system with the z0axis going through the maximum point X. The new coordinate system ðx0;y0;z0Þis rotated with respect to the original ðx;y;zÞsystem by an angle h/C3around the fixed y0¼yaxis (see Fig. 3). The spherical angles of the new system will be denoted by h0and/0. The advantage of the rotated system comes from the fact that the perturbed orbits are close to h0¼const circles everywhere, except in the vicinity of h0¼p. For the region of interest 0<h0/C20p=2 it is possible to define the shape of the orbit CðeÞas a perturbation h0ð/0;eÞ¼h0ðeÞþh?dhð/0;eÞaround the circles of constant h0. To prove the geometrically intuitive statement of the preceding paragraph one has to invert the equation eðh0;/0Þ¼e. The form of energy function in the new spheri- cal angles if calculated in Appendix B. Up to the first order inh?we find e¼e0ðh0Þþbe1ðh0;/0Þþ…; (18) where the small parameter is b¼h? h0/C0x0(19) and e0ðh0Þ¼/C0x0 2cos2h0þh0cosh0; e1ðh0Þ¼x0ð1/C0cosh0Þsinh0cos/0:(20) As expected, the zeroth order term is given by the unper- turbed energy as a function of the new polar angle h0. The same would have happened in the original coordinates, but there is an important difference between the new and original coordinates, which manifests itself in the behavior of e1at small values of h0. Note that the second term in approxima- tion (18) grows with h0slower than the first one, e1/C24h03, while e0/C0e0ð0Þ/C24h02. As a result, the inequality be1/C28e0/C0e0ð0Þholds uniformly in h0. We will see in a moment that it is precisely this uniformity that is important.In the original coordinates as h!0 at an arbitrarily small but fixed bthe second term exceeds the first one, and the uni- formity is violated. The orbit equation h0¼h0(/0,e) is found form the con- stant energy condition e0ðh0Þþbð1/C0cosh0Þsinh0cos/0¼e; which has to be solved for h0. The solution is searched in the form of a power series in b, h0¼h0ðeÞþbh1ð/0;eÞþ…; where h0(e) is the inverse function of e0(h), as was already discussed above. Up to the first order in bwe find h0¼h0ðeÞ/C0be1ðh0ðeÞ;/0Þ ðde0=dhÞjh¼h0ðeÞþ…: (21) In the “dangerous” limit of small h0(e) near the North pole the second term in Eq. (21) is proportional to h2 0eðÞthus being a small correction. Due to b/C281 it remains a small correction for the energy values up to near the equator e/C250. This algebraically proves the geometrically intuitive conclu- sion made above: the perturbed orbits are close to theh 0¼const lines. Using the approximations (18) and(21) for the energy and orbit shape, the integrals (15) and(16) can be evaluated up to the first order in b. The details are given in Appendix C. Substituting the results into Eq. (17) we get a differential equation de dt¼/C0a@e0 @h/C18/C192 h¼h0ðeÞþO ð b2Þ: (22) Up to the first order terms in bthis is the same equation as one would have for magnetic switching in the unperturbedcase with h ?¼0. We conclude that in the rotated coordinates one can approximate the switching time by the expression for the unperturbed case. The latter27–31is reviewed in Appendix Dand gives the switching time as a function of the starting angle hinand the cut-off angle hsw. To find the time s2spent by the moment in Stage II we just have to set hinto be the value of h0at the end of Stage I, and hswto be the value of h0at the selected switching moment given by e¼0. Thus hin¼h0(nx(sh),ny(sh)) and hsw¼h0(h¼p/2, u¼p/2)¼p/2. The substitution is performed in Appendix D and gives s2¼1 2a1 h0/C0x0lnh0/C0x0coshin h0ð1/C0coshinÞ/C18/C19/C26 /C01 h0þx0lnh0/C0x0coshin h0ð1þcoshinÞ/C18/C19 /C27 :(23) C. Total switching time The total switching smtime is given by the sum of the contri- butions from Stages I and II and equals sm¼shþs2. To use the expression (23) fors2we need the value of hin. This FIG. 3. Rotated reference frame and the definition of h*. Unperturbed equi- potential orbits are shown as dashed lines and the actual orbits are given by the solid lines. Gray arrows show the transformation of the representative orbits as the bias field h?is turned on. Point Xis the energy maximum posi- tion in the presence of bias field.063920-6 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27angle is given by the distance between the end point of Stage I {nx(sh),ny(sh)} and the position of the energy maximum point Xgiven by (- h?/(h0/C0x0), 0) on a unit sphere. Since the point { nx(sh),ny(sh)} and the point X are both close to þz, we can approximately write hin¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnxðshÞþh?=ðh0/C0x0ÞÞ2þnyðshÞ2q : (24) A long expression for hincan be obtained by substituting the formulae (12) into the equation above. As already discussed, the expressions (12) represent nx andnyas a sum of two terms, where the first decreases and the second increases with time. Calculations are substantially simplified when the increasing term dominates, which turns out to be true in a large part of the parameter space. Even forsmall values of athis is guaranteed for 1/( x 0þh0) /C28ffiffiffiffiffiffiffiffiffiffiffi sh=h0p or x0sh/C29x0h0 ðx0þh0Þ2: (25) Since we already assumed that x0sh/C291, and the right hand side of Eq. (25) is always less than 1/4, this inequality is automatically satisfied whenever we can use the steepestdescent approximation (12) for Stage I. Next, we assume that one can also ignore h ?/(h0-x0)i n the first term of Eq. (24) compared with nx(sh) given by the dominant term of Eq. (12). This condition is satisfied for 1/(h0-x0)/C28ffiffiffiffiffiffiffiffiffiffiffi sh=h0p or x0sh/C29x0h0 ðh0/C0x0Þ2: (26) As a result, leaving only the dominant terms we obtain hin/C25h?ffiffiffiffiffiffiffipsh h0r eaDu: (27) Using the smallness of hin, we approximate sin hin/C25hinand coshin/C251 in the expression (23) and rewrite s2as s2¼1 2a1 h0/C0x0ln2ðh0/C0x0Þ h0h2 in !( /C01 h0þx0lnh0/C0x0 2h0/C18/C19 /C27 ¼lnð1=h2 inÞ 2aðh0/C0x0ÞþsR;(28) where sRða;h0;x0Þ¼1 2a1 h0/C0x0ln2ðh0/C0x0Þ h0/C18/C19/C26 /C01 h0þx0lnh0/C0x0 2h0/C18/C19 /C27 (29) is independent of shandh?. Substituting hinfrom Eq. (27) into Eq. (28), we produce the first principal result of our paper, a formula for the switching timesm¼shþln½h0=ph2 ?sh/C138/C0aD/ðshÞ 2aðh0/C0x0ÞþsR ¼3h0þx0 4h0shþln½h0=ph2 ?sh/C138 2aðh0/C0x0ÞþsRða;h0;x0Þ:(30) The obtained sm(sh) dependence indeed has a minimum. It is reached at the optimal field sweep time s/C3 h¼1 2aðh0/C0x0Þ4h0 3h0þx0(31) that is independent of the bias field. This formula is our sec- ond main result. The independence of s/C3 hofh?is a result of the logarithmic dependence in the second term of Eq. (30) and ultimately stems from the logarithmic dependence of s2 on the initial deviation angle. The minimal switching time sms/C3 h/C0/C1 corresponding to the optimal field sweep time is equal to smðs/C3 hÞ¼1þlnaðh0/C0x0Þð3h0þx0Þ 2ph2 ?/C18/C19 2aðh0/C0x0ÞþsR: It does depend on h?, which is a quite natural since the initial deviation from the easy axis is controlled by thebias field. Expression (30) has its limits of applicability discussed in the next section. In particular, it is not applicable for smalls hwhere the steepest descent approximation (11) is invalid. Nevertheless, one can easily calculate sm(0) since in this case there is no motion in Stage I, and, according to Figs. 1 and3, the initial angle for Stage II is simply hin¼huþh/C3¼h? h0þx0þh? h0/C0x0: Using this value of hinin Eq. (28) we get smð0Þ¼1 aðh0/C0x0Þlnh2 0/C0x2 0 2h0h?/C18/C19 þsR: (32) The drop of switching time from sh¼0 to the minimal value is given by a formula smð0Þ/C0smðs/C3 hÞ¼lnpðh0/C0x0Þðh0þx0Þ2 2ah2 0ð3h0þx0Þ ! /C01 2aðh0/C0x0Þ: Note that this difference is again independent of the bias field h?, as long as the approximation (30) is valid. The frac- tional change ( sm(0) -sm(s/C3 h))/sm(0) will depend on the bias field, being an increasing function of h?. V. VALIDITY REGIONS OF THE ANALYTIC APPROXIMATION A number of approximations were made in our deriva- tion and the final expressions can only be used in the region on their validity.063920-7 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27The approximations made in our treatment of Stage I are as follows: (a) The steepest descent method employed to evaluate the integrals (9)has to be sufficiently accurate. The required conditions [see Eq. (A4) in Appendix A] read ffiffiffiffiffish h0r /C291 h0/C0x0>1 h0þx0; (33) where the second inequality holds automatically. (b) Next, we assumed that the inequality h? h0þx0/C28h?ffiffiffiffiffiffiffipsh h0r is satisfied, allowing one to neglect the first terms on the right hand sides of Eqs. (12). However, this requirement is not new because it is already contained in Eq. (33). (c) In order to make approximations in Eq. (24) we required the inequality (25) to hold. This inequality is also con- tained in Eq. (33) and does not add new conditions. (d) Finally, the inequality (14) should be satisfied to ensure small deviation of nfromþz. Overall, the inequalities h? h0/C0x0/C28h?ffiffiffiffiffish h0r /C28e/C0aDuðshÞ<1( 3 4 ) summarize the requirements for Stage I. For our treat- ment of Stage II we assumed the following: (e) The Gilbert damping constant should be small, a/C281, to be able to use the orbit averaged equation of motion (17). (f) The parameter describing the orbit deformation (21) should be small, b¼h?/(h0-x0)/C281. But this inequal- ity follows from requirements (34) and thus brings no additional restrictions. The requirements (34) discussed above can be equiva- lently presented as conditions on shthat have to be satisfied at fixed bias field h?. In this form they read h0 ðh0/C0x0Þ2/C28sh/C28sðþÞ h; (35) where sþðÞ his a solution of ffiffiffiffiffish h0r e/C0aDuðshÞ¼1 h?: We can now check when does the optimal field sweep time s/C3 hlie in the region of validity of our approximation. Using our result (31) we can write s/C3 h/C251 2aðh0/C0x0Þ; and substitute it into the requirement (33). We get a/C28h0/C0x0 2h0:The right hand side of this inequality is always smaller than unity, thus it automatically implies a/C281. We also have to satisfy the condition (14) atsh¼s/C3 h. For the exponent aDuone can write aDuðs/C3 hÞ¼aðh0/C0x0Þ2 4h0s/C3 h/C25h0/C0x0 8h0/C201 8/C281; and thus condition (14) simplifies to h?ffiffiffiffiffi s/C3 h h0r /C281: The above inequalities on aandh?can be combined into a single requirement h2 ? h0ðh0/C0x0Þ/C28a/C28h0/C0x0 2h0: (36) If the inequalities (36) are satisfied, the optimal sweep time s/C3 hoccurs inside the interval (35) and can be calculated using formula (31). VI. COMPARISON OF ANALYTIC AND NUMERIC RESULTS Comparisons of the analytic approximation and exact numeric results were performed in Ref. 2and shown a very good agreement between the two. First, we compared the numerically calculated switching times with the expression (30). When the inequalities (36) were well satisfied, the quality of approximation was verygood (Fig. 4). As one approached the limits of the approxi- mation’s validity by, e.g., increasing a, the errors grew larger. Second, the numeric results for optimal field sweep time s /C3 hwere compared with the formula (31). The correspon- dence was generally good (Fig. 5), although some visible deviations existed. They were attributed to the fact that the accuracy of the determination of s/C3 his lowered by a flat shape of the sm(sh) curve minimum. Because of the shallow minimum, small errors in smproduce much larger errors ins/C3 h. In general, the analytic expression approximated the sm(sh) dependence up to a 10% accuracy in a surprisingly wide range of parameters. Such accuracy is certainly suffi- cient for the estimates related to the device design. When shis outside of the validity region of the results (30) and (31), exact expressions (9)and(24) for the inte- grals and the initial angle hincan be used. As long as the deviation from the þzdirection in Stage I remains small, they provide a good approximation for sm. Consider for example the case of small sh. Approximation (30) does not work for sh!0 predicting an infinite increase of sm, while the actual limit sm(0) is finite and given by formula (32). It was checked that using the exact expressions (A3) for the integrals C and S and the formula (24) forhin, one can obtain an excellent agreement between the two-stage063920-8 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27theory and the no-approximation numeric simulations in this limit. VII. PHYSICAL PICTURE OF THE BALLISTIC- ASSISTED SWITCHING We now discuss the physical reason for the existence of the minimum of the function smðshÞ.R e f . 2identified it as the contribution of ballistic switching. We start by noticing that the bias field has two roles in the switching process. First, it provides the initial deviation from the easy axis. Second, italters the equations of motion for n(t). The first role of h ? manifests itself in our formulas in two ways, by providing the first terms in expressions (12) and by introducing the term h?/(h0-x0) into the formula (24). In our derivation of the switching time expression (30) we have found that bothcontributions are negligible. Therefore within our approxima- tion only the second role of the bias field is important. Let us proceed by discussing this role qualitatively. Recall that in the absence of anisotropy and other fields thetorque H ?/C2M0ndue to the bias field would rotate the unit vector nfromþzto/C0zalong a meridian of the unit sphere (dashed line in Fig. 6), in a “ballistic” or “precessional” fash- ion. In our case a weak bias field is applied on top of the strong uniaxial anisotropy and switching field, which to- gether induce a fast orbital motion of vector n(t) along the parallel circles (line C in Fig. 6). The bias field still attempts to move nalong the meridians, but now its action has to be averaged over the period of orbital motion. As illustrated inFig. 6, in constant fields Hk^z ?averaging gives zero due to the cancellation of the contributions from the diametrically opposed infinitesimal intervals dl1anddl2of equal lengths. This way ballistic contribution of the bias field is quenched. However, the contribution of H?does not average to zero for a variable switching field H(t). In this case the velocity of nchanges along the orbit, the times spent in the intervals dl1 anddl2are different, and the contributions of the two do not cancel each other. We conclude that in the presence of atime dependent field H(t) ballistic contribution of the perpen- dicular bias field is partially recovered. FIG. 5. Numeric (points) and approximate analytical (solid lines) dependen- cies of the optimal field sweep time s/C3 hon the system parameters. (A) Fixed aandh?, (B) fixed h0/x0andh?, (C) fixed h0/x0anda. When not varied, the parameter values are h0¼2.2x0,a¼0.01, h?¼0.005. FIG. 4. Dependencies sm(sh) calculated for h0¼2.2x0,h?¼0.001 x0, and variable aindicated on each panel. As aincreases, the theoretical fit gets poorer due to the violation of the strong inequality (36).063920-9 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27Based on the discussion above, we may expect to find that the largest contribution of the ballistic switching will happen when the change of velocity is biggest. The measureof the velocity change is Dx/x, where Dxis the change of the velocity as one precesses from the interval dl 1to the interval dl2. We can estimate Dx x/C24_xT x/C24_x x2/C24€/ _/2; where T¼2p/xis the instantaneous period. We see that the estimate diverges at the point _/¼0. But this is exactly the stationary phase point tmthat gives the largest contribution to the integrals (9)in our approximation. We have thus established a one-to-one correspondence between the quali-tative description of the ball istic contribution and our deri- vation of the analytic approximation (30). To illustrate the fact that the action of h ?is only important near the station-ary phase point tm, we have performed a numeric experi- ment with pulsed bias field h?(t) changing in time as shown in the inset of Fig. 7.I ti sk e p tc o n s t a n tf o r t<0, switched off at t¼0, and then switched on again for a short interval of time Dt, centered around the tmpoint. With the pulsed bias field the ballistic contribution is only present duringthe interval Dt. It follows from the steepest descent calcula- tion of Appendix A that formulas (11) would be valid al- ready for Dt> /C242ffiffiffiffiffiffiffiffiffiffiffi sh=h0p , and when this inequality is satisfied our theory would predict the same switching time (30) for pulsed and constant bias fields. The results of the numeric experiment (Fig. 7) are completely consistent with this prediction. As the pulse width approaches the value of 2ffiffiffiffiffiffiffiffiffiffiffi sh=h0p , the switching time drops to the value obtained earlier at constant h?. The minimum of the sm(sh) function can be understood as follows. Ballistic contribution helps to move vector n fromþzto -zand is thus responsible for the initial decrease ofsm. As the sweep time grows larger, the change of the or- bital velocity during the precession period decreases and the ballistic contribution averages out progressively better. Thehelping effect of ballistic switching is lost and s mstarts to increase as it normally would. Finally, we want to remark that ballistic contribution to switching can be also viewed as a phenomenon complimentary to the magnetic resonance and RF-assisted switching.20–26In the case of RF-field application the external field His constant, while the bias field H?(t) is time-dependent. As a result, the contributions of the RF bias field torque on the intervals dl1 anddl2in Fig. 6do not cancel each other because the torque itself changes with time. This leads to a nonzero average of the bias field contribution on a prece ssion orbit and thus creates a helping effect responsible for the RF assistance to switching.From this point of view the magnetic resonance and the time dependence of the axial switching field are two different ways to achieve the same goal: a non-vanishing average contributionof the bias field torque on an orbit. VIII. CONCLUSIONS We have identified and investigated the phenomenon of ballistic contribution to the conventional damped magneticswitching by a time-dependent field. An analytic approxima- tion is derived for the ballistic-assisted switching time in a constant perpendicular bias field. It is also shown that, ifpractical, a constant bias can be substituted by short pulse of bias field applied near the stationary phase time point. Our results provide a convenient approximation for the optimalfield sweep time, an important parameter in the device design. The expressions obtained in this study can be used as a starting point for the investigations of the switching time ingranular media, where each grain can be modeled by a single moment and bias field is produced by the other grains or by the spread of grain orientations. ACKNOWLEDGMENTS Ya. B. Bazaliy is grateful to B. V. Bazaliy for illuminat- ing discussions. This work was supported by the NSF GrantNo. DMR-0847159. FIG. 6. Average ballistic contribution of the bias field. Vector n(t)o r b i t s around a parallel circle C on a unit sphere. The torque due to h?pushes n along the meridians of the sphere. In the constant switching field the torque contributions from the diametrically opposed elements dl1and dl2cancel each other. For variable H(t) the cancellation does not happen (see text). FIG. 7. Numerically calculated witching time with pulsed bias field. The parameters are set to h0¼2.2x0,h?¼0.001 x0,a¼0.01 (cf. Fig. 4). Field sweep time is fixed at sh¼50/x0, close to the optimal sweep time s/C3 h¼48.2/x0. The time dependence of the pulsed bias field h?(t) is shown in the inset. As the width of the pulse Dtapproaches the theoretical target of 2ffiffiffiffiffiffiffiffiffiffiffi sh=h0p /C2510=x0, the switching time approaches the value obtained at h?¼const (shown by a horizontal line).063920-10 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27APPENDIX A: STEEPEST DESCENT APPROXIMATION FORC(T) AND S(T) Here we evaluate the integrals (9) SðshÞ¼ðsh 0eauðtÞsinuðtÞdt; CðshÞ¼ðsh 0eauðtÞcosuðtÞdt; with the phase /(t) given by a quadratic function u¼ðx0þh0Þt/C0h0t2 sh: The integrals in question can be obtained from the real and imaginary parts of a complex integral I¼ðsh 0e/C0luðtÞdt¼C/C0iS; l¼i/C0a:(A1) By completing the square we can rewrite uðtÞ¼ðh0þx0Þ2 4h0sh/C0h0 shðt/C0tmÞ2; where tm¼shx0h0ðÞ =2h0is the point of maximum phase, 0<tm<sh.N o w I¼elðh0þx0Þ2 4h0sh/C1J; J¼ðsh 0e/C0lðh0=shÞðt/C0tmÞ2dt:(A2) Changing variables to z¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi lh0=shp ðt/C0tmÞ, we can write down Jas J¼ffiffiffiffiffiffiffish lh0rð Ce/C0z2dz; where Cis a straight line in the complex plane going between the points z1¼/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi lh0=shp tmand z2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi lh0=shp ðsh/C0tmÞ. Due to our definition of z,l i n e Ccrosses the complex zero point. Since the integrand of Jis a regular function, integration can be performed along any contour connecting z1andz2. The integral can be expressed in terms of the error function of complex variable ErfðzÞ¼ð 2=ffiffiffippÞÐz 0expð/C0z2Þdzas33 J¼ffiffiffiffiffiffiffish lh0rffiffiffipp 2Erfðz1ÞþErfðz2Þ ðÞ : (A3) The above is an exact formula. The steepest descent approxi- mation corresponds to the case of large absolute valuesjz 1j/C291,jz2j/C291. Due to the smallness of aone has jlj/C251, so these conditions translate to ffiffiffiffiffi h0 shr tm/C291;ffiffiffiffiffi h0 shr ðsh/C0tmÞ/C29 1; or equivalentlyðh0þx0Þ2 4h0sh/C291;ðh0/C0x0Þ2 4h0sh/C291: (A4) Using33ErfðzÞ! 1 forjzj!1 we find the approximation J/C25ffiffiffiffiffiffiffipsh lh0r /C25ffiffiffiffiffiffiffipsh h0r eip=41/C0ia 2/C18/C19 ; where we have also expanded in small a. Substituting this back into Eq. (A2) we get I/C25eaumffiffiffiffiffiffiffipsh h0r eiðp=4/C0umÞ1/C0ia 2/C18/C19 ; (A5) where um¼utmðÞ. The real and imaginary parts of Igive C andSaccording to Eq. (A1) C/C25eaumffiffiffiffiffiffiffipsh h0r cos/m/C0p 4/C17/C16 /C0a 2sin/m/C0p 4/C17/C16 hi ; S/C25eaumffiffiffiffiffiffiffipsh h0r sin/m/C0p 4/C17/C16 þa 2cos/m/C0p 4/C17/C16 hi : APPENDIX B: ENERGY IN ROTATED COORDINATES The relationship between the projections of nandhin the primed and original coordinate system are given as nz¼n0 zcosh/C3þn0 xsinh/C3 ¼cosh0cosh/C3þsinh0cos/0sinh/C3 nx¼/C0n0 zsinh/C3þn0 xcosh/C3 ¼/C0 cosh0sinh/C3þsinh0cos/0cosh/C3 ny¼n0 y and h0 z¼hcosh/C3/C0h?sinh/C3 h0 x¼hsinh/C3þh?cosh/C3 One can now rewrite Eq. (4)through the angles ( h0,/0) e¼/C0x0 2n0 z2/C0n0 xh0x/C0n0 zh0z ¼/C0x0 2ðcosh0cosh/C3þsinh0cos/0sinh/C3Þ2 /C0hðcosh0cosh/C3þsinh0cos/0sinh/C3Þ /C0h?ðsinh0cos/0cosh/C3/C0cosh0sinh/C3Þ:(B1) The above is the exact expression. We are looking at the case h¼/C0h0andh?!0. Up to the first order in h?, sinh/C3/C25h? h0/C0K;cosh/C3/C251: Expanding the energy up to the first order in h?we get e/C25e0ðh0Þþbe1ðh0;/0Þ; (B2) where the small parameter is063920-11 Y a. B. Bazaliy J. Appl. Phys. 110, 063920 (2011) [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: 130.209.6.50 On: Sun, 21 Dec 2014 05:40:27b¼h? h0/C0x0(B3) and e0ðh0Þ¼/C0x0 2cos2h0þh0cosh0; e1ðh0Þ¼x0ð1/C0cosh0Þsinh0cos/0:(B4) The first term in the expansion (18) is the energy unperturbed by the bias field, evaluated at the new polar angle h0. APPENDIX C: INTEGRALS ALONG THE PERTURBED ORBITS First, we calculate the approximate value of j@e=@nj. Taking the identity @e @n/C12/C12/C12/C12/C12/C12/C12/C12¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @e @h0/C18/C192 þ1 sin2h0@e @/0/C18/C192s and expanding it in small bup to the first order we find @e @n/C12/C12/C12/C12/C12/C12/C12/C12¼@e0 @h0þb@e1 @h0þ…: (C1) Next, we need the element of the orbit length jdnj. Using jdnj¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2hd/02þdh02p and calculating up to the first order in bwe get jdnj¼ sinh0/C0bcosh0e1ðh0;/0Þ ðde0=dhÞjh¼h0ðeÞ ! d/0: (C2) Using the form of e1Eq.(B4) we rewrite it as jdnj¼ sinh0/C0bAðeÞcos/0ðÞ d/0: We will further use the notation e1ðh0ðeÞ;/0Þ¼BðeÞcos/0: To perform the integrals (15) and(16) we use the expansions (C1) and(C2), and, expanding up to the first order in b, get þ C@E @n/C12/C12/C12/C12/C12/C12/C12/C12dn¼ð2p 0@e0 @hsinh0þb@B @hsinh0/C0/C18@e0 @hA/C19 cos/0/C20 þ/C1/C1/C1/C21 d/0: The integral of the term proportional to bvanishes for any A andB, and þ C@E @n/C12/C12/C12/C12/C12/C12/C12/C12dn¼2p@e0 @hsinh0þO ð b2Þ: Performing a similar calc ulation for the integral (16)we get þ Cdn @e=@njj¼2psinh0 @e0=@hþO ð b2Þ:According to (17) the differential equation on e(t) reads de dt¼/C0a@e0 @h/C18/C192/C12/C12/C12/C12/C12 h¼h0ðeÞþO ð b2Þ: (C3) APPENDIX D: SWITCHING TIME IN THE UNPERTURBED CASE The problem of switching time of a uniaxial particle in the absence of perpendicular field was probably first solved by Kikuchi27for the case of K¼0. The derivation was gen- eralized to the arbitrary values of Kby many authors.28–31 Here we re-derive this result for the completeness of the pre- sentation. In the case of h?¼0 one can find the switching time either by solving Eq. (22) [same as Eq. (C3)] truncated to zeroth order, or by a direct inspection of the system (2), (3). Since e0depends only on h, it is enough to consider the first equation which reads _h¼/C0a@e @h¼/C0aðx0coshþhÞsinh: Integrating we get /C0at¼ðhsw hindh sinhðx0coshþhÞ: A variable change x¼coshgives t¼1 aðxsw xindx ð1/C0x2Þðx0xþhÞ ¼/C01 2a1 hþx0ln1/C0x x0xþh/C18/C19/C26 /C01 h/C0x0ln1þx x0xþh/C18/C19 /C27/C12/C12/C12/C12xsw xin:(D1) In our problem hsw¼p/2 (xsw¼0) and h¼/C0h0which gives t¼1 2a1 h0/C0x0lnh0/C0x0coshin h0ð1/C0coshinÞ/C18/C19/C26 /C01 h0þx0lnh0/C0x0coshin h0ð1þcoshinÞ/C18/C19 /C27 :(D2) 1A. Stankiewicz, paper presented at the APS March Meeting, Portland, Oregon, 15–19 March 2010, abstract H33.11, Bulletin of the APS, Vol. 55. 2Ya. B. 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Mailly, Nature Mater. 2, 524 (2003). 21W. T. Coffey, D. S. F. Crothers, J. L. Dormann, Yu. P. Kalmykov, E. C. Kennedy, and W. Wernsdorfer, Phys. Rev. Lett. 80, 5655 (1998). 22J. L. Garca-Palacios and F. J. Lazaro, Phys. Rev. B 58, 14937 (1998).23G. Bertotti, C. Serpico, and I. D. Mayergoyz, Phys. Rev. Lett. 86, 724 (2001). 24K. Rivkin and J. B. Ketterson, Appl. Phys. Lett. 89, 252507 (2006). 25Z. Z. Sun and X. R. Wang, Phys. Rev. B 74, 132401 (2006). 26W. Scholz, T. M. Crawford, G. J. Parker, T. W. Clinton, T. Ambrose, S. Kaka, and S. Batra, IEEE Trans. Magn. 44, 3134 (2008). 27R. Kikuchi, J. Appl. Phys. 27, 1352 (1956). 28J. C. Mallinson, Magnetic Properties of Materials , Inter-University Electron- ics Series 13. edited by J. Smit (McGraw-Hill, New York, 1971), p. 252. 29J. C. Mallinson, IEEE Trans. Magn. 36, 1976 (2000). 30Y. Uesaka, H. Endo, T. Takahashi, Y. Nakatani, N. Hayashi, and F. Fukushima, Phys. Status Solidi 189, 1023 (2002). 31G. Bertotti, I. Mayergoyz, C. 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1.1852191.pdf

Magnetic normal modes of nanoelements R. D. McMichael and M. D. Stiles Citation: Journal of Applied Physics 97, 10J901 (2005); doi: 10.1063/1.1852191 View online: http://dx.doi.org/10.1063/1.1852191 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin mode calculations in nanometric magnetic rings: Localization effects in the vortex and saturated states J. Appl. Phys. 103, 083910 (2008); 10.1063/1.2887921 Normal mode splitting in interacting arrays of cylindrical permalloy dots J. Appl. Phys. 99, 08C701 (2006); 10.1063/1.2150806 Effect of the classical ampere field in micromagnetic computations of spin polarized current-driven magnetization processes J. Appl. Phys. 97, 10C713 (2005); 10.1063/1.1853291 Micromagnetic simulation of thermal ripple in thin films: “Roller-coaster” visualization J. Appl. Phys. 93, 8020 (2003); 10.1063/1.1556094 Magnetization pattern of ferromagnetic nanodisks J. Appl. Phys. 88, 4437 (2000); 10.1063/1.1289216 [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: 128.103.149.52 On: Tue, 07 Oct 2014 06:36:57Magnetic normal modes of nanoelements R. D. McMichaela!and M. D. Stiles National Institute of Standards & Technology, Gaithersburg, Maryland 20899 sPresented on 9 November 2004; published online 13 May 2005 d Micromagnetic calculations are used to determine the eigenfrequencies and precession patterns of some of the lowest-frequency magnetic normal modes of submicron patterned elements. Twoexamples are presented. For a Permalloy-like ellipse, 350 nm 3160 nm 35 nm thick in zero field, the lowest frequency normal mode at 4 GHz corresponds to precession in the “ends” of the ellipse.Other resonant frequencies are compared with the frequencies of spinwaves with discrete wavevectors. For a normally magnetized 50 nm diameter 315 nm thick cobalt disk, the calculated eigenfrequencies increase linearly with applied field, mimicking the behavior of the experimentalcritical current for spin transfer instabilities in an experimental realization of this disk.fDOI: 10.1063/1.1852191 g I. INTRODUCTION Knowledge of the magnetic normal modes is valuable for understanding the thermal noise behavior of small mag-netic elements such as those in sensors or magnetic memorycells. A number of experimental investigations of normalmodes have been carried out on micrometer-scale patternedelements including squares in remnant states, 1,2on squares3,4 and circles3,5–7in vortex states and on thin strips.8–10There are relatively few theoretical investigations of normal modesin magnetic patterned elements, mostly due to the difficultyof dealing with nonuniform magnetostatic fields. 11,12Grims- ditchet al.have used micromagnetic techniques to examine the normal modes of a small rectangular block, comparingthe computed frequencies to frequencies calculated from in-finite film dispersion relations with discrete wave vectors. 13 While normal modes can describe dynamics only for lin- ear dynamics, they are also useful for understanding the in-stabilities that lead to nonlinear phenonomena such asswitching and large amplitude oscillations driven by appliedfields or spin transfer torques. While spin transfer effects are most frequently observed in systems with two magnetic layers, they have been beenobserved 14–16and calculated17,18in systems with a single magnetic layer. In single films, a proper description of spintransfer instabilities requires a tight integration of a transportcalculation with a micromagnetic calculation. Calculationsdone to date 17,18have focused on the transport calculation at the cost of using an oversimplified treatment of the micro-magnetic interactions. The most important simplification isthat the samples are treated as infinite layers with magneto-statics either ignored or treated as a uniaxial anisotropy. Tounderstand the consequences of this oversimplification, wehave computed the normal modes of some of the measuredsamples. 16In these samples, the ferromagnetic layer is more like a disk than a thin film so that the modes are very differ-ent from thin film modes. This article explores the normal modes of twonanometer-scale patterned bits, one a thin-film ellipsoid that illustrates the computational technique and the other is ashort cylinder that models the magnetic nanoelement in aspin-transfer torque experiment. 16The micromagnetic calcu- lations were performed using the NIST micromagnetic testcode OOMMF.19Starting with a minimum energy state, the magnetization was excited by a short, strong field pulse sothat magnetic moments were rotated a maximum of approxi-mately 10°. In general this field pulse was not uniform. Fieldpulses with different symmetries were used to excite normalmodes with corresponding symmetries. After the field pulse,the evolution of the magnetization was calculated using theLandau–Lifshitz equations of motion with Gilbert dampingand a=0.01. The evolution of the magnetization during the ringdown was captured by saving the magnetization configu-rationMsr i,tjdat uniform time intervals. In many cases, the ringdown appears as a complicated wiggling of the magne- tization when viewed as an animation. However, whenviewed in the frequency domain, the apparently complicatedbehavior can be understood as the superposition of a fewnormal modes. For each point r iin the magnetic element, the ringdown record contains a time series of the magnetization at thatpoint. Local power spectra of the magnetization are con-structed by performing a discrete Fourier transform S xsri,fd=Uo jMxsri,tjdei2pftjU2. s1d To obtain an overall view of the magnetization behavior, the power spectra are summed over ri S¯xsfd=o iSxsri,fd. s2d Note that S¯is very different from the power spectrum of the spatially averaged magnetization. We find that plots of S¯sfd exhibit many peaks corresponding to oscillations at eigenfre- quencies of the magnetization. At these peak frequencies,Ssr i,fdgives a map of the precession amplitude for the ex- cited mode.adAuthor to whom correspondence should be addressed; electronic mail: rmcmichael@nist.govJOURNAL OF APPLIED PHYSICS 97, 10J901 s2005 d 0021-8979/2005/97 ~10!/10J901/3/$22.50 97, 10J901-1 [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: 128.103.149.52 On: Tue, 07 Oct 2014 06:36:57II. THIN ELLIPSE We have calculated linear magnetization dynamics for a 160 nm 3350 nm 35 nm ellipse of isotropic Permalloy in zero applied field using 2.5 nm 32.5 nm 35 nm cells. The relaxed state of this ellipse has nearly uniform magnetizationaligned with the long axis of the ellipse in the central regionbut nearer to the edges, the magnetization tends to follow theedge contour, except at the ends where the magnetizationpoints normal to the edge. Field pulses with different spatial symmetries were used to highlight different normal modes. The modes are shownalong the top of Fig. 1. The modes excited by a uniform fieldpulse include the lowest frequency mode corresponding tomotion near the “ends” of the ellipse where the magnetiza-tion has a large component normal to the edge. Also excitedare a mode with large amplitude in the center and a series ofmodes with even numbers of nodal lines running parallel andperpendicular to the symmetry axes of the sample. To exciteother modes, we have used excitation pulses with other sym-metries, including field pulses with field amplitudes propor-tional to xoryas measured from the center of the sample. The end modes have nearly the same resonant frequencywhether the precession in the ends is in phase suniform pulse dor out of phase soddypulse d. This indicates that the ends of the ellipse interact very weakly. We have attempted to explain the observed spectra using analytical models that include the effects of applied field,exchange, and magnetostatic interactions. The magnetostaticinteractions can be approximated two ways: s1dby calculat- ing demagnetization factors for uniform magnetization or s2d by using the dispersion relation for spin waves in infinite thinfilms and selecting discrete wave vectors appropriate for thesample geometry. Because the sample is not an ellipsoid, the magnetostatic fields are not uniform. We calculate spatially averaged de- magnetization factors for the ellipse using E d=1 2Vsm0NaMa2 for uniform Mpointing in the x,y, andzdirections to yield Nx=0.0515, Ny=0.0182, and Nz=0.931. For an ellipsoid with the same demagnetization factors, the precession frequency for uniform magnetization is v=gm0Ms˛sNx−NydsNz−Nydorf=4.91 GHz, marked as “ U” in Fig. 1. The gyromagnetic ratio, g=2.11 3105m/Asand Ms=83105A/m. To estimate frequencies for nonuniform modes, we use the dispersion relation for an infinite thin film. At zero ap-plied field with the magnetization in plane Svskd gD2 =FHd+Mss1−Nkd+2A m0Msk2G 3FHd+MsNkkx2 k2+2A m0Msk2G. s3d Here,Nk=f1−exp s−kddg/kdis ak-dependent demagnetiza- tion factor for a film of thickness d,20Ais the exchange stiffness parameter, and A=13 pJ/m. If we naively chose discrete wave vectors ksuch that kx=nxp/s160 nm dandky =nyp/s350 nm dfor integer nxandny, the frequencies given by Eq. s3dare plotted along the bottom of Fig. 1. The agreement between the eigenfrequencies determined from the dispersion relation and the eigenfrequencies deter-mined from the full micromagnetic calculation is qualitativeat best. The discrete values of kcorrespond more closely to rectangular samples than to the elliptical shape. For the re-sults shown, we have used the average static demagnetiza-tion field H dis −NyM=−14.6 kA/m, but there is some am- biguity in this choice. The static micromagnetic calculationyields a field of only H d=−6.22 kA/m at the center of the ellipse where the precession is strongest for many of themodes, but when this value is used, agreement with the dy-namic micromagnetic results is worse.Afinal shortcoming ofthe simple models is that they do not predict modes corre-sponding to the end modes. III. NANODISK For spin transfer instabilities in single films, calculations show that the stability or instability of particular modes ofthe magnetization depend on the competition between thecurrent induced torque and the damping. For the current in-duced torque to drive a mode of the magnetization in a singlelayer toward instability, two things are required. 17,18First, the mode must be laterally nonuniform so that spins diffusing ina lead from one part of the magnetic layer to another willexert a torque. Second, the two lead-magnet interfaces needto be asymmetric, otherwise the torques on both interfacescancel. This asymmetry can arise in two ways: the leads onthe two sides can be asymmetric 17or the mode that becomes unstable can be nonuniform through the thickness of thelayer. 18 We have used the micromagnetic spectral mapping tech- nique to look at the lateral nonuniformity and interface sym-metry of the eigenmodes of a normally magnetized disk ofisotropic cobalt, 50 nm in diameter and 16 nm thick.Experiments 16have shown that the magnetization of this nanodot becomes unstable when the current passing throughit exceeds a field-dependent critical current. In the computedminimum energy state, the magnetization was nearly satu-rated in the zdirection normal to the circular faces of the disk in applied fields ranging from 2 to 4 T. Because thethickness of the disk is comparable to the diameter, this is a FIG. 1. Eigenmode images stopdand spatially averaged power spectra smiddle dfor a 160 nm 3350 nm 35 nm ellipse of Permalloy in zero applied field.The spectra and images were obtained from three time series followingexcitation field pulses with three different symmetries. The calculated modefrequencies scenter dare compared with frequencies calculated from a spin- wave dispersion relation sbottom d. The end mode frequency is labeled “ E” and the uniform precession frequency for an ellipse is labeled “ U.”10J901-2 R. D. McMichael and M. D. Stiles J. Appl. Phys. 97, 10J901 ~2005 ! [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: 128.103.149.52 On: Tue, 07 Oct 2014 06:36:57three dimensional s3Ddcalculation, and we used 1 nm cubic cells. Precession was excited by applying a field pulseH psrd~xyzonly in the “corner” region x.0,y.0 andz .0. Modes even and odd in zwere determined by comput- ing power spectra of Mx+sr,td=Mxsx,y,d/2,td+Mxsx,y, −d/2,tdfor even modes and Mx−sr,td=Mxsx,y,d/2,td −Mxsx,y,−d/2,tdfor odd modes. The power spectra and mode images are shown in Fig. 2. Calculations that ignore magnetostatics18show that the lowest threshold for instability is for a mode odd in zand with a wavelength comparable to the dot diameter. From ourmicromagnetic results, we identify this mode as the secondodd mode shown in Fig. 2 sbd. Even though other modes have lower frequency and therefore less damping, as describedbelow, the greater current induced torque due to the asym-metry in the n z=1 mode gives it the lowest critical current. One of the prominent features of the experimental data is that the critical current extrapolates to zero as a functionof the magnetic field applied normal to the disk. 16The instability occurs when the spin transfer torques smodulo mode geometry dequal the damping. The damping Gis inde- pendent of the current, and for Gilbert damping, G =2svi/m0gda]vi/]H. We have computed the field depen- dence of the frequencies several of the normal modes, and we find that ]vi/]His very nearly constant ssee Fig. 3 d. Therefore, we claim that the critical current is very nearlyproportional to mode frequency for these modes. In agreement with experiment, we find that the mode frequencies sand therefore critical currents dextrapolate to zero frequency at a field close to zero. However, this agree-ment is coincidental rather than fundamental and may not bethe explanation for the experimental results. In summary, we have used dynamic micromagnetic tech- niques to determine the lowest frequency eigenmodes for amagnetic ellipse in zero field and for a perpendicularly mag- netized disk. The eigenfrequencies are only in qualitativeagreement with simple models, which fail to predict local-ized modes at the end of the ellipse. The field dependence ofthe eigenmode frequencies in the disk mimics the field de-pendence of the critical currents in a spin transfer torqueexperiment. 1S. Tamaru, J. A. Bain, R. J. M. Van de Veerdonk, T. M. Crawford, and M. H. Kryder, J. Appl. Phys. 91, 8034 s2002 d. 2M. Belov, Z. Liu, R. D. Sydora, and M. R. Freeman, Phys. Rev. B 69, 094414 s2004 d. 3J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. B 67, 020403 s2003 d. 4H. Stoll, A. Puzic, B. van Waeyenberge, P. Fischer, J. Raabe, M. Buess, T. Haug, R. Höllinger, C. Back, and G. Denbeaux, Appl. Phys. Lett. 84, 3328 s2004 d. 5Y. Acremann, C. H. Back, M. Buess, O. Portmann, A. Vaterlaus, D. Pes- cia, and H. Melchior, Science 290, 492 s2000 d. 6Y. Acremann, A. Kashuba, M. Buess, D. Pescia, and C. Back, J. Magn. Magn. Mater. 239, 346 s2002 d. 7M. Buess, R. Höllinger, T. Haug, U. Krey, D. Pescia, M. R. Scheinfein, D. Weiss, and C. H. Back, Phys. Rev. Lett. 93, 077207 s2004 d. 8J. P. Park, P. Eames, D. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. Lett. 89, 277201 s2002 d. 9C. Bayer, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Appl. Phys. Lett.82, 607 s2003 d. 10C. Bayer, J. P. Park, H. Wang, M. Yan, C. E. Campbell, and P. A. Crowell, Phys. Rev. B 69, 134401 s2004 d. 11K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rev. B 66, 132402 s2002 d. 12K. Y. Guslienko, R. W. Chantrell, and A. N. Slavin, Phys. Rev. B 68, 024422 s2003 d. 13M. Grimsditch, G. K. Leaf, H. G. Kaper, D. A. Karpeev, and R. E. Cam- ley, Phys. Rev. B 69, 174428 s2004 d. 14Y. Ji, C. L. Chien, and M. D. Stiles, Phys. Rev. Lett. 90, 106601 s2003 d. 15T. Y. Chen, Y. Ji, C. L. Chien, and M. D. Stiles, Phys. Rev. Lett. 93, 026601 s2004 d. 16B. Özyilmaz, A. D. Kent, J. Z. Sun, M. J. Rooks, and R. H. Koch, Phys. Rev. Lett. 93, 176604 s2004 d. 17M. L. Polianski and P. W. Brouwer, Phys. Rev. Lett. 92, 026602 s2004 d. 18M. D. Stiles, J. Q. Xiao, and A. Zangwill, Phys. Rev. B 69, 054408 s2004 d. 19M. J. Donahue and D. G. Porter, OOMMF User’s Guide , Version 1.0, Interagency Report NISTIR 6376 sNational Institute of Standards and Technology, Gaithersburg, MD, 1999 d. 20K. J. Harte, J. Appl. Phys. 39,1 5 0 3 s1968 d. FIG. 2. Normal modes and spectral density for a 16 nm thick, 50 nm diam cobalt disk for sadmodes with even zsymmetry and sbdmodes with odd z symmetry. Because the second odd mode is nonuniform both in plane andnormal it is most likely to be driven unstable by a spin current. FIG. 3. Applied field dependence of normal mode frequencies. Crosses arefor evenz-symmetry modes and circles are for odd z-symmetry modes. The line shows the extrapolation of the second odd mode to zero field.10J901-3 R. D. McMichael and M. D. Stiles J. Appl. Phys. 97, 10J901 ~2005 ! [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: 128.103.149.52 On: Tue, 07 Oct 2014 06:36:57

1.367690.pdf

A variational approach to exchange energy calculations in micromagnetics M. J. Donahue Citation: Journal of Applied Physics 83, 6491 (1998); doi: 10.1063/1.367690 View online: http://dx.doi.org/10.1063/1.367690 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/83/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic calculation of the magnetization process in nanocontacts J. Appl. Phys. 97, 10C508 (2005); 10.1063/1.1850856 Multiscale versus micromagnetic calculations of the switching field reduction in FePt ∕ FeRh bilayers with perpendicular exchange spring J. Appl. Phys. 97, 10J101 (2005); 10.1063/1.1844931 Stiffness analysis for the micromagnetic standard problem No. 4 J. Appl. Phys. 89, 7600 (2001); 10.1063/1.1355355 Exchange-bias systems with compensated interfaces Appl. Phys. Lett. 75, 3995 (1999); 10.1063/1.125517 Role of boundary conditions and dimensions on the micromagnetics of a cobalt point contact J. Appl. Phys. 85, 6196 (1999); 10.1063/1.370219 [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: 145.116.150.241 On: Wed, 03 Dec 2014 11:29:50A variational approach to exchange energy calculations in micromagnetics M. J. Donahue National Institute of Standards and Technology, Gaithersburg, Maryland 20899 This article presents a magnetization interpolation method for micromagnetic exchange energy calculations using a variational procedure to relax spins on a supplemental ~refined !lattice. The approximationsimplicitinstandardmicromagneticdiscretizationschemesfailwhenanglesbetweenneighboring spins in the model become large, but the proposed approach effectively reduces theangle between neighboring spins, alleviating many of the associated problems. Moreover, thismethod does not introduce excessive discretization-induced vortex pinning observed with somelarge angle exchange energy formulations. This article includes details on proper post-interpolationexchange torque calculation, bounds on nearest-neighbor angles for interpolated lattices, a simplemodel predicting discretization-induced Ne ´el wall collapse, and an example of a collapsed ~1 cell wide!domain wall that can be restored by the proposed technique. @S0021-8979 ~98!33311-3 # I. INTRODUCTION Many difficulties arise in micromagnetic simulations when angles between neighboring spins become large. None-theless, computational limitations often prevent many inter-esting micromagnetic problems from being discretized at ascale fine enough to resolve all the details of the magnetiza-tion structure. In particular, models of thin magnetic filmsoften contain vortices and crossties with unresolved coresmeasuring only a few nanometers across. As discussed be-low, such undersampled core regions can collapse duringmodel evolution into one cell wide 180° domain walls, evenin settings where the Ne ´el wall width is many cells wide. Once formed, these structures are stable, because they tendto be supported by magnetostatic and crystalline anisotropyfields, and the usual exchange energy formulation provideszero torque across 180° spins. One can introduce an ex-change energy formulation modified for large angles, butsimple approaches result in strong artificial pinning of vorti-ces to the computation grid. 1 One solution to these problems is to base the exchange energy formulation on a continuous interpolation of the mag-netization that respects the constraint that the reduced mag-netization imi[1. In this article, an interpolatory ‘‘supple- mental’’ lattice is introduced, and a variational procedure isused to relax the spins on this lattice to achieve a smoothinterpolation. It is shown that a simple half-step interpolationsuffices to avoid the aforementioned one cell wide 180° do-main walls, without introducing excessive false pinning ofvortices. II. NE´EL WALL COLLAPSE Figure 1 shows an example of a situation where an un- derresolved structure produces errors at a larger scale. This isa simulation of the first mMag standard problem,2a2 0n m thick, 1 mm32mm rectangle of Ni 80Fe20~Ms58.0 3105A/m,A51.3310211J/m,Ku5500 J/m3!. The weak uniaxial magnetocrystalline anisotropy is directed along thelong axis of the film. The computation cells are 25 nmsquares, 20 nm thick, with 3D spins. The exchange energy isgiven by the eight-neighbor dot product formula E i5(A/3)(n518(12mimn), detailed in Ref. 1, though similar results are obtained using the more common four-neighborexpression. The magnetostatic fields are calculated via anFFT-based scalar potential method on an offset grid, de-scribed in Ref. 3. The magnetization is relaxed using heavilydamped Landau–Lifshitz–Gilbert equations of motion. Formore details on the calculation technique, see Ref. 4. The configuration in Fig. 1 is the relaxed state just past the coercive point, after saturation to the left along the longaxis of the film. The 180° domain wall in the lower right-hand portion of Fig. 1 was formed in an intermediate ~non- relaxed !state as part of a vortex 1crosstie pair. The vortex drifted upward ~behind the inset, but symmetric with the vor- tex in the opposite corner !, and the crosstie flattened out into the observed 1-cell wide domain wall. It is difficult to predicthow wide this wall should be, given the restricted spatialdimensions and the complicated magnetic structure, but onecertainly expects it to be wider than a single 25 nm cell. Figure 2 presents a simple illustrative 1D model of a coarsely discretized Ne ´el wall. ~See Refs. 5 and 6 for more on 1D wall models, and Ref. 7 for a numerical study of 2Dwall structures. !In this 4 cell model, each cell is a constant magnetization region, infinite along the yaxis, with width a FIG. 1. Simulation results of the mMag first standard problem, using 25 nm square, 20 nm thick calculation cells ~434 subsample !. This is a relaxed state with an applied field of m0H54.5 mT directed towards the right. The inset displays all the calculation spins in the dashed box region, showing acollapsed Ne ´el wall.JOURNAL OF APPLIED PHYSICS VOLUME 83, NUMBER 11 1 JUNE 1998 6491 [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: 145.116.150.241 On: Wed, 03 Dec 2014 11:29:50as shown, and a thickness tsmall enough to force the spins to lie in the xyplane. Material parameters are saturation magnetization Msand exchange constant A. Magnetocrystal- line anisotropy is ignored. The outer spins m1andm4are held fixed and antiparallel as shown, while the inner spinsm 2andm3are allowed to rotate, with udenoting the angle between the inner spins and their outside neighbor. We makethe simplifying assumption that the inner spins are symmet-ric about the midpoint, as illustrated, because then there areno free poles along the center line, greatly reducing the mag-netostatic energy. ~This alignment of the center of a Ne ´el wall between discretization nodes is observed in practice. ! We also assume that u,90°. Magnetic poles collect along the infinite strip between the two leftmost cells, and between the two rightmost cells.This produces a field at spin m 2that acts against the ex- change torque produced on m2fromm3. If we include the exchange torque at m2fromm1, and solve for 0 torque, we find a unique energy minimum at secu522m0a2Ms2$arctan ~t/a!1arctan @t/~3a!#% 2Ap, provided the right-hand side is .1. Note that as a!0,u !60° as expected. If the right-hand side is <1, then the antiparallel state ( u50) is the only stable configuration, and the Ne´el wall collapses completely. If Ms,A, andtare fixed, then for asufficiently large the wall will collapse. For aN i80Fe20film with t520 nm, this works out to alarger than about 7.1 nm, lending credence to the conjecture that amechanism of this sort is responsible for the wall collapseobserved in Fig. 1. It seems likely that the under-resolved crosstie formed during the evolution to the relaxed state of Fig. 1 produces alocal condition not unlike that modeled in Fig. 2, and seedsthe collapse of the entire wall. A similar situation can alsoarise through grid refinement. Regardless of its origins, the observed Ne ´el wall col- lapse is made possible by the disappearance of the exchangetorque in the antiparallel state. One can try a modified ex-change field formulation appropriate for large angles, butsimple attempts yield unacceptably strong vortex pinning. 1 More sophisticated interpolations of mbetween grid points are made difficult by the apparent importance of the imi 51 constraint, and the need to produce an interpolation that is consistent across neighboring discretization cells.A different approach is to interpolate the coarse grid spinsm1,...,mNwith a differentiable function m(x,y,z) that minimizes the variational integral E~m!5AE~¹mx!21~¹my!21~¹mz!2dV, ~1! subject to some constraints. A discrete version of this is de- veloped in the next section. But let us first examine howlarge an interpolated spin angle can be. As a simple estimate,suppose we are trying to align an interpolating spin m ˜be- tween neighboring spins m1,...,mn. Consider all of these spins as points in S2, the unit sphere in R3. If the angle betweenm˜and spin miis to be less than u, thenm˜must lie outside the circular disk symmetrically opposite to mionS2 with diameter 2 p22u. The area of such a disk is 2 p(1 1cosu). This is true for each i, so if the total area of nsuch disks is less than the total area of the sphere, then there existsam˜that is no farther than ufrom each of the spins m1,...,mn. Solving for uwe find u<arccos( 2112/n). As an example, if we are trying to fit m˜between four fixed spins, then there is a direction for m˜that is at most arccos (2112/4)5120° from each of the fixed spins. III. THEORY We now develop a discrete analog to ~1!. Given the coarse grid spins m5(m1,...,mN), we want to find interpo- lating spins m˜5(m˜1,...,m˜N˜) solving min m˜F~m;m˜!subject to F~m˜!50, ~2! where F5(fk) is a collection of constraints, k51,...,K.~In Sec. IV we will use fk(m˜)5im˜ki21.!We will assume that both the objective function Fand the constraints are differ- entiable. Let us assume for the moment that the interpolated spins m˜are differentiable with respect to m, and use the extended discretization set ~m,m˜!to evaluate the exchange energy E(m)5E˜@m,m˜(m)#. To relax our solution over m, whether by integrating the Landau–Lifshitz–Gilbert equations, or through direct energyminimization, we needed to know ]E/]m: ]E ]mi5]E˜ ]mi1( j51N˜ ]E˜ ]m˜j]m˜j ]mi. ~3! We have only an implicit relation for m˜in terms of m,s ot h e last term above is difficult to evaluate. However, suppose we useE˜as the objective function Fin~2!. It follows from the theory of Lagrange multipliers that if ( ]fk/]m˜j)k,jhas full rankK,N˜, then at a local minimum m˜we can write ]E˜/]m˜ as a linear combination of ]fk/]m˜, i.e., ]E ]mi5]E˜ ]mi1( k51K lk( j51N˜ ]fk ]m˜j]m˜j ]mi. If the constraints Fare independent of m, then the last sum is zero, and we get the simple relation FIG. 2. Illustration of a simple 1D model to study discretization-induced Ne´el wall collapse. The outer spins m1andm4are fixed and antiparallel.6492 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Michael J. Donahue [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: 145.116.150.241 On: Wed, 03 Dec 2014 11:29:50]E~m! ]m5]E˜@m,m˜~m!# ]m. The one difficulty is that we cannot guarantee the differ- entiability of m˜with respect to m. More work needs to be done to identify and handle those spin configurations forwhich differentiability is lost, but in practice such occur-rences appear to be relatively uncommon. IV. RESULTS AND CONCLUSIONS To test this interpolation technique, we introduced a supplemental lattice to the simulation described in Sec. II.The supplemental lattice interpolated the main grid at halfthe cell dimension, i.e., with 12.5 nm square cells. The eight-neighbor dot product exchange energy formulation was mini-mized to determine the spins on the supplemental lattice~holding fixed the spins on the original lattice !, subject to the constraint imi51 for all spins. ~For this initial study, we employed a simple gradient descent minimization algorithm,which required computation time comparable to that of thedemagnetization calculation. We expect a sophisticated mini-mization algorithm will be much faster. !The refined lattice is used only for the exchange energy and exchange torquecalculations. The magnetization configuration in Fig. 1 is not a stable state under the new scheme, but using it as an initial stateand allowing the simulation to evolve to a new energy mini-mum yields Fig. 3. Note that the interpolation has allowedthe crosstie to reform, and the domain wall is now a resolvedNe´el wall. These results are similar to those obtained using the standard exchange scheme and a ‘‘real’’ refinement with12.5 nm cells. Conversely, using the proposed method theNe´el wall does not collapse even with 50 nm cells ~a n da2 5 nm supplemental lattice !. As another test, we repeated the vortex pinning simula- tions detailed in Ref. 1, and found no increase in the vortexpinning field.It is important to distinguish this interpolation technique from a straightforward grid refinement. In the proposed tech-nique the interpolated spins affect only the exchange energy,and at each step the interpolated spins are relaxed completelyto an exchange energy minimum ~holding fixed the spins on the coarse mesh !. Because of this, the angle between neigh- boring spins on the refined mesh cannot collapse to 180°, asdescribed in Sec. II. Instead, this technique effectively pro-duces an exchange energy formulation that does not breakdown in the case of large angles between neighboring spins,yet does not increase vortex pinning. 1M. J. Donahue and R. D. McMichael, Physica B 233, 272 ~1997!. 2Round-robin results for a 2 313.02mmN i80Fe20computational problem are available at http://cobalt.nist.gov/mumag/prob1/prob1report.html 3D. V. Berkov, K. Ramsto ¨ck, and A. Hubert, Phys. Status Solidi A 137, 207~1993!. 4R. D. McMichael and M. J. Donahue, IEEE Trans. Magn. 33, 4167 ~1997!. 5A. Aharoni, J. Appl. Phys. 37, 3271 ~1966!. 6W. F. Brown, Jr. and S. Shtrikman, Phys. Rev. 125, 825 ~1962!. 7K. Ramsto ¨ck, W. Hartung, and A. Hubert, Phys. Status Solidi A 155, 505 ~1996!. FIG. 3. Simulation results using the described interpolation technique, with Fig. 1 as the initial state ( m0H54.5 mT). The collapsed wall in that figure has expanded into a crosstie and a resolved Ne ´el wall.6493 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Michael J. Donahue [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: 145.116.150.241 On: Wed, 03 Dec 2014 11:29:50

1.3596808.pdf

Correlation of surface domain structure and magneto-impedance in amorphous microwires M. Ipatov, A. Chizhik, V. Zhukova, J. Gonzalez, and A. Zhukov Citation: Journal of Applied Physics 109, 113924 (2011); doi: 10.1063/1.3596808 View online: http://dx.doi.org/10.1063/1.3596808 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Giant magneto-impedance effect in amorphous ferromagnetic wire with a weak helical anisotropy: Theory and experiment J. Appl. Phys. 113, 243902 (2013); 10.1063/1.4812278 Circular domains nucleation in magnetic microwires Appl. Phys. Lett. 102, 202406 (2013); 10.1063/1.4807595 Stress-induced magnetic hysteresis in amorphous microwires probed by microwave giant magnetoimpedance measurements J. Appl. Phys. 113, 17A326 (2013); 10.1063/1.4798278 Enhanced giant magneto-impedance effect in soft ferromagnetic amorphous ribbons with pulsed laser deposition of cobalt ferrite J. Appl. Phys. 113, 17A323 (2013); 10.1063/1.4795802 Domain structure in Joule-heated CoFeSiB glass-covered amorphous microwires probed by magnetoimpedance and ferromagnetic resonance J. Appl. Phys. 94, 4539 (2003); 10.1063/1.1608474 [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: 129.127.200.132 On: Mon, 02 Feb 2015 01:31:37Correlation of surface domain structure and magneto-impedance in amorphous microwires M. Ipatov,1,a)A. Chizhik,1V. Zhukova,1J. Gonzalez,1and A. Zhukov1,2 1Department de Fı ´sica de Materiales, Universidad del Paı ´s Vasco, San Sebastia ´n, Spain 2IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain (Received 23 February 2011; accepted 2 May 2011; published online 15 June 2011) The correlation between surface domain structure (SDS) and high frequency magneto-impedance (MI) in amorphous microwires has been systematically studied. First, we applied the magneto-optical polarizing microscopy to determine the SDS and found that it is considerably different in unstressedmicrowire and in microwires to which tensile and torsional stress were applied. Then, we measured the longitudinal and off-diagonal MI in these microwires and also observed quite different MI dependencies. We analyzed the experimental MI curves and their dependence on the SDS with thehelp of a simple model that nevertheless yields good qualitative agreement with experiment. We have demonstrated that the analysis of the MI dependencies, especially the off-diagonal one, can reveal the SDS in the microwires. The obtained results can also be useful for magnetic and stresssensing applications. VC2011 American Institute of Physics . [doi: 10.1063/1.3596808 ] I. INTRODUCTION The giant magneto-impedance (GMI) effect is under- stood as a significant change of the impedance of a magneti- cally soft conductor upon the application of an external magnetic field due to the change of skin-depth of the conduc-tor. This phenomena has received much attention since its dis- covery 1,2due to its potential for small-size, high-performance and low-cost magnetic field sensor applications. As a result ofintensive investigation of the MI effect in the past decade, the magnetic field detection resolution as high as 1 lOe has been achieved 3and the first low-cost multi-axis sensors in micro- chip with claimed nano-Tesla sensitivity that can operate over a wide temperature range have recently appeared on the market.4 These industrial sensors utilize amorphous microwire5 as sensing element and are realized on so-called off-diagonalimpedance 6–10which exhibits higher sensitivity and linearity as compared with the longitudinal diagonal impedance of the conductor. The off-diagonal impedance is observed in mag- netically soft wires with circumferential or helical magneticanisotropy and is related to the cross-magnetization process m z=hu:9If the static magnetization is helical (as a result of application of external longitudinal magnetic field, for exam-ple), then the precision of the circular magnetic field h ucre- ated by the ac current igives rise to appearance of the nonzero axial magnetization mzwhich, in accordance with the Faraday’s law, induces a voltage in the pick-up coil wounded on the wire. In this way, the off-diagonal imped- ance can be detected. The off-diagonal impedance, besides the higher sensitiv- ity to magnetic field comparing with the longitudinal imped- ance, is also more sensitive to the particularities of SDS. Forexample, if the SDS divides into domains with opposite cir- cular magnetization direction (bamboolike SDS), the voltageresponse averages over domains and the observed off-diago- nal MI effect is very small and irregular, 9while the longitu- dinal MI effect in the same sample, that we experimentally demonstrate further, is rather high. Recently, the interest in MI effect in microwires has been reinforced due to the proposal to use them in tunable epsilon negative11–14and double negative15,16composite metamaterials. The use of ferromagnetic microwires with high MI or stress impedance effects makes it possible to sen- sitively tune the composite electromagnetic characteristicsby changing the magnetic structure of the wire with external magnetic, mechanical or thermal stimuli. The possibility to control or monitor the electromagnetic parameters (andtherefore, scattering and absorption) of composite metamate- rials is of great interest for large-scale applications such as remote nondestructive testing, structural health monitoring,tunable coatings and absorbers. For example, a material with self-monitoring properties could be able to evidence invisi- ble structural damages, defects, excessive loadings, localstress and temperature distribution, thus, considerable facili- tating the in situ health monitoring of large scale objects such as infrastructure (bridges, buildings, etc.). A detailed understanding of the influence of SDS on MI properties is essential for further progress in the development of stress sensitive composite materials and sensors. In thispaper we investigate the correlation between the SDS, which was modified by the application of stresses, and high fre- quency MI in amorphous microwires with circumferentialand helical anisotropy. We propose a rather simple method for the determination of SDS from MI measurements. The paper is organized as follows. First, we acquired images by magneto-optical polarizing microscopy (MOPM) of SDS of microwire in unstressed state and with applied ten- sile and torsional stresses and determined the SDS in eachcase. Then, we measured the longitudinal and off-diagonal MI in the wires to which the same stress were applied and we observed a considerable transformation of MI curves. a)Electronic mail: mihail.ipatov@ehu.es. 0021-8979/2011/109(11)/113924/6/$30.00 VC2011 American Institute of Physics 109, 113924-1JOURNAL OF APPLIED PHYSICS 109, 113924 (2011) [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: 129.127.200.132 On: Mon, 02 Feb 2015 01:31:37Finally, we applied a rather simple model to analyze the ex- perimental MI curve and reveal the correlation between theSDS and MI. The model, in spite of its simplicity, describes qualitative well the experimental data. II. MAGNETO-OPTICAL EXPERIMENT In the present paper we have investigated the surface do- main structure and magneto-impedance in amorphous micro-wire fabricated by Taylor-Ulitovsky method with nominal composition Co 67Fe3:8Ni1:4Si14:5B115Mo 1:7with amorphous core radius aof 10.7 lm and the glass coating thickness of 2.4lm. In this Section, we present the investigation of the effect of tensile and torsional stresses on domain structure obtainedwith MOPM in longitudinal configurations. The images of SDS have been obtained for microwire to which different stresses were applied as given in Table I. Figure 1shows the SDS of the sample in unstressed state (a) and with applied tensile stress (b). One can see that the SDS is a bamboolike in the unstressed wire which transformsto a mono-domain after applying tensile stress. Figure 2shows the SDS of the twisted wire that, as it can be seen, resulted in the appearance of mono-domain statewith helical anisotropy. The anisotropy angle can be eval- uated from the image in zero field [Fig. 2(a)] as the image contrast is determined by the magnetization direction. Appli-cation of the axial magnetic field H Eleads to rotation of the magnetization as shown in Figs. 2(b) and2(c). III. MAGNETO-IMPEDANCE EXPERIMENT We have experimentally investigated the impedance dependencies in three pieces of the same microwire to whichdifferent stresses were applied in the same way as in the magneto-optical experiment and given in Table I. The longitudinal Z zzand off-diagonal Zuzimpedance components dependencies on external axial magnetic fieldHEwere measured in 6 mm long pieces of amorphous glass- coated microwire. The microwires were placed in a specially designed microstrip cell. One wire end was connected to theinner conductor of a coaxial line through a matched micro- strip line while the other was connected to the ground plane. The impedance components Z zzand Zuzwere measured simultaneously using vector network analyzer in the fre- quency range 10 /C0300 MHz. The longitudinal impedance of the sample Zw¼Zzzl, where lis the wire length, was obtained from reflection coefficient S11and the off-diagonal impedance Zuzwas measured as transmission coefficient S21 as a voltage induced in a 2-mm long pick-up coil wounded over the wire. The static bias field HBwas created by the dc current IBapplied to sample through the bias-tee element. The other experimental details are given in Ref. 17. The measurements of the real parts of longitudinal Z0 zzand off- diagonal Z0 uzimpedances at frequency of 30 MHz are shown in Fig. 3. The similar dependencies were observed in the whole frequency range. However, at higher frequencies we observed the increase of Zzzcomponent. The Zuzimpedance, on the contrary, decreases that is probably related to the re- actance of the pick-up coil. The graphs in Fig. 3show both ascending and descending branches of the field dependenciesso that the magnetic hysteresis can be evaluated. When no additional stresses and no bias field are applied to the microwire, as we know from the magneto-opticalexperiment, the SDS is the bamboolike [Fig. 1(a)]. In this wire, the longitudinal impedance Z 0 zzðHEÞ[solid curve in Fig. 3(a)] exhibits a rather high symmetrical double-peak depend- ence with maxima at 180 A/m. The impedance changes from 35.6 to 101.5 Xs that gives MI effect of 185%. From the other hand, the observed off-diagonal impedance Z0 uz[solid curve in Fig. 3(d)] is very small and irregular, that can be expected for a wire with bamboolike SDS in which the con- tribution of the domains with opposite magnetization aresubtracted in the voltage induced in the pick-up coil. When the dc bias field H B(created by dc current IB¼10 mA) is applied [dashed curves in Figs. 3(a) and 3(d)], the longitudinal impedance Z0 zzdependence demon- strates a lower sensitivity to magnetic field HEwith maxima increasing to 230 A/m. The effect of bias field HBis much more pronounced for the off-diagonal impedance, it becomes much higher with high degree of antisymmetry that is aTABLE I. Description of the measurements. # Applied stress Domain structure Anisotropy 1 no applied stresses bamboo circumferential 2 tensile mono-domain circumferential3 torsional mono-domain helical FIG. 1. MOPM photograph of the surface domain structure of microwire in unstressed state (a) and with applied tensile stress r(b). FIG. 2. MOPM photograph of the surface domain structure obtained of microwire subjected to torsional stress. The arrows show the magnetization direction obtained from the image contrast.113924-2 Ipatov et al. J. Appl. Phys. 109, 113924 (2011) [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: 129.127.200.132 On: Mon, 02 Feb 2015 01:31:37result of a single domain state formation in the surface layer of the sample. The sample 2 was soldered in a slightly pulled state that resulted in the appearance of a tensile stress in the sample and transformation of the bamboolike SDS to the mono-do-main one [Fig. 1(b)]. The longitudinal impedance Z 0 zzðHEÞ measured at zero bias field, as one can see from Fig. 3(b), exhibits a slightly higher MI effect comparing with the sam-ple 1: the impedance changes from 29.0 to 102.5 Xs that gives MI effect of 253%. The applied tensile stress gave rise to increasing of anisotropy field H Aand, consequently, the reduced field HE=HAbecame lower and the maxima of MI curve increased up to 320 A/m. This is a manifestation of a well-known stress-impedance effect. The effect of tensilestress on the off-diagonal impedance, shown in Fig. 3(e) is even more pronounced. Now, as the SDS is a mono-domain even at zero bias field, the dependence Z 0 uzis considerably higher comparing with sample 1 and hysteretic. When the dc bias field HBis applied [dashed curves in Figs. 3(b) and3(e)], the longitudinal MI effect, similarly to the sample 1, became lower, although in a less degree, with maxima increasing to 330 A/m. The lower effect of bias field on MI is also related to the increased anisotropy field as thereduced bias field H B=HAdecreases. After the application of bias field HB, the off-diagonal impedance, as in the sample 1, becomes higher and antisymmetric. The sample 3 was twisted when being soldered that induced a torsional stress in the sample and transformed the SDS into mono-domain one with helical anisotropyeasy axis [Fig. 2(a)]. Figures 3(c) and3(f)show the imped- ance components Z 0 zzandZ0 uzof the sample. In our previouswork17we have investigated the MI effect in microwire with helical anisotropy and found the similar MI behavior:hysteresis of MI curves which become anhysteretic and asymmetric when the bias field is applied. As in the sample 2,jZ 0 uzjis high when HB¼0 that is caused by a mono- domain state. The application of axial magnetic field HE makes the magnetization rotate [Figs. 2(b) and 2(c)] and, consequently, the impedance changes. The dashed curvesshow the effect of dc bias field. When a sufficiently high bias field H Bis applied ( IB¼10 mA or more), the depend- ence becomes anhysteretic and asymmetric. In contrast tothe previous two samples, the induced voltage is not zero asu 06¼0a t HE¼0. It is required to apply HEas high as 125 A/m to make the magnetization lies in the transversalplane ( S 21crosses zero) that is caused by helical anisotropy. From the measurement shown in Fig. 3(f), we found the following parameters of the third sample as it is describedin Ref. 17:H A¼265 A/m, a¼35/C14and HB¼155 A/m with I B¼10 mA. Recently, we have investigated the MI sensitivity to dc bias field HBand found that in the microwires with helical anisotropy the longitudinal impedance depends on dc bias field HB, while in the microwires with circumferential anisot- ropy there is no such dependence.18To compare, we have also measured the off-diagonal impedance Z0 uzðHBÞas a function of the bias field HBatHE¼0 in all three samples. Similarly to the longitudinal impedance, the off-diagonal im- pedance is not sensitive to the bias field HBin the microwires with circumferential anisotropy (samples 1 and 2), and itsensitively changes with bias field H Bin the wire with helical anisotropy (sample 3), as shown in Fig. 4. FIG. 3. (Color online) Experimental dependencies of the real parts of longi- tudinal impedance Zzz¼Zw=l(a)–(c) and off-diagonal impedance Zuz/S21 (d)–(f) on axial magnetic field HEat fre- quency f¼30 MHz for samples to which no stress were applied, (a) and(d), and with applied tensile, (b) and (e), and torsional, (c) and (f), stresses. The inserts show the sketch of domain struc- ture obtained by magneto-optical polar- izing microscope at H E¼0 and HB¼0.113924-3 Ipatov et al. J. Appl. Phys. 109, 113924 (2011) [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: 129.127.200.132 On: Mon, 02 Feb 2015 01:31:37IV. DISCUSSION Further, we will consider a simple model of MI at high frequency in amorphous microwire that qualitatively repro- duces the experimental results. For a wire geometry and the high-frequency approximation (strong skin effect) the longi-tudinal Z zzand off-diagonal Zuzimpedance components can be expressed as:7 Zzz¼ð1/C0iÞðffiffiffiffiffiffiffiffiffiffiffi elþ1p sin2uþcos2uÞRdca d0; (1) Zuz¼ð1/C0iÞðffiffiffiffiffiffiffiffiffiffiffi elþ1p /C01Þsinucosu; (2) where uis the angle between the magnetization and the transversal plane, Rdcis the wire dc resistance, ais the wire radius, d0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q=l0xp is the nonmagnetic skin depths, xis the angular frequency, and elis the effective transverse permeability:17 el¼x2 M x2 0/C0iagxx M/C0x2; (3) where xM¼cl0MS,x0is ferromagnetic resonance fre- quency, agis the Gilbert damping constant.If the surface layer, where the high frequency current is concentrating, is in a mono-domain state (we have ascer- tained from magneto-optical experiment that this is the casefor the samples 2 and 3), then the equilibrium angle u 0, under the axial external magnetic field HEand dc bias field HB, can be determined from the equation of energy minimization:17 0:5HAsin 2ðu0þaÞ/C0HEcosu0þHBsinu0¼0;(4) where HAis the surface anisotropy field and ais the angle between the anisotropy easy axis and the transversal plane. We neglect the demagnetizing fields and the contribution ofthe exchange energy. As we have mentioned above, the discussed model of MI assumes a mono-domain SDS. Then it cannot be applieddirectly for wires with bamboolike SDS like that shown in Fig.1(a). The right-hand side of Eq. (2)for the off-diagonal impedance averages over domains and tends to zero due tothe factor cos u 0that has opposite signs in the domains with the opposite magnetization.9This effect, experimentally demonstrated in Fig. 3(d), allows the determination of the SDS type (bamboolike or mono-domain) from the measured off-diagonal MI curve at HB¼0. Figures 5(a) and5(b) shows the solution of Eq. (4)as a function u0ðHEÞ, and Figs. 5(c)–5(f)shows the longitudinal Z0 zzand off-diagonal Z0 uzimpedance dependencies on HEcal- culated through Eqs. (1)–(4)for different angles aandHB, We have selected two values of angle athat corresponds to circumferential anisotropy ( a¼1/C14) and helical anisotropy (a¼35/C14that we obtained for the sample 3 from experimen- tal data). The effect of dc bias field is also demonstrated for two values: at zero field ( HB¼0) and at a field sufficiently high to suppress the hysteresis ( HB>HAsina). In the calcu- lations the following typical parameters of a cobalt-based microwire were used: MS¼640 kA/m, c¼0.035 MHz/ [A/m] (Ref. 19) and ag¼0.012 (Ref. 17). As one can see both in experimental [Figs. 3(a)and3(b)] and in theoretical [Figs. 5(d) and5(f)] graphs, the application FIG. 4. (Color online) Experimental dependencies Re Zuzon bias field HB in sample 3, HE¼0. FIG. 5. (Color online) Modeling of the equilibrium angle u0and the real parts of impedance components Z0 zzandZ0 uzdependencies on external magnetic field HEwithaandHBas parameters, f¼30 MHz.113924-4 Ipatov et al. J. Appl. Phys. 109, 113924 (2011) [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: 129.127.200.132 On: Mon, 02 Feb 2015 01:31:37of the static circumferential bias field HBresults in increasing of magnetic ‘hardness’ of the conductor, similarly to the effect of increasing the anisotropy field, a higher magneticfield H Eis required to rotate the magnetization in axial direc- tion. As it seen from Eqs. (1),(2)and Figs. 5(c)–5(f), the lon- gitudinal impedance Z0 zzis at minimum while the off-diagonal Z0 uzcross zero, that was observed experimentally in Figs. 3(a),3(b)and3(e). As one can see from Fig. 5(a),u0ðHEÞmay exhibit hys- teresis even at aclose to zero. If ais exactly zero, the func- tion u0ðHEÞhas two equiprobable branches. In the calculation, we set a¼1/C14. This does not affect the calcu- lated curves (in contrast with much higher values of a) but allows avoiding this ambiguity. The off-diagonal impedance, as a function of sin u0cosu0, exhibits hysteresis at any ani- sotropy angle a>0 as shown in Fig. 5(e) and was observed experimentally [Figs. 1(e) and1(f)], to suppress which the application of bias field HB>HAsinais required. When the anisotropy is circumferential with small value of a(sample 1 and 2), at HE¼0, the angle u0¼0 and the off-diagonal im- pedance Zuzis zero. This gives the possibility to estimate the type of anisotropy (circumferential or helical) from off- diagonal MI curve. It is interesting to note that at HE¼0, the experimental longitudinal impedance Z0 zzin sample 1 decreases from 35.6 Xto 29.2 X(18%) [Fig. 3(a)] when the bias field is applied while the theoretical curve do not show any change of im-pedance with bias field H BatHE¼0. On the other hand, the change of impedance Z0 zzin sample 2 which is in the mono- domain state, is considerably lower. We believe that the rea-son for this disagreement is the 180 /C14domain walls between domains with opposite magnetization that exist in the unstressed wire but are not taken into account by the model.Inside the domain walls, the magnetization direction is not circumferential, and, therefore, the impedance of these regions is higher and their contribution to MI effect is lowercomparing to those regions where magnetization lies in the circumferential direction. When a sufficiently high bias field H Bor tensile stress is applied, the bamboolike SDS trans- form to mono-domain one and, consequently, the measured wire impedance Z0 zzis lower. As it seen from Eq. (2), the off-diagonal impedance Z0 uz vanishes at u¼0 (magnetization vector lies in the transver- sal plane) and at u¼p=2 (saturation state, magnetization lies along the wire axis). In the wire with circumferential ani-sotropy, the off-diagonal impedance practically falls to zero when H Eexceeds HAandHB¼0. The application of the dc bias filed HB, as it is seen from Fig. 3(e) and Fig. 5(d), besides the suppressing the hysteresis, also considerably broadens the range of nonzero off-diagonal impedance with the maximum of jZ0 uzjmoving to higher fields exceeding HA. A peculiar characteristic of the wire with helical anisotropy is an asymmetrical curve with the nonzero off-diagonal impedance Zuzwhen both HEand HBare equal to zero. The anisotropy angle acan be found from Eq. (4)as a¼0:5 arcsin ð2H0 E=HAÞwhere H0 Eis the axial magnetic field at which Zuzcrosses zero and u0¼0. The investigated stress effect on SDS and MI can be used in different applications. As the MI curves depend onSDS (multi-domain bamboolike or mono-domain) and also on the anisotropy angle aand anisotropy field HA, this can be used in stress or torque sensors as well as in stress-sensitivecomposite materials. Also, we have found that the MI effect is higher in wires where SDS is a mono-domain one, which can be induced by applied bias field or stress. This observa-tion can be useful for improving the magnetic sensors sensitivity. V. CONCLUSIONS We have investigated the correlation between the SDS and MI at high frequency in amorphous microwires with cir-cumferential and helical anisotropy. First, we investigated the effect of stress on SDS by magneto-optical polarizing micros- copy. We have found that the bamboolike SDS exists in theunstressed microwire. The application of the tensile stress resulted in the disappearance of the domain structure. In both cases the anisotropy was circumferential. In the third case wetwisted the microwire and observed the transformation of the SDS from bamboolike with circumferential anisotropy to mono-domain with helical anisotropy. Then, knowing the SDS in the unstressed microwire and in the microwire with applied tensile or torsional stresses, we investigated the longitudinal and off-diagonal MI behaviorin each case. In the MI experiment, we observed a consider- able transformation of the MI curve as a result of modifica- tion of the SDS. In the microwire with bamboolike SDS, theobserved off-diagonal impedance Z 0 uzwas very small and irregular while the longitudinal impedance Z0 zzwas rather high. If the SDS is a mono-domain and circumferential, Z0 uz becomes considerably higher. In this microwire we also observed the increase of longitudinal MI effect that is explained by the disappearance of domain walls. In themicrowire with helical anisotropy, the MI curves are asym- metric with hysteresis. The presented analysis of the MI dependence, especially the off-diagonal one, can be used for determination of the SDS in microwires. Additionally, the obtained results can be useful for further development of the magnetic field sensors,and the stress and torque sensors as well as the stress-sensi- tive composites with embedded microwires, as we demon- strated the sensitivity of impedance to the tensile andtorsional stresses. ACKNOWLEDGMENTS This work was supported by EU ERA-NET program under projects “DEVMAGMIWIRTEC” (MANUNET-2007- Basque-3) and “SoMaMicSens” (MANUNET-2010-Basque-3), by EU under FP7 “EM-safety” project and by Spanish Ministry of Science and Innovation, MICINN under Project MAT2010-18914. 1L. Panina and K. Mohri, Appl. Phys. Lett. 65, 1189 (1994). 2R. S. Beach and A. E. Berkowitz, Appl. Phys. Lett. 64, 3652 (1994). 3K. Mohri, T. Uchiyama, L. P. Shen, C. M. Cai, and L. V. Panina, J. Magn. Magn. Mater. 249, 351 (2002). 4Seehttp://www.aichi-mi.com . 5V. Zhukova, M. Ipatov, and A. Zhukov, Sensors 9, 9216 (2009).113924-5 Ipatov et al. J. Appl. Phys. 109, 113924 (2011) [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: 129.127.200.132 On: Mon, 02 Feb 2015 01:31:376N. Usov, A. Antonov, and A. Lagar’kov, J. Magn. Magn. Mater. 185, 159 (1998). 7A. Antonov, I. Iakubov, and A. Lagarkov, J. Magn. Magn. Mater. 187, 252 (1998). 8D. P. Makhnovskiy, L. V. Panina, and D. Mapps, Phys. Rev. B. 63, 144424 (2001). 9S. I. Sandacci, D. P. Makhnovskiy, L. V. Panina, K. Mohri, and Y. Hon-kura, IEEE Trans. Magn. 35, 3505 (2004). 10M. Ipatov, V. Zhukova, J. M. Blanco, J. Gonzalez, and A. Zhukov, Phys. Status. Solidi. A 205, 1779 (2008). 11O. Reynet, A.-L. Adenot, S. Deprot, O. Acher, and M. Latrach, Phys. Rev. B66, 094412 (2002). 12D. P. Makhnovskiy, L. V. Panina, C. Garcia, A. P. Zhukov, and J. Gonza- lez,Phys. Rev. B 74, 064205 (2006).13L. V. Panina, M. Ipatov, V. Zhukova, A. Zhukov, and J. Gonzalez, J. Appl. Phys. 109, 053901 (2011). 14M. Ipatov, G. Aranda, V. Zhukova, L. Panina, J. Gonzalez, and A. Zhukov, Appl. Phys. A , (2011). 15J. Carbonell, H. Garcı ´a-Miquel, and J. Sa ´nchez-Dehesa, Phys. Rev. B 81, 024401 (2010). 16H. Garcı ´a-Miquel, J. Carbonell, and J. Sa ´nchez-Dehesa, Appl. Phys. Lett. 97, 094102 (2010). 17M. Ipatov, V. Zhukova, A. Zhukov, J. Gonzalez, and A. Zvezdin, Phys. Rev. B 81, 134421 (2010). 18M. Ipatov, V. Zhukova, A. Zhukov, and Gonzalez, Appl. Phys. Lett. 97, 252507 (2010). 19M. Phan and H. Peng, Prog. Mater. Sci. 53, 323 (2008).113924-6 Ipatov et al. J. Appl. Phys. 109, 113924 (2011) [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: 129.127.200.132 On: Mon, 02 Feb 2015 01:31:37

1.3276165.pdf

Precessional dynamics of Ni 45 Fe 55 thin films for ultrahigh frequency integrated magnetics Jeffrey F. Godsell, Santosh Kulkarni, Terence O’Donnell, and Saibal Roy Citation: Journal of Applied Physics 107, 033907 (2010); doi: 10.1063/1.3276165 View online: http://dx.doi.org/10.1063/1.3276165 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/107/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Influence of a Dy overlayer on the precessional dynamics of a ferromagnetic thin film Appl. Phys. Lett. 102, 062418 (2013); 10.1063/1.4792740 Effects of radio-frequency noise suppression on the microstrip line using FeCoNiB soft magnetic thin films J. Appl. Phys. 113, 043922 (2013); 10.1063/1.4789607 Soft magnetic properties of (Ni80Fe20)1−x(Ni0.5Zn0.5Fe2O4)x films for high frequency applications J. Appl. Phys. 109, 07A308 (2011); 10.1063/1.3535550 Combination of ultimate magnetization and ultrahigh uniaxial anisotropy in CoFe exchange-coupled multilayers J. Appl. Phys. 97, 10F910 (2005); 10.1063/1.1855171 Thickness effects on magnetic properties and ferromagnetic resonance in Co–Ni–Fe–N soft magnetic thin films J. Appl. Phys. 91, 8462 (2002); 10.1063/1.1453950 [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: 147.188.128.74 On: Wed, 03 Jun 2015 09:06:26Precessional dynamics of Ni 45Fe55thin films for ultrahigh frequency integrated magnetics Jeffrey F. Godsell, Santosh Kulkarni, Terence O’Donnell, and Saibal Roya/H20850 Microsystems Centre, Tyndall National Institute, University College Cork, Lee Maltings, Cork, Ireland /H20849Received 29 July 2009; accepted 19 November 2009; published online 4 February 2010 /H20850 Future monolithic point of load switched mode power supplies will be expected to meet the energy requirements of miniaturized, high functionality electronic devices. Recently, Ni 45Fe55has emerged as a potentially important material choice for use as a soft magnetic core material within highfrequency integrated passive magnetic components. The operating frequency range of the integratedpassives which form a key part of the point of load power supply must increase to allow forinductor/transformer miniaturization to become monolithic with power integrated circuits. In thiswork, an analysis of the high frequency permeability spectra of an electroplated Ni 45Fe55thin film has been carried out to quantitatively analyze the material’s high frequency performance. Complexpermeability spectra of the film have been investigated at frequencies up to 9 GHz to identify boththe film’s spectroscopic splitting factor /H20849g/H20850and its effective dimensionless damping parameter /H20849 /H9251/H20850. The Kittel equation is utilized to identify gas 2.128, while /H9251is determined to be approximately 0.045. The critically damped condition for the film is also examined to extract /H9251in the critically damped case under a range of externally applied bias fields. It is concluded that for monolithicpower inductors, improved device performance can be achieved when the ferromagnetic core is inan underdamped state up to a critical frequency. © 2010 American Institute of Physics . /H20851doi:10.1063/1.3276165 /H20852 I. INTRODUCTION Magnetic thin films having high saturation magnetiza- tion, high permeability, and low losses at high frequencieshave been intensively studied due to their significance in arange of applications. Examples include, integrated induc-tors, transformers, magnetic random access memory/H20849MRAM /H20850elements, magnetic recording heads, and rf cir- cuits. The ultrahigh frequency soft magnetic properties ofcore materials constitute some of the key requirements forthin film inductors, 1coupled inductors,2and transformers3to achieve increased efficiencies. This increases their applica-bility for use in monolithic voltage converter topologies em-bedded in miniaturized handheld electronic devices. In thiswork, we examine the high frequency permeability spectrumof Ni 45Fe55thin films to examine the limit of their usable frequency range. This information will become increasinglyimportant as the material is pushed to operate at ever greaterfrequencies in such applications as monolithic on-chip induc-tors and transformers. To allow for the passive magnetic cir-cuit elements to be integrated onto a silicon substrate, a re-duction in the component’s physical size is needed. This sizereduction necessitates an increase in the component’s oper- ating frequency. Ni 45Fe55represents a potentially important alloy compo- sition due to its higher saturation magnetization, higher anis-tropy field and lower conductivity when compared to Per-malloy /H20849Ni 80Fe20/H20850. These material qualities make Ni 45Fe55a more suitable choice for use as core material in monolithic on-chip microinductors or transformers. ElectroplatedNi45Fe55was first introduced for use in hard disk write heads by IBM in 1997 and was adopted as a standard in the thinfilm write head industry after Permalloy, 4highlighting the material’s potential. Electroplated Ni 45Fe55thin films have also previously been incorporated into microinductors oper-ating up to 100 MHz. 5While ferromagnetic resonance /H20849FMR /H20850studies on Ni 80Fe20films have been widely reported,6,7to date there have been very limited reports on losses in Ni 45Fe55thin films at gigahertz frequencies. At lower frequencies eddy currents and hysteresis loss domi-nate, however as the frequency is increased, further lossesdue to FMR may become important. For example, converterswith switching frequencies as high as 480 MHz have alreadybeen reported 8and even higher operating frequencies may follow in the near future. It is known that the relaxationprocesses within a magnetic film determine the maximumpermissible switching speed of the material. These processesbecome increasingly important for the design of passivemagnetic circuit components as they are considered for op-eration into the gigahertz frequency range. The magnetic dimensionless damping parameter /H20849 /H9251/H20850for ferromagnetic thin films quantifies one of the limiting factors in the high speed switching of magnetic thin films at elevatedfrequencies. In this study, the effective dimensionless damp-ing constant /H20849 /H9251/H20850of an electroplated Ni 45Fe55thin film is determined experimentally at operating frequencies up to 9 GHz. The spin dynamics of the material are described by the Landau–Lifshitz–Gilbert /H20849LLG /H20850equation of motion /H208491/H20850, where Mthe magnetization, /H9253the gyromagnetic ratio, H/H6023the effective field, and /H20849/H9251/H20850the phenomenological effective di- mensionless damping parameter are input terms.9 a/H20850Electronic mail: saibal.roy@tyndall.ie.JOURNAL OF APPLIED PHYSICS 107, 033907 /H208492010 /H20850 0021-8979/2010/107 /H208493/H20850/033907/8/$30.00 © 2010 American Institute of Physics 107, 033907-1 [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: 147.188.128.74 On: Wed, 03 Jun 2015 09:06:26/H11509M/H6023 /H11509t=−/H9253M/H6023/H11003H/H6023eff+/H9251eff Ms/H20873M/H6023/H11003/H11509M/H6023 /H11509t/H20874. /H208491/H20850 The spectroscopic splitting factor /H20849commonly referred to as the Landé g-factor /H20850additionally provides information on the separation of the contributions from orbital and spin mo-ments. The g-factor is important in FMR studies and is ex- perimentally determined using the Kittel equation /H208492/H20850 10/H20849as outlined later /H20850, where fFMRis the ferromagnetic resonant fre- quency, /H9253is the gyromagnetic ratio, /H92620is the permeability of free space, Msis the saturation magnetization, Hkis the an- isotropy field, and Hbiasis the bias field. fFMR=/H20849/H9253/2/H9266/H20850/H92620/H20881Ms/H20849Hk+Hbias/H20850/H20849 2/H20850 The use of electroplating for thin film deposition has the advantage of full integration compatibility with currentcomplementary metal-oxide semiconductor fabrication tech-nologies. This thin film deposition technique is also fast, in-expensive, and lends itself readily to the three dimensionaltopology of many device structures. By performing complexpermeability measurements on the as electroplated samplesand by deconvoluting the resulting spectrum into its real andimaginary components, the spectroscopic splitting factor ofthe as deposited films is identified. The effective dampingparameter /H20849 /H9251/H20850is subsequently determined by comparing the experimentally measured permeability spectrum to a simu- lated spectrum, derived from a combined LLG and eddy cur-rent based model. Finally, critically damped conditions havebeen determined for a range of bias fields. II. EXPERIMENT Ni45Fe55films were galvanostatically deposited using a dc plating technique; details of the plating process have pre- viously been reported.11The films were electrodeposited at a constant current density of 3 mA /cm2. Pt/Ti pieces were used as anode and silicon wafer pieces /H20849/H110112c m2area /H20850with a sputtered Ti /H2084920 nm /H20850/Cu/H2084950 nm /H20850seed layer were used as the cathode /H20849subsequently referred to as Si/Ti/Cu substrates /H20850.I n determining the appropriate seed layer thickness a compro-mise was met: thin enough to minimize eddy current contri-butions from the seed layer while also being thick enough tofacilitate the electrodeposition at the required current density.The films were deposited in the presence of a magnetic field/H1101113.5 kA /m/H20849/H11011170 Oe /H20850, in order to induce a uniaxial an- isotropy in the film. The electrodeposited film was visually smooth, shiny, and showed good adhesion to the substrate.The surface morphology and compositions of the plated filmwas characterized using scanning electron microscopy andenergy dispersive x-ray /H20849EDX /H20850diffraction, respectively. The film composition was confirmed to be close to Ni45%Fe55% by EDX analysis. The static magnetic properties of a square sample were measured in an SHB Instruments hysteresis loop tracer/H20849model: Mesa 200HF /H20850. The use of a square sample mini- mized the effects of shape anisotropy. The complex perme-ability was measured using a wide band /H208491 MHz–9 GHz /H20850 Ryowa permeameter /H20849model: PMM-9G1 /H20850, operating on the vector network analyzer FMR technique /H20849VNA-FMR /H20850. Thefilm thickness was determined using a dual beam focused ion beam /H20849FIB /H20850/H20849model: FEI Nova 600 /H20850, as shown in Fig. 1. The figure shows the thickness of the Ti/Cu seed layer to beapproximately 29 nm Ti and 46 nm Cu with the NiFe elec-trodeposited thin film being approximately 390nm. III. THEORETICAL FRAMEWORK FOR INDUCTOR DESIGN When designing a thin film inductor core, the general premise is to find an optimum film thickness. One must bal-ance the need for maximum inductance enhancement on theone hand and minimum eddy current loss on the other. In-ductance is proportional to the thickness of the magnetic coreas shown in the following equation: 5 L=/H9262N2tlcore lm, /H208493/H20850 where Lis the inductance in Henrys /H20849H/H20850,tis the thickness of the core in metres /H20849m/H20850,/H9262is the permeability of the core material /H20849H/m /H20850,Nis the number of turns of the inductor, lcore is the core length /H20849m/H20850, and lmis the magnetic path length /H20849m/H20850. By incorporating a ferromagnetic core material into an inte-grated inductor, an inductance enhancement is achieved overa simple air core inductor. Coreless /H20849air core /H20850inductors are popular for many applications in the rf range because of theirhigh efficiencies at low power levels. These inductor coresperformed quite well with high Q values /H20849/H1101160/H20850in the GHz frequency range without the need for a ferromagnetic core; however, due to their low energy density, air core inductorslack suitability for power applications. For these applicationsferromagnetic cores are needed to increase the inductance ona given area footprint. In powering integrated circuits, mono-lithic inductors will generally be incorporated into buck con-verter or similar topologies. FIG. 1. Film thickness determined using a dual beam FIB. From the bottom up, the film layers are Si, Ti, Cu, NiFe, and Pt /H20849Pt was only deposited locally prior to FIB milling to ensure accurate layer thickness measurements /H20850.033907-2 Godsell et al. J. Appl. Phys. 107, 033907 /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: 147.188.128.74 On: Wed, 03 Jun 2015 09:06:26The duty cycle of such a buck converter is given by D=Vo Vin=Ton Ton+Toff, /H208494/H20850 where Dis the duty cycle, Vois the output voltage for the buck converter, Vinis the input voltage to the buck converter, Tonis the on-time of the semiconductor switch in the con- verter, and Toffis the off-time of the switch. The required inductance to maintain the output voltage for a given ripplecurrent across the inductor is dependent on the switching rateof the converter through Eq. /H208495/H20850as shown below L=V oToff /H9004i, /H208495/H20850 where /H9004iis the ripple current across the inductor. To de- crease the value of inductance required to meet the needs ofthe integrated buck converter, the value of T offmust be de- creased as both Voand/H9004iare fixed as constants by the ap- plication. The duty cycle too is fixed by the required outputvoltage according to Eq. /H208494/H20850. An increase in the switching frequency of the buck converter will therefore be required. The inductance in henrys /H20849H/H20850is related to the physical size of the inductor as shown by Eq. /H208493/H20850. Increasing the switching frequency to decrease the physical size of an inte-grated passive magnetic component will bring with it an in-crease in the switching related loss mechanisms within thecore such as eddy current losses. The cutoff frequency asshown in Eq. /H208496/H20850is the frequency at which the skin depth is equal to the film thickness. f c=/H9267 t2/H9262r/H9262o/H9266, /H208496/H20850 where /H9267is the resistivity, /H9262ois the permeability of free space, and/H9262ris the relative permeability. By targeting a particular operating frequency and using the following material prop-erties of Ni 45Fe55,/H9267=45/H9262/H9024cm and /H9262r=0.9 t−0.5for a thin film, Eq. /H208496/H20850can be rearranged as shown in Eq. /H208497/H20850. The relation /H9262r=0.9 t−0.5is an empirically derived relationship for Ni45Fe55thin films as previously reported.5 t=/H208813/H20875/H9267 /H208490.9/H20850fc/H92620/H9266/H208762 . /H208497/H20850 Equation /H208497/H20850can then be used to give the optimum film thickness at a particular operating frequency. For this study,an operating frequency of 500 MHz was targeted, whichyielded a desired film thickness of 400 nm. This thicknessrepresents an acceptable compromise core thickness betweeninductance enhancement and eddy current loss. Hence, aNi 45Fe55film with an average thickness of /H11011390 nm was electrodeposited using the technique described earlier.IV. RESULTS AND DISCUSSION The as measured static magnetic properties of the Ni45Fe55thin film are listed in Table I, while a typical hys- teresis loop measurement for the easy and hard axes are dis-played in Fig. 2. FMR occurs when the frequency of an applied micro- wave field matches the characteristic precessional frequencyof the atomic magnetic spins within the film. Both the peakfrequency /H20849f p/H20850of the imaginary component of the complex permeability spectrum /H20849/H9262r/H11033/H20850and the zero crossing frequency /H20849fo/H20850of the real component of the complex permeability spec- trum /H20849/H9262r/H11032/H20850are generally considered to be a good estimate of the FMR frequency of the film /H20849as shown in Fig. 3/H20850.A s evident in Fig. 3and again later in the analysis in Fig. 6,a clear FMR peak is present across all the bias fields exam-ined. The VNA-FMR technique was employed in the charac- terization of the permeability spectrum. This technique al-lows for operation over extended ac frequencies concurrentwith the application of orthogonal static longitudinal biasfields. The calculation of complex permeability is hence en-abled by the measurement of the standard microwaveS-parameters on the VNA. The ac field is applied along thehard axis, while a range of static bias fields are applied in thesame plane but perpendicular to the ac field, and parallel tothe easy axis /H20849see Fig. 4/H20850. By incrementally increasing the static magnetic field strength parallel to the easy axis whilesweeping the ac field in the plane of the hard axis, the FMRfrequency is observed to increase in line with the static fieldas illustrated in Figs. 5and6. TABLE I. Ni45Fe55thin film properties. Hc easy axis /H20849A/m /H20850Hc hard axis /H20849A/m /H20850Hk /H20849A/m /H20850Bs /H20849T/H20850Thickness /H20849nm/H20850 232 192 674 1.6 390 FIG. 2. /H20849Color online /H20850Hysteresis loop measurement for Ni45Fe55thin film sample as measured on SHB hysteresis loop tracer. FIG. 3. /H20849Color online /H20850Typical graph of relative complex permeability for a 9950 A/m /H20849/H11011125 Oe /H20850bias field applied along the easy axis of the film.033907-3 Godsell et al. J. Appl. Phys. 107, 033907 /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: 147.188.128.74 On: Wed, 03 Jun 2015 09:06:26The increase in the FMR frequency with field strength is due to the increased precessional frequency of the spins withlarger externally applied field strengths consistent with thegyromagnetic effect. This effect describes how an increase inapplied field strength will act to raise the frequency of pre-cession, but in the theoretical undamped case will not act tobring the moment and field into parallel alignment. Hencethe change in /H9262/H20849moment of the atoms /H20850is perpendicular to both/H9262and to B/H20849magnetic field /H20850. Rather than turning /H9262 toward B, the magnetic field causes the direction of /H9262to precess around B, the larger the value of Bthen the greater the speed of the precession. In Sec. IV A below the Kittelequation together with the measured FMR frequencies areutilized to determine the film’s g-factor. A. Spectroscopic splitting factor „g-factor … Apart from magnetic x-ray dichorism and magnetic neu- tron scattering, measurement of the g-factor is the only ex- perimental method which provides information on the sepa-ration of the contributions from the orbital and spin momentsto the total magnetic moment. 12Due to the innate difficulties associated with the calculation of Landé g-factors, an experi- mental approach to their determination is usually adopted.Since /H9253from Eq. /H208492/H20850can be shown to be equal to /H9253=g/H9262B//H6036, /H208498/H20850 where gis the g-factor, /H9262Bis Boltzmann’s constant, and /H6036is h/2/H9266where his Planck’s constant. The g-factor can subse- quently be determined from the experimental results throughan isolation of the gyromagnetic ratio /H9253. By rewriting Eq. /H208492/H20850 in the form of the equation of a straight line /H20851as shown in Eq. /H208499/H20850/H20852and plotting the square of the experimentally measured FMR frequencies against the applied dc bias fields /H20849HBias/H20850/H20849as shown in Fig. 7/H20850, the value of gcan be determined from the slope of the resulting line. fFMR2=/H92532/H926202Ms 4/H92662HBias+/H92532/H926202MsHk 4/H92662. /H208499/H20850 The square of the FMR frequencies can hence be exploited to identify the g-factor of the material in the as deposited thin film. The experimentally determined values of gusing this approach are shown in Table IIbelow. The slight difference between the g-factor calculated from the fpandfomay be accounted for by the variation in fpandfo, which in turn has been reportedly attributed to the possible existence of higherorder magnetostatic modes within the film. 10Theg-factor is, to a first order approximation, expected to range from 1 to 2. FIG. 4. /H20849Color online /H20850Schematic representation of the experimental setup in a commercial Ryowa PMM-9G1. The sample is slid into the detection coil/H20849into the plane of the page /H20850which is situated within the rf cavity which in turn is positioned within the center of a dc field coil. The field coil is onesingle coil and slides over the rf cavity and the sample after the sample isloaded. FIG. 5. /H20849Color online /H20850/H9262/H11032vs frequency for a range of bias fields applied along the film’s easy axis. FIG. 6. /H20849Color online /H20850/H9262/H11033vs frequency for a range of bias fields applied along the film’s easy axis. FIG. 7. /H20849Color online /H20850Graph showing square of FMR frequency vs applied bias field, squares and dots are from the measured values of foand fp respectively, the solid line is the best fit line to the experimentally derived data points.033907-4 Godsell et al. J. Appl. Phys. 107, 033907 /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: 147.188.128.74 On: Wed, 03 Jun 2015 09:06:26When there is no resultant orbital contribution to the overall moment g=2, whereas g=1 when there is no resultant spin contribution to moment /H20849i.e., the spins cancel /H20850.g-factors greater than 2 exist due to the fact that in a crystal lattice,atoms interact through the crystalline field, which in turnquench the orbital moments. This quenching, however, canbe seen to slightly lessen when spin orbit interactions areconsidered as the spins may carry an amount of orbital mo-ment with them. Hence, if parallel orientations of the spinand orbital moments are favored by the sign of the spin orbitinteraction, then gwill exceed 2 and the total magnetic mo- ment will be greater than that of spin alone. After the experi-mental measurement of the g-factor, all of the relevant ma- terial constants have been identified, allowing for theeffective dimensionless damping constant /H20849 /H9251/H20850to be deter- mined using the approach as outlined in the subsequent sec- tion. B. Dimensionless effective damping constant /H9251 With the identification of the effective damping constant /H9251, a material’s dynamic high frequency response can be fully predicted allowing for future passive magnetic circuit com-ponent’s design, prototyping and analysis. The FMR peak inthe /H20849 /H9262/H11033/H20850spectrum /H20849see Fig. 6/H20850represents a loss within the material due to the conversion of the energy associated with the oscillatory motion of the electron spins to heat within thematerial. An analysis of the material’s damping is used toquantify this effect; if there is zero damping then the mag-netization will never align with the applied field and willsimply precess around the applied field direction. Converselyif the damping is very large, then the magnetization willalign itself with the applied field; however, this will occurrelatively slowly. Between the two extremes lies criticaldamping which results in the fastest permissible magneticswitching rate. In this study, the effective dimensionless damping pa- rameter /H9251is identified by fitting a simulated spectrum, which is derived from a coupled LLG and eddy current model tothe measured microwave permeability spectrum. Eddy cur-rent effects often complicate the analysis of the FMR phe-nomenon; however, their effects are included in the effectivedamping parameter reported here in Fig. 11. This is evident in a broader tail on the lower frequency side of the FMRpeak on the /H9262/H11033curve as opposed to the sharper tail on the higher frequency side of FMR peak as apparent in Fig. 6.I n general, due to eddy current complications of the spectrum,the net cumulative effects of eddy currents are often dealtwith by assuming quasistatic conditions or that the currentsare accounted for in the damping parameter of the LLGequation itself. In this study, a built-in LLG and eddy currentsimulator available with the Ryowa permeameter was em-ployed. This software combined the effects of eddy currentsand FMR in a method similar to that previously reported, 13,14 which simultaneously solve Ampère’s /H20851Eq. /H2084910/H20850/H20852and Fara- day’s laws /H20851Eq. /H2084911/H20850/H20852with the LLG Eq. /H208491/H20850 /H11612/H11003H/H6023eddy /H11015/H9268E, /H2084910/H20850 /H11612/H11003E/H6023=−/H92620/H20875/H11509H/H6023applied /H11509t+/H11509M/H6023 /H11509t/H20876. /H2084911/H20850 The following as measured input parameters were used in the simulation: film thickness=390 nm, Ms=1.65 T, Hk =674.2 A /m, resistivity /H20849/H9267/H20850=45/H9262/H9024cm,g=2.128, while /H9251 was allowed to vary as the free parameter. Figures 8–10 show an example fitting of measured data sets to simulatedspectra /H20849critical damping curves are also included as dis- cussed later /H20850, while Fig. 11shows the both the value of /H9251and that of /H9251critical resulting from a range of applied longitudinal dc bias fields. The good fitting of the simulated to the measured spectra as illustrated in Figs. 8–10validates the assumption that FMR and eddy current effects are the dominant loss mecha-nisms. Any minor deviations between the spectra is likelyattributed to such effects as, magnetic inhomogeneities, two-magnon contributions, scattering of the uniform precessionmode, and various local FMR frequencies, etc. While theseparasitic effects are relatively small in comparison to theTABLE II. Experimentally determined g-factors based on fpandfo. g-factor extracted from fpdatag-factor extracted from fodataAverage g-factorAverage /H9253 2.154 2.102 2.128 1.87 /H110031011 FIG. 8. /H20849Color online /H20850Measured and simulated complex permeability spec- tra for 9947 A/m /H20849/H11011125 Oe /H20850bias field applied. FIG. 9. /H20849Color online /H20850Measured and simulated complex permeability spec- tra for 13 925 A/m /H20849/H11011175 Oe /H20850bias field applied.033907-5 Godsell et al. J. Appl. Phys. 107, 033907 /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: 147.188.128.74 On: Wed, 03 Jun 2015 09:06:26dominant loss mechanisms, their effects are likely to contrib- ute toward the slight line broadening as illustrated in Figs.8–10below. By running the experiments with relatively large applied dc bias fields, the model was seen to consistently fit quitewell with the measured permeability spectra. It was, how-ever, noted that at lower bias fields the modeled and mea-sured spectra were seen to be divergent regardless of thevalue of /H9251. While the reasons for this lack of agreement between simulated and measure spectra remain unclear, itmay be due to changes in film anisotropy with bias field or itmay possibly be attributed to very large eddy current effectswhich are reported to significantly broaden the line widths,particularly for devices with thick Fe elements. 15However these large eddy current effects became progressivelydamped out at larger applied bias fields which could be simi-lar to a high bias scenario in integrated inductors for powerapplications. The classical eddy current power loss equation 16in W /cm3is given by P=/H92662t2B2f2 /H9267/H208496/H110031016/H20850, /H2084912/H20850 where tis the thickness in centimeters, Bis the flux ampli- tude at the surface in gauss, fis the frequency in hertz, and /H9267 is the resistivity in ohm centimeters. It can be seen from Eq./H2084912/H20850above that eddy current power loss is proportional to B2. Bin turn is related to the external bias field strength through the permeability /H9262, which will itself decrease with increased dc bias fields applied as illustrated in Fig. 5. The latter is attributed to the fact that when the bias field is aligned withthe easy axis then the permeability can be written in firstorder approximation as /H9262r=/H20851Ms//H20849Hk+Hbias/H20850/H20852+1, as B=/H9262H =/H9262r/H9262oH, increasing Hbiasmeans that Bwill approach a roughly constant value. Figure 2shows that the films are effectively saturated at a 2000A/m applied bias field. Assuch, all FMR experiments on the films are conducted withapplied bias field strengths of several times that required tosaturate the film. Eddy current contributions to the effectivedamping parameter will therefore decrease only marginallyover the range of bias fields studied. Hence, the slight varia-tion in /H9251as shown in Fig. 10cannot be quantifiably attrib- uted to any physical mechanisms, as the slight upward trendwas small enough to be attributed to the inherent error asso-ciated with the experimental procedure. Figures 5and6show that within the range of externally applied bias fields studied, the film’s permeability remainsalmost constant up to the point where the effects of FMRdominate. As previously stated, the value of /H9251is very impor- tant in modeling the high frequency dynamic response of thefilm as used in monolithic magnetic applications. Subsequentto the determination of the phenomenological value of /H9251for the Ni 45Fe55film, the resulting numeric value can be com- pared to the film’s theoretical optimum critical damping/H20849 /H9251critical /H20850value. It has previously been reported that the damping constants of thin magnetic films will appreciably change the bandwidth of the permeability spectra. For ex-ample, if the damping constant of CoFeN is increased five-fold from 0.02 to 0.1 then the spectral bandwidth of thepermeability is also reported to increase fivefold. 17If the film is critically damped then this traditionally represents the op-timum condition for high speed switching, where the systemwill reach equilibrium in the shortest possible time. The phe-nomenologically derived critical damping condition for a fer-romagnetic thin film which results in the fastest permissibleswitching speed has previously been reported. Assuming auniaxial anisotropy and small signal excitations, the criticaldamping condition is given by Eq. /H2084913/H20850below. /H9251critical =2/H208811 /H92730, /H2084913/H20850 where /H92730=Ms Hk+Hbias/H2084914/H20850 is the relative dc susceptibility10andHbiasis the dc bias field applied along the easy axis. It is known that the damping will affect the manner in which the magnetization will relax toward equilibrium. Forcertain applications such as hard disk magnetic write heads,MRAM, band-stop filters, or read sensors, an increaseddamping may be desirable as it will reduce unwanted oscil-lations associated with ringing which will accompany highspeed switching. For other applications such as notch filters, FIG. 10. /H20849Color online /H20850Measured and simulated complex permeability spectra for 17 904 A/m /H20849/H11011225 Oe /H20850bias field applied. FIG. 11. /H20849Color online /H20850Effective dimensionless damping parameter plotted against applied bias field for the thin film, both experimentally derived andthe theoretical critically damped values are shown.033907-6 Godsell et al. J. Appl. Phys. 107, 033907 /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: 147.188.128.74 On: Wed, 03 Jun 2015 09:06:26a small damping is required. For the application under inves- tigation /H20849planar inductors and transformers /H20850where operating frequencies may potentially reach into the gigahertz fre-quency range, a rudimentary analysis of the effects of damp-ing is warranted. For the NiFe film, at zero applied bias field the value of critical damping is calculated to be 0.0446. Here, /H9251critical is calculated to range between 0.179 for an applied bias field of9947 A/m up to 0.237 for an applied bias field of 17904 A/m;simulations of the critically damped film’s real and imagi-nary permeability spectra are included in Figs. 8–10. The measured permeability spectra are overlaid onto the simu-lated spectra for comparative purposes. It is seen that the filmis in a strongly underdamped state, in fact since the value of /H9251is less than 0.18 for the plated thin film over all of the bias fields examined, the film remains at all times in the under-damped state. The latter is not uncommon and in fact, mostmagnetic thin films are reported to show underdampedbehavior. 17In the underdamped case a ferromagnetic film will experience a classical underdamped ringing in the timedomain upon switching. It can be seen in Figs. 8–10that in the underdamped situation the permeability roll-off of thereal part of the permeability spectra /H20849 /H9262r/H11032/H20850occurs at higher frequencies than in the critically damped case. Conversely it can be seen that above a certain frequency the losses in the film /H20849as evident in the /H9262r/H11033curve /H20850reach the largest magnitudes in the underdamped condition. For an in-ductor application both a high permeability roll-off fre-quency and a low film loss are desirable. With this in mindthe underdamped condition appears to be more desirablethan the critically damped case provided the device only op-erates within the range of frequencies where the device’s realpermeability /H20849 /H9262r/H11032/H20850remains constant. For example, in the case of the 9947 A/m applied bias field at frequencies of up to 1.5 GHz the underdamped condition provides a higher real per-meability /H20849 /H9262r/H11032/H20850and a lower imaginary permeability /H20849/H9262r/H11033/H20850 than the critically damped cases. For a power inductor application where a lower power loss is more important than a faster switching speed, theunderdamped condition will be more desirable than the criti-cally damped one, as the lower the damping the lower thelosses in the system up to a critical frequency /H20849f critical /H20850as shown in Fig. 8. The critical damping value will increase as the in-plane net magnetic field along the easy axis is in-creased, hence, a more underdamped condition may beachieved by changing the external magnetic field strength.By increasing the net in-plane magnetic field strengththrough, for example, the use of shape or induced anisotropyeffects, the possibility exists to increase the value of criticaldamping within the film, thereby leaving the film in a pro-gressively more underdamped state. This can be explainedthrough an examination of Eqs. /H2084913/H20850and /H2084914/H20850, the anisotropy field of a sample can be modified by changing the shape ofthe material. This in turn can change the critical dampingvalue of the sample, in the lower damping case /H20849more under- damped /H20850, the real part of the complex permeability spectra will in general display a higher roll-off frequency as illus-trated in Fig. 5. From Fig. 11it would appear that /H9251experimental is relatively invariant with bias field applied. The latterwould imply that the changes in the permeability spectra associated with H/H20849HbiasorHk/H20850applied are primarily attrib- uted to the reduction in /H9273o/H20849and hence /H9262/H20850with increasing H as outlined in Eq. /H2084914/H20850and also to the increase in FMR frequency with Has outlined in Eq. /H208492/H20850. Control of these bias/anisotropy field effects will allow for a tailoring of thefilm’s frequency domain permeability response to achieveimproved characteristics for the final device at elevated fre-quencies. Further work may focus on optimizing the films to achieve the most desirable damped state, one which has bothlower film losses and a constant value of /H9262r/H11032over the oper- ating frequency range of the device. This has been shown tobe characteristic of the underdamped case at low to subgiga-hertz frequencies. The design optimization of a film’s damp-ing state is likely to become of increased importance in thefuture for monolithic inductor applications as ferromagneticmaterials are incorporated into devices who’s frequencybandwidth will increasingly extend up into the gigahertzrange. V. CONCLUSIONS In this work the g-factor, complex permeability spectra and the effective damping constant have been reported for aNi 45Fe55thin film electrodeposited onto Si/Ti/Cu substrates. The simulated contribution of the eddy current effects towardthe dynamic behavior of the film, coupled with the LLGequation were employed in correlating simulated and mea-sured results. This correlation was in turn utilized to identifythe key material parameter dictating the system’s spin dy-namics, principally the effective damping parameter /H9251. The losses due to eddy currents were shown to be relatively in-variant within the range of bias fields of interest. The filmdisplayed FMR frequencies above 2 GHz highlighting thismaterial’s potential for application in future integrated induc-tors operating up well into the hundreds of megahertz fre-quency range. A discussion on the damping present in thefilm postulated that an underdamped film state is likely to bethe more desirable state for a monolithic inductor or trans-former application. ACKNOWLEDGMENTS This work has been financially supported by Science Foundation Ireland, Grant No. SFI–PI–06/IN.1/I98. Enter-prise Ireland Grant for the competence centre in microelec-tronics /H20849MCCI /H20850No. PHMIS CC-2008-2403 is further ac- knowledged. 1N. Wang, T. O’Donnell, R. Meere, F. M. F. Rhen, S. Roy, and C. O’Mathuna, IEEE Trans. Magn. 44,4 0 9 6 /H208492008 /H20850. 2S. Prabhakaran, T. O’Donnell, C. R. Sullivan, M. Brunet, S. Roy, and C. O’Mathuna, J. Magn. Magn. Mater. 290–291 , 1343 /H208492005 /H20850. 3N. Wang, T. O’Donnell, S. Roy, S. Kulkarni, P. Mccloskey, and C. O’Mathuna, IEEE Trans. Magn. 43,2 7 1 9 /H208492007 /H20850. 4P. C. Andricacos and N. Robertson, IBM J. Res. Dev. 42,6 7 1 /H208491998 /H20850. 5T. O’Donnell, N. Wang, S. Kulkarni, R. Meere, F. M. F. Rhen, S. Roy, S. C. O’Mathuna, J. Magn. Magn. Mater. /H20849in press /H20850. 6Y. Zhuang, M. Vroubel, B. Rejaei, and J. N. Burghartz, J. Appl. Phys. 97, 10N305 /H208492005 /H20850. 7N. Inaba, H. Asanuma, S. Igarashi, S. Mori, F. Kirino, K. Koike, and H. Morita, IEEE Trans. Magn. 42, 2372 /H208492006 /H20850.033907-7 Godsell et al. J. Appl. Phys. 107, 033907 /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: 147.188.128.74 On: Wed, 03 Jun 2015 09:06:268G. Schrom, P. Hazucha, J. Hahn, D. S. Gardner, B. A. Bloechel, G. Dermer, S. G. Narendra, T. Karnik, and V. De, Proceedings of the IEEEPower Electronics Specialists Conference /H20849PESC ‘04 /H20850, Aachen, Germany, 2004, in PESC Record, p. 4702. 9K. Seemann, H. Leiste, and A. Kovas, J. Magn. Magn. Mater. 320, 1952 /H208492008 /H20850. 10N. X. Sun, S. X. Wang, T. J. Silva, and A. B. Kos, IEEE Trans. Magn. 38, 146 /H208492002 /H20850. 11S. Roy, A. Connell, M. Ludwig, N. Wang, T. O’Donnell, M. Brunet, P. McCloskey, C. O’Mathúna, A. Barman, and R. J. Hicken, J. Magn. Magn. Mater. 290–291 ,1 5 2 4 /H208492005 /H20850.12J. Pelzl, R. Mechenstock, D. Spoddig, F. Schreiber, J. Pflaum, and Z. Frait, J. Phys.: Condens. Matter 15, S451 /H208492003 /H20850. 13G. Hrkac, T. Schrefl, O. Ertl, D. Suess, M. Kirschner, F. Dorfbauer, and J. Fidler, IEEE Trans. Magn. 41, 3097 /H208492005 /H20850. 14L. Torres, L. Lopez-Diaz, E. Martinez, and O. Alejos, IEEE Trans. Magn. 39, 2498 /H208492003 /H20850. 15Y. V. Khivintsev, B. K. Kunar, I. Harward, R. E. Camley, and Z. Celinski, J. Appl. Phys. 99, 08P512 /H208492006 /H20850. 16B. D. Cullity and C. D. Graham, Introduction to Magnetic Materials ,2 n d ed. /H20849Wiley, New York, 2009 /H20850,p .4 4 3 . 17N. X. Sun and S. X. Wang, IEEE Trans. Magn. 40, 253 /H208492004 /H20850.033907-8 Godsell et al. J. Appl. Phys. 107, 033907 /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: 147.188.128.74 On: Wed, 03 Jun 2015 09:06:26

1.1854333.pdf

Spin-transfer-induced magnetic excitation: The role of spin-pumping induced damping Jonathan Z. Sun, Barbaros Özyilmaz, Wenyu Chen, Maxim Tsoi, and Andrew D. Kent Citation: Journal of Applied Physics 97, 10C714 (2005); doi: 10.1063/1.1854333 View online: http://dx.doi.org/10.1063/1.1854333 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Remanent states and magnetization reversal of nanopatterned spin-valve elements using off-axis electron holography J. Appl. Phys. 105, 07D517 (2009); 10.1063/1.3058626 Linear magnetic field response spin valve with perpendicular anisotropy ferromagnet layer J. Appl. Phys. 103, 07E906 (2008); 10.1063/1.2830968 Spin-transfer-induced magnetic domain formation J. Appl. Phys. 100, 073906 (2006); 10.1063/1.2357002 Spin-transfer-induced excitations in bilayer magnetic nanopillars at high fields: The effects of contact layers J. Appl. Phys. 99, 08G511 (2006); 10.1063/1.2166596 Quantitative studies of spin-momentum-transfer-induced excitations in Co/Cu multilayer films using point-contact spectroscopy Appl. Phys. Lett. 82, 1260 (2003); 10.1063/1.1556168 [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: 131.94.16.10 On: Sun, 21 Dec 2014 15:05:30Spin-transfer-induced magnetic excitation: The role of spin-pumping induced damping Jonathan Z. Sun IBM T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598 Barbaros Özyilmaz and Wenyu Chen Department of Physics, New York University, New York, New York 10003 Maxim Tsoi Physics Department, University of Texas at Austin, Austin, Texas 78712 Andrew D. Kent Department of Physics, New York University, New York, New York 10003 sPresented on 10 November 2004; published online 5 May 2005 d Spin-transfer-induced magnetic excitation in large magnetic field applied perpendicular to the thin film junction surface reveals both a current threshold Icand a voltage threshold. The current threshold follows the Slonczewski-type of magnetic field dependence fJ. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 s1996 dg. The voltage step at IcisDVwhich appears to scale with the applied field with a prefactor of the order of 2 mB/e, suggesting a threshold to magnetic excitation. Furthermore, experimentally it is observed that DV<IcdR, where dRis the magnetoreistance between the parallel and the antiparallel states. This apparent coincidence can be unified when oneincludes the effect of spin-pumping-related nonlocal damping. The spin-pump damping relatesmagnetic instability threshold I ctodR, producing sdIc/dHddRthat is about 2 mB/e, explaining the origin of the coincidence. © 2005 American Institute of Physics .fDOI: 10.1063/1.1854333 g A simple geometry to study spin-transfer-induced mag- netic excitation in thin film spin valve is to have a largemagnetic field sH@4 pMs, whereMsis the spin-valve free- layer’s magnetization dapplied perpendicularly to the current- perpendicular sCPPdspin-valve thin film surface.1It makes the precession axis symmetric, avoiding complexities relatedto easy-plane and easy-axis shape anisotropy. The large fieldalso makes the magnetic layer’s macrospin precession dy-namics less susceptible to thermal fluctuation. This simplegeometry is ideal for measuring spin-transfer threshold cur-rent’s magnetic field dependence, I csHd, and for quantitative comparison with theories. The zero-temperature threshold current for spin-torque induced magnetic instability, according to the simplestmodel, 2–5can be expressed as IcsHd=s2e/"dsa/hdmsH−4pMsd, s1d where ais Gilbert damping,6h=sI1−I2d/sI1+I2dis the spin- polarization factor for the current, with I1,2being the major- ity and minority spin current, with the spin axis defined bythe fixed layer moment. m=sabtdM sis the magnetic moment of a free layer having aandbin lateral size and tin thickness. The junctions here are of the type4iCuuCouCuuCouCui, where one of the cobalt layers is kept thin susually below 4n m dand the other thick sfixed at 12 nm d. They are called the free layer and the fixed layer, respectively, for their rolein magnetic excitation under spin current. Junctions in thisstudy were fabricated using a stencil substrate approach. 7,8 The full junction stack, from bottom to top, reads i150 Cu u20Ptu10 Cu u2.5 Co u10 Cu u12 Co u250 Cu u10 Pt i, where numbers are thicknesses in nanometers. Experimentally, the I–Vcharacteristics exhibit a step- wise voltage increase sDVdat a threshold current defined as Ic, which has been interpreted either as a consequence of magnetic reversal of the free layer,9or as a result of magnon emission-related magnetic excitation.10 Below we show the observed Icdepends linearly on ap- plied field for H@4pMs<15 kOe for a cobalt free layer. The voltage step height also follows a linear magnetic fielddependence, with a slope ,2 mB/esmB=e"/2mecbeing a Bohr magneton where meis the electron mass d. Interestingly, the two observed thresholds, IcandDV, are related by the relation DV<IcdRwhere dRis the magnetoresistance sMRd change between parallel and antiparallel configurations. Weargue that such a correlation is a natural consequence whenthe free layer is sufficiently thin for its total Gilbert dampingto be dominated by the spin-pump-induced dissipation. A representative I–Vis shown in Fig. 1. A linear back- groundI–Vis subtracted from the measured voltage.Acon- tour plot of this junction’s dV/dIas a function of bias current Iand applied field His shown in Fig. 2, from which the magnetic field dependence of the threshold current I csHdcan be readily determined. The high-field linear dependence es- timated from these data is IcsHd=4.80 310−7sH+31.5 3103dat 5 K. Here Icis in ampere and His in Oersted. The low-field MR for this junction stack is measured using in- plane field geometry to be about 4.2% at 15 K. The voltage step at Icin Fig. 1 depends linearly on H, giving a high-field slope about 4.5 mB/eforT=5 K as shown Fig. 3 sad. This voltage step also corresponds to a resistanceJOURNAL OF APPLIED PHYSICS 97, 10C714 s2005 d 0021-8979/2005/97 ~10!/10C714/3/$22.50 © 2005 American Institute of Physics 97, 10C714-1 [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: 131.94.16.10 On: Sun, 21 Dec 2014 15:05:30change of about 3.0%–3.6% at 5 K for fields between 20 and 50 kOe, slightly below and asymptotically approaching thevalue of low-field MR, as shown in Fig. 3 sbd. This says DV<s2 mB/edHis related to Icin a simple form of DV <IcdR, where dRis the spin-valve’s MR. This similarity between DVandIcdRis not limited to one particular sample or even one particular material’s com-bination. In Table I three sets of experimental results arelisted from recent publications by other groups that show dRsdIc/dHdto scale with field Hwith a slope around 2 mB/e. The linear high-field dependence of IconHis consistent with the spin-transfer model described by Slonczewski2and co-workers, as is summarized by Eq. s1d. The slope of IcsHd contains the total Gilbert damping aas well as the spin po- larization factor h. The presence of a dc voltage accompanying magnetic excitation has been predicted by Berger.13For large ampli- tude excitation in a highly asymmetric magnetic stack wherethe free magnetic layer is much thinner than the thick one,the asymptotic limit of the voltage rise due to precession iswhat we will hereafter call the Berger voltage V b =s"v/2edh. Replacing v=2mBH/"in our case gives Vb =hsmB/edHwhich follows from energy conservation for maintaining the nanomagnet precession at large cone angleu. This dissipation power, when supplied by a transport cur- rent ofI=Ic, gives rise to a voltage difference DVsatisfying IcDV=−dUsud/dt=agmH2sin2u. For maximum spin-wave excitation, u=p/2; hence esDVd=hmBH=eVb. Here g =gmB/"<2mB/"andUsud=−mHcosu. The threshold Vbis independent of the details of mag- netic damping a. The model gives approximately the right field dependence. It, however, does not naturally explain theobserved relationship between the voltage step and the valueofV c, defined as Vc=IcdR. The puzzle is resolved by examining the origin of mag- netic damping a. In a spin valve, the total magnetic damping of the free layer is a combination of the material’s intrinsicdamping a0and the circuit environment-determined spin- pumping dissipation a8.14–16The total a=a0+a8a0.a0, in- dependent of thickness t, gives a dissipation proportional to the volume of the free layer. a8is, in thin free-layer limit, inversely proportional to thickness. For permalloy sNi80Fe20d the crossover thickness for a8to dominate dissipation is about thickness t,10 nm.14Our samples may also fall within the region where the spin-pump-induced damping a8 dominates a0. The field-dependent slope of Vcis understood when one realizes that the same set of spin-dependent con-ductance matrices that determines a8shenceIcdsRef. 16 d also determines dR.14 FIG. 1. The I–Vcharacteristics measured for a 0.05 30.20 mm2junction.A linear background of 0.935 Vis subtracted from data to make threshold behavior clearly visible. The I–Vcharacteristics for applied field values of 20, 30, 40, and 50 kOe are shown. The curvature in I–Vis commonly seen in such CPP junctions, and is believed to be a result of Joule heating. Thearrow shows the direction of current sweep. Data were taken at 5 K. FIG. 2. Contour plots of the junction shown in Fig. 1.The switching currentI csHdcan be readily identified in these plots as the bright peak in dV/dI. For smaller features at larger currents, one explanation is in Ref. 11. FIG. 3. sadThe magnetic field dependence of DVat threshold current in Fig. 1. As the applied magnetic field increases, so does Ic, and DVatIc. The solid line on the bottom is the voltage corresponding to Zeeman energy s2mB/edsH−4pMsdfor comparison. sbdThe apparent resistance change at critical current IcsHddefined as dR/Rwhere dR=DV/Ic. The dashed lines indicate low-field RsHdmeasurement-based MR at 15 K.10C714-2 Sun et al. J. Appl. Phys. 97, 10C714 ~2005 ! [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: 131.94.16.10 On: Sun, 21 Dec 2014 15:05:30To quantitatively calculate Vc=IcdRusing magnetoelec- tronic circuit theory,14–16assume a macrospin suniform pre- cession dynamics dfor the free layer. Consider the structure N1uF1uN2uF2uN3swhereNis a nonmagnetic layer such as Cu andFa magnetic layer such as Co dwhich is contacted either to real reservoirs to the left or right, or to materialswith a high spin-flip probability se.g., Pt dwhich is not part of the spin-active regions. Take a set of typical values for thespin-dependent conductance values for Co–Cu system fromRefs. 17 and 18 and include the interface resistances due toFermi surface mismatching, then in the limit if a thin freelayer is compared to the fixed one we have sdI c/dHddR=2sGeffmixdR/kds1+a0/a8dsmB/ed, s2d whereGeffmixandkare the mixing conductance and a dimen- sionless constant as defined in Ref. 19. For not too asymmet-ric junction stacks, k,1 andG effmixdR,1. For highly asym- metric junctions numerical evaluation is required. Whenincorporating the stack designs listed in Table I and Ref. 7for the special stencil geometry, 20with typical resistance pa- rameters from Refs. 17 and 18 and an a0,0.005, one finds in Table II the calculated sdIc/dHddRthat are within about factor of 2 of experimental values. These agreements show spin pumping can explain why the slope of sdIc/dHddRwould be around 2 mB/efor a mac- rospin model with a complete magnetic reversal at Ic. This shows spin pumping as an important factor in the under-standing of current-induced magnetic excitation and switch-ing. It also demonstrates the ability of magnetoelectronic cir-cuit theory for simultaneous quantitative predictions of themagnetoresistance, the enhanced Gilbert damping, and theswitching current threshold. In conclusion, the reason the junction can simulta- neously satisfy a threshold current relationship I csHdand a threshold voltage relationship IcdR=Vc,Vb,s2mB/edHis that the junction is within the thin free layer limit where the effective magnetic damping is dominated by spin-pumpdamping. The spin-pump damping relates to magnetoresis-tance dRas well as to threshold current Icsthrough a8d, giving rise to an IcdRthat is in the neighborhood ofsmB/edH, similar to the Berger voltage. The final state for I .Icmay be persistent large angle precession, a magnetic reversal, or a combination involving internal magnetic de-grees of freedom beyond macrospin limit; the resulting volt-age step at I cwill still be of the order of smB/edHas long as the free-layer is sufficiently thin for spin-pumping-induced damping to dominate a. ACKNOWLEDGMENTS The authors wish to thank the valuable support from the IBM MRAM team and also thank Arne Brataas for pointingout the importance of spin-pumping-related damping in theprocess, as well as for contributing to the spin-pumping-related numerical calculations. Research at NYU was sup-ported by NSF-FRG Grant No. 0101439 and NSF-DMRGrant No. 0405620, and was conducted under an NYU-IBMjoint study program. 1M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 s1998 d. 2J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 s1996 d. 3J. Z. Sun, J. Magn. Magn. Mater. 202, 157 s1999 d. 4J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 s2000 d. 5J. Z. Sun, Phys. Rev. B 62, 570 s2000 d. 6E. M. Lifshitz and L. P. Pitaevskii, Magnetism sWheaton Exeter, 1981 d, Vol. II, Chap. 7, p. 285. 7J. Z. Sun, D. J. Monsma, M. J. Rooks, and R. H. Koch, Appl. Phys. Lett. 81,2 2 0 2 s2002 d. 8J. Z. Sun, D. J. Monsma, T. S. Kuan, M. J. Rooks, D. W. Abraham, B. Oezyilmaz, A. D. Kent, and R. H. Koch, J. Appl. Phys. 93, 6859 s2003 d. 9B. Özyilmaz, A. D. Kent, D. Monsma, J. Z. Sun, M. J. Rooks, and R. H. Koch, Phys. Rev. Lett. 91, 067203 s2003 d. 10M. Tsoi, J. Z. Sun, M. J. Rooks, R. H. Koch, and S. S. P. Parkin, Phys. Rev. B69, 100406 sRds2004 d. 11B. Özyilmaz, A. D. Kent, M. J. Rooks, and J. Z. Sun, e-print cond-mat/ 0407210. 12S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, M. Rinkoski, C.Perez, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 93, 036601 s2004 d. 13L. Berger, Phys. Rev. B 59, 11465 s1999 d. 14Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 s2002 d. 15A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. 84,2 4 8 1 s2000 d. 16Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 67, 140404 sRds2003 d. 17Q. Yang, P. Holody, R. Loloee, L. L. Henry, W. P. Pratt, Jr., P. A. Schroeder, and J. Bass, Phys. Rev. B 51,3 2 2 6 s1995 d. 18J. Bass and W. P. Pratt, J. Magn. Magn. Mater. 200, 274 s1999 d. 19J. Manschot, A. Brataas, and G. E. W. Bauwer, e-print cond-mat/0403760. 20The stencil opens up after 35 nm of the top Cu layer. In addition, the lip of the stencil contains extended magnetic films.Thus the effective top copperlead thickness would be <35 nm and not the full copper film thickness.TABLE I. A list of measured field dependent slope of Vc=IcdRfrom results in literature. The numbers preceding element names in junction stack indicate layer thicknesses in nanometers. Here the lateral sizes are in nm2,dRis in V,dIc/dHin A/Oe, sdIc/dHddRinmB/e, andTin Kelvin. Junction stack Size dRd Ic/dH sdIc/dHddRT Reference s1di120 Cu u10 Cu u6C u u2.5 Cu u15 Cu u3P tu60 Au i 1003100 0.075 2.90 310−73.76 300 4 s2di10 Cu u3C o u10 Cu u12 Co u300 Cu u10 Pt i 903140 0.03 4.51 310−72.34 4.2 9 s3di80 Cu u40 Co u10 Cu u3N i8 0F e2 0 u2C u u30 Pt i 703130 0.129 2.16 310−74.82 4.2 12 s4di150 Cu u20 Pt u10 Cu u2.5 Co u10 Cu u12 Co u250 Cu u10 Pt i 503200 0.039 4.80 310−73.23 5 This work TABLE II. Spin-pumping theory calculation results. Sample numbers are the same as in Table I. Sample dRsVd a8 sdIc/dHddRinsmB/ed 2 0.03 0.02 7 4 0.04 0.02 710C714-3 Sun et al. J. Appl. Phys. 97, 10C714 ~2005 ! [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: 131.94.16.10 On: Sun, 21 Dec 2014 15:05:30

1.2747171.pdf

Ferromagnetic resonance force microscopy on a thin permalloy film E. Nazaretski, I. Martin, R. Movshovich, D. V. Pelekhov, P. C. Hammel, M. Zalalutdinov, J. W. Baldwin, B. Houston, and T. Mewes Citation: Applied Physics Letters 90, 234105 (2007); doi: 10.1063/1.2747171 View online: http://dx.doi.org/10.1063/1.2747171 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/90/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Localized ferromagnetic resonance force microscopy in Permalloy-cobalt films J. Appl. Phys. 106, 046103 (2009); 10.1063/1.3204029 Temperature-dependent magnetic resonance force microscopy studies of a thin Permalloy film J. Appl. Phys. 101, 074905 (2007); 10.1063/1.2715761 Effect of conformal roughness on ferromagnetic resonance linewidth in thin Permalloy films J. Appl. Phys. 97, 10A721 (2005); 10.1063/1.1860271 Intrinsic damping and intentional ferromagnetic resonance broadening in thin Permalloy films J. Appl. Phys. 93, 6903 (2003); 10.1063/1.1543884 Ferromagnetic resonance frequency of a thin Mo-permalloy film J. Appl. Phys. 87, 5846 (2000); 10.1063/1.372542 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: 128.59.222.12 On: Fri, 22 Aug 2014 03:37:09Ferromagnetic resonance force microscopy on a thin permalloy film E. Nazaretski,a/H20850I. Martin, and R. Movshovich Los Alamos National Laboratory, Los Alamos, New Mexico 87545 D. V. Pelekhov and P . C. Hammel Department of Physics, Ohio State University, Columbus, Ohio 43210 M. Zalalutdinov SFA Inc., Crofton, Maryland 21114 J. W. Baldwin and B. Houston Naval Research Laboratory, Washington, D.C. 20375 T . Mewes Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama 35487 /H20849Received 17 November 2006; accepted 11 May 2007; published online 7 June 2007 /H20850 Ferromagnetic resonance force microscopy /H20849FMRFM /H20850offers a means of performing local ferromagnetic resonance. The authors have studied the evolution of the FMRFM force spectra in acontinuous 50 nm thick permalloy film as a function of probe-film distance and performednumerical simulations of the intensity of the FMRFM probe-film interaction force, accounting forthe presence of the localized strongly nonuniform magnetic field of the FMRFM probe magnet.Excellent agreement between the experimental data and the simulation results provides insight intothe mechanism of FMR mode excitation in a FMRFM experiment. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2747171 /H20852 Magnetic resonance force microscopy /H20849MRFM /H20850offers a very high sensitivity approach to detection of magnetic reso-nance and has demonstrated three dimensional imaging withexcellent spatial resolution. Proposed by Sidles, 1it has been used for the detection of electron spin resonance,2nuclear magnetic resonance;3recently Rugar et al. reported detection of a force signal originating from a single electron spin,4 emphatically demonstrating MRFM sensitivity. Incorporat-ing basic elements of magnetic resonance imaging /H20849MRI /H20850, MRFM can provide much higher spatial resolution than con-ventional MRI. Electron spin density images with microme-ter scale resolution in an arbitrarily shaped sample can bedeconvolved from MRFM spatial force maps. 5MRFM im- age deconvolution requires a thorough understanding of theunderlying interaction between the MRFM probe and theobject imaged. This deconvolution process, analyzed for thecase of noninteracting spins in a paramagnetic sample isgiven in Ref. 6. Recently, ferromagnetic resonance /H20849FMR /H20850 has been detected by MRFM in yttrium iron garnet /H20849YIG/H20850 bar, 7YIG dot,8,9YIG film,10and permalloy dots.11The role of the probe magnet in FMRFM is dual: it both perturbs theFMR modes and detects the force signal. However, in Refs. 7–9the FMR modes were only weakly modified by the tip field. In this letter we focus on the regime when the effect of the tip is nonperturbative: the field inhomogeneity due to thetip field strongly modifies the resonance modes as well asleads to the formation of the local resonance under the tip. We report FMRFM spectra from a continuous, 50 nm thickpermalloy film, performed with a cantilever with a nearlyspherical micron-size magnetic tip. We report the evolutionof the FMRFM spectra as a function of the tip-sample spac-ing and propose a model which describes the observed behavior. The cantilever is mounted on top of a double scanning stage, comprised of a three dimensional attocube scanner 12 for coarse scanning and a piezotube for fine scanning. The optical feedback control of the attocube allows to positionand move the cantilever stage with an accuracy better than250 nm. The microwave power is used to manipulate thesample magnetization and is generated by the Giga-tronics12000A synthesizer at a frequency /H9275rf/2/H9266=9.55 GHz, 79 mW of power, and amplitude modulated with a modula-tion depth of 70%. It is fed into a strip line resonator with thebroad resonant characteristics and allows to record FMRFMspectra in the frequency range between 9 and 11.5 GHz. Amore detailed description of the microscope can be foundelsewhere. 13Contrary to MRFM experiments where the state-of-the-art ultrasensitive cantilevers are required toachieve high sensitivity 4in FMRFM due to much stronger signals commercially available cantilevers can be used. Weuse a silicon nitride cantilever with a fundamental resonantfrequency /H9275c/2/H9266/H110158.06 kHz and a spring constant k/H1101110 mN/m. The magnetic tip is a 2.4 /H9262m diameter spheri- cal Nd 2Fe14B particle14shown in the inset to Fig. 2.W e removed the original tip of the cantilever by focused ionmilling and manually glued the magnetic sphere to the can-tilever with Stycast 1266 epoxy in the presence of an align-ing magnetic field of a few kiloersteds. The 50 nm thickpermalloy film was deposited on a 20 nm thick Ti adhesionlayer on a 100 /H9262m thick silicon wafer. The permalloy was capped with a protective of 20 nm Ti layer. An approxi-mately 2 /H110032m m 2sample was glued to the strip line resona- tor and the film plane was oriented perpendicular to the di-rection of the external magnetic field H ext. In Fig. 1, we show themagnitude of the FMRFM spectra recorded as a function of the probe-sample distance at a constant temperaturea/H20850Electronic mail: evgnaz@lanl.govAPPLIED PHYSICS LETTERS 90, 234105 /H208492007 /H20850 0003-6951/2007/90 /H2084923/H20850/234105/3/$23.00 © 2007 American Institute of Physics 90, 234105-1 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: 128.59.222.12 On: Fri, 22 Aug 2014 03:37:09T=11.000±0.005 K. Each spectrum displays two distinctive features: the main resonance signal, which occurs at approxi-mately H ext/H1101514.54 kOe, and the secondary resonance structure at lower fields. The structure of the FMRFM signalis reminiscent of those observed in permalloy micro-structures. 11The intensities of both features decrease as the probe magnet is moved away from the surface of the sample.It is important to note that retraction of the probe does notchange the position of the main resonance peak significantly,and at the same time the width of the secondary featurechanges substantially. The quality factor Qof the cantilever decreases from /H1101111 000 to /H110116000 as the probe approaches the sample. The change in Qis due to tip-sample interactions /H20849other than magnetic resonance /H20850and is consistent with pre- vious reports. 15–17We measured Qby two methods /H20849ring- down technique and swept frequency through resonance /H20850at each probe-film spacing and subsequently calculated themagnitude of the force signal acting on the cantilever, Fig. 1. In general, the force Fdetected in a MRFM experiment is a convolution of /H9254m/H20849r,t/H20850/H20849the change in sample magneti- zation due to rf manipulation /H20850with the field gradient /H11612Htip/H20849r/H20850of the magnetic tip. The force is given by the fol- lowing volume integral: F=/H20848Vs/H20849/H9254m/H20849r,t/H20850·/H11612/H20850Htip/H20849r/H20850dr.I no u r experiments, the spherical shape of the probe magnet allows analytical calculation of its magnetic field profile Htip/H20849r/H20850 /H20849Ref. 18/H20850and is used to provide precise knowledge of the magnetization term /H9254m/H20849r,t/H20850needed to interpret FMRFM spectra correctly. The total magnetic field inside the sample is Htot=Hext +Htip+Hd, where Hextis a uniform external magnetic field, Htipis a nonuniform magnetic field of the probe magnet, and Hdis the demagnetizing field. The exact spatial profile of Htotdepends on the total magnetic moment of the probe magnet, the probe-film spacing, and the relative orientationofH extwith respect to the orientation of probe magnet mag- netization /H20849in our case they are parallel /H20850. The well defined shape of the magnetic tip allows us to schematically dividethe sample into two regions according to the magnitude of thezˆcomponent of the total magnetic field H totz. The first,region I, is a circular region directly under the magnetic tip where its field is significant and positive. The area of thisregion is determined by the distance between the center ofthe probe tip and the sample. In region II the field of theprobe is much weaker and negative and its area encompassesthe remaining sample area of /H110112/H110032m m 2. The schematics of two regions is shown in the inset to Fig. 2. For a conventional FMR experiment the expected reso- nance field for the uniform FMR mode is Hresu=/H9275rf//H9253 +4/H9266Ms, where we neglect the anisotropy contribution,.17,19 For our experimental parameters /H9253/2/H9266=2.89±0.05 GHz/kOe /H20849Ref. 17/H20850and/H9275rf/2/H9266=9.55 GHz, we obtain the value of Hresu=14.6 kOe, which agrees within the error with the observed resonance field for the main peak /H20849dotted line in Fig.1/H20850; we thereby attribute it to the resonance originating from region II of the sample and representing its large area.The main resonance peak in Fig. 1can be understood as the fundamental FMR mode observed in conventional FMRexperiments, 20,21modified by the tip field. Analytical deriva- tion of the exact profile of such a modified mode is difficult,so we have performed micromagnetic simulations based onthe numerical solution of the Landau-Lifshitz-Gilbertequation. 22For simulation we used a damping constant /H9251=0.01, an exchange constant A=1.4/H1100310−6erg cm−1, and values of 4 /H9266Ms=13.2 kG for the probe magnet and 4 /H9266Ms =11.3 kG for the permalloy film were measured indepen-dently by the superconducting quantum interface devicemagnetometry. 23Simulations indicate that the FMR mode FIG. 1. /H20849Color online /H20850Magnitude of the FMRFM signals as a function of the probe-film spacing. The dotted line indicates the position of the mainresonance peak, independent of the probe-film distance. The arrows markthe onset of the lower field resonance feature. The sharp dip in the magni-tude spectrum indicates the change of sign of the force acting on the canti-lever and approximately separates local resonance /H20849in the region of the positive tip field /H20850from the main resonance /H20849region of the weak negative tip field/H20850. Experimental parameters: /H9275rf/2/H9266=9.55 GHz, T=11 K. FIG. 2. /H20849Color online /H20850Upper panel: force map of the FMRFM probe-film interaction. The force acting on a cantilever due to an elementary ring-shaped area is calculated as a function of radius of the ring and thecantilever-film spacing. The force map is normalized to a maximum positiveforce value at each probe-film distance. Force contribution from region I ofthe sample is set to zero because the fundamental resonant mode whichoccurs at H resdoes not significantly penetrate into region I, where Htotz/H11022/H9275rf//H9253. The inset shows schematically the probe-sample arrangement and two regions of the sample contributing to the FMRFM signal. Lowerpanel: integrated probe-film interaction force as a function of the cantilever-film spacing for the main resonance peak at H ext/H1101114.54 kOe. The solid symbols represent experimental points and the solid line is the result ofcalculations based on the exclusion of region I. The dashed line shows theforce if region I was included in integration. In this case there is no forceexerted by a uniformly magnetized infinite film on a spherical probe tip.Two insets show scanning electron microscopy micrographs of the cantile-ver tip.234105-2 Nazaretski et al. Appl. Phys. Lett. 90, 234105 /H208492007 /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: 128.59.222.12 On: Fri, 22 Aug 2014 03:37:09excited in region II of the sample at the resonant field Hres does not penetrate significantly into spatial region I, where Htotz/H11022/H9275rf//H9253. We will present details of the analysis elsewhere.24 We assign the lower field feature in the FMRFM spectra shown in Fig. 1to the resonance contributions originating from the localized FMR excitations spatially confined ap- proximately to region I of the sample. In this region, theresonance occurs at lower values of H extthan that of the main peak /H20849Fig.1, dotted line /H20850. The frequency shift of local- ized FMR is determined by two factors: /H208491/H20850the strength of the tip field Htipat the sample surface and /H208492/H20850the effect of mode confinement to the spatial region I with characteristicdimensions defined by the tip-sample distance, which furtherincreases the local mode frequency relative to the bulk reso-nance by a value /H9004 /H9275conf/H20849r/H11032/H20850. Both effects cause the local resonance to occur at the external field value that is lower than that of the bulk resonance by the amount /H9004Hext /H11015−Htip/H20849r/H11032/H20850−/H9004/H9275conf/H20849r/H11032/H20850//H9253. From numerical simulations,24for a confinement within a disk of radius r/H11032/H1101110/H9262m, /H9004/H9275conf/H20849r/H11032/H20850//H9253/H1101530 Oe, which combined with the estimated value of Htip/H20849r/H11032/H20850/H1101520 Oe, results in a total shift consistent with experimental findings. Both the tip field Htipand /H9004/H9275conf/H20849r/H11032/H20850decrease as the probe magnet is retracted away from the film surface, and local modes merge into the main resonance /H20849Fig.1, dotted line /H20850. The non-Lorentzian and broad shape of the signal possi- bly indicates the presence of multiple modes contributing tothe resonance. While numerical simulations are required todetermine the possibility of such modes, in particular,geometry/materials, we believe that their appearance is ge-neric in thin-film soft magnets and is induced by the localfield inhomogeneity. We will present the detailed theoreticalanalysis elsewhere. 24 The normalized FMRFM spatial force map obtained from the uniform FMR mode modified by the tip field isshown in Fig. 2. The semicircular region of the plot where the tip-sample interaction force is set to zero corresponds toregion I. In the lower panel of Fig. 2we show the intensity of the main resonance signal as a function of the tip-sampledistance and compare simulations with the experiment. In conclusion, we conducted FMRFM experiments in a thin permalloy film. We performed quantitative analysis ofthe force exerted by the fundamental mode and observedlocally excited FMR. We conducted simulations and deter-mined two distinctive regions of the sample contributing tothe FMRFM spectra. We find clear evidence for local modi-fication of the FMR mode structure by the probe tip, provid-ing insight into the interaction of the probe tip with ferro- magnetic samples in FMRFM. The work performed at Los Alamos National Laboratory was supported by the U.S. Department of Energy, Center forIntegrated Nanotechnologies, Contract No. W-7405-ENG-36at Los Alamos National Laboratory and Contract No. DE-AC04-94AL85000 at Sandia National Laboratories. Thework at Ohio State University was supported by the U.S.Department of Energy through Grant No. DE-FG02-03ER46054. The work at Naval Research Laboratory /H20849NRL /H20850 was supported by the Office of Naval Research through theInstitute for Nanoscience at NRL. 1J. A. Sidles, Appl. Phys. Lett. 58,2 8 5 4 /H208491991 /H20850. 2D. Rugar, C. S. Yannoni, and J. A. Sidles, Nature /H20849London /H20850360,5 6 3 /H208491993 /H20850. 3D. Rugar, O. Züger, S. Hoen, C. S. Yannoni, H. M. Vieth, and R. D. Kendrick, Science 264, 1560 /H208491994 /H20850. 4D. Rugar, R. Budakian, H. J. Mamin, and W. Chui, Nature /H20849London /H20850430, 329/H208492004 /H20850. 5O. Züger and D. Rugar, Appl. Phys. Lett. 63, 2496 /H208491993 /H20850. 6A. Suter, D. V. Pelekhov, M. L. Roukes, and P. C. Hammel, J. Magn. Reson. 154, 210 /H208492002 /H20850. 7Z. Zhang, P. C. Hammel, and P. E. Wigen, Appl. Phys. Lett. 68,2 0 0 5 /H208491996 /H20850. 8V. V. Naletov, V. Charbois, O. Klein, and C. Fermon, Appl. Phys. Lett. 83, 3132 /H208492003 /H20850. 9V. Charbois, V. V. Naletov, J. Ben Youssef, and O. Klein, Appl. Phys. Lett. 80, 4795 /H208492002 /H20850. 10R. Urban, A. Putilin, P. E. Wigen, S.-H. Liou, M. C. Cross, P. C. Hammel, and M. L. Roukes, Phys. Rev. B 73, 212410 /H208492006 /H20850. 11T. Mewes, J. Kim, D. V. Pelekhov, G. N. Kakazei, P. E. Wigen, S. Batra, and P. C. Hammel, Phys. Rev. B 74, 144424 /H208492006 /H20850. 12www.attocube.com, models ANPx /H20849z/H20850100/LIN 13E. Nazaretski, T. Mewes, D. Pelekhov, P. C. Hammel, and R. Movshovich, AIP Conf. Proc. 850, 1641 /H208492006 /H20850. 14http://www.magnequench.com/assets/download/MQP-S-11-9.pdf 15I. Dorofeyev, H. Fuchs, G. Wenning, and B. Gotsmann, Phys. Rev. Lett. 83, 2402 /H208491999 /H20850. 16B. C. Stipe, H. J. Mamin, C. S. Yannoni, T. D. Stowe, T. W. Kenny, and D. Rugar, Phys. Rev. Lett. 87, 277602 /H208492001 /H20850. 17E. Nazaretski, J. D. Thompson, M. Zalalutdinov, J. W. Baldwin, B. Houston, T. Mewes, D. V. Pelekhov, P. Wigen, P. C. Hammel, and R.Movshovich, J. Appl. Phys. 101, 074905 /H208492007 /H20850. 18J. D. Jackson, Classical Electrodynamics 3rd ed. /H20849Wiley, New York, 1999 /H20850, p. 194. 19Z. Frait, Physic aB&C 86-88 , 1241 /H208491977 /H20850. 20C. Herring and C. Kittel, Phys. Rev. 81, 869 /H208491951 /H20850. 21M. Sparks, B. R. Tittmann, J. E. Mee, and C. Newkirk, J. Appl. Phys. 40, 1518 /H208491969 /H20850. 22T. L. Gilbert, Phys. Rev. 100, 1243 /H208491955 /H20850; L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Statistical Physics , 3rd ed. /H20849Butterworth-Heinemann, Oxford, England, 1980 /H20850Vol. 9, Part 2, Chap. 7, p. 284. 23E. Nazaretski, J. D. Thompson, D. V. Pelekhov, T. Mewes, P. E. Wigen, J. Kim, M. Zalalutdinov, J. W. Baldwin, B. Houston, P. C. Hammel, and R.Movshovich, J. Magn. Magn. Mater. 310,9 4 1 /H20849E/H20850/H208492006 /H20850. 24D. V. Pelekhov /H20849to be published /H20850.234105-3 Nazaretski et al. Appl. Phys. Lett. 90, 234105 /H208492007 /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: 128.59.222.12 On: Fri, 22 Aug 2014 03:37:09

1.5030341.pdf

Low magnetic damping and large negative anisotropic magnetoresistance in half- metallic Co 2-xMn1+xSi Heusler alloy films grown by molecular beam epitaxy Mikihiko Oogane , Anthony P. McFadden , Kenji Fukuda , Masakiyo Tsunoda , Yasuo Ando , and Chris J. Palmstrøm Citation: Appl. Phys. Lett. 112, 262407 (2018); doi: 10.1063/1.5030341 View online: https://doi.org/10.1063/1.5030341 View Table of Contents: http://aip.scitation.org/toc/apl/112/26 Published by the American Institute of Physics Articles you may be interested in Study on measurement technique for magnetization dynamics of thin films Applied Physics Letters 112, 252403 (2018); 10.1063/1.5032305 Spin-wave-induced lateral temperature gradient in a YIG thin film/GGG system excited in an ESR cavity Applied Physics Letters 112, 212401 (2018); 10.1063/1.5022452 Negative spin Hall magnetoresistance in antiferromagnetic Cr 2O3/Ta bilayer at low temperature region Applied Physics Letters 112, 232404 (2018); 10.1063/1.5026555 Magnetic field dependence of antiferromagnetic resonance in NiO Applied Physics Letters 112, 252404 (2018); 10.1063/1.5031213 Correlation between domain wall creep parameters of thin ferromagnetic films Applied Physics Letters 112, 262402 (2018); 10.1063/1.5026702 Proximity effect induced enhanced spin pumping in Py/Gd at room temperature Applied Physics Letters 112, 262403 (2018); 10.1063/1.5033418Low magnetic damping and large negative anisotropic magnetoresistance in half-metallic Co 22xMn11xSi Heusler alloy films grown by molecular beam epitaxy Mikihiko Oogane,1,a)Anthony P . McFadden,2Kenji Fukuda,1Masakiyo Tsunoda,3 Y asuo Ando,1and Chris J. Palmstrøm2,4 1Department of Applied Physics, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan 2Department of Electrical and Computer Engineering, University of California, Santa Barbara, California 93106, USA 3Department of Electronic Engineering, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan 4Materials Department, University of California, Santa Barbara, California 93106, USA (Received 20 March 2018; accepted 10 June 2018; published online 27 June 2018) Co2/C0xMn 1þxSi films with various composition xwere epitaxially grown using molecular beam epitaxy (MBE). High crystallinity and atomic ordering in the prepared Co 2/C0xMn 1þxSi films were observed, and their magnetic damping and anisotropic magnetoresistance (AMR) effect weresystematically investigated. An ultra-low magnetic damping constant of 0.0007 was obtained in the Co 2/C0xMn 1þxSi film with a valence electron number ( NV) of about 29.0. Additionally, a relatively large negative AMR effect was observed in the Co 2/C0xMn 1þxSi films that had a NVof about 29.0. This low damping and the large negative AMR effect indicate that epitaxial Co 2/C0xMn 1þxSi films with high atomic ordering grown by MBE possess a high-spin polarization. Published by AIP Publishing. https://doi.org/10.1063/1.5030341 Half-metals have potential for enhancing the perfor- mance of various kinds of spintronic devices, such as mag- netic sensors and magnetic random access memory (MRAM) devices, because the conduction electrons are completelyspin-polarized due to the Fermi level ( E F) being in the energy gap of one of the spin channels. Among the half-metals, Co-based Heusler alloys are expected to show half-metallicproperties at room temperature due to their high Curie tem- perature. 1–3In the past decade, high tunnel magnetoresistance (TMR) and current-perpendicular-to-plane giant magnetore-sistance (CPP-GMR) have been experimentally demonstrated using Co-based Heusler alloy electrodes. 4–10However, fur- ther improvement of their half-metallic properties is requiredto obtain improved performance in magnetic tunnel junctions (MTJs) and CPP-GMR devices. One fingerprint of half-metallicity is the negative aniso- tropic magnetoresistance (AMR) effect. According to a theory based on the two-current model with s-d scattering, the AMR ratio in half-metals should be negative. 11,12 Recently, research on the AMR effect in Co-based Heusler alloys has been actively pursued.13–17Yang et al. investi- gated the AMR effect in Co 2FexMn 1/C0xSi films and observed that it changed from negative to positive at x>0.8.13These results suggested the disappearance of half-metallic proper- ties at x>0.8 and were consistent with previous studies on TMR and CPP-GMR using Co 2FexMn 1/C0xSi electrodes.18,19 Sakuraba et al. reported systematic results for research on the influence of the valence electron number ( NV) on the AMR ratio in various Co-based Heusler alloy thin films and found a clear positive correlation between the magnitude ofthe negative AMR ratio and the GMR output in CPP-GMR devices using Heusler alloy electrodes.15 Another fingerprint of half-metallic properties is mag- netic damping. Low magnetic damping in some Co-basedHeusler alloys was predicted by calculation usingKambersky’s torque correlation model, 20considering the spin-orbit interaction in combination with the first principles calculation of the electronic band structures.21–23According to this research, the magnetic damping constant can vary with the density of states (DOSs) at the EF. Half-metals do not have DOSs of one spin channel at the EF, so their mag- netic damping should be low. In addition, Co-based Heusleralloys can exhibit low magnetic damping because of their small spin-orbit interaction and low orbital magnetic moment. In previous studies, low magnetic damping in Co-based Heusler alloys has been experimentally reported. 24–30 In a previous work, Oogane et al. reported low magnetic damping in Co 2FexMn 1/C0xSi Heusler alloy films in which x¼0.0–0.6 and a significant increase in the damping at x>0.8.24This also indicates the half-metallic properties of Co2FexMn 1/C0xSi when x<0.6 and the disappearance of those properties at x>0.8. Although there are many reports on AMR and magnetic damping in Co-based Heusler alloy films prepared by the sputtering method, there are few reports on films prepared using molecular beam epitaxy (MBE). Recently, Andrieuet al. reported that Mn-rich Co 2/C0xMn 1þxSi films fabricated by MBE exhibit a very low damping constant of 0.0007 at x¼0.1.31They also investigated the electronic structure of these films using spin-resolved photoemission spectroscopyand confirmed their half-metallicity. In addition, the enhancement of the spin polarization and the tunnel magne- toresistance (TMR) in Mn-rich Co 2MnSi was reported, anda)E-mail: oogane@mlab.apph.tohoku.ac.jp 0003-6951/2018/112(26)/262407/5/$30.00 Published by AIP Publishing. 112, 262407-1APPLIED PHYSICS LETTERS 112, 262407 (2018) the mechanism of enhancement was considered to be the suppression of the creation of Co anti-sites32,33and/or sur- face states.31However, the relationship between the actual film composition and the half-metallic properties of theCo 2/C0xMn 1þxSi films has not been investigated in detail. In this work, we fabricated Co 2/C0xMn 1þxSi films prepared by MBE in which the composition was well controlled. We alsosystematically investigated their structural and magnetic properties, magnetic damping, and AMR effect. Our findings on the effect of film composition on the magnetic damping and the AMR effect suggest excellent half-metallic proper- ties of Co 2/C0xMn 1þxSi films around Nv¼29.0. The 10-nm-thick Co 2/C0xMn1þxSi films ( x¼/C00.1, 0.0, 0.1, 0.2, and 0.3) were deposited by MBE on MgO(001)-substrates and 20-nm-thick MgO buffer layers in a growth chamber undera base pressure of <5/C210 /C011Torr. The MgO substrates were annealed at 800/C14C in an oxygen-ambient furnace, followed by 700/C14C ultra-high-vacuum (UHV) annealing. The MgO buffer layers were deposited by e-beam evaporation and annealed at 500/C14C for 1 h in UHV to obtain a smooth surface. The Co2/C0xMn1þxSi films were grown at ambient temperature by co-evaporating the elemental source materials using standard effusion cells. All fluxes were calibrated ex situ by measuring the elemental atomic areal density of calibration samples byRutherford backscattering spectrometry (RBS). After the depo- sition of the Co 2/C0xMn1þxSi films, in-situ annealing was con- ducted at 600/C14C for 15 min. After the cooling of the films, the 3-nm-thick AlO xfilms were deposited by e-beam evaporation as a capping layer. The surface of the films was monitored by reflection high-energy electron diffraction (RHEED) duringand after the growth of each layer. The structural properties of the Co 2/C0xMn 1þxSi films were characterized by Cu K aX-ray diffraction (XRD). The saturation magnetization Mswas measured using a Quantum Design MPMS XL superconducting quantum interference device (SQUID). The magnetic damping constant awas measured at RT by ferromagnetic resonance spectrometry (FMR) using an X-band microwave source ( f¼9.4 GHz) and a TE011 cavity. The samples were fixed on a quartz rod anda goniometer was used to measure the effect of the out- of-plane angle ( h H) on the resonance field and the linewidth of the FMR spectra. The linewidth was analyzed using a pre-viously reported method 24to obtain an intrinsic damping constant while excluding the effects of the inhomogeneous magnetic properties and the surface roughness of the films.The AMR effect was measured at 300 K using a conven- tional DC four-terminal method. The composition of the Co 2/C0xMn 1þxSi films was qualitatively measured by X-ray fluorescence (XRF) spectrometry. Subsequently, the magni- tude of the composition measured by XRF was calibrated with the results of inductively coupled plasma (ICP) spec-troscopy of the Co 2MnSi ( x¼0.0) film. Figures 1(a) and 1(b) show the film composition and the estimated valence electron number NVfor the prepared Co 2/C0xMn 1þxSi films. Although the prepared Co 2/C0xMn 1þxSi films were slightly Co rich, the film composition was fairly well controlled. Figure 2(a) shows a RHEED image of the Co 2MnSi (x¼0.0) film along the Co 2MnSi [110] direction. Secondary streaks indicated with white arrows in the RHEED image are consistent with L2 1ordering of the Co 2MnSi film, aspreviously reported.34Because all of the prepared Co 2/C0x Mn 1þxSi films showed the secondary streaks, we concluded that the films had L2 1-ordered surfaces regardless of their x value. Figure 2(b)shows a typical XRD (2 h-scan) pattern for the Co 2MnSi ( x¼0.0) film. The prepared Co 2/C0xMn 1þxSi films exhibited only the (200) and (400) peaks of Co 2MnSi, except for the peaks from the MgO buffer layer and the sub- strate, regardless of their xvalue. We confirmed that the pre- pared Co 2/C0xMn 1þxSi films have good (001)-orientation and do not have other crystal phases. The inset in Fig. 2(b)shows a pole-figure of the (111) peak for the Co 2MnSi ( x¼0.0) film. The four-fold peak was clearly observed in the pole-figures of all the prepared Co 2/C0xMn 1þxSi films. These results indicate that the Co 2/C0xMn 1þxSi films were epitaxially grown on the MgO buffers and substrates and contained a L2 1- ordered structure. Figure 2(c)shows the lattice parameter of thec-axis calculated with the (400) peaks. The resulting FIG. 1. (a) Film composition as determined by XRF and ICP and (b) the estimated valence electron number in prepared Co 2/C0xMn 1þxSi films. Dashed lines indicate the expected composition and the valence electron numberfrom estimated elemental deposition fluxes. FIG. 2. (a) RHEED image and (b) XRD patterns for the Co 2MnSi film (x¼0.0) and the effect of composition on (c) the c-axis lattice constant and (d) L2 1- and B2-ordered parameters of prepared Co 2/C0xMn 1þxSi films.262407-2 Oogane et al. Appl. Phys. Lett. 112, 262407 (2018)lattice constants are close to the bulk value of 0.5654 nm, except in the case of the Co-rich Co 2.1Mn 0.9Si (x¼/C00.1) film. The small c-axis lattice parameter of the Co 2.1Mn 0.9Si film indicates an expansion of the a-axis due to the lattice mismatch between Co 2MnSi and MgO. Figure 2(d) shows the effect of the film composition on both the L2 1-(SL21) and B2-(SB2) ordered parameters. SL21and SB2were, respec- tively, evaluated by the integrated peak intensity ratio of [(111)/(220)] exp./[(111)/(220)] theory in the pole-figures and [(200)/(400)] exp./[(200)/(400)] theory in the 2 h-scans, taking into account Lorentz polarization absorption corrections. Here, [(111)/(220)] exp.and [(200)/(400)] expare experimental integrated peak intensity ratios and [(111)/(220)] theory (¼0.047) and [(200)/(400)] theory(¼0.338) are theoretical values for stoichiometric Co 2MnSi alloy. The evaluated atomic order- ing parameters include some error because the atomic numbers of Co and Mn atoms are close, and we used a theoretical inten- sity ratio even for nonstoichiometric Co 2/C0xMn1þxSi films. Figure 3(a) shows the typical FMR spectra measured at RT with various out-of-plane angles ( hH)f o rt h eC o 2MnSi (x¼0.0) film. Clear Lorentzian curves with small linewidths were observed for each hH.F i g u r e 3(b)is an example of a typi- cal analysis of the effect of hHon the FMR linewidth DHpp. We expressed the linewidth as DHpp¼DHppaþDHpp4pMeff þDHpphH.DHppais the linewidth resulting from the intrinsic damping and DHpp4pMeffandDHpphHare, respectively, line- widths resulting from the distribution of the magnitude andfrom the direction of the effective magnetic field in the films. Although we took the inhomogeneity of magnetic properties into account when analyzing the linewidths, the effect ofmagnetic inhomogeneity was small, as shown in Fig. 3(b). This means that the prepared Co 2/C0xMn 1þxSi films have uniform magnetic properties; the intrinsic damping was thedominant contribution to the observed FMR linewidth. Figure 3(c)shows 4 pM eff(¼4pMs/C0l0Hk?:Hk?, a perpen- dicular magnetic anisotropy field) estimated from the analy-sis of the FMR resonance field and 4 pM smeasured at 300 Kby the SQUID. The Mswas close to the bulk value for x¼0.0, 0.1, and 0.2. The observed Mswas higher than those for Co 2MnSi films prepared by sputtering,24which indicates that the films in this study have a highly B2-ordered struc- ture. In contrast, the Msdecreased slightly at x¼/C00.1 and 0.3 due to the atomic disorder. The 4 pMeffvalues were basically the same as those of 4 pMs because of the small magnetic anisotropy; only the Co-rich Co 2.1Mn 0.9Si film showed a relatively large 4 pMeff. Although the origin of large in-plane anisotropy in the Co 2.1Mn 0.9Si film is not clear, one possible reason for it is the lattice distortion,as shown in Fig. 2(c).F i g u r e 3(d) shows the effect of film composition on the intrinsic magnetic damping constant ain the Co 2/C0xMn 1þxSi films. The Co 2/C0xMn 1þxSi films in which x¼0.0, 0.1, and 0.2 exhibited a low aof<0.003. In particu- lar, the minimum aof 0.0007 for x¼0.1 was quite low, and this low magnetic damping indicates good half-metallic properties of the films. Conversely, aincreased in the Co2/C0xMn 1þxSi films in which x¼/C00.1 and 0.3. The dotted line shown in Fig. 3(d) indicates previously reported ain Co2/C0xMn 1þxSi films grown by MBE.31As Fig. 3(d) shows, our results quantitatively agree with previous studies,although the composition of the prepared Co 2/C0xMn 1þxSi films was slightly Co rich, as mentioned above. Figure 4(a) shows typical AMR curves in the Co2/C0xMn 1þxSi films in which x¼/C00.1, 0.0, and 0.3 mea- sured at 300 K. An external magnetic field of 2 T was applied in-plane to sufficiently align the magnetization along the direction of the field. An electric current flowed in theCo 2MnSi [110] direction, and the magnetic field was rotated in the (001) film plane. In Fig. 4(a), the change in resistivity [¼(q(h)/C0q?)/q?] is plotted as a function of h, which is defined as the angle between the magnetization and thecurrent; therefore, h¼0 means a parallel configuration of the electric current and magnetization. A clear negative AMR effect was observed in all the Co 2/C0xMn 1þxSi films regardless ofxvalue. Figure 4(b) shows the effect of composition on the AMR ratio measured at 300 K in the Co 2/C0xMn 1þxSi films. We found that the AMR ratio decreased as xdeviated from x¼0.0. Although a negative AMR effect in various Co-based Heusler alloys has already been reported in otherresearch, the observed negative AMR ratio >0.2% for x¼/C00.1, 0.0, and 0.1 was relatively high compared with previously reported values. 13–17This large negative AMR effect is also a signature of the half-metallic properties of theCo 2/C0xMn 1þxSi films grown by MBE. FIG. 3. (a) Typical FMR spectra measured at RT for various out-of-plane angles ( hH) and (b) fitting results for FMR linewidths in the Co 2MnSi film (x¼0.0). Bold line indicates the calculated total DHpp; dotted red, blue, and green lines show three components of calculated total DHpp, which are due to intrinsic damping ( a), distribution of the effective magnetic field (D4pMeff), and fluctuation of hH, respectively. The effect of composition on (c) 4pMeffand 4 pMsand (d) the magnetic damping constant aat RT in the prepared Co 2/C0xMn 1þxSi films. Dotted line in (c) shows the bulk value for stoichiometric Co 2MnSi and in (d) shows previous report.31 FIG. 4. (a) AMR curves measured at 300 K in Co 2/C0xMn 1þxSi with x¼/C00.1, 0.0, and 0.3 and (b) the effect of composition on the AMR ratio at 300 K.262407-3 Oogane et al. Appl. Phys. Lett. 112, 262407 (2018)Figures 5(a)and5(b) show the effect of Nvon the satu- ration magnetization and the Gilbert magnetic damping con- stant ( G¼acMs;cis the gyromagnetic ratio). The results from previous works on the magnetization and magnetic damping constant in Co 2MnAl 1/C0xSixand Co 2FexMn 1/C0xSi Heusler alloy films prepared by sputtering are shown by green dotted lines.24The films with ca.Nv¼29.0 prepared by MBE showed high Mscompared with those of the films prepared by sputtering. This high Msindicates that the films contain a large amount of the B2-ordered structure. The minimum Gof the films prepared in this study was 1.2 /C2107rad/s, a much lower value than that of the films prepared by sputtering, at around Nv¼29.0, which is the number for the stoichiometric Co 2MnSi. Gcalculated using Kambersky’s torque correlation model for L2 1-ordered Co2MnAl ( Nv¼28.0), Co 2MnSi ( Nv¼29.0), and Co 2FeSi (Nv¼30.0) is also shown in Fig. 5(b)by a blue dotted line.22 The calculation is consistent with our experiments, althoughthe magnitude of Gfor the experiment was larger than the calculated value. We infer that the deviation between the experiment and the calculation is caused by imperfect atomic ordering in the prepared Co 2/C0xMn1þxSi films, especially in the Co-rich film in which x¼/C00.1 and the Mn-rich film in which x¼0.3. Figure 5(b) shows the effect of Nvon the AMR ratio. The negative AMR effect showed a maximum around Nv¼29.0. This is consistent with previous results of Co-based Heusler alloy films prepared by sputtering (greendotted line), 15although the magnitude of the observed negative AMR ratio was greater than previous results. The magnetic damping is simply proportional to the DOSs at theE F, whereas the negative AMR effect is basically inversely proportional to the ratio of the DOSs for down- and up-spins (D#/D") at the E F, as discussed in a previous report.15 Therefore, the low magnetic damping and the high AMR are not necessarily compatible; however, we infer that a large negative AMR ratio also results from good half-metallicproperties of the Co 2/C0xMn 1þxSi films because of the highly atomic B2-ordering in the films prepared by MBE. For the nonstoichiometric Co 2/C0xMn 1þxSi films, strictly speaking, the rigid band model is not accurate because the electronic band structure can change due to the atomic disorder in the films.35,36However, we infer that the observed change in the Gand AMR ratio against the Nvcan be explained by the change in the DOS at the EFbased on the rigid band model, since the degree of the atomic disorderis small in the Co 2/C0xMn 1þxSi films with x¼0.0, 0.1, and 0.2. The previous study also suggested that the DOSs in the Co2/C0xMn 1þxSi films with x<0.3 changed to follow the rigid band model by varying x.31Additionally, the obvious improvement of the half-metallic properties in Mn-rich composition previously reported32,33was not observed since the films with stoichiometric composition ( NV¼ca. 29.0) showed a large negative AMR and low damping in this work. Further careful investigations are necessary to clarifythe influence of Mn enrichment on half-metallic properties of Co 2MnSi films. In summary, systematic investigations on the effect of film composition on magnetic damping and the AMR effect were performed in epitaxial Co 2/C0xMn 1þxSi films grown by MBE. We found that the prepared Co 2/C0xMn 1þxSi films pos- sessed high (001)-orientation and atomic ordering, althoughL2 1- and B2-ordering slightly decreased with deviation from the stoichiometric composition. A very low magnetic damp- ing constant and a large negative AMR were observed in theCo 2/C0xMn 1þxSi films around Nv¼29.0. These results suggest that the epitaxial Co 2/C0xMn 1þxSi films grown by MBE show excellent half-metallic properties, thanks to the high atomicordering in the Co 2/C0xMn 1þxSi films. In addition, the effect of the valence electron number on both magnetic damping and the AMR ratio was consistent with our expectations and withprevious work. These findings on the importance of controlling the valence electron number in Co-based Heusler alloy thin films will be useful for developing high-performance spin-tronic devices. This work was supported by the Center for Spintronics Research Network (CSRN), S-Innovation program, Japan FIG. 5. Effect of the valence electron number on (a) the saturation magneti- zation at 300 K, (b) the Gilbert magnetic damping constant Gat RT and (c) the AMR ratio at 300 K. Blue and green dotted lines in (a) show calcula-tions 22and previous experimental results,24respectively. Dotted line in (b) shows previous report.15262407-4 Oogane et al. Appl. Phys. Lett. 112, 262407 (2018)Science and Technology Agency (JST), and U.S. Department of Energy (DE-SC0014388). The MBE growth, X-ray diffraction, and magnetic characterization performed at the University of California Santa Barbara (UCSB) weresupported by the U.S. Department of Energy (DE- SC0014388). This research made use of shared facilities of the UCSB Materials Research Science and EngineeringCenter (NSF DMR 1720256), a member of the Materials Research Facilities Network. 1S. Ishida, S. Fujii, S. Kashiwagi, and S. Asano, J. Phys. Soc. Jpn. 64, 2152 (1995). 2I. Galanakis, P. H. Dederiches, and N. Papanikolaou, Phys. Rev. B 66, 174429 (2002). 3P. J. Webster, J. Phys. Chem. Solids 32, 1221 (1971). 4Y. Sakuraba, M. Hattori, M. Oogane, Y. Ando, H. Kato, A. Sakuma, T. Miyazaki, and H. Kubota, Appl. Phys. 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1.323131.pdf

Detection of stored momentum in magnetic bubbles by a bias jump effect A. P. Malozemoff and S. Maekawa Citation: Journal of Applied Physics 47, 3321 (1976); doi: 10.1063/1.323131 View online: http://dx.doi.org/10.1063/1.323131 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/47/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromechanical detection of magnetic resonance by angular momentum absorption Appl. Phys. Lett. 69, 3920 (1996); 10.1063/1.117570 Abstract: Detection of iron stores in the human body using magnetic susceptibility (invited) J. Appl. Phys. 52, 2581 (1981); 10.1063/1.329004 Bubble propagation direction effects on high frequency bias margin J. Appl. Phys. 50, 2280 (1979); 10.1063/1.327027 Magnetic bubble detection using waveguide fiber optics J. Appl. Phys. 44, 1801 (1973); 10.1063/1.1662452 ThinFilm Surface Bias on Magnetic Bubble Materials J. Appl. Phys. 42, 1360 (1971); 10.1063/1.1660252 [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: 128.59.171.71 On: Tue, 09 Dec 2014 06:56:25Detection of stored momentum in magnetic bubbles by a bias jump effect A. P. Malozemoff and S. Maekawa * IBM Thomas J. Watson Research Center. Yorktown Heights. New York 10598 (Received 24 February 1976) A pulse of uniform bias field is applied to an isolated -S-,...m bubble in a garnet film. The bubble (S= 0) was previously propagated by a strong gradient·field pulse. The bubble is observed to jump forward in the same direction as the previous gradient propagation. irrespective of the sign of the bias-field pulse. Subsequent bias pulses may cause further "bias jumps" of this type until a maximum displacement is reached, which can be many ,...m in typical cases. Dependence of the bias jumps on the strength and length of the bias-field pulses and also of the previous gradient-field pulses is reported. The shape of the bubbles, observed by high-speed photography during the bias pulse, is elliptical, indicating lower mobility on the sides of the bubble perpendicular to the over-all direction of motion. These results provide evidence for the existence of unwinding Bloch-line pairs which remain in the bubble wall at the end of a gradient propagation. These Bloch lines are shown to be tantamount to a stored bubble momentum, which is released when the bias pulses are applied. PACS numbers: 7S.60.Fk, 7S.60.Hn I. INTRODUCTION Various authorsl-3 have noted anomalous "turnaround" behavior in bubble propagation. For example, consider a bubble propagated in a pulsed bias-field gradient in the usual way. 4 When the polarity of the gradient is reversed on a subsequent pulse, the bubble often does not simply reverse its direction of motion. Instead it may jump sideways or even in the same direction as its original direction of motion, and it often takes several pulses before the new direction of motion is firmly established. Since the reverse pulses are usually ap plied seconds or minutes after the original pulse, the bubble apparently remembers its original direction of motion for a long time and somehow "resists" the turnaround. Recently Beaulieu and Voegeli5 have observed a re lated effect. By means of a pulse of current down a long strip line, they propagated a bubble to the potential well at the edge of the strip line, where the bubble ran out into a stripe. When the pulse ended, they found that not only would the stripe contract, but also the resulting bubble would jump forward. Apparently the bubble could remember its original direction of motion even when it stayed in a runout condition for msec. Furthermore, when the pulse terminated, the domain effectively ex perienced only a uniform bias field which caused the stripe to run back into a bubble. Somehow the bias-field pulse was able to propel the bubble forward. Slonczewski6 first proposed theoretically that a uni form bias-field pulse could cause bubble displacement, if the bubble contained a clump of Bloch lines, all of the same signature. More recently we have observed ex perimentally a kind of "ballistic overshoot" in the gradient propagation of bubbles. 7,B In this effect the bubble appears to move after the end of the drive pulse in the absence of any external forces. We proposed theoretically that this effect could be caused by the un winding of Bloch lines which had wound up in the wall during the drive pulse. B,9 In this case there are pre sumably two clumps of Bloch lines of opposite signature on the flanks of the bubble. We showed that the Bloch 3321 Journal of Applied PhYSics, Vol. 47, No.7, July 1976 lines can be viewed as giving rise to a "bubble momen tum". We also raised the possibility that the Bloch lines might not unwind completely during the overshoot and that some might remain metastably in the wall of the bubble. In this case the metastable Bloch lines are tantamount to a "stored bubble momentum". However, no direct experimental evidence for the existence of such a stored momentum was given. Viewed in this per spective, the "turnaround" effects and the Beaulieu-Voegeli overshoot effect offer the first experimental indication for stored bubble momentum. However, the interpretation of these experiments is complicated. In the turnaround effect, the bubble gen erally experienced both a bias-field pulse and a gradient pulse simultaneously. On the other hand in the Beaulieu-Voegeli experiment, the bubble becomes extremely distorted. In this paper we propose a simpler experiment to demonstrate the existence of stored bubble momentum. We apply a pulse of uniform bias field to a bubble which has previously been propagated by a gradient-field pulse. We find that the bubble con sistently jumps forward in the same direction as the gradient propagation. The jump direction is the same for both positive and negative pulses. Subsequent bias field pulses may cause further jumps until a maximum displacement is reached which may be many /lm for typical garnet bubble materials. We call this kind of bubble motion a "bias jump" because the bubble appears to jump in response to a bias pulse. In Sec. II we describe the experimental results on a garnet film, and in Sec. III we discuss the results in terms of the concept of stored bubble momentum. We conclude that these observations provide strong evidence for stored bubble momentum and that they confirm the Bloch-line wind-up model of ballistic overshoot. II. EXPERIMENTAL PROCEDURES AND RESULTS In an earlier paperB we have reported on the proper ties of a series of garnet films of different compositions supporting -5-/lm bubbles. We have observed anoma- Copyright © 1976 American Institute of Physics 3321 [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: 128.59.171.71 On: Tue, 09 Dec 2014 06:56:25A INNER STRIPLINES ~ CENTER I , WEAK GRADIENT PULSES .. I GRADIENT PROPAGATION BIAS 1 PULSE PULSES I ~ . . . ...... B C 0 E I I ~ FIG. ]. Schematic bubble positions in bias-jump effect. Bubble is brought to center of a stripline pattern by weak gradient pulses (A -B), then propagated by a stronger gradient pulse (B-C), then propagated by a first bias pulse (C-D) and addi tional bias pulses till the bubble becomes "inert" (D-E), then finally returned to start the procedure all over again (E-A). lous turnaround behavior in most of these films. We have performed a more thorough study of bubble motion due to a bias-field pulse in three of these films (sam ples 1 and 4 and the 6. 8-Mm sample of Fig. 9(a) of Ref. 8) and have found similar behavior in all cases. Here we report in detail on the results in one of these films (sample 1 of Ref. 8). It is a EUO.65Ga1.2 YIG film, with thickness h=4.3 Mm, magnetization 41TM=160 G, length parameter l = O. 77 Mm, uniaxial anisotropy Ku = 9050 erg/ cm3, gyromagnetic ratio y= 1. 21 X 107 Oe-1 sec-1, Gilbert damping parameter a = 0.05, wall width parameter ~=0.043-Mm and linear mobility y~a-l = 1050 cm/sec Oe. The [111} axis is well aligned with respect to the surface normal (~c -0.15°) and there is accordingly no measurable in-plane anisotropy energy10 (Kp < 150 erg/cm3) or anisotropy in dynamic properties for wall motion in different directions. 8 The transla tional saturation velocity is 800 ± 50 cm/ sec. For all the experiments to be described below, the dc bias field was adjusted to give a bubble diameter d= 2r= 5 Mm. The sample was placed face down on a strip-line array deposited on a glass slide. As described pre viouslY,7 the strip-li.112 array had four straight lines symmetrically arranged, two for creating a field gra dient and two for providing bias compensation. Cen tered underneath the glass slide was a hand-wound pancake coil with i. d. 1 mm. When pulsed, the coil created a bias field at the sample. The possibility of gradients in this bias field or in the sample itself was checked by pulsing "inert" bubbles (see below); no net average displacement was found, indicating that stray gradients are negligible. The experiment was performed as follows. An S = 0 bubble1 was chosen on the basis of its propagation perpendicular to the strip-line array. The bubble was brought (see A - B of Fig. 1) to its starting position B at the center of the strip-line array using weak gradient pulses. Next the bubble was propagated (see B - C in Fig. 1) by a pulse of gradient field of length Tg and strength Hg=rH' (where H' is the field gradient). The proper bias compensation7 was determined on the basis of the previously observed translational saturation velocity. The net bubble displacement BC was recorded 3322 J. Appl. Phys., Vol. 47, No. 7,July 1976 using a Filar micrometer or a TV micrometer. After a wait lasting from several seconds to several minutes (to record the data), a pulse of bias field was applied from the coil. The pulse had a length T b and strength Hb• The pulse length was generally kept below 150 nsec to prevent the bubble from collapsing in a positive (con tracting) pulse or running out in a negative (expanding) pulse. The bubble was almost always observed to move forward (C - D in Fig. 1) in the same direction as the preceding gradient propagation. This tendency was the same, irrespective of the sign of the bias pulse. The displacement CD was recorded and then the bubble was bias pulsed again. Often a second forward jump was observed, although the second jump was usually smaller than the first. Succeeding bias pulses would cause fur ther jumps of, on the average, decreasing size until the bubble finally became "inert" -that is, ceased jump ing forward and came to a stop at the final position E. Subsequent pulses could cause small displacements « 1 Mm), but these were generally like a random walk and caused only a small uncertainty (± 1 Mm) in deter mining the final pOSition E. Finally, weak gradient pulses were applied to resume bubble propagation and return it to its original pOSition (E -A). Often, on its return path, the bubble exhibited a skew deflection, 1 indicating that its S state had changed from O. Both positive and negative S-state changes were observed. When exactly this change oc curred was not determined: it could have occurred during the original propagation BC or during the sub sequent bias pulses CDE. One clue was the fact that while the motion i.n the original propagation and the subsequent bias pulses was often straight, perpendicular to the strip lines, it also occasionally began to skew during the bias pulse motion. In the data presented below, we recorded only the displacement perpendic- 10 ,------,-- STATE CHANGE? -,-----,-----, ~. FIRST JUMP { ~ = ~~S 8 x NET OF ALL {+-NO E BIAS JUMPS x -YES x ~ x x !:z 6 x x x * /~, + ~ q, x t ~,--~/ '1, d)ll -.oJ 4 r. x + /::5 I~ x 0 ' 3; I*-~----f' ;j:\)( I + 6Ft+ V-" *~4 2 ~~_(J ~ ~\ ,'!-4I ~ -'1 I : 1 ~-~~r 1- I 1 -~ -~ -~ 0 ~ ~ ~ FIG. 2. First bias jump (CD of Fig. 1) and net of all bias jumps (CE) for positive and negative bias pulse strengths Hb, at fixed bias pulse length Tb = 1 00 nsec and fixed prior gradient pulse (Tg=300 nsec, Hg=4.8 Oel. Sample is a EuGa garnet film supporting 5-!.Im bubbles. State changes determined at the end of the bias jump procedure (E-A) are also indicated. Solid lines connect average first jump displacements and dashed lines connect average net jump displacements. A.P. Malozemoff and S. Maekawa 3322 [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: 128.59.171.71 On: Tue, 09 Dec 2014 06:56:2510 STATE CHANGE? '------..,.---1' FIRST JUMP 0 -Y S + {e -N~ 8 ET OF ALL {x-NO * BIAS JUMP + -YES ~ 'i6 x f-z If x x w ::::E + + X w 4 x +;k x ~ x / ' ...J ',,~ + 'x_-xt a... en \ C5 2 \ 150 100 150 '----- ~ ___ I ,'-___ ~ __ _ Tb (nsec)~Hb=-1750e Tb (nsec),Hb=+1750e FIG. 3. First bias jump (CD) and net of all bias jumps (CE), as a function of bias pulse length Tb, for fixed bias pulse strength Hb = ± 17.5 Oe and fixed prior gradient pulse (Tg =300 nsec, Hg=4.8 Oel. Sample and state changes and dashed and solid lines as in Fig. 2. ular to the strip lines, but we took note of those sequences in which a state change was observed in the return propagation. No systematic differences in the perpendicular displacements of the sequence were ob served between cases with and without a state change. The response of the bubble to bias pulses was also checked prior to the original propagation BC in a separate experiment. We found a small tendency of the bubble to jump forward by a net of order 1 /J.m after a large number of pulses. Thus the bubbles at the start of the sequence were almost, though not quite, inert. Figure 2 shows the size of the first bias jump (CD of Fig. 1) and the net bias-jump displacement (total of all bias jumps, CE of Fig. 1), as a function of the strength and polarity of the bias field Hb, for a fixed bias pulse length Tb = 100 nsec and a fixed gradient propagation pulse (Tg=300 nsec, Hg=4.8 Oe). Figure 2 shows the remarkable fact that both positive and negative bias pulses tend to move the bubble forward. It is also note worthy that the net bias displacements are large; in this case we observe up to 8 /J.m, while the gradient prop agation displacement (BC) is only 4.5 ± 0.5 /J.m. Fur thermore, the size of the first jump increases mono tonically with I Hbl and is somewhat larger for positive (contracting) than negative (expanding) pulses of the same strength. On the other hand, the net displacement due to all bias jumps is, considering the large scatter, roughly independent of pulse strength for either positive or negative pulses. This means that it generally takes more bias pulses to make the bubble inert when the pulse is weak than when it is strong, and typical num bers of bias pulses run from one to ten. Figure 3 shows similar data as a function of the length of the bias pulse Tb, for fixed bias pulse strength Hb = ± 17. 5 Oe and a fixed gradient propagation pulse (Tg=300 nsec, Hg=4. 8 Oe). The displacement in creases with pulse length, and tends to be larger for positive than for negative pulses of the same strength. 3323 J. Appl. Phys., Vol. 47, No.7, July 1976 -Tl S~? 4 FIRST BIAS {. -NO JUMP 0-YES • • • e3 • 0 • ~ • • ::t. 0 I I-0 Ii • • 0 • 6 1 :z i ~ • ~2 • • 0 • 0 • w • «> 0 ro • ~ • • • 0 -' • e;1 § Ci , 0 • _.1 0 I I I ~_~ __ --.-J 0 2 3 4 5 6 7 8 Hq =-rH' (De) FIG. 4. First bias jump (CD) as a function of strength Hg of prior gradient pulse, for fixed gradient pulse length Tg = :WO nsec and a fixed bias pulse IJIb = 17.5 Oe and Tb = 1 00 nsee). Sample and state changes as in Fig. 2. Once again the final displacements exhibit considerable scatter but are roughly independent of bias pulse length. Figure 4 shows the size of the first bias jump CD, as a function of the strength Hg of the preceding gradient pulse, for a fixed gradient pulse length Tg = 300 nsec and a fixed bias pulse (Hb = 17. 5 Oe, Tb= 100 nsec). For Hg above about 3 Oe, the jump size is constant. By contrast Fig. 5 shows the gradient propagation dis placement BC and the net over-all displacement BE resulting from both the gradient displacement and the subsequent bias jumps, as a function of gradient pulse strength Hg for the same conditions as Fig. 4. The gradient displacement data shows a plateau or "satura tion" in the gradient jump size BC for moderate drive, E -=-18~-~-~ S~?_--, __ ,~_,----/--, 16 14 12 10 4 2 CllADlENT {A -NO ffia'AGATION e,. -YES JUMPS PLUS x -NET OF ALL { NO GRADIENT + -YES X CD( fxatO) + t x + + x x x x ~ + + 1;. A A ~ x + x x + x x + 1< x x *" A - )L~1&~--±2--!;-3 --±4----c5!o-' ----';:-~ ---7 Hg=-rH'(Oe) x 8 FIG. 5. Gradient propagation (BC) and net of all jumps plus gradient propagation (BE) as a function of gradient pulse strength Hg, for fixed gradient pulse length Tg = 300 nsec and a fixed bia§. pulse IJIb =17.5 Oe and Tb =100 nsec). Theoretical line X~ (1/J,..., = 0) is a plot of Eq. (3-7) of the text. Sample and state changes as in Fig. 2. A.P. Malozemoff and S. Maekawa 3323 [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: 128.59.171.71 On: Tue, 09 Dec 2014 06:56:25STATEC~ I I 4 FIRST BIAS~' JUMP 0 -YES - • • E 3f--• ~ 8 ::I.. , , 0 • ~ • z • w 2 • ::::ii: • 0 -w I 0 0 • ~ • --J ~ !7; B I I o 100 200 300 400 500 600 Tg (nset) FIG. 6. First bias jump (CD) as a function of length Tg of prior gradient pulse, for fixed gradient pulse strength Hg = 4.8 Oe 18 ,-----,---STATE CHANGE? .----r-,---, .~.-~. GRADIENT 16 PRO~GATION NET OF ALL JUMPS PLUS 14 GRADIENT _12 E ::I.. ~ 10 w ::::ii: w ~ 8 --J a.. en C5 6 + + x {A -NO b. -YES {X -NO + -YES X ~ + + + ~ b. + X i I --i and a fixed bias pulse (lIb = 17.5 Oe and Tb =100 nsee). Sample 4 ~ x A A , and state changes as in Fig. 2. as reported before. 8 Figure 5 also shows that the net bias-jump displacement CE increases with increasing gradient drive. Figure 6 shows the size of the first bias jump CD as a function of the length TIf of the preceding gradient pulse, for a fixed gradient pulse strength H, = 4.8 Oe and a fixed bias pulse (Hb= 17.5 Oe, Tb = 100 nsec). The jump size is roughly constant. By contrast, Fig. 7 shows the gradient propagation displacement BC and the net over -all displacement BE as a function of gradient pulse length Tg, for the same conditions as in Fig. 6. The gradient displacement data increases mono tonically with TIf, as does the net bias-jump displacement. The shape and position of the bubbles during the bias jump was also investigated by high-speed photogra- phy. 7,11 The bubbles were propagated by a 4. 8-0e 300- nsec gradient pulse and then pulsed by either a positive or a negative 13. 5-0e 100-nsec bias pulse. The laser was timed to fire at the end of the bias pulse. The average transient bubble shape and position at this time is shown in Fig. 8(a), relative to the initial (C) and final (D) positions of the bubble. Because of experi mental uncertainties, the relative position of the center of the transient bubble shape is accurate to only ± O. 4 j.Lm. Figure 8(a) shows that the transient bubble shape is consistently elliptical, with its major axis in the direc tion of motion for an expanding pulse but perpendicular to the direction of motion for a contracting pulse. These results indicate that the flanks of the bubble respond slowly with a velocity of, in this case, 300 cm/sec. The front and back of the bubble behave asymmetrically with respect to the initial position of the bubble, with one side moving at a velocity of order 1400 cm/sec and the other at a much lower velocity. These velocities may be compared to the radial saturation velocity which was previously determined to be symmetric in all directions for "inert" S = 0 bubbles and of magnitude 950 cm/ sec. 8 III. DISCUSSION In this section we present a semiquantitative discus sion of the experimental results in terms of a theory 3324 J. Appl. Phys., Vol. 47, No.7, July 1976 t 2 200 3OO'-~4~OO~-c5~O=0-~600 Tg (nsec) FIG. 7. Gradient propagation (BC) and net of all jumps plus gradient propagation (BE) as a function of gradient pulse length TIf, for fixed gradient pulse strength Hg=4.8 Oe and a fixed bias yulse (lIb = 1 7. 5 Oe and Tb = 1 00 nsee). Thea retical line X", (ljJx'" = 0) is a plot of Eq. (3-7) of the text. Sample and state changes as in Fig. 2. of stored bubble momentum. The theory can explain (1) why the bubbles jump forward for both positive and negative bias pulses, (2) why the net bias-jump dis placement (CE) is roughly independent of the bias pulse parameters Hb and Tb, (3) why the net jump displace ment is larger for positive than negative bias pulses, (a) EXPERIMENT ~ CONTRACTING r?r)) PULSE ~ I NITIAL TRANSIENT FINAL EXPANDING ~ PULSE ~ (a) (b) THEORY 0(3-@) V --~ (b) FIG. 8. (a) Average relative positions and shapes of bubbles before, during, and after the first bias pulse of strength Hb = 13. 5 Oe and length Tb = 100 nsec. The transient position and shape were observed by high-speed photographs at the end of the 100 nsec pulse, for both contracting (positive) and expand ing (negative) pulses. The bias pulse followed a gradient prop agation with Hg=4.8 Oe and Tg=300 nsec. (b) Schematic posi tions, signatures, and directions of motion of Bloch lines at the beginning of the bias jump of (a). A.P. Malozemoff and S. Maekawa 3324 [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: 128.59.171.71 On: Tue, 09 Dec 2014 06:56:25(a) M o '-.--J 2 BlOCH LINES (a) ~ =0 x (b) o (b) H--+--+X ;j!'/) :/.. t+2 BLOCH LINES - . D sinO" 1/1. =n5101-',0-x 0" FIG. 9. (a) Schematic "inert" S= 0 bubble, with Ifx= 0 andtwo Bloch lines of negative signatu re, illustrating definition of f3 and l/J. (1:,) Schematic S = 0 bubble with a positive stored momen tum in the form of !n unwinding Bloch-line pairs. Positive Bloch lines have a-counterclockwise rotation of l/J 1j3) as one moves counterclockwise around the bubble circumference, assuming the domain magnetization polarities shown. and (4) why the net bias-jump displacement depends monotonically on the gradient pulse parameters HK and TK• However, we do not yet have a theory which can account satisfactorily for the size of the individual bias jumps. A. Stored bubble momentum in gradient propagation We begin with a discussion of the quantityB,9 ~ x = -(rrrh)_l J dS 1PU3, z) cos{3 + ~xO' (3-1) which is a generalized wall-magnetization precession angle. Here X is the direction of bubble motion, r is the bubble radius (the bubble is assumed to be cylindri cal at all times), dS is an integral over the bubble's wall surface, (3 is an angle specifying positions along the bubble wall as shown in Fig. 9(a), z is the coordi nate normal to the film plane, 1P({3, z) is the angle of the in-plane wall magnetization measured from a fixed direction in the sample, and iJ;xo is an arbitrary con stant. The relation of iJ;x to some possible Bloch line states is illustrated in Figs. 9 and 10. If the bubble has just two Bloch lines (the minimum number required for an S= 0 bubble), we define ~x to be 0 by fixing iPxo. On the other hand, if the bubble has two clumps of Bloch lines of opposite signature (i. e., unwinding pairs) sym metrically disposed on either side of the bubble, as shown in Fig. 9(b), then one can show from Eq. (3-1) that f" = (n sin{3o sinaVa, (3-2) where n is the absolute number of Bloch lines (ignoring the extra two mentioned above), f30 is the angular posi tion of the clumps, and a is one-half their angular extent. In this calculation we have assumed Hubert's uniform rotation model, 12 so that a= f'irrAn/4r, where A= (A/~Af!)1/2 is the Bloch-line width parameter. The sign of IPx depends on the signature of the Bloch lines and the assumed direction of + x and is positive for the case indicated in Fig. 9(b). Equation (3-2) shows that if the angular extent of the Bloch-line clumps is small (a-O, sina/a-l), then ~x is simply nsintlo• Thus if 3325 J. Ap.pl. Phys., Vol. 47, No.7, July 1976 the Bloch lines are on the flanks of the bubble ({3o = trr), iPx is precisely the total number of Bloch lines E. It is also useful to note that as (3o changes from trr, IPx de creases whether f30 increases or decreases. As will be seen below, this fact is helpful in understanding why the jump direction is independent of the sign of the bias pulse. An alternative way of writing Eq. (3-2) is (3-3) where sl is the Bloch-line signature as defined in Fig. 9 and Y I is the coordinate of the ith Bloch line perpendic ular to the direction of motion. Here we consider the Bloch lines as discrete entities. Once again, if the angular extent of the Bloch lines is small and they are located on the flanks of the bubble, then Y I -± r for the positive and negative Bloch lines, and Eq. (3-3) reduces to ~" = n as before. As discussed in Refs. 8 and 9, IP" is proportional to the x component of bubble momentum Px as Px=2rrMy-1rh~x. (3-4) The equation of motion for Px and hence for IP" is de termined by dP j dt= Fx' where F" is the force in the x direction. Thus we find the important equation, r-1 d(rlP) = r(H _ H sgnX -WIX) dt g cb , (3-5) which is a generalization of previous treatments, since it allows for radius changes. Here X is the bubble prop agation velocity in the positive x direction, as illus trated in Fig. 9. Hg = -rW is the effective drive field resulting from the applied gradient W« 0) and repre senting a positive force on the bubble in the x direction. HCb is the bubble coercivity for translational motion, 6 which depends upon the radial motion';. HCb is related to the plane wall coercivity H by H b= 41T-1H [1 -(y! • •• c. c. c X)2]1/2for Irl<XorHcb=Ofor Irl>X, andHcbap- proaches unity for ;X-1 -O. The third term of the right hand side of Eq. (3-5) represents the viscous drag of the Bloch wall; we have ignored extra damping due to the Bloch lines. Equation (3-5) allows one to predict the bubble propagation behavior if the dependence of X on ~x is known. In one publication it was assumed th;lt9 ~x=O ~x-2 iiiX-4 ~x-IO tE--~ ~m--= A STACKINGOO--- ~.:- --~--- ------- :-:::----"" ------- Z A + PUNCH-THROUGH -I/y [iE--tij----i]---~x ____ '-__ V -- ,__ _-:: FIG. 10. Schematic bubble Bloch-line states and their corre sponding Ifx values, illustrating the options of stacking and punch-through in Bloch-line wind up. Punch-through, if it oc curs, initiates at points A in the bubble with iJix ~ 2. A.P. Malozemoff and S. Maekawa 3325 [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: 128.59.171.71 On: Tue, 09 Dec 2014 06:56:2510 E ::t. ~ 8 w 2l ~ 6 -' a.. (j) is 4 2 STATE CHANGE? ~ro=O ~ GRADIENT {"'-NO PROPAGATION 6-YES 4 5 6 9 Hg = -rH' (Oe) FIG. 11. Gradient propagation data as in Fig. 5, with theoret ical plots of X~ as a function of Hg fo r different values of ~x~' according to Eq. (3-8), X=O, X=Vs, l/ix>O, (3-6) and that r was constant. Applied to the case of gradient propagation, this model predicted a saturation velocity in bubble translation and a ballistic overshoot after the termination of the gradient pulse, in qualitative agree ment with experiment. However, the predicted final position of the bubble, 9 (3-7) lay considerably above the observed final bubble dis placement X~. This situation is illustrated in Fig. 11 where the gradient displacements of Fig. 5 are replotted along with a grid of lines labeled by the parameter ~~. The line ~x~ = 0 corresponds to the prediction of Eq. (3-7); clearly it lies far above the actual data pOints. A more detailed consideration of the mechanism of velocity saturation in the gradient propagation of bubbles has led to the following picture8: When the gradient drive is applied, Eq. (3-5) predicts that ~x will increase. This increase is achieved by the nucleation of Bloch lines on the front and back sides of the bubble, as iirst proposed by Hagedorn13 and as shown in Fig. 10. These Bloch lines expand and push to the flanks of the bubble. The theory is unclear as to whether the Bloch lines "stack" at the surfaces, or whether they "punch through" to the surface at points A in Fig. 10 (see diagram for ~x-2) and give rise to vertical Bloch lines which subsequently move to the flanks of the bubble. In the first case overshoot occurs because after the gradient pulse turns off the Bloch lines retract under the influen~e of their line tension, and the resulting reduction in l/ix causes a continued forward motion according to Eq. (3 -5). However in the second case, overshoot is partially eliminated because the line tension which sustains the velocity is cut. For example, if the gradient drive terminated at just the moment when ~x -4 in Fig. 10, the bubble in the "punch- 3326 J. Appl. Phys., Vol. 47, No.7, July 1976 through" case would remain as is, with two pairs of unwinding Bloch lines on its flanks, whereas the bubble in the "stacking" case would unwind back to ~x = 0 and exhibit overshoot. In the earlier studyB, comparison of theory to the experiments, which show an intermediate amount of overshoot, indicates that some mix of stacking and punch-through must occur. At the end of the gradient motion (X = X~), one would thus expect to be left with a certain number of unwinding Bloch line pairs in two clumps on the flanks of the bubble (/30 = h in Fig. 9). Clearly the smaller X~ is the larger is the number of unwinding Bloch-line pairs left behind. The actual num ber may be computed from Eq. (3-5) if one assumes a constant saturation velocity during th~ overshoot. The result may be expressed in terms of l/ix~ which is the value of ~x at the end of the motion14; it is (3-8) lPx~ is related to the number of Bloch lines remaining in the bubble by means of Eq. (3-2), assuming that the Bloch lines remain in two clumps. As discussed above, it is equal to the number of Bloch lines if /30 = h and a -O. The number of Bloch lines left after a given ex periment may be found by a graphical procedure indi cated in Fig. 11: Equation (3-8) is solved for X~ and plotted as a function of Hg with ~x'" as a parameter. Then ifx'" can be determined from the intersection of one of the diagonal lines with the X~ data points. For exam pIe, in the case of the 4. 8-0e drive. there are roughly ten Bloch lines (i. e., ~x~ -ten, or five unwinding pairs) remaining in the bubble which propagated 5 /lm. The maximum ~x'" shown in Fig. 11 is about 18 for the 8-0e drive. These results may be compared to the maximum number of Bloch lines (21Tr//21TA) which can be ac commodated around the circumference of the bubble according to Hubert's uniform rotation model. 12 In our case this maximum number is 27. Evidence for the existence of Bloch lines on the flanks of the bubble after propagation may be seen in the high-speed photography results of Fig. B(a). Theory indicates6 that the sections of the bubble wall with Bloch lines should have lower velocity than the sections without Bloch lines on the front and the back. Indeed, during the ensuing bias pulse the average velocities of the front and back of the bubbles in Fig. 8 are more than double those of the sides, leading to an elliptical bubble shape with the major axis parallel to the motion in the case of the expanding pulse and perpendicular in the case of the contracting pulse. In summary we see that the unwinding clumps of Bloch lines remaining on the flanks of the bubble after propagation imply a value of ~x= 'iP~ which is nonzero at the end of the gradient propagation. Since ~x is proportional to the bubble momentum, iF.", must cor respond to a "stored bubble momentum" which remains in the bubble at the end of the motion. 13 In the usual context of mechanics, such a concept is unintuitive, since momentum is usually considered to be a property of a moving object, but in the bubble analogy, momen tum must be generalized to include both moving and static bubbles. A.P. Malozemoff and S. Maekawa 3326 [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: 128.59.171.71 On: Tue, 09 Dec 2014 06:56:25B. Release of stored bubble momentum in bias jumps In Sec. IlIA, evidence for unwinding clumps of Bloch lines in the bubble wall has been based partly on bubble shapes during bias pulsing and partly on a theoretical deduction from X~, the measured size of the gradient displacement. In this section we propose that the bias jump effect gives confirming experimental evidence of the fact that these Bloch lines exist and constitute a stored bubble momentum. Intuitively this is obvious because the experiments described in Sec. II show that the bubble tends to move in a given direction, as if the bias pulses had somehow "released" the stored momentum. To understand the bias-jump effect, one must consider the relations between f,X, and the Bloch- line positions. The detailed equations of motion are given in the Appendix. Here we base our discussion on Eq. (3-5) as applied to the case of the bias jump. It is easy to see that application of the bias field induces spin precession and that therefore the Bloch lines move along the circumference as shown schematically in Fig. 8(b). For a contracting pulse the Bloch lines move forward in the same direction as the gradient motion which originally generated them, and for an expanding pulse they move backward. However, in the Appendix we argue that in both cases the unwinding Bloch lines move toward each other: (3-9) The essential reason is that the tangential motion of the Bloch lines is faster than the radial expansion of the bubble. According to Eq. (3-3), Eq. (3-9) implies that 1Px decreases, or in other words that the momentum is being released. Substituting Eq. (3-3) into Eq. (3-5) we find (3-10) where we have neglected the coercivity and taken Hlf = 0 since there is no field gradient. This equation shows that the Bloch-line motion leads to a translational motion of the bubble. The motion may be understood as the effect of the gyrotropic force of the Bloch-line motion. Substituting Eq. (3-9) in Eq. (3-10), we find that the bubble motion is forward (X> 0) irrespective of the sign of the bias pulse. In summary, the mechanism of the bias jump may be visualized as a Bloch-line motion during the bias pulse which propels the bubble forward via a gyrotropic force. When the bias pulse ends, radial equilibrium has not in general been reached, and thus the reverse Bloch li'ne motion when the bubble relaxes to its static size is minimal. Detailed calculations based on the model of the Appendix have, however, failed to account for the size of the individual bias jumps. We attribute this difficulty partly to the distortion of the bubble shape shown in Fig. 8(a) and partly to the nucleation of new Bloch lines on that side of the bubble where the velocity is large. On the other hand, let us consider the net displacement after all bias jumps and assume that the inert bubble is free of unwinding Bloch lines. Then there is no net contribution to the over -all momentum change from new 3327 J. Appl. Phys., Vol. 47, No.7, July 1976 Bloch lines nucleated during the bias pulses because these lines as well as the original lines are eventually annihilated. Furthermore Eq. (3-10) is approximately correct for elliptical domain shapes provided r is in terpreted as the semiaxis perpendicular to the direc tion of motion. Integrating Eq. (3.10) and utilizing Eq. (3-3), we find the net displacement of all bias jumps: Xb= l1yl(rtl6 SlY! I = l1yl(rol(r»)1P~. (3-11) Here ro is the static bubble radius and (r) is an ap propriate average of the perpendicular semiaxis during the bias jump. The radius factor r 01 (r) explains the tendency for the net bias displacement to be larger for contracting (ro(r)-1 > 1) than for expanding (ro(r)-l < 1) pulses. For example, Figs. 2 and 3 show an average net jump of 3. 5 11m for expanding and 5 11m for con tracting pulses; the ratio of these results is close to the ratio of the contracted (2.85 11m) and expanded (3.7 11m) perpendicular diameter of Fig. 8(a). Adding Xb of Eq. (3-11) to the original distance X~ determined by Eq. (3-8), we find the same formula as in Eq. (3 -7), namely Xtot = I1Hb Tg, provided we neglect coercivity and the radial changes. This remarkable result shows that the sum of the gradient displacement plus all the bias-jump displacements is the distance which would have been achieved in the original gradient displacement if no punch-through had occurred at all. Intuitively, we may view this distance as a potential maximum displacement characteristic of the original gradient pulse. That is, if the bubble did not actually go the full distance during or immediately after the gradient pulse, the theory indicates that the bubble still retains the potential to go the rest of the way. Our theory proposes that the bias pulses release the stored momentum and permit the bubble to attain this full distance. The experimental data in Figs. 2,3,5 and 7 support this simple picture reasonably well. The concept of a potential maximum displacement which is characteris tic of the gradient pulse explains the approximate inde pendence, shown in the dashed lines of Figs. 2 and 3, of the net bias-jump distance CE on the bias pulse parameters. Figures 5 and 7 show plots of Eq. (3-7); the over-all displacements BE (see Fig. 1) of the bubbles approach these theoretical lines except at the highest drives. ConSidering the scatter in the data, we have not attempted to correct these theoretical lines for the more detailed model of coercivity discussed in the Appendix or for the radial effects calculated in Eq. (3-11). In conclusion we have demonstrated the existence of a stored bubble momentum and have offered a semi quantitative theory to account for the effect. These re sults also help confirm the earlier theory of bubble overshoot,9 in particular the phenomenon of nucleation and punch-through of Bloch lines, which determines the size of the bubble propagation displacement. In related work, similar concepts are being applied to explain phenomena such as the turnaround effect. 15 A.P. Malozemoff and S. Maekawa 3327 [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: 128.59.171.71 On: Tue, 09 Dec 2014 06:56:25ACKNOWLEDGMENTS The authors thank J. Co DeLuca, who made the first experiments on the bias-jump effect, J. C. Slonczewski for extensive discussions and permission to quote theo retical results, and B. E. Argyle and P. Dekker for discussions and comparisons of experimental data. APPENDIX • In. this appendix, the coupled dynamical equations for r, X, and e (8=h-i3o, see Fig. 9) will be derived, following Slonczewski's formalism6 and neglecting the extra two Bloch lines of the S = 0 bubble. We assume that the Bloch lines remain clustered at a constant arc separation. Taking the energy derivatives, the static forces on the bubble are iJW ar = 4ITrhMH e' (A1) where He represents the combined effect of an applied bias field and the effect of the r-dependent bubble po tential. The equations of motion are obtained by balanc ing the static forces against the dynamic reaction forces: 2yHc a 0 • 1 '" [ (SI ) yH +--E=--r+X-L; cos -+ 8 e IT A 2r I r 4yH 0 __ c_ COSE sgnX IT = -~X -rlL; cos(~ + 8\ + e L;sin(SI + 8) "" r I r 'J I r ' o=n~+x~ sin(~ + 8), where E = arcsin(~/ I X I). (A2) (A3) (A4) (A5) SI is the arc distance of the ith Bloch line from the center of the clump in which this Bloch line is located, as defined by Slonczewski. 6 Equation (A2) shows that the the Bloch line motion is induced by the bias pulse. Equation (A3) is essentially the same as Eq. (3-5) since L:1sjYj=rL:1cos(S/r+ 8). Equation (A4) originates from the fact that the Bloch-line motion is tangential to the circumference of the cylindrical bubble, and it reduces to o • r= -CXcosi3 o, (A6) where 3328 J. Appl. Phys., Vol. 47, No.7, July 1976 Equation (A6) tells us that the radial velocity is smaller than the translation velocity during a bias field pulse ({3o * hand C < 1). Equation (A5) determines the bubble coercivity in the case of simultaneous radial and trans lational motion, as described earlier by Slonczewski. 6 The COSE factor in Eq. (A3) shows that the translational coercivity is reduced by the radial motion; in particular, for C = lone finds that E --8; so that as the Bloch- line clusters approach each other, the translational coercivity goes to zero. Next, we write the solutions only for times shortly after the onset of a bias pulse, when 8 is small. With this condition the solutions of Eqs. (A2)-(A4) without coercivity are expressed ;= -(2A/ a)yHeC2(j2, X= (2A/ a)yHeC8, 8 = 2YHet/n. (A7) (AB) (A9) Since 8 is small in Eqs. (A7)-(A9), we find that the radial motion is small compared with the others. In particular, L:j SlY 1 can easily be seen to be negative irrespective of the sign of~. Also, substituting Eq. (A9) into Eq. (AB), we find that X depends on the square of He and so is independent of the sign of the bias pulse. The full solutions of Eqs. (A2)-(A4) with a small number of Bloch lines will be given elsewhere. *On leave from Tohoku University, Sendai, Japan. IA.P. Malozemoff, J. Appl. Phys. 44, 5080 (1973). 2H. Nakanishi and C. Uemura, Jpn. J. Appl, Phys. 13, 191 (1974). 3R.M. Josephs and B.F. Stein, AlP ConL Proc. 24, 598 (1975). 4G. P. Vella-Coleiro and W.J. Tabor, Appl. Phys. Lett. 21, 7 (1972). 5T.J. Beaulieu and O. Voegeli. AlP Conf. Proc. 24, 627 (1975). 6J. C. Slonczewski, J. Appl. Phys. 45, 2705 (1974). 7A.P. MalozemoffandJ.C. DeLuca, Appl. Phys. Lett. 26, 719 (1975). sA.P. Malozemoff, J.C. Slonczewski, and J.C. DeLuca, AIPConf. Proc. 29, 58 (1976). 9A.P. Malozemoff and J.C. Slonczewski, IEEE Trans. Magn. MAG-H, 1091 (1975). IOA.P. MalozemoffandJ.C. De Luca, J. Appl. Phys. 45, 4586 (1974). l1G. Zimmer (private communication) has reported to us his independent observation, by high-speed photography, of bubble ellipticity during a bias pulse. 12A. Hubert, AlP Conf. Proe. 18, 178 (1973). 13F.B. Hagedorn, J. Appl. Phys. 45, 3129 (1974). 14J. C. Slonczewski (private communication). 15B.E. Argyle, P. Dekker, S. Maekawa, and J.C. Slonczewski (private communication). A.P. Malozemoff and S. Maekawa 3328 [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: 128.59.171.71 On: Tue, 09 Dec 2014 06:56:25

1.1487564.pdf

AIP Conference Proceedings 622, 447 (2002); https://doi.org/10.1063/1.1487564 622, 447 © 2002 American Institute of Physics.Liquid Sodium Experiments: The Effect of Turbulence and Lorentz Forces Cite as: AIP Conference Proceedings 622, 447 (2002); https://doi.org/10.1063/1.1487564 Published Online: 19 June 2002 Daniel R. Sisan , Woodrow L. Shew , and Daniel P. Lathrop Liquid Sodium Experiments: The Effect of Turbulence and Lorentz Forces Daniel R. Sisan, Woodrow L. Shew, and Daniel P. Lathrop Dept. of Physics, University of Maryland, College Park College Park, MD 20742 USA Abstract. We are pursuing several liquid sodium experiments in order to understand magnetic field generation in astrophysical bodies. In one experiment, pulse decay techniques using small externally applied magnetic fields quantify the system's approach toward a dynamo. Using the same apparatus, a large constant external magnetic field was applied modifying the flow through the Lorentz force. The induced field fluctuations indicate four distinct regimes that were uniquely quantified by the Elasasser number, the ratio of the Lorentz force to the advective force. Two future experiments are also described briefly. INTRODUCTION There is a large body of theoretical and numerical research suggesting that Earth and other astrophysical bodies with magnetic fields are dynamos [1]. Indeed, the equations governing flowing liquid conductors in the presence of magnetic fields (specifically, the Navier-Stokes equation and the Maxwell equations) have been known for years, and many researchers have demonstrated analytical and numerical self-generating solutions [2,3,4]. However, the parameter values used to obtain these self-generating solutions differ substantially from the parameter values of astrophysical dynamos. Approximations are necessary in theoretical and numerical models because, at relevant parameter values, the fluid motion is highly turbulent, making the solutions to these equations very difficult to resolve to appropriate length scales. Experimental models, on the other hand, do not have this resolution restriction—"calculations" can be performed in seconds with highly turbulent experiments that would take decades, or even centuries, to perform with current computer technology. So liquid metal experiments are an excellent complement to theoretical and numerical work, and they have the potential to provide a better understanding of magnetic field generation. Nonetheless, only a few research groups have attempted, or are attempting, this type of experiment. (A significant proportion of the groups are, in fact, represented at this conference.) In this paper we report results of, and plans for, several liquid sodium experiments at the University of Maryland. The results come from a 30 cm diam. mechanically forced experiment. In this system, measurements were taken using pulsed external magnetic fields (100 G) and constant external magnetic fields (up to 2000 G). Our CP622, Experimental Chaos: 6th Experimental Chaos Conference, edited by S. Boccaletti et al. © 2002 American Institute of Physics 0-7354-0071-7/02/$ 19.00 447 future plans include a 60 cm dia. rotating convection experiment, and a 3 m dia. experiment that will be mechanically forced and rotating. Hall probes coils FIGURE 1. Cross-section of the 30 cm dia. experiment. Rotating shafts attached to propellers drive the flow. Hemholtz coil pairs supply external magnetic fields along, or perpendicular to, the rotation THE APPROACH TO SELF-GENERATION The device of the 30 cm dia. experiment (Fig. 1) is a hollow stainless steel sphere filled with liquid sodium [5]. Two propellers, powered independently by 7.5 kW motors, drive the flow. Hall effect probes outside the sphere measure the magnetic field. This system does not self-generate, but, using a pulse decay technique, a shift toward self-generation is observed. In this technique, an external magnetic field is applied and then rapidly turned off. The magnetic field decays exponentially due to induced currents in the liquid sodium (see Fig. 2). With no fluid motion, the decay constant was within 3% of the theoretical value for field diffusion in a conducting sphere. With the propellers spinning, however, there may be fluid motion that enhances field growth, in which case the decay constant becomes less negative. If the fluid motion enhances the field strongly enough to overcome resistive damping, then the system would self-generate. The competition between resistive decay and the stretching properties of the fluid motion is quantified by the magnetic Reynolds number, Rem = UL/r], where U is a characteristic velocity, L is a characteristic length, and 77 is the magnetic diffusivity, the inverse of the product of conductivity and magnetic permeability. We define U as the tip speed of the propellor, and L as the radius of the sphere. A typical applied magnetic pulse is shown in Fig. 2(a). The decay constant, measured as in Fig. 2(b), depends on the magnetic Reynolds number. For external fields aligned with the axis of rotation, the decay constant shifts toward self-generation, as seen in Fig. 2(d). Available power limits the magnetic Reynolds number that can be obtained practically. 448 B(G) 10 20 30 40 50 60 70 FIGURE 2. A pulse decay measurement estimates the largest decay rate of the system and shows a trend toward self-generation for external fields aligned with the shafts. A Hall probe signal shows the external magnetic field pulse (a) used to estimate the decay rate (the slope of the semilog plot in (b)). (c) shows the same data as in (b) divided by the exponential function. The mean decay rate is plotted in (d). The solid line, showing shifts toward self-generation, is for fields aligned with the shafts and the dotted line, showing shifts away from self-generation, is for fields perpendicular to the shafts. The error bars indicate shot-to-shot fluctuations in the decay constants. The distribution of decay constants is shown in the inset of (d); the narrow distribution is for no flow and the broad distribution is for Rem=42. The decay rates have been normalized to yield -1 for the purely resistive case (no flow). The distribution of decay constants shown in the inset plot of Fig. 2(d) leads to a prediction that the transition to self-generation will exhibit intermittent bursts in magnetic fields. This would occur as the mean decay rate nears zero (at larger Rem)— flow states associated with the positive side of this distribution should lead to growth of the magnetic field. As the field grows the Lorentz force will cause a back-reaction on the flow, suppressing the growth, if the system is non-hysteretic. Thus, just below transition the system will have short bursts of growing fields followed by periods of decaying fields [6]. FLUCTUATIONS While the external field is applied, fluctuations are observed in the induced field (Fig. 3). This external field (100 G) is too small to cause a back reaction on the flow, and the magnetic field is a passive vector modified by the fluid. Thus, the fluctuations in the induced field reflect the fluctuations of the velocity in the underlying fluid turbulence. Using Taylor's hypothesis, Fig 3(b) shows that, indeed, the system 449 contains Kolmogorov turbulence—not surprising for our system which has a (hydrodynamic) Reynolds number of order 107. (a) (b) B(t) (G) f(Hz) FIGURE 3. Typical field fluctuations. A time series is shown in (a) with the mean applied field (120 G) and the mean induced field (-17 G) subtracted. The power spectra (b) shows features of broadband turbulence including a power law decay at low frequency. The offset line represents a best fit exponent, r , consistent with Kolmogorov scaling. THE EFFECT OF THE LORENTZ FORCE The field produced by the coils used in our pulse decay measurements is too small to have a significant back reaction on the fluid motion. In order to better understand the saturation process we installed a magnet system that produces fields (2 kG) with magnetic energy comparable to the kinetic energy of the flow, allowing the Lorentz force to influence the fluid motion. As we increased the field strength, we observed qualitatively distinct changes in the system's response, as shown in Fig. 4. For the Lorentz force to affect the system it needs to overcome the inertia of the flow, thus a logical parameter quantifying the relative strength of the field is the Elsasser number, the ratio of the Lorentz force to the advective force (sometimes called the Interaction parameter). We define the Elsasser number as: A = - (1) where B0 is the externally applied field, £1 is the rotation rate, b is the sphere radius, a is the propellor radius, \io is the permeability of free space (and sodium), p the density of sodium, and r\ is the magnetic diffusivity of sodium. The Elsasser number for Earth is thought to be order one [4]. The Elsasser number for a general self-generating system depends on the effect the Lorentz forces have on the dynamics, which, with this experiment, we are examining. 450 A=l FIGURE 4. The induced field passes through four distinct regimes as the externally applied magnetic field is increased. For A=l, the external magnetic field is too weak to alter the fluid motion and the magnetic field is a passive vector; the induced field fluctuations reflect the underlying fluid turbulence. For A=4, the magnetic field suppresses turbulence in the system, seen in the decreased magnitude of the induced field fluctuations. For A=9, large oscillations emerge; the frequency of oscillation increases linearly with applied field strength. Finally, for A=20 the oscillations decrease again in magnitude and become cleaner; the oscillations consist of three sharp frequencies—the dominant frequency at l/2 the propellor rotation rate and two sidebands near 1A and 3/4 the rotation rate—that are independent of applied field strength. FUTURE EXPERIMENTS We are close to completing construction of a new 60 cm dia. rotating convection experiment, shown in Fig. 5. This experiment has a spherical geometry that is similar to the geometry in the Earth's outer core. Sodium fills the region between the smaller inner sphere, which is cooled with hexane, and the larger spherical shell, which is heated from the outside with halogen heat lamps. Centrifugal forces and the radial temperature difference drive the flow. We are collaborating with Gary Glatzmaier at the University of California, Santa Cruz, who is numerically simulating the experiment. This comparison between simulation and experiment will serve as a benchmark for the simulation, which, as with simulations of dynamos in general, includes necessary approximations. 451 Another, much larger experiment (3 m diameter), is in the planning stages. The linear trend of Fig. 2(d) suggests that self-generation may occur at larger values of Rem for a similar geometry. For this planned experiment, Rem up to 400 can be obtained. FIGURE 5. Basic structures of the 60 cm diam. rotating convection experiment. The smaller sphere in the photo fits inside the larger shell to create the geometry in the cross-sectional drawing on the left. The system will have a maximum rotation rate of 100 RPS and a maximum heat flux of 10 kW. ACKNOWLEDGMENTS This work was funded by the NSF Earth Sciences-Geophysics and the NSF Instrumentation Development programs, the Research Corporation, and the University of Maryland. We gratefully acknowledge assistance from Thomas Antonsen, A.B. Cawthorne, James Drake, Khurrum Ghulani, Adil Hassam, Donald Martin, Edward Ott, and Nicholas L. Peffley. REFERENCES P.H. Roberts, "Fundamentals of Dynamo Theory" In Lectures on Solar and Planetary Dynamos, ed. M.R.E. Proctor, A.D. Gilbert. 1-58. Cambridge, UK: Cambridge Univ. Press, 1994. F. H. Busse, "Homogeneous Dynamos in Planetary Cores and in the Laboratory," Annu. Rev. Fluid Mech. 2000, 32:383-408. M.L. Dudley and R.W. James, "Time-dependent kinematic dynamos with stationary flows," Proc. R. Soc. Lond. A 425, 407 (1989). E. Dormy, J.-P. Valet, and V. Courtillot, "Numerical models of the geodynamo and observation constraints," Geochem. Geophys. Geosys., vol. 1 (2000). N.L. Peffley, A.B. Cawthorne, and D.P. Lathop, "Toward a Self-generating Magnetic Dynamo: the Role of Turbulence," Phys. Rev. E 61, 5287 (2000). D. Sweet, E. Ott, J.M. Finn, T.M. Antonsen, Jr. and D.P. Lathrop, "Blowout Bifurcations and the Onset of Magnetic Activity in Turbulent Dynamos," Phys. Rev. E 63, 6621 (2001). 452

1.5139098.pdf

Appl. Phys. Lett. 116, 072406 (2020); https://doi.org/10.1063/1.5139098 116, 072406 © 2020 Author(s).Detecting current-induced quantum magnetization fluctuations with a spin- torque nano-oscillator Cite as: Appl. Phys. Lett. 116, 072406 (2020); https://doi.org/10.1063/1.5139098 Submitted: 19 November 2019 . Accepted: 12 February 2020 . Published Online: 20 February 2020 Lianwei Wang , Yong Wang , and Ke Xia Detecting current-induced quantum magnetization fluctuations with a spin-torque nano-oscillator Cite as: Appl. Phys. Lett. 116, 072406 (2020); doi: 10.1063/1.5139098 Submitted: 19 November 2019 .Accepted: 12 February 2020 . Published Online: 20 February 2020 Lianwei Wang,1,2 Yong Wang,1,a) and Ke Xia2,b) AFFILIATIONS 1School of Physics, Nankai University, Tianjin 300071, China 2Department of Physics, Beijing Normal University, Beijing 100875, China a)Electronic mail: yongwang@nankai.edu.cn b)Electronic mail: kexia@bnu.edu.cn ABSTRACT Interactions between conduction electrons and quantum fluctuations of ferromagnetic order have seldom been observed in magnetoelectronic devices. We show that current-induced quantum magnetization fluctuations can be detected using a spin-torque nano-oscillator by measuring its linewidth at different temperatures. The relative linewidth in a special dynamic region of the device can distin-guish quantum magnetization fluctuations from their thermal counterparts, which is important in understanding magnetization dynamics beyond the mean-field level in magnetoelectronic devices. Published under license by AIP Publishing. https://doi.org/10.1063/1.5139098 Ferromagnetic (FM) materials play a fundamental role in magne- toelectronic devices, where interactions between conduction electrons and FM order affect their respective dynamics. 1,2For example, in a spin valve, where a nonmagnetic (NM) material layer is sandwichedbetween two FM layers [see Fig. 1(a) ], the conduction electrons become spin-polarized when they flow through the first FM layer. 1,2 Furthermore, the resistance of the spin valve varies significantly whenthe relation between the directions of magnetization of the two FMlayers is tuned by an external magnetic field; this is the giant magneto-resistance (GMR) effect. 3,4On the other hand, the spin-polarized electrons can transfer their spin angular momentum to the FM layer and affect its magnetization dynamics in what is called the spin trans-fer torque (STT) effect. 5,6Theoretically, the FM order is described by the magnetization vector field, whose dynamics is given by the Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equation,7–10while the electrons move in the spin-dependent potential created by the FMorder, as governed by the Schr €odinger equation. 9,10Such a semiclassi- cal picture provides the standard theoretical basis for modern magne- toelectronic techniques.1,2 From a microscopic viewpoint, the emergence of FM order in materials is attributed to the strong electron–electron interactiongoverned by the laws of quantum mechanics. 11If the effects of quan- tum fluctuations become important in a magnetoelectronic device, themean-field description of FM order may no longer be sufficient and may even break down completely. Recent measurements of differential resistance in a spin-valve device at low temperatures suggested that quantum fluctuations therein are enhanced by spin transfer,12in a process already anticipated in a full quantum theory of STT13–16and further analyzed theoretically.17–19Despite these developments, further experimental evidence is lacking to directly demonstrate the interplay between quantum fluctuations of FM order and conduction electronsunder nonequilibrium conditions in magnetoelectronic devices.Unlike their thermal counterparts, current-induced quantum magneti- zation fluctuations depend on the current amplitude, the degree of electron spin polarization, and the angle between the magnetizationvector and electron spin (see the supplementary material ). 13,14In this Letter, we show that these features can be detected by measuring the linewidth of a spin-torque nano-oscillator (STNO) at low temperatures. An STNO is a spintronic device with a spin-valve structure as shown in Fig. 1(a) . The magnetization of one FM layer is driven into precession motion by the spin-polarized electrons, and this then gen- erates microwave signals.20–33Here, the STT effect can compensate the intrinsic damping effect and sustain periodic magnetizationdynamics, while the microwave power is generated from the oscillating magnetoresistance of the device. Since the precession orbital perturbed Appl. Phys. Lett. 116, 072406 (2020); doi: 10.1063/1.5139098 116, 072406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplby any magnetization fluctuation will induce a finite linewidth in the microwave spectrum,34–50this should be an ideal platform to detect quantum fluctuations of FM order. The performance of an STNO device is characterized by its power spectral density (PSD), which is the Fourier transform of the autocorrelation function of the microwave signal or, equivalently, the oscillatory electric current IoðtÞoutput from the STNO. IoðtÞis in principle stochastic owing to either its intrinsic shot noise or the ran- dom magnetoresistance caused by magnetization fluctuations. Therefore, the PSD of an STNO can be calculated as SðxÞ/C24ð1 /C01IoðtÞeixtdt/C12/C12/C12/C12/C12/C12/C12/C12 2*+ ; (1) where h /C1/C1/C1 i represents the average over the statistical ensemble fIoðtÞg. To take quantum magnetization fluctuations into account, we apply the quantum trajectory method13(also see the supplementary material ) to calculate the PSD of a simple STNO, which is modeled by the precession motion of a single-domain magnet driven by spin- polarized electrons. Here, the magnet is described by a macrospincoherent state jJ;H;Uiinstead of the magnetization vector M.T h e effective magnetic field and the damping effect cause unitary evolution of the state, while the spin-polarized electrons are injected in sequence to randomly change the state of the magnet, 13as shown schematically inFig. 1(a) . The magnet is assumed to be an elliptically shaped FM film in the y–zplane, and the electrons are transported along the xdirec- tion [see Fig. 1(a) ]. The effective magnetic field is then given as Heff¼Hkmz/C0HdmxþHa.7,51Here, m is the normalizedmagnetization vector, and Hk,Hd,a n dHadescribe the uniaxial anisot- ropy, the shape anisotropy, and the external magnetic field Ha, respec- tively. The Gilbert damping coefficient is denoted by a.D u r i n gt h e simulations, the cross-sectional area and the thickness of the film are set as A¼810 nm2andd¼1 nm, and the saturated magnetization is Ms¼1:27/C2106A/m. In addition, we set Hk¼0.03 T, Hd¼1.6 T, anda¼0:02. To calculate the scattering matrix for the quantum magnet and the electrons, the electron velocity is set as vg¼3:5 /C2107cm/s, and the spin-dependent potential barrier is set as V6 ¼0:760:6e V .13,14,51T h ee x t e r n a lm a g n e t i cfi e l d Haand the injected current Ican be tuned to control the dynamics of the magnet. The ensemble of quantum trajectories of the magnet is obtained by simulating the stochastic Schr €odinger equation (see the supplemen- tary material ),13which gives the average trajectory and quantum fluc- tuation of the magnetization dynamics.13InFig. 1(b) , one quantum trajectory of the precessing magnet is presented (green line), whichrandomly deviates from the deterministic orbit (red line) obtained from the LLGS equation at zero temperature. The external magnetic field and the injected current amplitude are set as H a¼ð0;0;0:1ÞT andI¼0.5 mA, respectively. Meanwhile, the states of electrons after scattering can also be collected to give the output current IoðtÞof the STNO,13as shown in Fig. 1(c) . Here, “1” corresponds to the current pulse due to each transmitted electron. The spin-polarization direction of the electrons is along the zaxis, with the degree of spin polarization p¼1. The PSD SðxÞof the STNO is then obtained by Eq. (1)and is shown in Fig. 1(d) . Here, the simulation time of each quantum trajec- tory is set as 200 ns to obtain a frequency resolution of 5 MHz for the spectra, and 1000 quantum trajectories are generated to sample the statistical ensemble in Eq. (1). The oscillation frequency fand the line- widthDfcan then be retrieved from the calculated PSD. In particular, the finite linewidth here originates from the quantum fluctuations of the magnet caused by the spin-polarized electrons. The dynamical phase diagram of the studied STNO device is obtained by sweeping the injected spin-polarized current Iand the external magnetic field Ha¼ð0;0;HÞin the zdirection, as shown in Fig. 2(a) . When there is no spin-polarized current ( I¼0), the magnet remains static and its magnetization direction is determined by the applied magnetic field. If H>0, the magnetization direction is along theþzdirection when I¼0, which is parallel to the spin-polarization direction of the injected electrons. This parallel (P) state can still be stable for small current, but will become unstable and evolve into an in-plane precession (IPP) mode around the zaxis [see Fig. 2(b) ]a sIis increased. When Iis increased further, the amplitude of the IPP mode becomes so large that the two endpoints of the precession orbital become glued together [see Fig. 2(c) ], forming a “gluing bifurcation” (GB) mode in the phase diagram. Beyond this GB region, a larger cur- rent will drive the magnetization dynamics into an out-of-plane pre- cession (OPP) mode around the xaxis [see Fig. 2(d) ]. The amplitude of the OPP mode will decrease and finally disappear when the current is continuously increased. The magnet will then become static again and will be switched to the— zdirection by a large enough current. In the case H<0, the magnetization direction will be along the— zdirec- tion when I¼0, i.e., in the antiparallel (AP) state. When the spin- polarized current is applied, the magnet can be driven into the OPP mode. Thus, the quantum trajectory method here has reproduced the dynamical modes of an STNO already revealed by the classical LLGS FIG. 1. (a) Schematic diagram of the FM—NM—FM sandwich structure of an STNO device and the quantum scattering process therein. The macrospin coherent state jJ;H;Uiof the free layer (FM2) is continuously changing owing to the scat- tering processes with the injected electrons, which have been spin-polarized by thefixed layer (FM1). Meanwhile, the electrons are either reflected or transmitted. (b) Trajectory of the magnetic dynamics obtained from the stochastic Schr €odinger equation (Q) and the LLGS equation (C). (c) A series of electron states after scat-tering. Here, “0” represents the reflected state and “1” the transmitted state. (d)Power spectral density of the STNO calculated from the ensemble of scattered electrons generated by the quantum trajectory method.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 072406 (2020); doi: 10.1063/1.5139098 116, 072406-2 Published under license by AIP Publishingequation.7,28,29Furthermore, the results suggest that the quantum magnetization fluctuations will induce an extended GB region in thedynamical phase diagram at zero temperature. In contrast, in classicalmicromagnetic simulations, the GB region can only appear when ther-mal fluctuations at a finite temperature are included. 28 We further calculate the PSD of the STNO by sweeping the cur- rent amplitude Ifrom 0.09 to 0.9 mA and fixing the external magnetic fieldHa¼ð0;0;0:1ÞT. The degree of spin polarization pof the conduction electrons is set as 1. As suggested by the dynamical phase diagram in Fig. 2(a) , the magnetization dynamics first undergo the transition from the IPP mode to the GB mode, and finally reach theOPP mode. The current-dependent frequency fand linewidth Dfof the STNO are then retrieved from the obtained PSD (see the supple- mentary material ). As shown in Fig. 3(a) , the oscillation frequency f Qof the IPP mode decreases from 24.5 GHz to about 7.8 GHz when the current amplitude Iis increased from 0.09 mA to 0.19 mA, where the preces- sion trajectory gradually becomes longer (see the supplementary mate- rial);fQof the OPP mode then continuously increases from 7.8 GHz to 42 GHz when Igrows from 0.19 mA to 0.9 mA, where the precession trajectory becomes shorter (see the supplementary material ). The min- imum frequency 7.8 GHz is reached at the critical GB mode. For com-parison, we also show the current-dependent frequency f Cof the STNO at finite temperatures 5 K and 10 K based on the stochastic LLGS equation (see the supplementary material ), and no difference in the frequencies is found for the three cases. Therefore, frequency mea-surement of the STNO cannot distinguish the quantum magnetizationfluctuations from their thermal counterparts. The current-dependent linewidth Df Qof the STNO is shown in Fig. 3(b) . Here, DfQcan be as large as 1.0 GHz when the STNO oper- ates in the GB mode. In the presence of magnetization fluctuations,the dynamics near the GB mode can be viewed as randomly hopping processes between the IPP and OPP modes. Since the trajectory of theIPP mode is about twice as long as that of the OPP mode, the oscilla- tion frequency can vary widely, which results in a large linewidth. Otherwise, the dynamics will be either IPP mode or OPP mode, andDf Qwill follow the same trend as the oscillation frequency fQ.B e s i d e s , thermal fluctuations at 5 K and 10 K can also result in similar features in the current-dependent linewidths DfCð5KÞandDfCð10 KÞ.S i n c e the thermal magnetization fluctuations are constant for a given tem-perature (see the supplementary material ), the results here imply a strongly nonlinear dependence of the linewidth on the current ampli- tude. Thus, the current-dependent linewidth Df Qis also unable to identify the quantum magnetization fluctuations directly, since thelinewidth Df Cinduced by the thermal fluctuations has similar features. However, the quantum magnetization fluctuations can manifest their existence in the relative linewidth of the STNO, if there exists aregion in the dynamical phase diagram where the linewidth Dfis pro- portional to the magnetization fluctuation W, i.e.,Df¼bðI;HÞW. Even though the coefficient bðI;HÞis an unknown function of the current amplitude Iand magnetic field H, one can still obtain W Q ¼ðDfQ=DfCÞWCin this special region. For a constant thermal magne- tization fluctuation WCat fixed temperature, the quantum magnetiza- tion fluctuation WQwill then be proportional to the relative linewidth DfQ=DfC. Experimentally, this dynamical region can be found by mea- suring the temperature-dependent linewidth (see the supplementary material ), or by measuring the relative linewidth DfCðT1Þ=DfCðT2Þat two different temperatures T1andT2. When the thermal magnetiza- tion fluctuation is dominant, one should have DfCðT1Þ=DfCðT2Þ ¼T1=T2in this dynamical region. Figure 3(c) shows the current- dependent relative linewidths DfCð5KÞ=DfCð10 KÞandDfQ=DfC ð10 KÞ.O n ec a ns e et h a t DfCð5KÞ=DfCð10 KÞ¼0:5f o rc u r r e n t amplitudes I2½0:25;0:7/C138mA, which suggests that the linewidth will be proportional to the magnetization fluctuation in this range. ThisFIG. 3. (a) Current-dependent frequency from the quantum trajectory method at 0 K and the classical LLGS equation at 5 K and 10 K. (b) Current-dependent linewidth from the quantum trajectory method at 0 K and the classical LLGS equation at 5 Kand 10 K. (c) Current-dependent relative linewidths Df Q=DfCð10 KÞandDfCð5KÞ= DfCð10 KÞ. (d) Spin-polarization-dependent PSD with fixed I/C2p¼0:5 mA at zero temperature. The inset shows the dependence of DfQ;DfCð5KÞ, andDfCð10 KÞ on the degree of spin polarization p. FIG. 2. (a) Dynamical phase diagram of the STNO device. I, current amplitude; H, amplitude of the magnetic field along the zaxis; P , parallel state; AP, antiparallel state; SP, stable point; IPP, in-plane mode; OPP , out-of-plane mode; and GB, glue bifurcation mode. (b) Trajectory of an IPP mode. (c) Trajectory of a GB mode. (d) Trajectory of an OPP mode.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 116, 072406 (2020); doi: 10.1063/1.5139098 116, 072406-3 Published under license by AIP Publishingcan also be confirmed by the linewidth-temperature curves in the sup- plementary material .M e a n w h i l e ,t h el i n e a ri n c r e a s ei n DfQ=DfCð10 KÞ with Iin this region indicates that the quantum magnetization fluctua- tions are proportional to the current amplitude, which coincides with previous theoretical predictions.13,14 In addition, the quantum magnetization fluctuations can also be enhanced by the imperfect polarization of the electron spin.13,14InFig. 3(d), we show the PSD calculated for the spin-polarized current with different p. The current amplitudes Ihave been set to satisfy the rela- tion I/C2p¼0:5 mA, since the same precession dynamics should be exploited for comparison. Here, the peak in the PSD is broadened when pis decreased from 1.0 to 0.2. Correspondingly, the linewidth DfQis inversely proportional to p,a ss h o w ni nt h ei n s e to f Fig. 3(d) . By contrast, the linewidth DfCdue to thermal magnetization fluctua- tions is irrelevant to the degree of spin polarization. Therefore, thequantum magnetization fluctuations caused by spin-polarized elec- trons can also be detected by measuring the dependence of the line- width on the electron spin polarization at extremely low temperaturesif the thermal fluctuations are suppressed. We have also investigated the variations in frequency and line- width when the external magnetic field is tuned with fixed currentamplitude. Figure 4(a) presents the results for an OPP mode with I¼0.5 mA and p¼1. Here, the oscillation frequency increases mono- tonically from about 16.5 GHz–39.3 GHz when the field His increased from 0 T to 0.3 T since the orbital gradually becomes smaller (also see thesupplementary material ). Once again, frequency measurements alone cannot distinguish quantum magnetization fluctuations from thermal magnetization fluctuations. In addition, all the linewidths Df Q;DfCð5KÞ,a n d DfCð10 KÞincrease monotonically with the increasing magnetic field and also cannot be utilized to detect the quantum magnetization fluctuations directly. However, the different features of quantum and thermal magne- tization fluctuations can be revealed by the relative linewidths DfQ=DfCð10 KÞandDfCð5KÞ=fCð10 KÞ.Figure 4(a) shows that DfCð5KÞ=fCð10 KÞis nearly equal to 0.5 for fields H2½0;0:3/C138T,which means that the linewidth is proportional to the magnetization fluctuation in this dynamical region (see the supplementary material ). Therefore, the dependence of DfQ=DfCð10 KÞonHinFig. 4(a) sug- gests that the quantum magnetization fluctuation will decrease with increasing magnetic field H. This phenomenon is attributed to the angle-dependent property of the current-induced quantum magnetiza-tion fluctuations (see the supplementary material ), 13,14which are pro- portional to 1 þmzin our model here. This is further verified by calculating the average value hmzi, which does indeed decrease when the orbital evolves with increasing magnetic field (see the supplemen- tary material ). We further consider the STNO working in the IPP mode with I¼0.13 mA, as shown in Fig. 4(b) . When the magnetic field H increases from 0.1 T to 0.6 T, the oscillation frequency fand linewidth Dfalso monotonically increase, since the orbital shrinks (see the supplementary material ). Nevertheless, no dynamical region can meet the condition DfCð5KÞ=fCð10 KÞ¼0:5( a l s os e et h e supplementary material ), and the quantum magnetization fluctuations cannot be quantitatively measured from the relative linewidth DfQ=fCð10 KÞ. However, the opposite trend of DfQ=fCð10 KÞandDfCð5KÞ=fCð10 KÞ suggests that the quantum magnetization fluctuations cannot be con-stant like the thermal magnetization fluctuations and, in fact, will decrease with the increasing magnetic field. This feature can also be explained by the angle-dependent property of the current-inducedquantum magnetization fluctuations (see the supplementary material ). In conclusion, we have shown that current-induced quantum magnetization fluctuations can be detected by measuring the relativelinewidth of the power spectral density of an STNO device. Owing to its highly nonlinear oscillatory dynamics, the measurement should be performed in a dynamical region where the linewidth is proportionalto the magnetization fluctuation. 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1.1900939.pdf

Ultrafast magnetization dynamics of epitaxial Fe films on AlGaAs (001) H. B. Zhao, D. Talbayev, Q. G. Yang, G. Lüpke, A. T. Hanbicki, C. H. Li, O. M. J. van ’t Erve, G. Kioseoglou, and B. T. Jonker Citation: Applied Physics Letters 86, 152512 (2005); doi: 10.1063/1.1900939 View online: http://dx.doi.org/10.1063/1.1900939 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/86/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Static and dynamic magnetic properties of epitaxial Fe1.7Ge thin films grown on Ge(111) J. Appl. Phys. 111, 07D502 (2012); 10.1063/1.3672396 Structural and magnetic properties of epitaxial L 2 1 -structured Co 2 ( Cr , Fe ) Al films grown on GaAs(001) substrates J. Appl. Phys. 97, 103714 (2005); 10.1063/1.1888050 Mesofrequency switching dynamics in epitaxial CoFe and Fe thin films on GaAs(001) J. Appl. Phys. 97, 053903 (2005); 10.1063/1.1854205 Magnetic properties of epitaxial NiFe/Cu/Co spin-valve structures on GaAs(001) J. Appl. Phys. 87, 5947 (2000); 10.1063/1.372575 Sweep rate-dependent magnetization reversal in epitaxial Fe/GaAs(001) and Fe/InAs(001) thin films J. Appl. Phys. 87, 5926 (2000); 10.1063/1.372569 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: 133.1.198.126 On: Tue, 02 Dec 2014 05:58:15Ultrafast magnetization dynamics of epitaxial Fe films on AlGaAs 001 H. B. Zhao, D. Talbayev, Q. G. Yang, and G. Lüpkea! Department of Applied Science, College of William & Mary, Williamsburg, Virginia 23185 A. T. Hanbicki, C. H. Li, O. M. J. van ’t Erve, G. Kioseoglou, and B. T. Jonker Naval Research Laboratory, Washington, D.C. 20375 sReceived 2 December 2004; accepted 1 March 2005; published online 8 April 2005 d Uniform magnetization precessions are generated by ultrafast optical excitation along the in-plane easy axis f100g, as well as along the hard axis f1-10g, in epitaxial Fe films grown onAlGaAs s001d over a wide range of applied magnetic fields. From the temporal evolution of the coherentmagnetization precession, we determine the magnetic anisotropy constants and damping parameterswhich are crucial in designing fast magnetic switching devices and spintronics devices. © 2005 American Institute of Physics .fDOI: 10.1063/1.1900939 g The process of magnetization reversal in thin films is of considerable importance in magnetic and magnetooptical re-cording, and in the context of magnetoelectronics. 1These applications require very small ferromagnetic elements withuniaxial anisotropy for storing binary information in twostable states. Epitaxial growth of a ferromagnetic metal on asemiconductor provides an approach to realize high-densityarrays of magnetic elements by using intrinsic in-planeuniaxial anisotropy instead of shape anisotropy. A ferromag-netic Fe film grown epitaxially on a GaAs s001dsubstrate is a particularly promising system because of its small latticemismatch and strong uniaxial magnetic anisotropy. 2–4The large magnetoresistance shown in magnetic tunneljunctions 5,6and the achievement of high spin injection effi- ciency into GaAs at room temperature via a Schottky barrier contact7illustrate the importance of this system. Several techniques have been applied to characterize magnetization dynamics in ferromagnetic films, includingferromagnetic resonance sFMR dsRef. 8 dand Brillouin light scattering. 9Juet al.10studied spin wave excitations in exchange-biased NiFe/NiO layers by laser-induced unpin-ning of the magnetization. Acremann et al. 11produced a lo- cal transient magnetic field in a CoFe film by optical currentgeneration at a Schottky barrier. Recently, Van Kampen et al. 12developed an optical pump-probe technique to excite and detect spin waves in magnetic layers by exploiting thetemperature dependence of the magnetic anisotropy.The pre-cessional motion of the excited spins was measured on apicosecond time scale, allowing for the quantitative determi-nation of damping and anisotropy parameters. In this letter, we report on time-resolved magneto-optical Kerr-effect sTRMOKE dexperiments which exploit the tem- perature dependence of the in-plane uniaxial magnetic aniso-tropy to launch coherent magnetization precession in thin Fefilms grown epitaxially onAlGaAs s001d.Auniform magne- tization precession is excited around both the hard and easyaxes over a wide range of applied magnetic fields. The de-pendence of spin wave frequency on sample orientation andmagnetic field at low excitation is used to determine themagnetic anisotropy and damping constants. Fes001dfilms with a thickness of 10 nm are grown on AlGaAs s001dby molecular-beam epitaxy. 7The equilibriummagnetic properties are verified ex situby standard magneto- optical Kerr-effect sMOKE dmeasurements and vibrating sample magnetometry sVSM d. The magnetization exhibits a hard axis out-of-plane along s001d, and uniaxial in-plane an- isotropy superimposed on a four-fold cubic anisotropy, lead-ing to an in-plane easy axis along f100gand hard axis along f1-10gdirection.3,8,13–15TRMOKE experiments are per- formed with a 150 fsTi:Sapphire amplifier system at 800 nmwavelength. A modulated pump beam with 15 mJ pulse en- ergy is focused to a spot of 1 mm in diameter on the sampleas illustrated in Fig. 1 sad. In equilibrium, the magnetization is along an effective field H eff, which is a sum of the applied field, the demagnetization field and the anisotropy field. Thepump beam instantaneously heats up the Fe film. The instan-taneous lattice expansion changes the anisotropy of the filmand induces a transient magnetic field H tr. The magnetization adElectronic mail: luepke@as.wm.edu FIG. 1. sadExperimental configuration with longitudinal geometry, sbdsche- matic diagram of the coherent excitation process of uniform magnetizationprecession, scdtransient Kerr signal, and sddFourier transforms of scdafter picosecond excitation of a 10 nm thick epitaxial Fe film with magnetic fieldH=560 Oe applied along f110g,f100g, and f1-10gdirections.APPLIED PHYSICS LETTERS 86, 152512 s2005 d 0003-6951/2005/86 ~15!/152512/3/$22.50 © 2005 American Institute of Physics 86, 152512-1 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: 133.1.198.126 On: Tue, 02 Dec 2014 05:58:15Mstarts to precess around Htr, as illustrated in Fig. 1 sbd. WhenHtrhas vanished si.e., the film has returned to equilib- rium d, the vector Mis away from its original equilibrium orientation along Heff. Therefore, it starts to precess around Heff. The effect on the magnetization is measured by a much weaker s,1mJd, time delayed probe beam using the MOKE technique. In the longitudinal geometry at an incidence angle of 45°, we detect both in-plane and out-of-plane componentsof the magnetization M. By varying the time delay Dtbe- tween pump and probe, the magnetization precession is mea-sured as a function of time after excitation. Figure 1 scdshows typical results for the magnetization evolution after excitation of the 10 nm thick Fe film for anapplied magnetic field of 560 Oe. The Fourier spectra of thetransient MOKE signals reveal two coherent excitations withdistinct frequencies fFig. 1 sddg. The high-frequency oscilla- tion at 42.5 GHz is independent of the magnetic field. Thismode corresponds to a transverse acoustic phonon which isgenerated in the GaAs substrate by the instant lattice expan-sion of the Fe film at Dt=0 ps. The coherent TA phonon is coupled to the photon by conserving the momentum throughbackward Raman scattering causing oscillation of the re-flected probe beam intensity. 16The second oscillation at a lower frequency s5–15GHz ddepends on the magnitude and orientation of the magnetic field. Figure 1 scdshows that the film relaxes back to quasi-equilibrium after approximately50 ps, and therefore the measured precession occurs in theoriginal anisotropy field, thus revealing the equilibrium mag-netic properties. 12As shown below, the anisotropy and de- magnetization field of the coherent magnetization precessionagrees well with the parameters of the uniform FMR modemeasured in a 96 Å thick Fe film on GaAs s001d. 8 As shown in Figs. 1 scdand 2 sad, large spin waves are generated along the f1-10gandf100gdirections, while only a weak spin wave excitation is observed along the f110gdirec- tion where the magnetization is less canted. For a magneticfield of 560 Oe applied along the f1-10gorf100gdirection, the magnetization is canted away from the applied field be-cause of the uniaxial character of the magnetic anisotropy ofthe Fe film. The cubic magnetocrystalline anisotropy alsocontributes to the transient anisotropy field, but this effect issmall. This has been verified with a 50 nm thick epitaxial Fefilm which exhibits no uniaxial anisotropy. In this controlsample, the precession amplitude is five times smaller. Fur-thermore, the oscillating MOKE signal shows the same phase for opposite applied magnetic fields. This excludes theexistence of an out-of-plane transient magnetic field.We notethe large spin wave amplitude at a small magnetic field ap-plied along the in-plane hard f1-10gaxis. An even larger rotation angle can be achieved at a higher excitation leveland larger uniaxial magnetic anisotropy, as for example inthinner Fe films. This would lead to a large out-of-planecomponent of magnetization and the resultant demagnetiza-tion field could trigger an out-of-plane precession. This pro-cess could then be utilized for fast magnetization reversal. The orientation dependence of the precession frequency clearly reveals uniaxial and cubic anisotropy as shown inFig. 2 sbd. We will show in the following that the magnetic anisotropy constants can be determined from the field depen-dence and anisotropy of the precession frequency. Figure3sadshows the frequency spectra for magnetic fields applied along the f100gandf1-10gdirections. The latter is the hard axis for the uniaxial and cubic magnetic anisotropies. Forsmall angle excitations, the frequency dispersion is well de-scribed by the model of the uniform FMR mode. 8Previous Brillouin scattering9and FMR studies8on thin Fe films sug- gest that surface anisotropy and spin pinning is negligible.The frequency behavior of the uniform magnetization pre-cession can be described by solving the Landau–Lifshitzequation including the shape-related demagnetization andanisotropy field. The frequency is given by v=gfsHecossd−fd+HadsHecossd−fd+Hbdg1/2,s1d where Ha=4pMs+2Kout Ms−Ku Msssinf−cos fd2+K1 Msf2 −sin2s2fdg, Hb=2K1 Mscoss4fd+2Ku Mssins2fd, and g=geg/2swith ge=1.76 3107Hz/Oe and g=2.09 dis the gyromagnetic ratio. fanddare the angles between the in-plane easy axis f100gand the directions of magnetization and applied magnetic field He, respectively. K1,Ku, andKout are the cubic anisotropy, in-plane uniaxial anisotropy, and out-of-plane anisotropy constants, respectively. Msis the saturated magnetization. FIG. 2. sadPrecession amplitude and sbdfrequency as a function of sample orientation at magnetic field H=560 Oe. The dashed line is a guide for the eyes. FIG. 3. sadPrecession frequency versus applied magnetic field along f100g and f1-10gaxes. The solid line is a fit by Eq. s1d.sbdGilbert damping parameter avs applied magnetic field along f100gandf1-10gdirections.152512-2 Zhao et al. Appl. Phys. Lett. 86, 152512 ~2005 ! 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: 133.1.198.126 On: Tue, 02 Dec 2014 05:58:15To accurately describe the field dependence of the pre- cession frequency, we need to know the magnetization angle fwhich is determined by the subtle balance of external and internal magnetic field caused by the magnetic anisotropy.The dependence of fon applied magnetic field is obtained from the hysteresis curves measured byVSM.The solid linesin Fig. 3 sadare the precession frequencies calculated from Eq.s1d. The cubic anisotropy K 1/Ms, uniaxial anisotropy Ku/Ms, and out-of-plane saturated magnetic field 4 pMs +2Kout/Msused in the calculation are 0.21 kOe, 0.09 kOe, and 17.5 kOe, respectively. The angle fcalculated from the above anisotropy constants reproduces the in-plane hyster-esis curve, and the out-of-plane saturated field is consistentwith independent MOKE measurements. For the easy f100g axis, the precession frequency increases monotonically withincreasing applied magnetic field. This behavior is typicalalong an easy axis. For the hard f1-10gaxis, the frequency initially increases due to the rotation of the magnetizationtoward the easy f100gaxis, then decreases due to further rotation toward the hard f1-10gaxis, and finally increases as the magnetization aligns along the applied magnetic fielddirection. ThesGilbert ddamping parameter ais another important factor in fast magnetic switching. Generally, the dampingparameter is treated as a constant. However, there is theoret-ical and experimental evidence 17,18thatacan vary with the magnetization angle relative to the field direction and filmnormal, with the magnitude of the applied field, and with theprecession frequency. By including the damping term in theLandau–Lifshitz equation, the uniform magnetization preces-sion can be described by an oscillating term and an exponen-tial decaying term exp s−Gtd, where G= agfsHecossd−fd+Had+sHecossd−fd+Hbdg 2s1+a2d. s2d The expression for Gcan be simplified as G=avin the case ofHa=Hband small a. The damping parameter ais calculated from Eq. s2dand parameters used in Eq. s1dfor the frequency calculation. Figure 3 sbdshows aas a function of the magnetic field applied along the f100gandf1-10gdi- rections.The damping parameter varies strongly with appliedfield along the hard f1-10gaxis, whereas aremains nearly constant along the easy f100gaxis. This behavior may be explained by the two-magnon scattering process where thedamping rate depends on the spin wave manifold.17For the hard axis, the magnetization rotates toward the f1-10gdirec- tion with increasing applied field. This will change the den-sity of accepting magnon states and therefore the dampingparameter. In contrast, for the easy f100gaxis, the magneti- zation direction will not significantly rotate with increasingapplied field and therefore aremains nearly constant. Furtherstudies are required to fully elucidate the damping mecha- nism. In conclusion, we have studied picosecond magnetiza- tion precession dynamics in ferromagnetic films with in-plane uniaxial anisotropy epitaxially grown on AlGaAss001d. Coherent magnetization precessions are generated by ultrafast optical excitation along all directions of the sampleover a wide range of applied magnetic field. This provides asensitive optical probe that can locally measure the dynamicmagnetic properties of very small ferromagnetic elements.From the decay of the FMR mode, we determine the aniso-tropy of the damping parameter which is crucial in designingfast magnetic switching devices and novel spintronics de-vices. This work is supported in part by the National Science Foundation, the Office of Naval Research, and the DARPASpins in Semiconductors Program. 1J. Ferré, in Spin Dynamics in Confined Magnetic Structures I ,TopicsAppl. Phys.Vol. 83, edited by B. Hillebrandt and K. Ounadjela sSpringer, Berlin, 2002 d, pp. 127–165. 2G. A. Prinz and J. J. Krebs, Appl. Phys. Lett. 39,3 9 7 s1981 d. 3E. Kneedler, B. T. Jonker, P. M. Thibado, R. J. Wagner, B. V. Shanabrook, and L. J. Whitman, Phys. Rev. B 56, 8163 s1997 d. 4J. S. Claydon, Y. B. Xu, M. Tselepi, J. A. C. Bland, and G. van der Laan, Phys. Rev. Lett. 93, 037206 s2004 d. 5W. H. Butler, X.-G. Zhang, X.-D. Wang, J. van Ek, and J. M. MacLaren, J. Appl. Phys. 81, 5518 s1997 d. 6P. Mavropoulos, N. Papanikolaou, and P. H. Dederichs, Phys. Rev. Lett. 85, 1088 s2000 d. 7A. T. Hanbicki, B. T. Jonker, G. Itskos, G. Kioseoglou, and A. Petrou, Appl. Phys. Lett. 80, 1240 s2002 d; A. T. Hanbicki, O. M. J. van’t Erve, R. Magno, G. Kioseoglou, C. H. Li, B. T. Jonker, G. Itskos, R. Mallory, M.Yasar, and A. Petrou, ibid.82, 4092 s2003 d; B. T. Jonker, Proc. IEEE 91, 727s2003 d. 8J. J. Krebs, B. T. Jonker, and G. A. Prinz, J.Appl. Phys. 61,2 5 9 6 s1987 d. 9M. Madami, S. Tacchi, G. Carlotti, G. Gubbiotti, and R. L. Stamps, Phys. Rev. B69, 144408 s2004 d; P. Grünberg, M. G. Cottam, W. Vach, C. Mayr, and R. E. Camley, J. Appl. Phys. 53, 2078 s1982 d. 10G. Ju, A. V. Nurmikko, R. F. C. Farrow, R. F. Marks, M. J. Carey, and B. A. Gurney, Phys. Rev. Lett. 82, 3705 s1999 d. 11Y. Acremann, M. Buess, C. H. Back, M. Dumm, G. Bayreuther, and D. Pescia, Nature sLondon d414,5 1 s2001 d. 12M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 s2002 d. 13M. Gester, C. Daboo, R. J. Hicken, S. J. Gray, A. Ercole, and J. A. C. Bland, J. Appl. Phys. 80, 347 s1996 d. 14M. Zolfl, M. Brockmann, M. Kohler, S. Kreuzer, T. Schweinbock, S. Miethaner, F. Bensch, and G. Bayreuther, J. Magn. Magn. Mater. 175,1 6 s1997 d. 15O. Thomas, Q. Shen, P. Schieffer, N. Tournerie, and B. Lépine, Phys. Rev. Lett.90, 017205 s2003 d; and reference therein. 16I. Bozovic, M. Schneider, Y. Xu, Roman Sobolewski, Y. H. Ren, G. Lüpke, J. Demsar, A. Taylor, and M. Onellion, Phys. Rev. B 69, 132503 s2004 d. 17R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F. Egelhoff, Jr., J.Appl. Phys.83,7 0 3 7 s1998 d. 18T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, J.Appl. Phys. 85, 7849 s1999 d.152512-3 Zhao et al. Appl. Phys. Lett. 86, 152512 ~2005 ! 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: 133.1.198.126 On: Tue, 02 Dec 2014 05:58:15

1.3231874.pdf

Magnetic tunnel junction based microwave detector X. Fan, R. Cao, T. Moriyama, W. Wang, H. W. Zhang, and John Q. Xiao Citation: Applied Physics Letters 95, 122501 (2009); doi: 10.1063/1.3231874 View online: http://dx.doi.org/10.1063/1.3231874 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/95/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Nonlinear ferromagnetic resonance induced by spin torque in nanoscale magnetic tunnel junctions Appl. Phys. Lett. 103, 082402 (2013); 10.1063/1.4819179 Microwave phase detection with a magnetic tunnel junction Appl. Phys. Lett. 97, 212501 (2010); 10.1063/1.3511328 Perpendicular spin torque promotes synchronization of magnetic tunnel junction based spin torque oscillators Appl. Phys. Lett. 94, 112503 (2009); 10.1063/1.3100299 The rf magnetic-field vector detector based on the spin rectification effect Appl. Phys. Lett. 92, 032504 (2008); 10.1063/1.2837180 Calibration methods of a 2 GHz evanescent microwave magnetic probe for noncontact and nondestructive metal characterization for corrosion, defects, conductivity, and thickness nonuniformities Rev. Sci. Instrum. 76, 054701 (2005); 10.1063/1.1900683 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: 138.251.14.35 On: Sun, 21 Dec 2014 14:22:57Magnetic tunnel junction based microwave detector X. Fan,1R. Cao,1T . Moriyama,1,a/H20850W. Wang,1,b/H20850H. W. Zhang,2and John Q. Xiao1,c/H20850 1Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA 2State Key Laboratory of Electronic Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, People’ s Republic of China /H20849Received 24 July 2009; accepted 27 August 2009; published online 21 September 2009 /H20850 We investigated the tunneling magnetoresistance change in magnetic tunnel junctions in the presence of external microwaves. The changing relative angle between the free layer and the pinnedlayer results in a rectification of the average resistance change. Due to its miniature size and itssensitivity to the microwave magnetic field, the magnetic tunnel junction could be utilized as amicrowave power sensor with the ability to detect microwave frequencies. Studying microwavepower and bias current dependencies reveals desired sensor features with linear responses andenhanced signal levels. © 2009 American Institute of Physics ./H20851doi:10.1063/1.3231874 /H20852 In recent years, development of dynamics in magnetism and spintronics has attracted intensive interest in microwaverelated applications. 1–8Studies have been carried out through the two following complementary approaches: /H20849i/H20850high den- sity dc current is injected into a hybrid magnetoresistivenanostructure to induce spin transfer torque, which precessesthe magnetic layer to generate microwaves, serving as a mi-crowave oscillator; 3,4/H20849ii/H20850microwaves are used to excite mag- netization precession in a magnetoresistive device, which converts microwaves to a dc voltage or resistance signal,2,6–9 serving as a microwave detector. In various experimental demonstrations of the microwave detectors, single ferromag-netic strip rectification effect has been vigorously investi-gated. The induced voltage signal arises from bothmicrowave-induced photoresistance and photovoltage ef-fects. The former is due to the rectification of resistance andthe latter is due to the coupling between rf current and alter-nating resistance. 8However, the signal level is relatively small due to a low anisotropic magnetoresistance value, typi-cally less than a few percent. In this paper, a magnetic tunneljunction /H20849MTJ /H20850coupled with a coplanar waveguide /H20849CPW /H20850is used to detect microwaves. Due to the high tunneling mag-netoresistance, the MTJ microwave sensor exhibits a muchhigher sensitivity to microwaves. A MTJ is fabricated on top of a CPW by magnetron sputtering followed by standard lithography and etching-down procedure. A MTJ composed of Cu 100 nm/IrMn 15nm/CoFe 6 nm/ AlO x2.3 nm/ Ni 80Fe20/H20849Py/H2085020 nm/Cu 50 nm/Au 50 nm is patterned on the center line of the CPW sothat microwaves can be efficiently fed into the NiFe layer.The size of the MTJ dot is 40 /H1100370 /H9262m2. Magnetization in the CoFe bottom layer is pinned by the antiferromagneticlayer IrMn along the z-axis, defined in Fig. 1. The free mag- netic layer NiFe also exhibits a stress-induced anisotropy inthe same direction. The experimental set up is shown in Fig.1. Vector network analyzer supplies a microwave modulated with a 430 Hz ac signal. The microwave is then amplifiedwith a nominal gain of 28 dB and fed into a CPW using acoplanar air probe. The CPW is shorted at the end in order to enhance the microwave magnetic field and minimize the mi-crowave electric field. 10Helmholtz coils generate a tunable dc magnetic field from /H11002100 to 100 Oe. A constant dc cur- rent bias is applied to the MTJ and a lock-in amplifier picksup the voltage signal corresponding to the rectification effectcaused by the coupling of microwaves and the MTJ. In the presence of microwaves, the magnetic free layer of the MTJ /H20849NiFe /H20850will precess and resonate at certain mi- crowave frequencies in an external dc magnetic field. Therelative angle between the magnetization of the free layerand the pinned layer deviates from parallel/antiparallel con-figurations, resulting in an increased/reduced average resis-tance. In order to model the resistance change, we take cer-tain approximations: for the MTJ used in the experiment, the6 nm CoFe layer pinned by an antiferromagnetic IrMn layershows a large damping due to the interface roughness in-duced two-magnon scattering. 11The precession angle of CoFe is negligible compared to the precession of the freelayer NiFe. Thus the time dependent resistance is determinedby the longitudinal magnetization of the free NiFe layer,R=R P+/H20851/H20849RAP−RP/H20850/2/H20852/H208511−Mz/H20849t/H20850/Ms/H20852for a parallel configu- a/H20850Present address: Department of Physics, Cornell University, Ithaca, New York, 14853, USA. b/H20850Present address: Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland, 21218, USA. c/H20850Electronic mail: jqx@udel.edu. FIG. 1. /H20849Color online /H20850Experimental setup of the MTJ-based microwave detection.APPLIED PHYSICS LETTERS 95, 122501 /H208492009 /H20850 0003-6951/2009/95 /H2084912/H20850/122501/3/$25.00 © 2009 American Institute of Physics 95, 122501-1 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: 138.251.14.35 On: Sun, 21 Dec 2014 14:22:57ration and R=RAP−/H20851/H20849RAP−RP/H20850/2/H20852/H208511+Mz/H20849t/H20850/Ms/H20852for an anti- parallel configuration,12where RAPandRPare resistances at antiparallel and parallel configurations, respectively, Msis the saturation magnetization of NiFe, Mzis the magnetiza- tion component along the external field direction, and Mz/H20849t/H20850=/H11006/H20881Ms2−Mx2/H20849t/H20850−My2/H20849t/H20850. Here the out-of-plane magne- tization precession component Myis much smaller than the in-plane precession Mxdue to the strong demagnetizing field.13With a microwave power of 10 mW, the microwave magnetic field generated in the CPW is estimated to be 1 Oe/H20849Ref.14/H20850and the precession angle is as small as 6°, assuming the susceptibility is about 1000.15Since Mxis much smaller than Ms, the changing voltage for parallel and antiparallel configurations can be written as /H9004V=/H11006Idc/H20849RAP−RP/H20850 8Ms2hrf2/H20841/H9273/H20851f,H/H20852/H208412, /H208491/H20850 where Idcis a dc current applied to the MTJ, /H9273is the micro- wave susceptibility of NiFe which can be derived from theLaudau–Lifshitz–Gilbert equation,16in which fis the micro- wave frequency and His the external dc field, /H9273=/H9253Ms/H20851/H9253/H20849H+Ha+Ms/H20850+i/H9251f/H20852 /H20853/H20851/H9253/H20849H+Ha+Ms/H20850+i/H9251f/H20852/H20851/H9253/H20849H+Ha/H20850+i/H9251f/H20852−f2/H20854,/H208492/H20850 where /H9253is the gyromagnetic ratio, which is typically 28 GHz/T, Hais the effective anisotropy field, and /H9251is the damping constant. Tunneling magnetoresistance /H20849TMR /H20850measured at 100/H9262A bias current is about 6%, as shown in Fig. 2/H20849a/H20850. Figure 2/H20849b/H20850shows the microwave-induced voltage spectrum with 1 mW microwave input at different frequencies. Thevoltage curve resembles the ferromagnetic resonance /H20849FMR /H20850 spectrum of NiFe and is proportional to /H20841 /H9273/H20851f,H/H208522/H20841, consistent with Eq. /H208491/H20850. The fitting based on Eqs. /H208491/H20850and /H208492/H20850for the voltage spectrum in parallel configuration are shown in greencurves, with following extracted parameters: M s=0.91 T,Ha=16.1 Oe, and/H9251=0.014 /H110060.001. The micro- wave magnetic fields from 1 mW nominal power input areextracted to be 0.38 Oe at 1.5 GHz, 0.37 Oe at 2 GHz, and0.35 Oe at 2.5 GHz. The difference in magnitude is due tothe frequency dependent loss and dimensional resonance inthe transmission line. Opposite polarities of resonance peaksare also observed in parallel and antiparallel configurations,as predicted by Mecking et al. 8This suggests that the ob-served spectrum is due to a microwave-induced resistance effect instead of the bolometric effect.17The voltage at reso- nance in an antiparallel configuration is slightly higher thanthe voltage in a parallel configuration. This may be due tothat the pinned CoFe layer is closer to resonance condition inthe antiparallel configuration. This mutual precession willdecrease the resistance change in a parallel configuration,since the relative cone angle is reduced, while increasing theresistance change in an antiparallel configuration, where therelative cone angle is enhanced. The microwave-induced voltage arising from the rectifi- cation effect is proportional to the square of the microwavemagnetic field and thus, linear to microwave power. Thismakes MTJ sensor a potential microwave detector. Unlike aconventional rf diode, which rectifies the microwave voltage,the MTJ microwave detector is sensitive to the intensity ofthe microwave magnetic field. Due to the nature of ferro-magnetic resonance, the MTJ sensor can distinguish the fre-quency of a microwave, although the frequency resolution isaffected by its FMR linewidth, which is around 300 MHzfor NiFe. 18 Microwave power dependence of induced voltage is measured at a frequency of 2 GHz and a fixed dc magneticfield of 44 Oe, which is the resonance field for NiFe at lowmicrowave power. The sample used in this investigation hassimilar structure as the one used in Fig. 2, but with smaller CPW size. In order to cancel out the background signal, in-duced voltage is measured at both +80 and −80 /H9262A bias current and the difference is taken. As shown in Fig. 3/H20849a/H20850, the microwave-induced voltage increases with low input micro- wave power, with a dependence of /H9004V/H11008Prf0.86, where Prfis the nominal power fed into the CPW. When the power in-creases, the induced voltage starts to level off due to the spinwave instability effect, in which the coherent precessionmode couples to a chaotic spin wave mode and effectivelyreduces the average precession cone angle. 19,20Examples of microwave-induced voltage spectra are shown in Fig. 3/H20849b/H20850. At higher powers, the spectrum is distorted and the peakvoltage shifts toward a higher dc magnetic field. According to Eq. /H208491/H20850, a high dc bias current could en- hance the microwave-induced voltage. However, the bias islimited by the breakdown voltage of a MTJ, which is typi-cally around 1 V. 21Higher bias current will also influence the tunneling rate in a MTJ and normally will result in a reducedTMR,22which decreases the term RAP−RPin Eq. /H208491/H20850,a s shown in Fig. 4/H20849a/H20850. The microwave-induced voltage differ- ence at parallel and antiparallel resonances has the same pro- FIG. 2. /H20849Color online /H20850/H20849a/H20850Magnetoresistance curve measured at 100 /H9262A bias current. /H20849b/H20850Microwave-induced voltage measured at different micro- wave frequencies at 100 /H9262A bias current. Green lines are fittings for the spectrums at parallel configuration. FIG. 3. /H20849Color online /H20850/H20849a/H20850Power dependence of microwave-induced volt- age. The red line is the fitting at low microwave power, which shows the dependence of Prf0.86./H20849b/H20850Microwave-induced voltage spectrum at different microwave powers.122501-2 Fan et al. Appl. Phys. Lett. 95, 122501 /H208492009 /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: 138.251.14.35 On: Sun, 21 Dec 2014 14:22:57file as the dc voltage difference, Fig. 4/H20849b/H20850. The discrepancy at a high bias current is due to the thermal effect as it nears thebreakdown voltage. In summary, it is demonstrated that a significant voltage signal can be generated from a biased MTJ irradiated bymicrowaves, due to the rectification effect. The MTJ has amicrowave-induced voltage proportional to microwavepower, making it a microwave power detector. The MTJ in-herits a ferromagnetic resonance feature from the magneticlayer, which is sensitive to the microwave frequency. On theother hand, it also results in a nonlinear high power responsedue to a spin wave instability effect. Bias current can betuned to enhance the voltage response, scarifying TMR andthermal stability. CoFeB/MgO/CoFeB based MTJ with muchhigher TMR and comparable free layer damping can replacethe Alumina/Py based MTJ and improve the sensitivity byapproximately ten times. 23,24Moreover, this detector senses the intensity of a microwave magnetic field, which can betuned by changing the dimension of feed in waveguide withthe same amount of power input. The detector should also beable to resist the high power microwave due to the absenceof microwave electric field during the detection. In a minia-ture circuit, the MTJ has a larger resistance due to the currentperpendicular to the plane configuration, which offers the advantage of a smaller thermal effect, compared to deviceswith current-in-plane configuration. This work was supported by NSF DMR Grant No. 08242249. 1T. Moriyama, R. Cao, X. Fan, G. Xuan, B. K. Nikolic, Y. Tserkovnyak, J. Kolodzey, and J. Q. Xiao, Phys. Rev. Lett. 100, 067602 /H208492008 /H20850. 2A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S. Yuasa, Nature /H20849Lon- don /H20850438, 339 /H208492005 /H20850. 3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850425,3 8 0 /H208492003 /H20850. 4F. B. Mancoff, N. D. Rizzo, B. N. Engel, and S. Tehrani, Nature /H20849London /H20850 437, 393 /H208492005 /H20850. 5A. Mourachkine, O. V. Yazyev, C. Ducati, and J. P. Ansermet, Nano Lett. 8, 3683 /H208492008 /H20850. 6S. T. B. Goennenwein, S. W. Schink, A. Brandlmaier, A. Boger, M. Opel, R. Gross, R. S. Keizer, T. M. Klapwijk, A. Gupta, H. Huebl, C. Bihler, andM. S. Brandt, Appl. Phys. Lett. 90, 162507 /H208492007 /H20850. 7Y. S. Gui, N. Mecking, A. Wirthmann, L. H. Bai, and C. M. Hu, Appl. Phys. Lett. 91, 082503 /H208492007 /H20850. 8N. Mecking, Y. S. Gui, and C. M. Hu, Phys. Rev. B 76, 224430 /H208492007 /H20850. 9J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 96, 227601 /H208492006 /H20850. 10K. Beilenhoff, H. Klingbeil, W. Heinrich, and H. L. Hartnagel, IEEE Trans. Microwave Theory Tech. 41, 1534 /H208491993 /H20850. 11S. M. Rezende, A. Azevedo, M. A. Lucena, and F. M. de Aguiar, Phys. Rev. B 63, 214418 /H208492001 /H20850. 12J. C. Slonczewski, Phys. Rev. B 39, 6995 /H208491989 /H20850. 13A. Kaya and J. A. Bain, J. Appl. Phys. 99, 08B708 /H208492006 /H20850. 14T. Moriyama, R. Cao, J. Q. Xiao, J. Lu, X. R. Wang, Q. Wen, and H. W. Zhang, Appl. Phys. Lett. 90, 152503 /H208492007 /H20850. 15A. Gerber, J. McCord, C. Schmutzl, and E. Quandt, IEEE Trans. Magn. 43, 2624 /H208492007 /H20850. 16T. L. Gilbert, IEEE Trans. Magn. 40, 3443 /H208492004 /H20850. 17Y. S. Gui, S. Holland, N. Mecking, and C. M. Hu, Phys. Rev. Lett. 95, 056807 /H208492005 /H20850. 18G. Counil, P. Crozat, T. Devolder, C. Chappert, S. Zoll, and R. Fournel, IEEE Trans. Magn. 42, 3321 /H208492006 /H20850. 19H. M. Olson, P. Krivosik, K. Srinivasan, and C. E. Patton, J. Appl. Phys. 102, 023904 /H208492007 /H20850. 20T. Gerrits, P. Krivosik, M. L. Schneider, C. E. Patton, and T. J. Silva, Phys. Rev. Lett. 98, 207602 /H208492007 /H20850. 21W. Oepts, H. J. Verhagen, R. Coehoorn, and W. J. M. de Jonge, J. Appl. Phys. 86, 3863 /H208491999 /H20850. 22J. Zhang and R. M. White, J. Appl. Phys. 83,6 5 1 2 /H208491998 /H20850. 23W. G. Wang, C. Ni, A. Rumaiz, Y. Wang, X. Fan, T. Moriyama, R. Cao, Q. Y. Wen, H. W. Zhang, and J. Q. Xiao, Appl. Phys. Lett. 92, 152501 /H208492008 /H20850. 24G. D. Fuchs, J. C. Sankey, V. S. Pribiag, L. Qian, P. M. Braganca, A. G. F. Garcia, E. M. Ryan, Z. P. Li, O. Ozatay, D. C. Ralph, and R. A. Buhrman,Appl. Phys. Lett. 91, 062507 /H208492007 /H20850. FIG. 4. /H20849a/H20850Bias dependence of dc voltage differences between parallel and antiparallel conditions. The deviation from linear dependence indicates thedecrease of TMR at high bias. /H20849b/H20850Bias dependence of microwave-induced voltage with input microwave at 2 GHz frequency and 1 mW nominalpower. Differences in microwave-induced voltage at resonance in paralleland antiparallel conditions are taken in order to eliminate the backgroundsignal.122501-3 Fan et al. Appl. Phys. Lett. 95, 122501 /H208492009 /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: 138.251.14.35 On: Sun, 21 Dec 2014 14:22:57

1.4815876.pdf

Current-induced distortion of the band structure and formation of pseudogaps in magnonic crystals N. I. Polushkin Citation: J. Appl. Phys. 114, 033908 (2013); doi: 10.1063/1.4815876 View online: http://dx.doi.org/10.1063/1.4815876 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i3 Published by the AIP Publishing LLC. 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 02 Aug 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsCurrent-induced distortion of the band structure and formation of pseudogaps in magnonic crystals N. I. Polushkina) Instituto Superior Tecnico and ICEMS, Av. Rovisco Pais, 1049-001 Lisbon, Portugal (Received 24 April 2013; accepted 27 June 2013; published online 17 July 2013) Using numerical simulations, we have studied how electric current, passing along the periodicity direction in a lateral magnetic superlattice with modulated saturation magnetization, affects the propagation of magnetostatic surface spin waves (MSSWs) across it. It is shown that when the current flows against the normal lattice modes excited by a built-in antenna, it mediates excitation ofnew MSSW modes. These current-assisted modes are found to be co-propagating with the normal lattice ones but travel with negative group velocities and their wave-packet dispersions opposite to those in the normal lattice modes. Surprisingly, their intensity is high enough to effectively interactwith the normal lattice modes under realistic parameters of the lattice and current. This intermode interaction gives rise to new frequency bands where the MSSW intensity is lowered but essentially nonzero (pseudogaps). The pseudogap positions can be shifted by several gigahertz either upwards ordownwards with respect to the bandgaps occurring at Brillouin zone edges in the absence of current. The pseudogap shifting depends on the strength of the current and on the lattice magnetization and period. VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4815876 ] I. INTRODUCTION The current-induced magnetization dynamics in small (nanoscale) structures1has become one of central topics in the field of spintronics for the last two decades since initial predic- tions of so-called spin-transfer torques.2,3It has been demon- strated during this while that a strong enough (106–109A/cm2) electric current is able to switc h the magnetization in multi- layers,4to move and switch domain walls in nanowires,5and to manipulate spin-wave amplitudes6and frequencies7,8in thin-film waveguides. All of th ese properties of nanoscale magnetic systems have a definite potential to serve as a basisfor the next generation of electronics and spintronics devices. 9 However, such properties might be not limited to those men-tioned above, as discussed in Ref. 1, for instance. We report on another nanomagnetic property that can be affected and manipulated with electric current via spin trans- fer. This property is the gaps for spin-wave propagation[magnonic bandgaps (MBGs)] 10that open up in magnetic periodic systems (so-called magnonic crystals) at the Brillouin zone (BZ) edges k¼lp/K, where kis the wave vec- tor,l¼/C0 1 ,þ1 are integers, and Kis the lattice period. The MBG formation has recently attracted considerable in- terest because of possible utilization of this property inmicrowave electronics, e.g., spin logic devices, filters, and waveguides operating in the microwave range. 11,12One of the critical issues in fabrication of magnetic microwave devi-ces is manipulating the allowed and forbidden magnonic bands with external magnetic and/or electric fields. The abil- ity of MBGs to be tuned by electric current would open newperspectives in this arena. As found previously, 13a uniform current passed across a ferromagnetic layer, in which the saturation magnetizationis modulated in the film plane, causes a frequency shift of the MBG for magnetostatic surface spin waves (MSSWs) excited by a source built into the lattice. This shift is the spin-wave Doppler effect in a periodic lattice, which is de-pendent on its period and given by D/C23 l¼/C23/C0uðk/C0lqÞ=2p; (1) where /C23is a frequency of the driving field at which the bandgap occurs, u¼jelB/jejM0is the drift velocity of the electron flow, jeis the current density, M0is the saturation magnetization averaged over K, and q¼2p/K. Equation (1) can be applied to the effective field in a ferromagnetic me- dium with a periodic lattice, which can be written as Heff;lðkÞ¼½xðkÞ/C0uðk/C0lqÞ/C138=c; (2) where x(k) is the dispersion law for spin waves and the sec- ond term is a pure effect of the current. This infinite set ofthe effective fields should provide, respectively, the infinite number of MSSW resonances. The aim of this paper is (1) to establish the current- induced MSSW resonances—with spin-wave wavelengths in the submicron-scale regime and (2) to highlight their role in arising new frequency bands where the MSSW intensity is low-ered but essentially nonzero. To understand the formation of these pseudogaps (PGs), we show that the current-induced MSSW resonances provide the modes that propagate across thelattice in a negative dispersion regime, 14i.e., their group veloc- ities ( @x/@k) and wave-packet dispersions are opposite to those of the normal lattice modes. As a result, the current-inducedmodes interfere destructivel y with the lattice ones, giving rise to the PGs. Importantly, t he effect of electric current depends on whether the direction of the electron flow is parallel[positive current direction ( u>0)] or antiparallel [negative current direction ( u<0)] to that of MSSW propagation. a)E-mail: nipolushkin@fc.ul.pt. Tel.: þ351-217500911. Fax: þ351-217500977. 0021-8979/2013/114(3)/033908/6/$30.00 VC2013 AIP Publishing LLC 114, 033908-1JOURNAL OF APPLIED PHYSICS 114, 033908 (2013) Downloaded 02 Aug 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsA reason for this non-reciprocity can be understood by ana- lyzing the current-induced effects in a homogeneous layer. II. MODEL SYSTEM A sketch of a hypothetic micr owave device and the geom- etry of calculation are shown in Fig. 1. As the magnetic film, one can use an Fe layer (4 pMs¼21.0 kG) of a thickness in a few tens of nanometers grown epitaxially on a GaAs substrate a n dh a v i n gt h ed a m p i n gc o n s t a n t a¼0.004.15Electric current can be injected into the device by short pulses with a current density up to je/C244/C2108A/cm2,16so that it could flow along thexaxis. Also, a dc magnetic field His applied along the z axis. We explore how the magne tic oscillations, excited locally by an ac magnetic field he(x,t)¼h0f(x)cos2 p/C23tx0, decay along the periodicity direction of a magnetic grating that can be pro-duced, for instance, by di rect laser patterning. 17An ac mag- netic field can be generated by the antenna of a width wthat can be built into the lattice, as shown in Fig. 1. In a one-dimensional geometry of the system, the diffu- sion term r/C1(je/C10n),18where n(r,t)¼Ms(r,t)/Ms(r,t) and r¼xx0þyy0þzz0, is reduced to the vector quantity @JM/@x, so that with neglecting spin-flip scattering,19the magnetiza- tion current flowing along the xaxis is JM(x,t)¼uMs(x,t). Such an approach within a phenomenological model for fer-romagnetism of itinerant electrons provides the term that describes the adiabatic process for the interaction of the non- equilibrium conduction electrons with the nonuniform mag-netization. 7,20With adding the diffusion term, the equation of motion for the magnetization written in the Landau- Lifshitz-Gilbert form reads13 @n=@t¼/C0c½n;Hef f/C138þa½n;@n=@t/C138þð1=MsÞ@JM=@x:(3) With a conserved jMsjproperty, the excited magnetization is Ms(x,t)¼Ms(x)z0þmx(x,t)x0þmy(x,t)y0,w h e r e mxandmyare the components of the dynamic magnetization. With neglect- ing nonuniform exchange,21the effective magnetic field is Heff(x,t)¼Hz0þhd(x,t)þhe(x,t), where hd¼hxx0þhyy0and hx(y)are the components of the dipole field. To study the magnetization dynamics, we find the dynamic susceptibility of the system by solving numericallyEq.(3)for a small modulation in M s. Details of the performedcalculations are given in Appendix. We emphasize that under the used approximation, all the dynamic fields oscillate har- monically with an angular frequency x¼2p/C23: (4) Then, after calculating and averaging the dipole fields,22the equations for the dynamic magnetization amplitudes, m0x(y)(x), can be reduced to the infinite set of linear algebraic equations with respect to the components of the dynamic sus-ceptibility tensor v xx¼4pm0x/h0(vxx/C17vx)a n d vyx¼4pm0y/h0 (vyx/C17vy)a sf u n c t i o n so f kandx[see Eq. (A7) in Appendix]. All the plots presented in Figs. 2–6result from solving numerically this set. III. RESULTS AND DISCUSSION A. Homogeneous layer: Non-reciprocity in MSSW excitation spectrum under current switching To understand how exactly the current modifies the ex- citation spectrum via the Doppler shift, first of all, we haveanalyzed the MSSW intensity of a homogeneous layer with a saturation magnetization M 0. In this particular case [Ms(x)/C17M0], the MSSW intensity can be obtained in its ana- lytical form given in Appendix [Eq. (A8)]. The ferromagnetic resonance occurs when the real part of the denominator in Eq. (A8) is equal to zero. This characteristic equation provides the dispersion law for MSSWs in the presence of current x¼ukþcHef fðkÞ; (5) where Heff¼([(Hþ4pM0(1/C0P)][Hþ4pM0P])1/2and P(k) are the dynamic demagnetizing factors given in Appendix.Atu¼0 and ks<2, where sis the layer thickness, the dis- persion law (5)is in good quantitative agreement with the behavior of the Damon-Eshbach mode. 23It shows up that the effect of electric current on the MSSW behavior is adding a linear dependence on k, which is according to the Doppler FIG. 1. (a) MSSW excitation and current injection into a thin-film (with a thickness s) lateral periodic lattice with modulated saturation magnetization, Ms. The lattice is infinite along the x(shown for x>0) and zdirections. A built-in antenna of a width w¼K/2 generates an oscillating magnetic field he¼h0f(x)cos2 p/C23tx0to excite MSSWs. The electron flow is directed along the periodicity direction to pass either in the same direction (positive current direc- tion) in which the MSSWs travel or against them (negative current direction). FIG. 2. Homogeneous layer: (a) 3D x-kdiagram at zero current ( u¼0), pos- itive ( u¼0.35lm GHz) and negative ( u¼0.35lm GHz) current directions; (b)–(d) Im vx(k) and Re vx(k) functions at u¼0.35lm GHz at different close to the cut off frequency, /C23c, that results from the coupling between L 0and C0modes. The MSSW intensity and dispersion at 21.0 GHz (d) are magni- fied by a factor of 7.0 by comparison to these quantities shown at 19.0 GHz (b) and 20.0 GHz (c). Dashed lines in (b)–(d) indicate a zero level of the MSSW intensity and dispersion.033908-2 Nikolay I. Polushkin J. Appl. Phys. 114, 033908 (2013) Downloaded 02 Aug 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionseffect. Figure 2(a)depicts dispersion curves at H¼0.55 kOe andw¼0.15lm for an Fe film having s¼0.03lm (Fig. 1). The characteristics are obtained by plotting /C0Imvx(bright- ness of the contrast) as a function of xandkfor zero ( u¼0), positive ( u¼þ0.35lm GHz24), and negative ( u¼/C00.35lm GHz) current directions. It is striking that, unlike the x-kdia- grams at u¼0a tu>0, a current flowing in negative direc- tion ( u<0) provides a nonmonotonous x(k) dependence with its maximum at ks/C250.5 and /C23c/C2521.5 GHz. The occur- rence of such a maximum can be interpreted by merging two branches noted as L 0and C 0. The existence of the two branches in the x-kdiagram can be understood by plotting the MSSW excitation spectrum at u<0 and different fre- quencies close to /C23c[Figs. 2(b)–2(d)], which is indicative of two maxima in /C0Imvx(k) functions. A lower- kpeak, L 0,i s the Damon-Eshbach mode. Of course, this mode is also excited at u¼0 and u>0. In the absence of current ( u¼0), the L 0peak appears to be in a frequency range between /C230/C259.7 GHz and /C23/C2532.0 GHz. As seen from Fig. 2(a), Eq. (A8) does not provide any dependence of /C230onu. A higher- k mode, C 0, induced by a current, flowing in negative direc- tion, appears to be at /C23>1.0 GHz and propagates in the dis- persion regime which is opposite to that of the L 0mode. Since their @k/@xhave opposite signs, while their phase velocities have the same sign ( k>0), these two peaks move, i.e., alter their k, to merge into each other at /C23c. When wave vectors of the wave packets become close enough each to other, so that Dk<0.1/s, transfer of the energy occurs from the L 0to weaker C 0mode25to effectively provide the destructive interference between these two modes via theiropposite wave-packet dispersions. 26The interference pattern of the wave packets with opposite dispersions, Re v(k), is close to an even function with respect to k¼kmaxat which /C0Imv(kmax) is maximal for k>0. This superposition [Fig. 2(d)] provides lowering the MSSW intensity, /C0Imv(kmax), which is according to the Kramers-Kronig relation. Such abehavior of the mathematic functions reflects that the mag- netic moments acquire different random phases, so that the dispersion is still nonzero, and the system breaks away fromthe ferromagnetic resonance at /C23 c. With a further increase in frequency above /C23c, not only Im vx(kmax)!0b u ta l s o RevxðkmaxÞ!0: (6) The latter reflects decaying the precession amplitude of indi- vidual magnetic moments, so a homogeneous medium doesnot support propagation of MSSWs with frequencies higher than that of cutting off ,/C23 c. It is interesting that in periodic lattices, the intermode coupling of such kind can provide theformation of bandgaps with a finite width. B. Current-induced effects in periodic lattice The dispersion characteristics and excitation spectrum are found to be strongly modified in a periodic lattice, even with a small modulation MsðxÞ¼M0/C02M1cosqx;M1/C28M0: (7) Results obtained for a periodic variation in Msare illustrated for a lattice with 4 pM1¼0.6 kG and K¼0.3lm, which is FIG. 4. Periodic lattice: 3D x-kdiagrams at zero current ( u¼0) and positive current direction ( u¼þ0.35lm GHz) (a), and negative current direction (u¼/C00.35lm GHz) (b). For u¼0, a real MBG appears to be exactly at the BZ edge, k¼p/K, as a result of merging the L 0and L 1:N modes. At u>0, the merger and gap are shifted downwards both in xandk. The effect of a current, flowing in negative direction, ( u<0) is different due to the current- assisted modes C l. FIG. 5. /C0Imvxand Re vxfor a few strongest modes at different /C23close to the MBG frequency at u¼0 (a) and u¼/C00.35lm GHz ( u<0) (b). These data indicate that the gaps result from the coupling of the modes that travel with opposite group velocities in opposite dispersion regimes. In the absence of current ( u¼0), a real MBG (a3) results from the coupling between the L 0 and L 1:N modes dominating in the spectrum. A current flowing in negative direction mediates the formation of pseudogaps under the coupling between the L 0and C 1modes (b2) and L 0and L 1:N modes (b4). FIG. 3. Periodic lattice: MSSW excitation spectrum under a negative current(u¼/C00.35lmG H z ) . /C0Imv x(k)a n dR e vx(k) functions are shown by red and blue dashed curves, resp ectively. Purely lattice L l:N and current-induced modes Clhave @k/@x<0. The L 1,C1,a n dC 0m o d e sa r es h o w nw i t hm a g n i fi c a t i o n s of 40, 16, and 8. The arrows show directions of the wave-vector shifting of dif- ferent modes with increasing /C23. For frequencies close to the lower edge of the excitation band, the spectra at u¼0a n d u>0 are similar to the spectrum shown here for negative current direction, excluding the current-induced modes.033908-3 Nikolay I. Polushkin J. Appl. Phys. 114, 033908 (2013) Downloaded 02 Aug 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionshypothetically formed in the same layer whose parameters are used for plotting in Fig. 2. We find that in such a lattice, the MSSW excitation spectrum becomes substantiallyricher—with adding pure lattice L land current-assisted modes C lðl/C2161Þ. Each of the L lmodes arising with k¼lq at/C230splits up into two kinds of submodes—L l:N and L l:P— that have opposite signs of their @k/@xand propagate in op- posite dispersion regimes. Therefore, despite all these modes are forward waves, i.e., they travel away from the excitationsource, wave vectors of the L 0/Ll:P and, on the other hand, Ll:N/C lmodes get their shifts in opposite directions. This shifting is indicated by arrows in Fig. 3which depicts Revx(k) and /C0Imvx(k) functions for several low- kmodes— L0,L1:N, L 1:P, C 0, and C 1—that are dominating in the spec- trum at u¼/C00.35lm GHz and /C23¼12.0 GHz. Also, MSSW excitation spectra at u¼0 and u>0 are similar to this one plotted at u<0, excluding the current-assisted modes, C l. Figure 4shows 3D x-kdiagrams at u¼0a n d u¼þ0.35lmG H z( a )a n da t u¼/C00.35lmG H z( b ) .A l lt h e x-kdiagrams presented indicate the opening of the gaps in MSSW intensity by breaking up the dispersion curves. We willshow that these gaps result from the wave-vector shifting of the modes having opposite signs of their @k/@xtowards each other, as shown by arrows in Fig. 3, and their subsequent merger in the following pairs: L 0/C1,L0/L1:N, L 1:P/L 2:N, and so forth. At u¼0a n d u>0, a real MBG results from the merger of the L 0 and L 1:N modes [Fig. 4(a)]. Ifu¼0, this merger occurs exactly at the BZ edge, k¼p/K, while the MBG shifts downwards both inxandkunder positive current direction ( u>0). The disper- sion characteristics bec omes strongly different at u<0d u et o the current-assisted modes [Fig. 4(b)] that provide the formation of new gaps under their merger with purely lattice modes. To understand the current-induced distortion of the MSSW bandstructure, we will consider tw o different kinds of MSSW behav- ior, occurring at u¼0( o r u>0) (1) and u<0( 2 ) . 1. Zero current (or positive current direction): MBG formation We find that, in the absence of current, the MBG opens owing to the destructive interference between the L 0andL1:N modes, L 1:P and L 2: Nm o d e s ,a n ds oo na tB Ze d g e s k¼(2lþ1)p/K. If a current flows in pos itive direction, it pro- vides the shifting of the MBG from the BZ edges,13which is seen from the x-kdiagram at u>0i nF i g . 4(a). In addition, we exemplify the MBG opening with one pair of modes, L 0and L1:N, which are dominating in the spectrum [Fig. 5(a)]. The superposition of these two mode s provides a nonzero dispersion [Revx(kmax)6¼0] at the maximum of MSSW intensity, jImvx(kmax)j. The plots shown in Fig. 5(a3) indicate a violation of the necessary condition for fe rromagnetic resonance, notably Revx(kmax)¼0 whenever jImvx(k)jreaches its maximum at kmax. An increase in the wave-packet dispersion jRevx(kmax)j leads to lowering the MSSW intensity, jImvx(kmax)j. This fact reflects that the magnetic moment s, oscillating synchronously at frequencies outside the gap, acquire different phases. As aresult, the system of coupled osc illators breaks away from the ferromagnetic resonance inside t he MBG. It is interesting that, unlike the intermode coupling in a homogeneous layer, Re v x(k) is essentially nonzero inside the MBG, while /C0Imvx(k)!0a t a!0. This means that the precessi on amplitudes of individual magnetic moments are still large, even if the system is out offerromagnetic resonance. In othe r words, the superlattice is able to support the magnetic oscillations at /C23that lie both inside the MBG and out of it. This intermode coupling in a periodic lat-tice with formation of a narrow ( /C241.0 GHz) rejection band con- trasts to the coupling in a homogeneous layer where the MSSW intensity cuts off above /C23 c. Mathematically, this difference in the MSSW behaviors means that the Cauchy principal value in- tegral P:v:Ð1 /C01Revxðk0Þðk/C0k0Þdk0diverges at /C23lying inside the MBG, so that Im vx(k) cannot be derived from the Kramers- Kronig relation. 2. Negative current direction: PG formation Figure 5(b) shows /C0Imvxand Re vxfor different /C23at u¼/C00.35lm GHz ( u<0). We find that a current, flowing in negative direction, by contrast with the case u>0, alters the excitation spectrum by adding a system of current- induced modes, C l. The strongest of them, C 0, is excited in a homogeneous layer (Fig. 2) via the Doppler shift, i.e., Dx¼x/C0ku. In a periodic lattice, multiple satellites C l appear to be in addition to the C 0mode, which is in accord- ance with Eq. (2). In Fig. 3, for instance, the two modes, C 0 and C 1, are shown. The distance between the fundamental mode, C 0, and its first satellite, C 1, is equal to the period of reciprocal lattice, q. Similarly to L l:N modes, all the current- induced modes have @k/@x<0, so they propagate either with the oscillation phases shifted by p(/C0Imv<0) (at lower /C23, as shown in Fig. 3) or with dispersions, Re v, opposite [at higher /C23; Fig. 5(b)] to those of L 0/Ll:P modes. It is interest- ing that there is a crossover between these two regimes forpropagation of C lmodes, which occurs as a result of their coupling with L 0/Ll:P modes. Another result of this coupling is arising a system of new bands that appear to be with low-ered MSSW intensity. For example, Fig. 5(b2) indicates low- ering MSSW intensity because of merging the L 0and C 1 modes that interfere destructively via their opposite disper- sions. In addition, we mark another rejection band, which is the MBG shifted by current. This band results from the FIG. 6. (a) The maximal MSSW intensity as a function of driving field fre- quency under zero current, negative ( u¼/C00.35lm GHz), and positive (u¼þ0.35lm GHz) current direction. The MBG at u¼0 and pseudogaps, PG1 and PG2, at u<0 are indicated in the inset. (b) PG2 shifting as a func- tion of ufor different lattice periodicities in the submicron regime.033908-4 Nikolay I. Polushkin J. Appl. Phys. 114, 033908 (2013) Downloaded 02 Aug 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionscoupling between the L 0and L 1:N modes [Fig. 5(b4) ]. Note, however, that the bands with lowered MSSW intensity occur- ring under a current, flowing in negative direction, are not realbandgaps (and even the MBG shifted by the current) because there are modes, e.g., C 0,L1:N, and C 1, that are excited and able to propagate with the frequencies lying inside these bands.For example, the L 1and C 0modes are still persistent in the spectrum under destructive interference between the L 0and C 1 modes. A number of the PGs depends on the strength of the current. For instance, at u¼/C00.35lmG H z ,t h e r ea r et w oP G s arising at 19.3 GHz and 20.5 GHz [Figs. 5(b2) and5(b4) ]. The same bands of the lowered MSSW intensity under negativec u r r e n tc a nb es e e ni nF i g . 6(a)where the maximal MSSW intensity, /C0Imv x(kmax), is plotted as a function of /C23atu¼0 (solid magenta curve), u¼/C00.35lm GHz (solid blue curve), and u¼þ0.35lm GHz (dashed curve). One of these new bands, noted as PG1 and shown in the inset of Fig. 6(a),o r i g i - nates from the MBG shifted by the current, while anotherpseudogap, PG2, results from the coupling with the current- assisted mode C 1. It is interesting that with increasing current, only the PG2 is persistent in th e amplitude-frequency charac- teristics, while the PG1 is dege nerated because of its merger with the edge of cutting off above /C23c. Finally, we have studied how the PG2 shifts in its frequency with increasing current for different Kin the submicron regime. As seen from plotting in Fig. 6(b), the smaller the K,t h el a r g e r the downward shift of PG2 frequency. This behavior is compati-ble with the equations, Eqs. (1)and(2), for the Doppler effect in a periodic lattice. Note, finally, that the PG2 shifting can be both upwards (positive shift) and downwards (negative shift) withrespect to the MBG opened at u¼0. The sign and magnitude of the PG2 shifting depend on the group velocity of the C 1mode, which in turn depends on the current strength. IV. SUMMARY We study effects of electric current on the MSSW band structure in a thin-film medium wi th a laterally modulated satu- ration magnetization. It is found that, due to the Doppler effect, a current flowing against the MSSWs mediates an additionalsystem of the modes propagating with negative group veloc- ities. Under realistic parameters of the lattice ( K¼0.3lm) and current ( j e/C24108–109A/cm2), the coupling of this current- mediated modes with the normal l attice ones gives rise to new frequency bands with lowered but essentially nonzero MSSW intensity. These pseudogaps can be shifted by a few GHz eitherupwards or downwards with respect to the MBGs occurring at BZ edges in the absence of current. The pseudogap shifting depends on the lattice period [according to Eqs. (1)and(2)] strength of the current and the lattice magnetization. A micro- wave device is proposed for testing experimentally the pre- dicted effects. Their experim ental verification will open new perspectives in the field of spintronics and related ones. ACKNOWLEDGMENT Work was supported via the program “Ciencia 2008” and research grant PTDC/FIS/121588/2010 funded by thePortugal Foundation of Science and Technology.APPENDIX: DYNAMIC-RESPONSE CALCULATION IN PRESENCE OF ELECTRIC CURRENT To describe the current-induc ed magnetization dynamics, we use here a phenomenological a pproach within Stoner ferro- magnetism and get Eq. (3)in which the current-induced term is identical to that described by the adiabatic spin torque7obtained within an “s-d” exchange model. If one uses the latter approach, the equation for the Ms(x,t) magnetization dynamics includes both the adiabatic and nonadiabatic parts of spin torque20 @Ms @t¼/C0c½Ms;Hef f/C138þða=MsÞ½Ms;@Ms=@t/C138þTAþTNA; (A1) where Ms(x,t)¼Ms(x)z0þmx(x,t)x0þmy(x,t)y0,mx(y)are the components of the dynamic magnetization, Heff(x,t)¼Hz0 þhd(x,t)þhe(x,t),hd¼hxx0þhyy0,hx(y)are the components of the dipole field, heis an ac external field, TA¼/C0 ð u=M2 sÞ ½Ms;½Ms;@Ms=@x]] is the adiabatic spin torque with u¼jelB=jejMs,TNA¼/C0 ð c=MsÞ½Ms;@Ms=@x/C138is the nona- diabatic part with c¼nu, and n/C281 being the ratio between exchange and spin-flip relaxation times.20Substituting Ms intoTAandTNA, we get for spin torques as follows TA¼/C0 ð u=MsÞðDxx0/C0Dyy0Þ; (A2a) TNA¼/C0 ð c=MsÞðDxy0þDyx0Þ; (A2b) where Dx¼/C0Msð@mx=@xÞþmxð@Ms=@xÞandDy¼Ms ð@my=@xÞ/C0myð@Ms=@xÞ. It is assumed within this calculation that the magnetiza- tion modulation is much smaller than M0. Under such an approximation, we neglect the Ms(x) dependence in the denominators of ½MsðxÞ/C138/C01@mxðyÞ=@xas well as in the TAand TNAterms containing @Ms=@x, so that MsðxÞ@mxðyÞ=@x/C29mxðyÞ@Ms=@x: (A3) Then, one can obtain the equations for the dynamic magnet- ization amplitudes affected by electric current are asfollows: iX4pm 0x¼/C0MsðxÞhhyiþ4pm0yXHþ4pu/C3ð@m0x=@xÞ /C04pnu/C3ð@m0y=@xÞ; iX4pm0y¼MsðxÞhhxi/C04pm0xXHþMsðxÞhefðxÞ þ4pu/C3ð@m0y=@xÞþ4pnu/C3ð@m0x=@xÞ;(A4) where X¼x=4pc,XH¼ðcHþiaxÞ=4pc,u/C3¼u=4pc.A s shown in Ref. 22, the dipole fields averaged on yare hhyðxÞ i¼/C0ð1 /C01Gðx;x0Þm0yðx0Þdx0; (A5a) hhxðxÞi ¼ /C0 4pm0xðxÞþð1 /C01Gðx;x0Þm0xðx0Þdx0;(A5b) where Gðx;x0Þ¼ð 2=sÞln½ðs2þðx/C0x0Þ2Þ=ðx/C0x0Þ2/C138with s being the film thickness. In the configuration where an in-plane dc magnetic field His perpendicular to the033908-5 Nikolay I. Polushkin J. Appl. Phys. 114, 033908 (2013) Downloaded 02 Aug 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionspropagation of magnetic oscillations along the xdirection, Eq.(A4) describe the propagation of MSSWs in the pres- ence of current. Importantly, as @m0y/@x/C24@m0x/@x, which are involved into both the adiabatic and nonadiabatic torques, and n/C240.01 (for metals),20the latter of these torques can be neglected in Eq.(A4). At least, this contribution is smaller than the compu- tation error in solving Eq. (A4). Therefore, Eq. (3)obtained in the main body of the paper can be used with a good accuracy. To solve Eq. (A4), the components of the dynamic mag- netization and excitation field were represented by the Fourier integrals m0xðyÞðxÞ¼1=ð2pÞ1=2ð1 /C01m0xðyÞðkÞexpðikxÞdk; fðxÞ¼1=ð2pÞ1=2ð1 /C01gðkÞexpðikxÞdk;(A6) while the saturation magnetization was expanded in a Fourier series, so that MsðxÞ¼X1 p¼/C01MpexpðipqxÞ; where q¼2p=K. Substituting all these expressions into Eq.(A4), multiplying the right- and left-side parts of the obtained equations by exp ð/C0ik0xÞ=ð2pÞ1=2, integrating them within the infinite limits on xand re-indexing the Fourier components, we get finally the infinite system of linear alge- braic equations with respect to the Fourier transforms of thecomponents of the dynamic susceptibility iR lðX/C0DX lÞdn;lvxðk/C0lqÞþRlBnlvyðk/C0lqÞ¼0 RlDnlvxðk/C0lqÞþiRlðX/C0DX lÞdn;lvyðk/C0lqÞ ¼RlMl/C0ngðk/C0lqÞ; (A7) where n;l¼/C0 1 ;…;1,DX j¼u/C3ðk/C0lqÞare Doppler shifts, dn,lis the Kronecker delta, Bn;l¼/C0Ml/C0nPðk/C0lqÞ/C0XHdn;l, Dnl¼Ml/C0n½1/C0Pðk/C0lqÞ/C138 þXHdn;l,a n d Pðk/C0lqÞ¼ð1/C0exp ½/C0ðjk/C0lqjs/C138Þ=jk/C0lqjsare the dynamic demagnetizing factors. For simplicity, we assume that the exciting field is f(x)¼1a t jxj<w=2a n d f(x)¼0a tjxj>w=2, so that gðkÞ¼ð2=pÞ1=2 sinðkw=2Þ=k. After cutting the number of the equations down to maximal values n;l/C2410, the matrix equation (A7) was solved numerically with finding vxðyÞðk;XÞ¼Dlðk;XÞ=Dðk;XÞ, where Dis the determinant of the left-hand side of Eq. (A7) andDlis the determinant obtained by replacing the lth column by the right-hand side of Eq. (A7). It is important that, for a small magnetization contrast ( DMs/Ms<0.1), the cutting even to n, l/C241 provides a sufficient computation accuracy (of the order of a few%).Finally, for a homogeneous layer with a saturation mag- netization M0the dynamic susceptibility can be obtained in its analytical form vxðkÞ¼½M0PðkÞþXH/C138M0 ½M0f1/C0PðkÞgþXH/C138½M0PðkÞþXH/C138/C0ðX/C0u/C3kÞ2gðkÞ: (A8) A similar equation can be derived for the y-component of the dynamic susceptibility. 1Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 2J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 3L. Berger, Phys. Rev. B 54, 9353 (1996). 4E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 (1999). 5A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, and T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004). 6S.-M. Seo, K.-J. Lee, H. Yang, and T. Ono, Phys. Rev. Lett. 102, 147202 (2009). 7Y. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B 57, R3213 (1998). 8V. Vlaminck and M. Bailleul, Science 322, 410 (2008). 9S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln /C19ar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488–1495 (2001). 10J. O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, and H. Puszkarski, Phys. Rev. B 54, 1043 (1996). 11V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 260301 (2010). 12B. Lenk, H. Ultichs, F. Grabs, and M. M €unzenberg, Phys. Rep. 507, 107 (2011). 13N. I. Polushkin, Appl. Phys. Lett. 99, 182502 (2011); ibid. 99, 229904 (2011). 14E. Feigenbaum, N. Kaminski, and M. Orenstein, Opt. Express 17, 18934 (2009). 15E. Schloemann, R. Tustison, J. Weissman, H. J. Van Hook, and T.Varitimos, J. Appl. Phys. 63, 3140 (1988). 16S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 17N. I. Polushkin, V. Oliveira, O. Conde, R. Vilar, Yu. N. Drozdov, A. Apolin /C19ario, A. Garc /C19ıa-Garc /C19ıa, J. M. Teixeira, and G. N. Kakazei, Appl. Phys. Lett. 101, 132408 (2012). 18L. L. Hirst, Phys. Rev. 141, 503 (1966). 19See our discussion of the spin-flip scattering effect (or nonadiabatic spin torque) in Appendix. 20S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). 21Nonuniform exchange can be neglected for spin waves with k/C282p/lex /C24103lm/C01, as a typical exchange length in ferromagnets is lex/C240.01lm. 22K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rev. B 66, 132402 (2002). 23R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961). 24A velocity of the electron flow, juj¼0.35 lm GHz, taken for this plotting corresponds to the current density je¼1.0/C2109A/cm2in an Fe film where the spin current polarization is assumed to be equal to unity. 25The effect of amplifying a weaker wave at the expense of the stronger onecan be described in terms of a general coupled-mode theory; see, for instance, M. I. Rabinovich and D. I. Trubetskov, Introduction to The Theory of Oscillations and Waves (Kluwer, Amsterdam, 1989), Chap. X. 26As follows from the textbooks on quantum dispersion theory, the wave- packet dispersion is negative (positive) if Re v<0 (Re v>0) at k<kmax and Re v>0 (Re v<0) at k>kmax, where kmaxis the wave vector at which /C0Imv(k) is maximal.033908-6 Nikolay I. Polushkin J. Appl. Phys. 114, 033908 (2013) Downloaded 02 Aug 2013 to 132.174.255.3. This article is copyrighted as indicated in the abstract. 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1.1935764.pdf

Spin waves in a periodically layered magnetic nanowire V. V. Kruglyak, R. J. Hicken, A. N. Kuchko, and V. Yu. Gorobets Citation: Journal of Applied Physics 98, 014304 (2005); doi: 10.1063/1.1935764 View online: http://dx.doi.org/10.1063/1.1935764 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/98/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dynamic control of spin wave spectra using spin-polarized currents Appl. Phys. Lett. 105, 112405 (2014); 10.1063/1.4896027 An effect of the curvature induced anisotropy on the spectrum of spin waves in a curved magnetic nanowire Low Temp. Phys. 39, 163 (2013); 10.1063/1.4792133 Reciprocal Damon-Eshbach-type spin wave excitation in a magnonic crystal due to tunable magnetic symmetry Appl. Phys. Lett. 102, 012403 (2013); 10.1063/1.4773522 Propagation and scattering of spin waves in curved magnonic waveguides Appl. Phys. Lett. 101, 152402 (2012); 10.1063/1.4757994 Formation and control of internal spin-wave channels in arrays of densely packed Permalloy nanowires J. Appl. Phys. 105, 07D302 (2009); 10.1063/1.3056151 [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: 133.1.198.126 On: Mon, 15 Dec 2014 06:31:24Spin waves in a periodically layered magnetic nanowire V. V. Kruglyaka/H20850and R. J. Hicken School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL, United Kingdom A. N. Kuchko Donetsk National University, 24 Universitetskaya Street, Donetsk, 83055, Ukraine V. Yu. Gorobets Institute of Magnetism of National Academy of Sciences (NAS) of Ukraine, 36-b Vernadskogo Boulevard, 03142 Kyiv, Ukraine /H20849Received 20 August 2004; accepted 27 April 2005; published online 1 July 2005 /H20850 We report a simple theoretical derivation of the spectrum and damping of spin waves in a cylindrical periodically structured magnetic nanowire /H20849cylindrical magnonic crystal /H20850in the “effective-medium” approximation. The dependence of the “effective” magnetic parameters upon the individual layerparameters is shown to be different from the arithmetic average over the volume of the superlattice.The formulas that are obtained can be applied firstly in the description of spin-wave dispersion inthe first allowed band of the structure and secondly in the design of a magnonic crystal with bandgaps in an arbitrary part of the spin-wave spectrum. © 2005 American Institute of Physics . /H20851DOI: 10.1063/1.1935764 /H20852 INTRODUCTION Materials with magnetic properties periodically modu- lated at the nanometer scale /H20851so-called magnonic crystals or magnetic superlattices /H20849MSLs /H20850/H20852have potential for applica- tions in magnetoelectronic devices. For example, a one-dimensional thin-film MSL is known to possess propertiesthat cannot be reduced to those of the separate layers. Also,phenomena such as giant magnetoresistance /H20849GMR /H20850, 1large out-of-plane magnetic anisotropy,2and magnetic-field- controlled photonic3and magnonic4–6band gaps have been observed. However, in other situations MSLs can be thoughtof as “effective media” with “effective” parameters 7that rep- resent an average of those of the constituent layers. A pat-terned recording medium consisting of a two-dimensionalperiodic array of nanosized magnetic elements 8represents another example of a periodic magnetic structure. Due to thedemand for greater recording densities, the element size andseparation are continuously decreasing, promoting interac-tions between elements via stray magnetic fields. Therefore,the dynamical response of an individual element is deter- mined by the spectrum of collective excitations of the entirearray. Moreover, current experimental methods do not yetprovide sufficient spatial resolution for the dynamical prop-erties of a single element to be studied directly. Conse-quently, the magnetic properties of the element must be de-duced from measurements made on the entire array. Arrays of cylindrical magnetic nanowires deposited elec- trochemically within porous membranes 9have attracted much attention due to their relative ease of fabrication andbecause of their potential for use as magnetophotonic crys-tals and recording media. 10In the latter case an element den- sity of 1 Tbit/in.2has recently been reported.11Arrays ofmultilayered nanowires can also be produced by this method,12,13providing an example of a three-dimensional /H208493D /H20850MSL. Such structures are important for the field of magnetophotonics, for magnetic recording technologythrough exploitation of the current perpendicular to the plane/H20849CPP /H20850GMR effect, and in designing patterned media with tunable effective magnetic properties. Finally, a 3D MSLmay also act as a 3D magnonic crystal, although the strongmagnetic damping present in metallic structures must be ad-dressed if they are to be used as media for spin-wave /H20849SW /H20850 propagation. As the speed of operation of magnetoelectronic devices containing MSL approaches the gigahertz regime, the dy-namical properties of the device are increasingly determinedby the SW spectrum of the MSL, which is strongly influ- enced by the presence of magnonic band gaps. Due to thecontinuous trend towards device miniaturization, one mustconsider MSL with very small characteristic dimensions. Aswe show below, the reduction of the period of the MSLpushes the SW band gaps to higher frequencies. The approxi-mate position of the band gaps and the dispersion of the SWmodes in the first allowed band, where the SW wavelength islong compared to the period of structural modulation, is thengoverned by the “effective-medium” parameters. It is there-fore important to know how the latter depend upon the pa-rameters of the MSL. In this paper we provide an analyticalderivation of the SW spectrum and damping in a single pe-riodic multilayer nanowire /H20849cylindrical MSL /H20850in the effective-medium approximation, and determine the corre-sponding effective material parameters. Since dynamicalproperties of such objects can be investigated by ferromag-netic resonance, 14Brillouin light scattering,15and magne- tooptical pump-probe experiments,16we anticipate that such studies will appear in the near future.a/H20850Electronic mail: V .V .Kruglyak@exeter.ac.ukJOURNAL OF APPLIED PHYSICS 98, 014304 /H208492005 /H20850 0021-8979/2005/98 /H208491/H20850/014304/4/$22.50 © 2005 American Institute of Physics 98, 014304-1 [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: 133.1.198.126 On: Mon, 15 Dec 2014 06:31:24THEORY AND DISCUSSION Let us consider an infinitely long cylinder of radius R consisting of two different alternating homogeneous layers.The layers have thicknesses d 1andd2, exchange constants /H92511 and/H92512, uniaxial anisotropy constants /H92521and/H92522, Gilbert damping parameters /H92611and/H92612, and gyromagnetic ratios g1 andg2/H20849gj/H110220,j=1,2 /H20850. The easy axis /H20849EA /H20850and the external bias magnetic field H0are aligned parallel to the axis of the cylinder, which is also an easy axis for the shape anisotropy.The value of the saturation magnetization M 0is assumed to be constant throughout the entire sample. In the static mag-netic state the sample is uniformly magnetized along H 0. The interfaces are assumed to be sharp and flat, and lie perpen-dicular to the EA. A cylindrical coordinate system is chosenso that the OZaxis is parallel to the EA. Hence, the spatial distribution of the material parameters can be described by /H9273/H20849z/H20850=/H20877/H92731,z2n/H11021z/H11021z2n+1, /H92732,z2n−1/H11021z/H11021z2n,/H20878z2n=nd,z2n+1=nd+d1, /H208491/H20850 where /H9273is one of the parameters /H9251,/H9252,/H9261, and g, the spatial period of the MSL is given by d=d1+d2, the coordinate of thenth interface is zn, and ntakes the values 0, ±1, ±2,.... To describe small perturbations from the ground state we use the Landau–Lifshitz–Gilbert equations /H11509Mj /H11509t=−gj/H20851Mj/H11003HE,j/H20852+/H9261j M0/H20875Mj/H11003/H11509Mj /H11509t/H20876, /H208492/H20850 where Mfis the magnetization. The effective magnetic field HE,jis given by HE,j=/H20849H0+/H9252jM0/H20850n+/H11509 /H11509r/H20873/H9251j/H11509Mj /H11509r/H20874+hm,j, /H208493/H20850 where nis an unit vector along OZ,hm,j=−/H11633/H9272jis the mag- netodipole field, and the magnetic potential /H9272jis determined from /H116122/H9272j−4/H9266/H20873/H11509Mj,x /H11509x+/H11509Mj,y /H11509y/H20874=0 . /H208494/H20850 Following the method described in Ref. 17, we write the magnetization as Mj/H20849r,t/H20850=nM0+Mj/H20849r,t/H20850, /H208495/H20850 where mjis a small perturbation from the ground state /H20849/H20841mj/H20841/H11270M0/H20850. Then, assuming a periodic dependence upon time, the latter becomes mj/H20849r,t/H20850=mj/H20849r/H20850exp /H20849−i/H9275t/H20850, and after some algebra one obtains /H20851/H9024j2−/H20849H˜j−/H9251j/H116122/H20850/H20849H˜j +4/H9266−/H9251j/H116122/H20850/H20852/H116122/H9272j+4/H9266/H20849H˜j−/H9251j/H116122/H20850/H115092/H9272j /H11509z2=0 , /H208496/H20850 where /H9024j=/H9275/gjM0,H˜j=h+/H9252−i/H9024j/H9261j, and h=H0/M0. This equation has solutions of the form/H9272j=Jm/H20849/H9260j/H9267/H20850exp /H20853i/H20849m/H9274+Gjz/H20850/H20854, /H208497/H20850 where Jmis the modified Bessel function of order m/H20849mis an integer /H20850,/H9267is the distance from the cylinder axis, and /H9274is the azimuthal angle. The axial and radial wave numbers Gjand /H9260jmust satisfy the following relation: /H9251j2/H20849Gj2+/H9260j2/H208503+/H9251j/H20849B˜j+H˜j/H20850/H20849Gj2+/H9260j2/H20850 +/H20849H˜jB˜j−/H9024j2−4/H9266/H9251jGj2/H20850/H20849Gj2+/H9260j2/H20850−4/H9266H˜jGj2=0 , /H208498/H20850 where B˜j=H˜j+4/H9266. In the absence of dissipation /H20849/H9261j=0 /H20850, this equation reduces to that derived in Ref. 17. In the general case, another relation between /H9024j,m,Gj, and /H9260jmust be found by application of appropriate boundary conditions onthe cylinder surface /H9267=R, but this requires numerical calcu- lations. An analytical solution may be obtained in the limit ofa thin nanowire R/H11021 /H9264ex, where /H9264exis the exchange length. In this case, the magnetization can be assumed to be uniformwithin the cross section of the cylinder /H20849 /H9260j=0 /H20850,17,18and the following dispersion relation for SW in a layer of type jis obtained: Gj=/H208811 /H9251j/H20851/H9024j−h−/H9252j−i/H9024j/H9261j/H20852. /H208499/H20850 To find the SW spectrum in the entire sample, we use the Bloch theorem and impose exchange boundary conditions/H20849without interface anisotropy /H20850at the interfaces at z n/H20849Ref. 19 /H20850 /H20841m1/H20841zn=/H20841m2/H20841zn,/H20879/H92511/H11509m1 /H11509z/H20879 zn=/H20879/H92512/H11509m2 /H11509z/H20879 zn, /H2084910/H20850 and finally arrive at the following expression for the SW spectrum in a thin cylindrical MSL: cos /H20849Kd /H20850= cos /H20849G1d1/H20850cos /H20849G2d2/H20850−1 2/H20873/H92512G2 /H92511G1+/H92511G1 /H92512G2/H20874 /H11003sin/H20849G1d1/H20850sin/H20849G2d2/H20850, /H2084911/H20850 where Re /H20849K/H20850=kis the SW quasiwave number and Im /H20849K/H20850 =/H9260˜is the inverse of the effective SW attenuation length. This equation is well known since it is identical to that obtainedfor a SW in a thin-film MSL in the short waveapproximation. 20,21This means that the effects predicted by these earlier calculations must also appear in the presentcase. However, the results discussed below, including ex-pressions for the effective-medium parameters, can also beapplied to a thin-film MSL when the exchange dominates. The spectrum and damping of a SW in the effective- medium limit are now derived from /H2084911/H20850by assuming that the SW wavelength is greater than the spatial period of theMSL /H20851Re/H20849G jd/H20850/H112701,Re /H20849Kd /H20850/H112701/H20852.20In this limit the SW will see the MSL as being quasiuniform with effective-medium parameter values corresponding to a certain average over theunit cell of the MSL. We obtain K eff/H11015/H208811 /H9251¯/H20875/H9275 M0g¯−h−/H9252¯−/H9275/H9261¯ M0g¯/H20876, /H2084912/H20850 where the effective-medium parameters are given by014304-2 Kruglyak et al. J. Appl. Phys. 98, 014304 /H208492005 /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: 133.1.198.126 On: Mon, 15 Dec 2014 06:31:24/H9252¯=/H92521d1+/H92522d2 d1+d2, /H2084913/H20850 /H9251¯−1=/H92511−1d1+/H92512−1d2 d1+d2,g¯−1=g1−1d1+g2−1d2 d1+d2, /H2084914/H20850 /H9261¯=/H92611g2d1+/H92612g1d2 g2d1+g1d2. /H2084915/H20850 The effective anisotropy constant /H9252¯is given by an arith- metic average over the volume of the MSL /H2084913/H20850,a so n e might expect. However, the effective exchange constant /H9251¯ and gyromagnetic ratio g¯are instead obtained in /H2084914/H20850from an arithmetic average of the quantities /H9251j−1andgj−1. As may be seen from Fig. 1, the rule for calculating the effectivevalue differs significantly from the arithmetic average /H2084913/H20850. The effective-medium Gilbert damping parameter is evenmore complicated /H2084915/H20850, depending also upon the values of the gyromagnetic ratio in the layers. The effective anisotropyand exchange parameters have been observed experimentallyin thin-film MSLs by microwave ferromagnetic resonance/H20849FMR /H20850. We therefore expect that such measurements may also be used for the experimental observation of the effectivedamping parameter and gyromagnetic ratio. In order to avoidconsideration of nonuniform demagnetizing fields, 22we have assumed that the saturation magnetization M0is not modu- lated in the structure. However, modulation of M0may be analytically considered in a thin-film MSL, in which case itis easy to see from /H208499/H20850and /H2084911/H20850that the dependence of the effective-medium parameters upon those of the layers wouldbe even more complicated. A uniform saturation magnetization also means that the magnetostatic fields outside the periodic multilayer nanowireare the same as those for a uniform nanowire. Hence, theformalism developed for SWs in an array of uniformnanowires 5,23could be generalized to describe the dynamics in an array of periodic multilayer nanowires. However, this isbeyond the scope of the present paper. In principle, a cylin-drical MSL in which the saturation magnetization is constantbut other magnetic parameters are modulated, could be madefrom Co–P alloys. 24,25The magnetic parameters of the latterare very sensitive to the phosphorous concentration due to transitions from the amorphous to the crystalline state andfrom a hexagonal to a cubic structure. Parameter values rep-resentative of the Co–P system are used in the graphs pre-sented here. In Fig. 2 we plot the SW spectrum of a cylindrical MSL /H2084911/H20850in which the uniaxial anisotropy is modulated, together with the spectrum of an effective medium /H2084912/H20850with the ef- fective anisotropy calculated from /H2084913/H20850using the same layer parameters. We see that band gaps emerge at effective-medium SW frequencies for which K eff=/H9266l d,l=1,2, ... . /H2084916/H20850 This may be understood by noting that the linearized Landau–Lifshitz equation without damping may be recast ina form identical to that of the Schrödinger equation, in whichthe periodically modulated anisotropy is analogous to a pe-riodic electronic potential. 26Modulation of the exchange constant and gyromagnetic ratio also leads to the formationof band gaps, although these quantities are not analogous tothe electron potential, and the corresponding band gaps shifttowards higher frequencies. Moreover, the width and posi-tion of the band gaps have a significant dependence upon theratio of the MSL layer thicknesses /H20849Fig. 3 /H20850. Nevertheless, condition /H2084916/H20850still provides a useful starting point for de- signing a MSL with the desired band-gap parameters. Furthertuning of the MSL parameters can be achieved using thegraphical technique described in Refs. 6 and 19. It is inter-esting to note that the position and the width of the band gapcan be tuned independently. Equation /H2084916/H20850shows that the position of the first band gap can be set by adjusting only theperiod of the MSL, which is probably the easiest parameterof the MSL to change. The width of the band can be variedby changing the magnetic parameters of the layers and theratio of their thicknesses, while leaving the effective param-eters /H2084913/H20850–/H2084915/H20850unchanged /H20849Fig. 3 /H20850. In Refs. 27 and 28 we considered /H20849in the short wave approximation /H20850SW in a thin-film periodic multilayer with interfaces of finite thickness, but without taking into accountmodulation of the gyromagnetic ratio. Those models are eas- FIG. 1. The ratio Sof an “effective-medium” parameter /H9273given by /H2084914/H20850to that given by /H2084913/H20850is plotted for the same values of the parameter in the layers /H9273i. FIG. 2. The spectrum of SW in a cylindrical MSL /H2084911/H20850with modulation of the constant of uniaxial anisotropy /H92522//H92521=5 and /H208491/H20850d=5 nm, /H208492/H20850d =10 nm, and /H208493/H20850d=20 cm. The dashed line represents the SW spectrum in a fine layered nanowire /H2084912/H20850.014304-3 Kruglyak et al. J. Appl. Phys. 98, 014304 /H208492005 /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: 133.1.198.126 On: Mon, 15 Dec 2014 06:31:24ily applied to a cylindrical MSL, and the results derived pre- viously remain valid. On the other hand, expressions /H208499/H20850and /H2084911/H20850–/H2084915/H20850, which are derived here after the inclusion of a periodically modulated gyromagnetic ratio, are easily gener-alized to those other models. SUMMARY The spectrum and damping of SW in a cylindrical MSL have been derived in the “effective-medium” approximation.It has been shown that the effective-medium parameters havean unexpected dependence upon those of the constituent lay-ers and, in general, are not given by an arithmetic averageover the volume of the MSL. The formulas that have beenobtained are useful in designing composite magnetic materi-als with the desired microwave properties.ACKNOWLEDGMENTS The authors thank Professor Yu. I. Gorobets for fruitful discussions. 1M. N. Baibich et al. , Phys. Rev. Lett. 61, 2472 /H208491988 /H20850. 2P. F. Carcia, A. D. Meinhaldt, and A. Suna, Appl. Phys. Lett. 47, 178 /H208491985 /H20850. 3I. L. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapov- alov, and T. H. Rasing, J. Phys. D 36, R277 /H208492003 /H20850, and references therein. 4J. O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, and H. Puszkarski,Phys. Rev. B 54, 1043 /H208491996 /H20850. 5H. Puszkarski and M. Krawczyk, Solid State Phenom. 94,1 2 5 /H208492003 /H20850. 6V . V . Kruglyak and A. N. Kuchko, Physica B 339, 130 /H208492003 /H20850, and ref- erences therein. 7V . V . Kruglyak and A. N. Kuchko, Fiz. Met. Metalloved. 92,3 /H208492001 /H20850 /H20851Phys. Met. Metallogr. 92,2 1 1 /H208492001 /H20850/H20852. 8A. Moser, K. Takano, D. T. Margulies, M. Albrecht, Y . Sonobe, Y . Ikeda, S. H. Sun, and E. E. Fullerton, J. Phys. D 35, R157 /H208492002 /H20850. 9P. Aranda and J. M. Garcia, J. Magn. Magn. Mater. 249, 214 /H208492002 /H20850,a n d references therein. 10M. Todorovic, S. Schultz, J. Wong, and A. Scherer, Appl. Phys. Lett. 74, 2516 /H208491999 /H20850. 11S. Shingubara, T. Shimizu, M. Nagayanagi, Y . Fugii, O. Yaegashi, R. G. Wu, H. Sakaue, and T. Takahagi, Talk CF-10 at the Ninth Joint Intermag-MMM Conference, California, USA 2004 /H20849unpublished /H20850. 12P. R. Evans, G. Yi, and W. Schwarzacher, Appl. Phys. Lett. 76,4 8 1 /H208492000 /H20850. 13A. Robinson and W. Schwarzacher, J. Appl. Phys. 93, 7250 /H208492003 /H20850. 14U. Ebels, J. L. Duvail, P. E. Wigen, L. Piraux, L. D. Buda, and K. Oun- adjela, Phys. Rev. B 64, 144421 /H208492001 /H20850. 15Z. K. Wang, M. H. Kuok, S. C. Ng, D. J. Lockwood, M. G. Cottam, K. Nielsch, R. B. Wehrspohn, and U. Gosele, Phys. Rev. Lett. 89, 027201 /H208492002 /H20850. 16R. J. Hicken, A. Barman, V . V . Kruglyak, and S. Ladak, J. Phys. D 36, 2183 /H208492003 /H20850. 17R. Arias and D. L. Mills, Phys. Rev. B 63, 134439 /H208492001 /H20850;66, 149903 /H20849E/H20850 /H208492002 /H20850. 18R. Skomski, M. Chipara, and D. J. Sellmyer, J. Appl. Phys. 93, 7604 /H208492003 /H20850. 19V . V . Kruglyak, A. N. Kuchko, and V . I. Finokhin, Fiz. Tverd. Tela /H20849S.- Peterburg /H2085046, 842 /H208492004 /H20850/H20851Phys. Solid State 46, 867 /H208492004 /H20850/H20852. 20R. P. van Stapele, F. J. A. M. Greidanus, and J. W. Smits, J. Appl. Phys. 57,1 2 8 2 /H208491985 /H20850. 21K. Y . Guslienko, Fiz. Tverd. Tela /H20849S.-Peterburg /H2085037, 1603 /H208491995 /H20850/H20851Phys. Solid State 37, 870 /H208491995 /H20850/H20852. 22A. Aharoni, J. Appl. Phys. 68,2 8 9 2 /H208491990 /H20850. 23R. Arias and D. L. Mills, Phys. Rev. B 67, 094423 /H208492003 /H20850. 24R. S. Iskhakov, S. V . Komogortsev, L. A. Chekanova, A. D. Balaev, V . A. Yuzova, and O. V . Semenova, Pis’ma Zh. Tekh. Fiz. 29,1 /H208492003 /H20850/H20851Tech. Phys. Lett. 29, 263 /H208492003 /H20850/H20852. 25R. S. Ishakov, G. V . Popov, and M. M. Karpenko, Fiz. Met. Metalloved. 56,8 5 /H208491983 /H20850. 26A. A. Abrikosov, Fundamentals of the Theory of Metals /H20849North-Holland, Amsterdam, 1988 /H20850. 27V . V . Kruglyak and A. N. Kuchko, Fiz. Met. Metalloved. 93,1 5 /H208492002 /H20850 /H20851Phys. Met. Metallogr. 93,5 1 1 /H208492002 /H20850/H20852. 28V . V . Kruglyak and A. N. Kuchko, J. Magn. Magn. Mater. 272–276 ,3 0 2 /H208492004 /H20850. FIG. 3. The spectrum of SW in a cylindrical MSL /H2084911/H20850with modulation of the constant of /H20849a/H20850the exchange parameter /H92512//H92511=5 and /H20849b/H20850the gyromag- netic ratio g2/g1=5. In both cases d=20 nm and /H208491/H20850d2/d1=2, /H208492/H20850d2/d1 =5, and /H208493/H20850d2/d1=10. The dashed lines represent the SW spectrum in a fine layered nanowire /H2084912/H20850.014304-4 Kruglyak et al. J. Appl. Phys. 98, 014304 /H208492005 /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: 133.1.198.126 On: Mon, 15 Dec 2014 06:31:24

1.4827808.pdf

Experimental investigation of spin Hall effect in indium tin oxide thin film Z. Qiu, T. An, K. Uchida, D. Hou, Y. Shiomi, Y. Fujikawa, and E. Saitoh Citation: Applied Physics Letters 103, 182404 (2013); doi: 10.1063/1.4827808 View online: http://dx.doi.org/10.1063/1.4827808 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/103/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effects of SR irradiation on tin-doped indium oxide thin film prepared by rf magnetron sputtering Rev. Sci. Instrum. 73, 1384 (2002); 10.1063/1.1423624 Effect of hydrogen partial pressure on optoelectronic properties of indium tin oxide thin films deposited by radio frequency magnetron sputtering method J. Appl. Phys. 86, 974 (1999); 10.1063/1.370834 Indium tin oxide thin films for organic light-emitting devices Appl. Phys. Lett. 74, 3444 (1999); 10.1063/1.124122 Effect of substrate absorption on the efficiency of laser patterning of indium tin oxide thin films J. Appl. Phys. 85, 4207 (1999); 10.1063/1.370332 Tin doped indium oxide thin films: Electrical properties J. Appl. Phys. 83, 2631 (1998); 10.1063/1.367025 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: 202.28.191.34 On: Fri, 20 Feb 2015 16:05:24Experimental investigation of spin Hall effect in indium tin oxide thin film Z. Qiu,1,a)T. An,2,b)K. Uchida,2,3D. Hou,1Y. Shiomi,2Y. Fujikawa,2and E. Saitoh1,2,4 1WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 3PRESTO, Japan Science and Technology Agency, Saitama 332-0012, Japan 4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan (Received 11 August 2013; accepted 7 October 2013; published online 30 October 2013) The inverse spin Hall effect (ISHE) induced by s pin pumping in indium tin oxide (ITO) films has been investigated quantitatively. We measured ferromagne tic resonance and electromotive force spectra with changing the ITO thickness in ITO/permallo y bilayer films. The intensity of the observed electromotive force changes systematically with th e ITO film thickness, which is consistent with the prediction of ISHE. By using an equivalent circuit model, the spin Hall angle and diffusion length of the ITO film are estimated. VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4827808 ] In the field of spintronics, the direct and inverse spin Hall effects (SHE and ISHE) have been extensively studied, since they can generate and detect a spin current, a flow ofelectron spins in a solid. 1–6SHE and ISHE are considered to be essential techniques for realizing spin based magnetic memories and computing devices in which the reciprocalconversion between a spin current and a charge current plays an important role. Improvement of the efficiency of the spin current generation and detection is important for the devel-opment of spintronics devices. Here, materials having a high conversion efficiency between spin currents and charge cur- rents are required. So far, metallic systems have been mainlystudied, which have been proved to have a good potential for experimental and theoretical studies of SHE and ISHE. 5–18 However, low cost and chemical stability are also expected for a real spintronic devices. Here, ISHE in indium tin oxide (ITO) has been investigated.19 In this letter, ISHE in ITO induced by spin pumping was investigated quantitatively. Ferromagnetic resonance (FMR) and electromotive force spectra were measured with various ITO thickness for ITO/permalloy (Py) bilayer films. Theelectromotive force induced by ISHE changes systematically with the ITO thickness, which is consistent with the equiva- lent circuit model. The spin Hall angle and the diffusionlength of ITO were determined by using a spin pumping model. 7–9,20,21 ITO films were deposited on thermally oxidized silicon wafers from a sintered ITO target at room temperature by using an RF-sputtering system. The thickness of the ITO films was changed from 17 to 175 nm. The conductivity ofthe prepared ITO films is 3 :4/C210 5X/C01m/C01, which was measured by a four-point probe method. Permalloy films with a fixed thickness of 17 nm were deposited on ITO filmsby an electron beam deposition technique, and the conduc- tivity was measured to be 1 :5/C210 6X/C01m/C01. Spin pumping was excited by using a waveguide sys- tem equipped with a network analyser (Fig. 1). Samples were cut to 1 /C25m m2in size and set near the shorttermination of a short-end coplanar waveguide (CPW), where is a low electric field and high magnetic field region.22Typical dimensions of the short-end CPW are shown in the inset of Fig. 1, and the characteristic imped- ance was designed to be 50 X. A static magnetic field H was applied to the sample. Magnetization precessionmotions, driven by FMR in the permalloy layer, pumped spin currents out of the permalloy layer into the ITO layer. The generated spin currents we re detected as electromotive forces in the ITO layer via ISHE, by using two electrodes attached to both ends of the ITO layer. In this work, the FMR and electromotive force spectra were measured simul-taneously at room temperature. Figure 2(a)shows the external magnetic field Hdepend- ence of the normalized electromotive force Vwith various microwave frequencies f. The sign of electromotive force V is reversed by reversing the direction of H. By changing the microwave frequency f, the FMR magnetic field varies systematically (Fig. 2(b)), which is consistent with Kittel’s formula: 2 pf¼l 0cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HFMRðHFMRþ4pMsÞp with c¼1:76 /C21011s/C01T/C01and 4 pMs¼0:750 T. Here, cand 4 pMsare the effective gyromagnetic ratio and the saturation magnet- ization of the permalloy film, respectively. Electromotive force Vspectra are reproduced by Lorentz and differential Lorentz functions. The symmetry Lorentz parts of Vspectra are normally attributed to the elec- tromotive force corresponding to the ISHE. However, wenote that the possible contribution from spin-rectification effects in permalloy layer is also discussed recently. 23–25 Here, we determine that the height of symmetry Lorentz peaks Vs¼VISHEþVsr, where VISHE and Vsrare the electromotive forces corresponding to ISHE and spin- rectification effects, respectively. Peaks in FMR spectra arereproduced by Lorentz functions. The heights of the Lorentz peaks are determined to be P ab, the microwave-absorption powers. The half width at half maximum (HWHM) Cof those Lorentz peaks are proportional to the microwave fre- quency f, as shown in Fig. 2(c)). The Gilbert damping con- stant a¼cC=2pfwas estimated by fitting the spectrum. The real part of the effective mixing conductance g"# r was estimated from the Gilbert damping of the permalloy single layer film aPyand ITO/Py bilayer film aITO=Pyasa)Author to whom correspondence should be addressed. Electronic mail: qiuzy@imr.tohoku.ac.jp b)Present address: RIKEN, Saitama 351-0198, Japan 0003-6951/2013/103(18)/182404/3/$30.00 VC2013 AIP Publishing LLC 103, 182404-1APPLIED PHYSICS LETTERS 103, 182404 (2013) 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: 202.28.191.34 On: Fri, 20 Feb 2015 16:05:24g"# r¼4pMsdPy glBðaITO=Py/C0aPyÞ; (1) where dPy,g, and lBare the thickness of the permalloy layer, the g factor, and the Bohr magneton, respectively. g"# ris esti- mated to be 1 :1ð60:2Þ/C21019m/C02in samples with a ITO layer thickness over 35 nm in the frequency range of 2.0–8.0 GHz, which is comparable to those for metal/metalinterfaces. 7–9 The spin current density j0 sat ITO/Py interface is given as j0 s¼g"# rc2h2/C22h½4pMscþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4pMsÞ2c2þð4pfÞ2q /C138 8pa2½ð4pMsÞ2c2þð4pfÞ2/C138;(2) where his the amplitude of the microwave magnetic field. h is related to Pabas26 Pab¼vl04pMsc 4ah2/C24pMscþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4pMsÞ2c2þð4pfÞ2q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4pMsÞ2c2þð4pfÞ2q2 643 75; (3) where vdenotes the volume of the region that is magneti- cally excited by the microwave in the permalloy layer. Experimental results of Vs=Pabas a function of fis shown inthe inset to Fig. 3. The slight decrease on increasing fis well reproduced by Eqs. (2)and(3)(inset to Fig. 3). For the bilayer spin pumping system, electric field corre- sponding to ISHE is expressed as EISHE¼hSHEktanhðdN=2kÞ dNrNþdFrF2e /C22h/C18/C19 j0 s; (4) where dN;dF;rN, and rFdenote the thickness dand the con- ductivity rof the nonmagnetic (N) and ferromagnetic (F) layers, respectively. hSHEandkare the spin Hall angle and the spin diffusion length of the nonmagnetic layer, respec-tively. Similarly, electric field due to spin-rectification is expressed as E sr¼jsr=ðdNrNþdFrFÞ. Here, jsris the equiv- alent current due to spin-rectification effects and should beproportional to P ab. FIG. 1. A schematic illustration of the experimental setup used in the present study. An enlargement of the sample and the short-end coplanar waveguide are shown in the lower part of the figure. Here, HandHrfrepresent the exter- nal magnetic field and the microwave magnetic field, respectively. The inset shows typical dimensions of the short-ended CPW. The characteristic im- pedance of the CPW was designed to be 50 X.FIG. 2. (a) The dependence of normalized dc electromotive forces Von the external magnetic field H. The transition of the line colors corresponds to the different microwave frequencies f. (b) The FMR magnetic field HFMRde- pendence of the microwave frequency f. The open squares represent the ex- perimental data. The solid curve is obtained using Kittel’s formula. (c) The microwave frequency fdependence of HWHM CofVISHEand/or Pspectra for the ITO/Py bilayer film (open squares) and the permalloy single layer film (open circles), and the linear fitting results (solid line). FIG. 3. The experimental results of microwave frequency fdependence of E=j0 s(open squares). The solid line is the mean value of E=j0 sfor all values off. The inset shows the experimental and fitting results of the microwave frequency fdependence of Vs=Pab.182404-2 Qiu et al. Appl. Phys. Lett. 103, 182404 (2013) 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: 202.28.191.34 On: Fri, 20 Feb 2015 16:05:24With a equivalent circuit as shown in the inset to Fig. 4, taking contribution from both ISHE and spin-rectificationeffects into consideration, the normalized whole electric field is expressed as E j0 s¼EISHEþEsr j0 s¼hSHEktanhðdN=2kÞ2e=/C22hþCsr dNrNþdFrF:(5) Here, E¼Vs/l, where lis the effective length of samples, and Csr¼jsr=j0 s. When dNis larger than the spin diffusion length k,j0 sshould be saturated and be proportional to Pab. Therefore, Csrcan be expected to be a constant for dN>k. The microwave frequency dependence of E=j0 sis shown in Fig.3, which is stable in range of f¼2.0–8.0 GHz. The open circles in Fig. 4show the ITO layer thickness (dITO) dependence of E=j0 s. The error bar for each point is esti- mated from experimental data in the range of 2.0–8.0 GHz forthree samples. By using Eq. (5), the best estimate of h SHEand kof our ITO are 0.0065 60.001 and 30 62 nm, respectively. Here, the contribution from Esris estimated to be about 5% for the sample with a 35 nm ITO layer. In this work, a normalized parameter g¼EISHE=j0 sis introduced for evaluation of the conversion between spincurrent density j 0 sand the generated electric field related to ISHE EISHE. Here, gdoes not depend on the shape of the de- vice but is a function of the sheet conductance R¼ rFdFþrPdPof the device. For a real application, the maxi- mum output power Pmaxfrom united spin current by ISHE is proportional to g2asPmax/E2 ISHE. By using Rdependence ofg, the transfer efficiency can be compared among different materials. In the inset to Fig. 4, calculations of Rdependence ofgare shown for a platinum/Py bilayer film. Here, the greatest reported value of hSHE¼0:078 is used, and k¼3 nm comes from same literature.13It can be seen that g of the ITO/Py bilayer films are comparable to that of Pt/Pybilayer films for some Rvalues. In this calculation, apermalloy layer having the same thickness and conductivity as that in the ITO/Py bilayer films used in this study has been considered in the Pt/Py bilayer system. In summary, the ITO thickness dependence of ISHE was measured for an ITO/Py bilayer spin pumping system. The mixing conductance at the ITO/YIG interface is esti- mated to be 1 :1/C210 19m/C02, which is very close to that of a metal/metal interface. The spin Hall angle and diffusion length were estimated to be 0.0065 and 30 nm, respectively. This work was supported by a Grant-in-Aid for Young Scientists (A) (25707029) from MEXT, Japan, a Grant-in- Aid for Scientific Research (A) (24244051) from MEXT,Japan, PRESTO-JST “Phase Interfaces for Highly Efficient Energy Utilization,” CREST-JST “Creation of Nanosystems with Novel Functions through Process Integration,” LC-IMRof Tohoku University, The Murata Science Foundation, The Mazda Foundation, and The Sumitomo Foundation. 1M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459 (1971). 2J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 3Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom, Science 306, 1910 (2004). 4J. Wunderlich, B. Kaestner, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 94, 047204 (2005). 5E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 6A. Azevedo, L. H. Vilela Leao, R. L. Rodriguez-Suarez, A. B. Oliveira, and S. M. Rezende, J. Appl. Phys. 97, 10C715 (2005). 7K. Ando, Y. Kajiwara, S. Takahashi, S. Maekawa, K. 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Lett. 102, 242402 (2013). 26R. Iguchi, K. Ando, R. Takahashi, T. An, E. Saitoh, and T. Sato, Jpn. J. Appl. Phys., Part 1 51, 103004 (2012).FIG. 4. The experimental result of dITOdependence of E=j0 sfor ITO/Py bilayer films (open circles). The solid curve is obtained using Eq. (5). The upper inset is equivalent circuit of ITO/Pt system in which both ISHE and spin-rectification effects are considered. The lower inset is the comparison ofR¼rFdFþrPdPdependence of g¼EISHE=j0 sfor ITO/Py and Pt/Py sys- tem with a same Py layer thickness.182404-3 Qiu et al. Appl. Phys. Lett. 103, 182404 (2013) 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: 202.28.191.34 On: Fri, 20 Feb 2015 16:05:24

1.4907332.pdf

Temperature-dependent structure of Tb-doped magnetite nanoparticles Katherine P . Rice,1,a),b)Stephen E. Russek,1,a),c)Roy H. Geiss,2,a)Justin M. Shaw,1,a) Robert J. Usselman,1,a)Eric R. Evarts,1,a)Thomas J. Silva,1,a)Hans T. Nembach,1,a) Elke Arenholz,3,a)and Yves U. Idzerda4,a) 1National Institute of Standards and Technology, Boulder, Colorado 80305, USA 2Colorado State University, Fort Collins, Colorado 80523, USA 3Lawrence Berkeley National Laboratory, Advanced Light Source, Berkeley, California 94720, USA 4Department of Physics, Montana State University, Bozeman, Montana 59717, USA (Received 30 November 2014; accepted 20 January 2015; published online 13 February 2015) High quality 5 nm cubic Tb-doped magnetite nanoparticles have been synthesized by a wet-chemical method to investigate tailoring of magnetic properties for imaging and biomedicalapplications. We show that the Tb is incorporated into the octahedral 3 þsites. High-angle annular dark-field microscopy shows that the dopant is well-distributed throughout the particle, and x-ray diffraction measurements show a small lattice parameter shift with the inclusion of a rare-earthdopant. Magnetization and x-ray magnetic circular dichroism data indicate that the Tb spins are unpolarized and weakly coupled to the iron spin lattice at room temperature, and begin to polarize and couple to the iron oxide lattice at temperatures below 50 K. Broadband ferromagnetic reso-nance measurements show no increase in magnetic damping at room temperature for Tb-doped nanoparticles relative to undoped nanoparticles, further confirming weak coupling between Fe and Tb spins at room temperature. The Gilbert damping constant, a, is remarkably low for the Tb-doped nanoparticles, with a¼0.02460.003. These nanoparticles, which have a large fixed moment, a large fluctuating moment and optically active rare-earth elements, are potential high- relaxivity T1 and T2 MRI agents with integrated optical signatures. VC2015 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4907332 ] Rare-earth ions have been incorporated into magnetic oxides1–4and magnetic metals5–7to modify crystalline struc- ture, magnetic anisotropy, optical properties, and magneticdamping. Rare-earth ions can have large spin and orbital moments and, since the moments reside in the f-shell, the or- bital moment is often not quenched when the ion is in a lat-tice. The strong spin-orbit coupling along with a large local moment then causes a large increase in anisotropy and mag- netic damping. 8Weak exchange or superexchange between Tb3þand neighboring magnetic ions can result in fluctuating spins embedded in a ferrimagnetic structure. Incorporation of rare-earth ions can also lead to strong temperature depend-ence of the magnetic properties near room and physiological temperatures. 9 While there have been extensive studies of rare earth dop- ing in bulk materials and thin films, the details of where the rare earth dopants are incorporated in nanoparticles and their effects on magnetic and optical properties are still poorlyunderstood. 3,4Here, we investigate Tb doped magnetite nanoparticles and show that the Tb3þions are incorporated in the octahedral sites, with some clustering and preference forthe surface. Tb 3þgoes into the magnetite lattice despite the large ionic radii mismatch: 78.5 pm for high-spin Fe3þand 106.3 pm for Tb3þ.10The octahedral site, which is larger than the tetrahedral site, can nominally accommodate cations up to /C2486 pm, which indicates Tb3þincorporation requires locallattice distortion. The ionic size favors incorporation of Tb into larger octahedral and surface sites. The addition of Tb-precur- sor changes the growth habit from spherical to cubic particles.In contradistinction with Tb-doped garnets and Tb-doped mag-netic metals, the anisotropy and magnetic damping are largely unaffected at room temperature, which indicates that the Tb spins are decoupled from the Fe spin lattice at room tempera-ture. Magnetization data show Tb spins begin to polarize and couple to the Fe spins at temperatures below 50 K. In a typical synthesis, 30.47 mmol Iron(III) acetylaceto- nate (Fe(acac) 3), 0.093 mmol Terbium(III) acetylacetonate hydrate (Tb(acac) 3), 1.12 mmol 1,2-hexadecanediol were mixed with 10 ml diphenylether and degassed for approxi-mately 45 min. The reaction was placed under a slight posi- tive nitrogen pressure and heated to 100 /C14C, where 1.0 mmol oleylamine was rapidly injected, followed by 1.0 mmol oleicacid. All chemicals were purchased from Sigma, 27and used as received. The reaction was refluxed at 260/C14C for 22 h. After cooling, the particles were cleaned from the reactionmixture by addition of excess ethanol and centrifugation, and redispersed in hexanes. High-resolution transmission electron micrographs (TEMs) of the Tb-doped and undoped nanoparticles are shown in Fig. 1. The cubic morphology of the Tb-doped par- ticles is clearly seen, and image simulations of the magnetitestructure with glide translations overlaid on the particlesshow excellent agreement with the particle lattice. 11 Previous work on ferrite nanoparticles12has shown that the growth habit can be controlled by growth rate, which ismodified by dopant incorporation. Histograms of particle size taken from TEM micrographs give mean sizes anda)Authors contributed equally to this work. b)Present address: Cameca Instruments, Madison, Wisconsin 53711, USA. c)Author to whom correspondence should be addressed. Electronic mail: stephen.russek@nist.gov 0003-6951/2015/106(6)/062409/4/$30.00 VC2015 AIP Publishing LLC 106, 062409-1APPLIED PHYSICS LETTERS 106, 062409 (2015) standard deviations of d ¼5.0 nm61.4 nm and d ¼4.6 nm 61.0 nm for Tb-doped and undoped nanoparticles, respec- tively. Fig. 2shows the X-ray diffraction spectra of the doped and undoped samples along with a model spectrum for ideal magnetite nanocubes. The X -ray data confirm high quality magnetite nanoparticles are pres ent (here, we do not differenti- ate between magnetite and c/C0Fe2O3) in the Tb-doped sample with a lattice constant of 0.8386 nm 60.0003 nm compared to 0.8365 nm 60.0003 nm for the undoped sample, showing an increase in the lattice paramet er of approximately 0.25%, con- sistent with the incorporation of the large radii Tb3þions. Some additional non-magnetite dif fraction lines are observed in the undoped control sample. These sharp peaks are from salt precipitates and are present wh en samples are not fully cleaned before drying. Single-particle energy dispersive X-ray spectra,13 taken in a 200 kV TEM, indicate a ratio of Tb:Fe of approxi- mately 1:6 within a nanoparticle, and no Tb on the surrounding grid. The measured Tb:Fe ratio is slightly less than the 1:5 molar ratio used in the reaction. High-angle annular dark-field (HAADF) images, taken in the scanning transmission mode of the TEM, are shown inFig. 3. The Tb atoms, which have a high atomic number, scatter the electrons to larger angles and appear as bright white spots in the image. The Tb appears to be dispersedthroughout the particles although some clustering and prefer- ence for the surface can be observed (higher intensity along the nanoparticle edges in the HAADF images).To confirm the location of the Tb in the magnetite lat- tice, x-ray magnetic circular dichroism (XMCD) spectrawere taken. XMCD spectra, taken at 300 K, for the doped and undoped samples are shown in Fig. 4. A field of 0.5 T was applied to orient the magnetic moment and the differ-ence spectra between right and left circularly polarized radia- tion are plotted for the Tb M4 and M5 edges (corresponding to excitations from the 3d to 4f states) in Fig. 4(a)and the Fe L2 and L3 edges (corresponding to excitations from 2p to 3d states) in Fig. 4(b). The Tb XMCD spectrum at 300 K shows a weak signal in comparison to the 60% theoretical valueexpected at 0 K. 14For the Fe L-edge data, a clear suppres- sion in the peak at 708 eV is observed. This peak is associ- ated with the Fe3þin the octahedral site, and the loss of signal in this position indicates Tb substitution at that site.15 FIG. 1. TEM images of (a) undoped and (b) Tb-doped magnetite particles. Insets show simulations of the magnetite crysta l structure. Scale bars represent 2 nm. FIG. 2. X-ray diffraction spectra using Cu K aradiation, for undoped, doped, and ideal magnetite nanoparticles. FIG. 3. HAADF image of Tb-doped nanoparticles. Tb atoms appear as bright white spots. FIG. 4. XMCD spectra at 300 K for (a) Tb M-edge in Tb-doped and refer- ence sample and (b) Fe L-edge in Tb-doped and undoped samples. The Fe L-edge spectra show suppression of signal from Fe in the octahedral 3 þ sites. The Tb M-edge data show weak polarization (a paramagnetic Tb 2O3 spectrum is shown for comparison).062409-2 Rice et al. Appl. Phys. Lett. 106, 062409 (2015)These XMCD data indicate that the Tb is incorporated into the crystalline lattice as opposed to a separate surface phase. The effect of Tb-doping on the magnetic structure of the magnetite nanoparticles was further determined using SQUID magnetometry, electron paramagnetic resonance (EPR), andbroad band ferromagnetic resonance (FMR). Figs. 5(a) and 5(b)show the magnetization vs. field curves for the nanopar- ticles dispersed in paraffin at 300 K and 1.8 K. There is nohysteresis at room temperature showing that both the undoped and doped nanoparticles are superparamagnetic. The onset of hysteresis occurs at blocking temperatures of 32 K and 18 Kfor the doped and undoped nanoparticles, respectively. The Tb-doped magnetization at 1.8 K does not saturate up to the maximum measurement field of 7 T, as shown in Fig. 5(c), indicating the presence of weak antiferromagnetic coupling of Tb with its nearest neighbor cations. Paramagnetic Tb 3þions at 1.8 K would show magnetic saturation well below 7 T asshown in Figs. 5(c). Fig. 5(d) shows the moment at 7 T versus temperature for the doped and undoped samples. The undoped nanoparticles show a saturation magnetization temperaturedependence that resembles bulk magnetite with the addition of a small paramagnetic component whose moment becomesappreciable at low temperatures, consistent with surface spins seen in many nanoparticle systems. 16A large fraction of the Tb-doped sample is unsaturated at room temperature and sub-sequently becomes magnetized below 50 K. Temperature-dependent XMCD data is shown in Fig. 6. The polarization of the Tb L-edge vs. temperature demon- strates a stronger, although still small, dichroism at tempera-tures below 50 K. The reference Tb 2O3powder demonstrated no changes vs temperature. These data indicate that the Tb isparamagnetic at room temperature and begins to align at lowtemperatures. Broadband FMR linewidths, measured using a co-planar wave guide technique 17,18at 300 K, on unoriented dried powders, are shown in Fig. 7. The linewidths were determined FIG. 5. (a) and (b) Magnetization ver- sus field for Tb-doped and undoped magnetite nanoparticles at 1.8 K and 300 K. (c) Moment versus field at 1.8 K for doped, undoped magnetite nanoparticles and Tb(acac) 3precursor showing that Tb-doped particles do not saturate at 1.8 K, 7 T. (d) Saturated moment at 7 T for doped and undoped samples versus temperature. FIG. 6. Tb L-edge polarization versus temperature showing a rapid increasein polarization below 50 K. FIG. 7. Magnetic resonance linewidths as a function of frequency for Tb-doped and undoped magnetite nanoparticles. The lines are fits in the sat- urated region, marked by vertical dotted lines. The inset shows a measured resonance at 40 GHz with a Voigt fit.062409-3 Rice et al. Appl. Phys. Lett. 106, 062409 (2015)by fitting the magnitude of the microwave transmission parameter S 12to a Voigt function (see inset in Fig. 7). Here, we assume that the ensemble is made up of non-interacting single-domain particles with the Gaussian component of thelinewidth, DB IN, due to the dispersion in magnetic anisotropy energies and axes and the Lorentzian component, DBLLG,d u e to the intrinsic Landau-Lifshitz-Gilbert (LLG) susceptibilities.The inhomogeneous broadening should be independent of fre- quency, f, while the LLG linewidths should increase with fre- quency as DB LLGffi2a c2pf,w h e r e ais the Gilbert damping parameter and cis the gyromagnetic ratio. Voigt fits confirm a relatively frequency independent Gaussian linewidth and a monotonically increasing Lorentzian linewidth.13At interme- diate frequencies, above magnetic saturation but whenDB LLG/C20DBIN, the measured Voigt linewidth is given by DBffi0:53DBLLGþDBIn.19The inhomogeneous linewidth, for the case of random uniaxial particles with an anisotropy field magnitude Bk, can be approximated by DBInffi3 2Bk, which is the difference in resonant field values for nanopar- ticles with their easy axis aligned with and perpendicular to the applied field.20,21The Gilbert damping constant, a,a n da v - erage anisotropy field magnitude, Bk, therefore, can be approximated from the slope and zero-frequency intercept of the linear fit to the linewidth data above the saturation fre-quency (defined as the frequency at which the resonant field induces a moment of 90% of its saturated moment). The satu- ration frequency/field for the Tb-doped and undoped nanopar-ticles are 22 GHz/0.78 T and 27 GHz/0.96 T, respectively. For fields below saturation, the linewidth narrows due to ther- mal fluctuations, an effect often referred to as “anisotropymelting.” 20–22 The measured 300 K inhomogeneous linewidths (the intercepts in Fig. 7) are 59 mT 64 mT and 74 mT 63m T corresponding to average anisotropy fields of 39 mT 68m T and 49 mT 610 mT for the Tb-doped and undoped magnetite nanoparticles, respectively. The Gilbert damping coeffi-cients, given by the slopes in Fig. 7, are 0.024 60.003 and 0.03160.002 for the Tb-doped and undoped magnetite nanoparticles, respectively. The changes in damping and ani-sotropy fields are rather small in contrast to rare earth doping in bulk garnets, 23which show rapid increase in damping and anisotropy with small amounts of Tb at room temperature.These data further support the conclusion that the Tb is in a paramagnetic state substantially decoupled from the Fe spin lattice at room temperature. The Gilbert damping constant,a, is quite small, considerably less than what is often reported for maghemite/magnetite in the literature. 24,25Our nanoparticle data are in good agreement with publisheddata 26showing that there is a strong thickness dependence of the damping in epitaxial magnetite films with the damping decreasing to a limiting value of a¼0.037060.001 as the film thickness goes to zero. In summary, we have described the synthesis of high quality Tb-doped magnetite nanoparticles with cubic mor-phology and have shown that the Tb is incorporated into theoctahedral 3 þsites. Magnetization, XMCD, and FMR data demonstrate that the Tb spins are weakly coupled to the Fespin lattice at room temperature and begin to polarize andcouple at low temperatures. Y.U.I. acknowledges the support of the National Science Foundation under grant CBET-0709358. TheXMCD work at the Advanced Light Source is supported byDOE. The authors thank Dr. Thompson Mefford and Dr.John Ballato for helpful discussions. K.P.R. and E.R.E.acknowledge funding support from the NRC-RAP program.We gratefully acknowledge the assistance of the NIST Precision Imaging Facility. 1R. F. Pearson, J. Appl. Phys. 33(3), 1236 (1962). 2R. Pauthenet, J. Appl. Phys. 29(3), 253 (1958). 3C. R. De Silva, S. Smith, I. Shim, J. Pyun, T. Gutu, J. Jiao, and Z. Zheng, J. Am. Chem. Soc. 131(18), 6336–6337 (2009). 4Y. Zhang, G. K. Das, R. Xu, and T. T. Yang Tan, J. Mater. Chem. 19(22), 3696 (2009). 5J. Russat, G. Suran, H. Ouahmane, M. Rivoire, and J. Sztern, J. Appl. Phys. 73(3), 1386 (1993). 6J. Russat, G. Suran, H. Ouahmane, M. Rivoire, and J. Sztern, J. Appl. Phys. 73(10), 5592 (1993). 7W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE Trans. Magn. 37(4), 1749–1754 (2001). 8A. H. Morrish, The Physical Principles of Magnetism (Wiley, New York, 1965). 9L.-N. Sun, J. Yu, H. Peng, J. Z. Zhang, L.-Y. Shi, and O. S. Wolfbeis,J. Phys. Chem. C 114(29), 12642–12648 (2010). 10R. Shannon, Acta Crystallogr., Sect. A 32(5), 751–767 (1976). 11CRISP2, in Calidris , Image simulation software, Vol. 2013. 12Q. Song and Z. J. Zhang, J. Am. Chem. Soc. 126(19), 6164–6168 (2004). 13See supplementary material at http://dx.doi.org/10.1063/1.4907332 for details on micro-characterization and data analysis. 14J. B. Goedkoop, B. T. Thole, G. Vanderlaan, G. A. Sawatzky, F. M. F.Degroot, and J. C. Fuggle, Phys. Rev. B 37(4), 2086–2093 (1988). 15J. S. Kang, G. Kim, H. J. Lee, D. H. Kim, H. S. Kim, J. H. Shim, S. Lee, H. Lee, J. Y. Kim, B. H. Kim, and B. I. Min, Phys. Rev. B 77(3), 035121 (2008). 16R. J. Usselman, M. T. Klem, S. E. Russek, M. Young, T. Douglas, and R. B. Goldfarb, J. Appl. Phys. 107(11), 114703 (2010). 17Y. Ding, T. J. Klemmer, and T. M. Crawford, J. Appl. Phys. 96(5), 2969 (2004). 18S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P.Kabos, T. J. Silva, and J. P. Nibarger, J. Appl. Phys. 99(9), 093909 (2006). 19J. J. Olivero and R. L. Longbothum, J. Quant. Spectrosc. Radiat. Transfer 17(2), 233–236 (1977). 20Yu. L. Raikher and V. I. Stepanov, Sov. Phys. JETP 75, 764 (1992). 21E. de Biasi, C. A. Ramos, and R. D. Zysler, J. Magn. Magn. Mater. 262(2), 235–241 (2003). 22R. J. Usselman, S. E. Russek, M. T. Klem, M. A. Allen, T. Douglas, M.Young, Y. U. Idzerda, and D. J. Singel, J. Appl. Phys. 112(8), 084701 (2012). 23J. Dillon and J. Nielsen, Phys. Rev. Lett. 3(1), 30–31 (1959). 24V. Castel, J. Ben Youssef, and C. Brosseau, J. Nanomater. 2007 , 1–16. 25P. C. Fannin, C. N. Marin, K. Raj, C. Couper, and P. Barvinschi, J. Magn. Magn. Mater. 324(21), 3443–3447 (2012). 26S. Serrano-Guisan, H.-C. Wu, C. Boothman, M. Abid, B. S. Chun, I. V. Shvets, and H. W. Schumacher, J. Appl. Phys. 109(1), 013907 (2011). 27Commercial equipment, instruments, or materials are identified only in order to adequately specify certain procedures. In no case does such identi-fication imply recommendation or endorsement by the National Instituteof Standards and Technology, nor does it imply that the products identified are necessarily the best available for the purpose.062409-4 Rice et al. Appl. Phys. Lett. 106, 062409 (2015)Applied Physics Letters is copyrighted by AIP Publishing LLC (AIP). Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. For more information, see http://publishing.aip.org/authors/rights- and- permissions.

1.4811750.pdf

Suppression of Walker breakdown in magnetic domain wall propagation through structural control of spin wave emission David M. Burn and Del Atkinson Citation: Applied Physics Letters 102, 242414 (2013); doi: 10.1063/1.4811750 View online: http://dx.doi.org/10.1063/1.4811750 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/102/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Investigation of dominant spin wave modes by domain walls collision J. Appl. Phys. 115, 243908 (2014); 10.1063/1.4885453 Spin wave based parallel logic operations for binary data coded with domain walls J. Appl. Phys. 115, 17D505 (2014); 10.1063/1.4863936 Fast domain wall dynamics in magnetic nanotubes: Suppression of Walker breakdown and Cherenkov-like spin wave emission Appl. Phys. Lett. 99, 122505 (2011); 10.1063/1.3643037 Spin wave assisted current induced magnetic domain wall motion Appl. Phys. Lett. 96, 242501 (2010); 10.1063/1.3446833 Magnetic domain-wall motion by propagating spin waves Appl. Phys. Lett. 94, 112502 (2009); 10.1063/1.3098409 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: 139.184.14.159 On: Tue, 11 Aug 2015 14:06:36Suppression of Walker breakdown in magnetic domain wall propagation through structural control of spin wave emission David M. Burn and Del Atkinsona) Department of Physics, Durham University, Durham DH1 3LE, United Kingdom (Received 7 March 2013; accepted 6 June 2013; published online 21 June 2013) The control of individual magnetic domain walls has potential for future spintronic memory and data processing applications. The speed and reliability of such devices are determined by the dynamic properties of the domain walls. Typically, spin precession limitations lead to Walker breakdown, limiting wall velocity resulting in low mobility. Here, we show the suppression ofWalker breakdown by the careful design of small amplitude periodic nanowire structuring to match the periodicity of domain wall spin structure transformations. This opens up a channel for energy dissipation via spin wave emission, allowing a domain wall to maintain its spin structure duringpropagation. VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4811750 ] Magnetic domain wall (DW) motion is significant in wide ranging applications, from spintronic technologies fordata storage, 1processing,2and sensing applications3to the manipulation of atoms4and nanoparticles5for nanoassembly or nanodelivery. Typically, DW velocity increases linearlywith field 3,6up to the Walker field, where periodic transfor- mations of the DW spin structure results in a dramatic reduc- tion in time-averaged DW velocity, known as Walkerbreakdown. 7,8This limits operation speeds and functional performance in both nanowires9and thin films.10 For device applications, fast DW motion over a broad field range with constant or monotonic field-dependent veloc- ity is desirable and various approaches have attempted to achieve this by suppressing DW breakdown behavior.Applying additional fields, either quasi-static perpendicular fields 11,12or oscillating axial fields,13is one approach, but they add technical complexity in both the device architectureand control protocol. Alternatively, modified nanowire designs incorporating either magnetic underlayers 14to intrinsically supply a bias field or large-scale lateral combstructures 15to repeatedly reset the DW structure show improved DW dynamics, but both approaches add fabrica- tional complexity and in the latter case significantly reduce de-vice packing density. In contrast, nanowire edge roughness has shown improved DW dynamics by distorting the vortex core nucleation process at the nanowire edges. 16,17The re- moval of the edges in simulations by forming tubular nano- wires with periodic boundaries has shown improved DW dynamics;18,19however, the geometrical complexity of tubular nanostructures is incompatible with conventional nanoscale lithographic fabrication, and therefore this approach is not suitable as a mode of control for technological applications. Here, we demonstrate the suppression of Walker break- down by the inclusion of small amplitude structural modula- tion to a planar nanowires’ geometry by careful matching ofthe modulation wavelength to the periodic length-scale of the DW structural transformations. This small amplitude structur- ing would not significantly affect the packing density butdoes make a significant improvement to the DW dynamics,by opening up a channel for the dissipation of DW energy via spin wave emission that disrupts the DW transformationprocess. Previous studies with periodic rectangular anti-notches 15 and triangular or sinusoidal edge modifications20show modi- fied DW dynamics,21but critically the linkage between the Walker breakdown periodicity and wire structuring wave- length has not been explored and the mechanism for sup-pressing Walker breakdown with periodic modulation has not been elucidated. This paper presents a detailed system- atic micromagnetic investigation 22,23of the dynamic behav- ior of DWs and shows how Walker breakdown suppression can be achieved by careful manipulation of nanowire edge modulation parameters. Planar nanowires with average width of 250 nm were modified by the introduction of small amplitude sinusoidal edge modulations characterized by their wavelength and am-plitude as illustrated in Figure 1(j). The DW velocity in these structures was investigated as a function of applied field. For an un-modulated wire structure (Figures 1(a)–1(e)), the ve- locity increases linearly with field up to the Walker field where the drop in time-averaged DW velocity results from Walker breakdown. Further increases in field bring about anincrease in velocity but show complex behavior resulting from multiple breakdown events. During Walker breakdown, transitions between trans- verse wall (Figure 1(a)) and vortex core (Figure 1(b)) micro- magnetic structures take place with a spatial periodicity shown by the open squares in Figure 1(e) and occur in this case with a minimum periodicity of 0.5 lm. This periodicity is dependent upon the material properties such as the Gilbert damping parameter, a, the saturation magnetisation, M s, and the exchange stiffness constant, A. At an edge modulation wavelength of k¼0.4lm, below the periodicity of DW structural changes, the effect of modula-tion amplitude on the DW motion can be studied. With 15 nm amplitude edge modulation (Figure S1 23), fast DW motion is observed at low fields, but at higher fields, the motion is inter-rupted by breakdown events resulting from complex DW dynamics. However, by increasing the amplitude to 25 nm (Figure 1(j)), the behavior changes and critically the breakdown a)Electronic mail: del.atkinson@durham.ac.uk 0003-6951/2013/102(24)/242414/4/$30.00 VC2013 AIP Publishing LLC 102, 242414-1APPLIED PHYSICS LETTERS 102, 242414 (2013) 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: 139.184.14.159 On: Tue, 11 Aug 2015 14:06:36events are suppressed and high DW velocity is maintained over a wide field range, starting below the Walker breakdown field experienced in the un-modulated wire. The structuralmodulation increases the de-pinning field compared to un- modulated structures, which for optimal device operation must be controlled and minimized in order to maximize thefunctional field range. A phase diagram showing the field dependence DW velocity as a function of structural wavelength, k, is shown in Figure 2(b), for k¼0.2lm to 0.8 lm with an amplitude of 25 nm and can be compared with the field dependent velocity behavior for the un-modulated wire 2(a). The wavelength ofthe edge modulation has a significant effect on the dynamic properties, with several regions of different behavior appear- ing on the diagram showing complex DW dynamics, sup-pression of Walker breakdown and pinning of DWs. At low fields, the DWs are pinned, showing no dynamic behavior up to the de-pinning field. This field depends on thestructural wavelength and shows a 1/ kdependence on the modulation, indicated by the solid line in Figure 2(b).A n increase in de-pinning field such as this can result from areduction of axial anisotropy due to wire modulation 21or increasing edge roughness.17 Above the de-pinning field, the DW dynamics depend strongly upon the modulation wavelength. For longer wave- length modulation, k>0.5lm, complex dynamic behavior occurs that includes additional suppression of Walker break-down but over a limited field range. In contrast, for kbelow 0.5lm, the DW velocity rises rapidly above the de-pinning field, becoming largely field invariant at higher fields. Here,the DW motion is consistent, where the transverse structure is maintained and only small variations in the velocity occur due to stretching and compressing of the domain wall as itpropagates through the modulated structure. FIG. 1. DW propagation at 30 Oe in a 250 nm wide (a)–(d) un-modulated and (f)–(i) modulated nanowire with 25 nm amplitude and 0.4 lm wavelength. (e) and (j) show the average DW velocity for these structures where Walker breakdown occurs in the plain wire with the periodicity shown by the open squa res. FIG. 2. (a) DW velocity as a function of applied field for a un-modulated 250 nm wide wire. (b) Phase diagram showing DW velocity as a function of edge modulation wavelength for a 250 nm wide wire with 25 nm amplitudemodulations. The white line indicates a 1/ kfit to the de-pinning field and the points represent the simulation conditions that have been investigated.242414-2 D. M. Burn and D. Atkinson Appl. Phys. Lett. 102, 242414 (2013) 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: 139.184.14.159 On: Tue, 11 Aug 2015 14:06:36This phase diagram shows that the edge modulation wavelength can control how DW propagation will be affected by Walker breakdown. Comparing Figures 2(a)and 2(b) shows how the edge modulation can improve the dynamic properties of DW propagation where high velocity can be achieved for conditions where Walker breakdown would occur in the un-modulated wire. The key to understanding and controlling Walker break- down is the physical mechanism for the transition between transverse and vortex core wall states. Suppression of break- down of a propagating DW results from disruption of the periodic DW structure transformations by the periodic edgemodulation. Magnetostatic effects make transverse walls more energetically favorable in narrower wires and vortex walls more favorable in wider wires. 24,25In the edge modu- lated nanowires, there is a spatial dependence to the effective wire width that gives rise to variations in the energy land- scape for the DW as it propagates the wire. The influence ofthe modulation on DW energy and thus suppression of Walker breakdown is then also to be expected to depend upon the amplitude, as mentioned earlier. Analysis hereshows that Walker breakdown is more reliably suppressed with larger amplitude modulation. The propagation of a transverse wall leads to the development of an increasingout-of-plane component of magnetic moment and it is the de- velopment of this component that initiates the transition from transverse to vortex core structure. Therefore, control-ling the growth of this out-of-plane component of DW mag- netization is key to suppressing Walker breakdown. When the modulation wavelength is larger than the vor- tex core nucleation period, the vortex core forms prior to the wall interaction with a constriction in the nanowire geome- try. As the modulation wavelength is reduced below the peri-odicity of vortex core nucleation, the transformation is prevented at the point where the vortex core would otherwise have nucleated. Therefore, when the structural modulationwavelength is shorter than the vortex core nucleation period, a DW arrives at a constriction before the vortex core can fully form, maintaining the transverse structure and allowingthe DW to continue at high velocity. Thus, Walker break- down is suppressed, giving rise to a field regime with fast DW motion in a region that would have experienced Walkerbreakdown without structural modulation. The mechanism for suppression is most readily explained by considering a modulated structure with a wave-length k¼0.8lm and an amplitude of 25 nm, which experi- ences all regimes of DW dynamics at different fields. Below 16 Oe, the DW motion shows uniform propagation behavior,by 18 Oe the wall motion displays Walker breakdown and around 24 Oe Walker breakdown is suppressed. Analysis of the time evolution of the out-of-plane component of magnet-ization, M z, as a function of position along the wire axis is shown for each of the three field amplitudes in Figure 3. The peaks in Mzshow the position of the DW and the movement of this peak in time shows the DW motion along the wire. The insets in each panel show snapshots of the DW structure at the 200 ps position and further images showing the timeevolution of the DW structure can be found in Figure S2. 23 At 16 Oe (Figure 3(a)), the DW has high velocity in the x direction and a stable Mzcomponent that does not developsufficiently to nucleate a vortex core, so the DW propagates consistently. In contrast, at 18 Oe (Figure 3(b)), a large Mz component is associated with the development of a vortex core at the edge of the nanowire as the wall approaches theconstriction in the wire. The vortex core propagates through the constriction and continues to traverse the wire, leading into Walker breakdown. Figure 3(c) shows the effect of a further increase in field up to 24 Oe; in this case, a vortex core nucleates at the edge in the wider section, propagates towards the narrower section, where it is annihilated on the FIG. 3. Time evolution of Mzas a function of position along the nanowire shown for 16 Oe, 18 Oe, and 24 Oe. At 24 Oe, the Mzdisturbance ejects a spin wave leading to Walker breakdown suppression. The x-y angle of the spins indicated by the colour wheel is shown in the top inset and Mzis shown in the bottom inset. The y position for the graphs is indicated by the dashed line in the inset.242414-3 D. M. Burn and D. Atkinson Appl. Phys. Lett. 102, 242414 (2013) 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: 139.184.14.159 On: Tue, 11 Aug 2015 14:06:36same side of the wire with the emission of a spin wave that travels in the negative x direction. The spin wave carries away magnetostatic energy from the DW so that the remain- ingMzcomponent of the wall remains as a disturbance of the propagating transverse wall structure. The distorted DW continues to propagate at high velocity in the positive x direction without Walker breakdown. At higher fields, fork¼0.8lm, the DW propagation resumes Walker breakdown behavior as the energy loss to the spin wave is insufficient to prevent a sustained vortex. This explains the complex sup- pression and return of Walker breakdown shown in the field dependent velocity behavior in Figure 2. For k<0.5lm, Walker breakdown is suppressed by this mechanism at all fields above the de-pinning field. The interaction between spin waves and DWs can affect the DW motion by the absorption 26–28and emission29of spin wave energy. In the wake of a propagating DW, the loss of energy through the emission of spin waves damps the DWmotion, which can stabilize fast dynamics in the limit of “huge anisotropies.” 30,31Here, we show that the triggering of the spin wave emission and stabilization process can beachieved in materials with negligible intrinsic anisotropy by structural modulation. Critically, by controlling the spatial synchronization of spin-wave emission with the nanowireedge modulation, consistent high velocity DW propagation behavior can be achieved. In summary, the phenomenon of Walker breakdown in the propagation of DWs in planar nanowires can be con- trolled by small amplitude ( /C2410%) periodic structuring. When the structural wavelength is shorter than the periodiclength-scale of the wall transitions causing Walker break- down, the breakdown is suppressed by the emission of spin waves. The comparatively small sinusoidal edge modulationhas a dramatic effect on the DW dynamic properties and pro- vides a realistic way to achieve reliable DW behavior in devi- ces in which dense packing of these structures is possible. We acknowledge E. Arac, L. K. Bogart, and A. T. Hindmarch for their helpful discussions. This work was sup- ported by a grant from the EPSRC. 1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 2D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 (2005).3T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, and T. Shinjo, Science 284, 468 (1999). 4A. D. West, K. J. Weatherill, T. J. Hayward, P. W. Fry, T. Schrefl, M. R. J. Gibbs, C. S. Adams, D. A. Allwood, and I. G. Hughes, Nano Lett. 12, 4065 (2012). 5G. Ruan, G. Vieira, T. Henighan, A. Chen, D. Thakur, R. Sooryakumar, and J. O. Winter, Nano Lett. 10, 2220 (2010). 6D. Atkinson, D. A. Allwood, C. C. Faulkner, G. Xiong, M. D. Cooke, and R. P. Cowburn, IEEE Trans. Magn. 39, 2663 (2003). 7N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). 8G. S. D. Beach, C. Nistor, C. Knutson, M. Tsoi, and J. L. Erskine, Nature Mater. 4, 741 (2005). 9H. Tanigawa, T. Koyama, M. Bartkowiak, S. Kasai, K. Kobayashi, T. Ono, and Y. Nakatani, Phys. Rev. Lett. 101, 207203 (2008). 10D. M. Burn and D. Atkinson, J. Appl. Phys. 108, 073926 (2010). 11K. Richter, R. Varga, G. A. Badini-Confalonieri, and M. Vazquez, Appl. Phys. Lett. 96, 182507 (2010). 12M. T. Bryan, T. Schrefl, D. Atkinson, and D. A. Allwood, J. Appl. Phys. 103, 073906 (2008). 13K. Weerts, W. Van Roy, G. Borghs, and L. Lagae, Appl. Phys. Lett. 96, 062502 (2010). 14J.-Y. Lee, K.-S. Lee, and S.-K. Kim, Appl. Phys. Lett. 91, 122513 (2007). 15E. R. Lewis Petit, D. Petit, L. O’Brien, A. Fernandez-Pacheco, J. Sampaio,A-V. Jausovec, H. T. Zeng, D. E. Read, and R. P. Cowburn, Nature Mater. 9, 980 (2010). 16Y. Nakatani, A. Thiaville, and J. Miltat, Nature Mater. 2, 521 (2003). 17E. Martinez, L. Lopez-Diaz, L. Torres, C. Tristan, and O. Alejos, Phys. Rev. B 75, 174409 (2007). 18M. Yan, C. Andreas, A. K /C19akay, F. Garc /C19ıa-S/C19anchez, and R. Hertel, Appl. Phys. Lett. 99, 122505 (2011). 19M. Yan, A. K /C19akay, S. Gliga, and R. Hertel, Phys. Rev. Lett. 104, 057201 (2010). 20J. Ieda, H. Sugishita, and S. Maekawa, J. Magn. Magn. Mater. 322, 1363 (2010). 21H. G. Piao, J. H. Shim, and D. Djuhana, IEEE Trans. Magn. 46, 224 (2010). 22M. J. Donahue and D. G. Porter, OOMMF User’s Guide Version 1.0 , NIST, Gaithersburg, MD (1999). 23See supplementary material at http://dx.doi.org/10.1063/1.4811750 for micromagnetic simulation details and additional results showing the do- main wall structure and amplitude dependence. 24R. D. McMichael and M. J. Donahue, IEEE Trans. Magn. 33, 4167 (1997). 25Y. Nakatani, A. Thiaville, and J. Miltat, J. Magn. Magn. Mater. 290–291 , 750 (2005). 26S.-M. Seo, H.-W. Lee, H. Kohno, and K.-J. Lee, Appl. Phys. Lett. 98, 012514 (2011). 27Y. -Le. Maho, J.-V. Kim, and G. Tatara, Phys. Rev. B 79, 174404 (2009). 28D.-S. Han Kim, S.-K. Kim, J.-Y. Lee, S. J. Hermsdoerfer, H. Schultheiss, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 94, 112502 (2009). 29X. S. Wang, P. Yan, Y. H. Shen, G. E. W. Bauer, and X. R. Wang, Phys. Rev. Lett. 109, 167209 (2012). 30R. Wieser, E. Y. Vedmedenko, and R. Wiesendanger, Phys. Rev. B 81, 024405 (2010). 31D. Bouzidi and H. Suhl, Phys. Rev. Lett. 65, 2587 (1990).242414-4 D. M. Burn and D. Atkinson Appl. Phys. Lett. 102, 242414 (2013) 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: 139.184.14.159 On: Tue, 11 Aug 2015 14:06:36

5.0032538.pdf

Appl. Phys. Rev. 8, 021308 (2021); https://doi.org/10.1063/5.0032538 8, 021308 © 2021 Author(s).Recent progress and challenges in magnetic tunnel junctions with 2D materials for spintronic applications Cite as: Appl. Phys. Rev. 8, 021308 (2021); https://doi.org/10.1063/5.0032538 Submitted: 08 October 2020 . Accepted: 15 March 2021 . Published Online: 15 April 2021 Lishu Zhang , Jun Zhou , Hui Li , Lei Shen , and Yuan Ping Feng COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Skyrmion battery effect via inhomogeneous magnetic anisotropy Applied Physics Reviews 8, 021402 (2021); https://doi.org/10.1063/5.0035622 Synthesis, engineering, and theory of 2D van der Waals magnets Applied Physics Reviews 8, 021301 (2021); https://doi.org/10.1063/5.0025658 Memory applications from 2D materials Applied Physics Reviews 8, 021306 (2021); https://doi.org/10.1063/5.0038013Recent progress and challenges in magnetic tunnel junctions with 2D materials for spintronic applications Cite as: Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 Submitted: 8 October 2020 .Accepted: 15 March 2021 . Published Online: 15 April 2021 Lishu Zhang,1,2 JunZhou,2 HuiLi,1,a) LeiShen,3,a) and Yuan Ping Feng2,4,a) AFFILIATIONS 1Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials, Ministry of Education, Shandong University, Jinan 250061, China 2Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore 3Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117575, Singapore 4Center for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore a)Authors to whom correspondence should be addressed: lihuilmy@hotmail.com ;shenlei@nus.edu.sg ;phyfyp@nus.edu.sg ABSTRACT As Moore’s law is gradually losing its effectiveness, the development of alternative high-speed and low-energy–consuming information tech- nology with postsilicon-advanced materials is urgently needed. The successful application of tunneling magnetoresistance (TMR) in magnetic tunnel junctions (MTJs) has given rise to a tremendous economic impact on magnetic informatics, including magnetoresistive random access memory (MRAM), radiofrequency sensors, microwave generators, and neuromorphic computing networks. The emergence of two-dimensional (2D) materials brings opportunities for MTJs based on 2D materials, which have many attractive characteristics and advantages.In particular, the recently discovered intrinsic 2D ferromagnetic materials with high spin polarization hold the promise for next-generationnanoscale MTJs. Various 2D materials, such as semimetallic graphene, insulating h-BN, semiconducting MoS 2, magnetic semiconducting CrI 3, magnetic metallic Fe 3GeTe 2, and some other recently emerged 2D materials, are discussed as the electrodes and/or central scattering materials of MTJs in this review. We discuss the fundamental and main issues facing MTJs; review the current progress made with 2D MTJs;briefly comment on work with some specific 2D materials and highlight how they address the current challenges in MTJs; and, finally, offeran outlook and perspective of 2D MTJs. Published under license by AIP Publishing. https://doi.org/10.1063/5.0032538 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 A. Spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 B. MTJs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 II. CURRENT STATUS AND ISSUES OF MTJS WITH 3D MATERIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 III. 2D MATERIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 A. Theoretical prediction and experimental synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 B. Advantages of 2D materials in solving current 3D MTJs problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 IV. 2D MATERIALS-BASED MTJS . . . . . . . . . . . . . . . . . . . . 7 A. Targeting high spin polarization . . . . . . . . . . . . . . . 7 B. Targeting effective spin injection . . . . . . . . . . . . . . . 11C. Targeting spin manipulation. . . . . . . . . . . . . . . . . . . 13 D. Targeting stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 V. SUMMARY AND PERSPECTIVES . . . . . . . . . . . . . . . . . . 18 I. INTRODUCTION Conventional silicon-based metal-oxide-semiconductor devices, which work on the manipulation of charges (one degree of freedom of electrons), will come to an end in the near future due to more funda- mental issues. Exploring advanced information technology with high- speed operation and low-energy consumption to replace existing silicon-based technologies is urgently demanded. So far, many strate- gies have been proposed, such as nanoelectronics, 1–4molecular elec- tronics,5spintronics,6–9and quantum information technologies.10–12 Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-1 Published under license by AIP PublishingApplied Physics Reviews REVIEW scitation.org/journal/areAmong them, spintronics has exhibited tremendous potential and, thus, has attracted lots of attention. Spintronics, which is based on manipulation of spins (another degree of freedom of electrons), prom-ises to integrate memory technology at the heart of information proc- essing units, such as classical and neuromorphic, which would be a big change in how architectures are designed toward in-memory comput- ing (currently, memory and logic are separate layers that need to com- municate). In addition, spintronics is more compatible withconventional electronics, compared to other strategies, so that many techniques applied in traditional electronics can be extended to spintronics. Even though information is processed using spin, it is desirable to manipulate spin or switch magnetization using electrical means with magnetoresistive materials. Magnetoresistive devices are con-structed based on magnetoresistive materials that, in general, exhibit a change in resistance with the application of a magnetic field. Early in their development, magnetoresistance (MR) devices made use of the anisotropic magnetoresistance effect. Recently, such devices are mainly based on the giant magnetoresistance (GMR) effect or tunneling mag-netoresistance (TMR). The discovery of magnetic tunnel junctions (MTJs) at room temperature dramatically increased storage den- sity. 13,14If an MTJ has TMR >100%, it can be used to make not only magnetic field sensors15and reading heads of hard disk drives16but also magnetoresistive random access memory (MRAM).17The more recent demonstration of spin-transfer torque (STT)18and spin–orbit torque (SOT) effects19,20makes MTJs more valuable for manufactur- ing multitudinous spintronics, including MRAM,21–23radiofrequency sensors,24,25microwave generators,26and even artificial neuromorphic networks.27STT allows change of the magnetization direction of a material by a spin-polarized current. By passing a current through a thick magnetic layer (the “fixed layer”), one can produce a spin- polarized current. If this spin-polarized current is directed into a sec-ond, thinner magnetic layer (the “free layer”), the angular momentum of charge carriers (such as electrons) can be transferred to this layer, changing its magnetization orientation. This can be used to excite oscillations or even flip the orientation of the magnet. Thus, the differ- ent resistive states, a low-resistance parallel (P) magnetic configurationand high-resistance state anti-parallel (AP) magnetic configuration, can be realized to represent the “0” and “1” state in STT memory, respectively. 28–30Thanks to the voltage-dependent switching ability for STT principle, the different logic gates can be reconstructed by the same MTJ structures with only one single-cycle operation. This prom- ising computing hardware application proves that MTJs have great potential for application in more aspects in the future. SOT is another approach of magnetization switching, an interconversion of chargeand spin current, which is a promising phenomenon that can be used to improve the performance of MRAM devices. 21–23The SOT effect has been achieved in heavy metal31and 2D topological insulator (TI) systems.32It essentially requires two functional layers, namely, one fer- romagnetic (FM) and one nonmagnetic (NM) layer with largespin–orbit coupling (SOC). The latter is to accumulate spin charges and inject it into the adjacent FM layer. The spin current then exerts a torque on the magnetic moment of the FM layer and revert it with anangle. The switching efficiency strongly depends on the strength of SOT. Furthermore, a large spin polarization in the nonmagnetic layer is necessary for efficient spin injection. Thus, heavy metals and 2D TIs with large spin Hall angles, such as Bi 33and 2D-TI a-Sn,34are used forSOT switching. Much progress has been made in SOT magnetic switching with heavy metals.35,36Recently, TIs, such as Bi 2Se3,h a v e attracted attention due to their spin-momentum locking property, which, in principle, is able to achieve an efficient SOT switching, even though the switching process needs to be further improved in terms of the switching hysteresis and its completeness.37 Advantages of MTJ devices include low-power consumption with high processing speed, nonvolatility, metal-oxide-semiconductor technology compatibility, and high integration density.38,39The most common materials used to fabricate MTJs are ferromagnetic metals and alloys, such as Fe and CoFeB, Heusler alloys, and dielectrics, such as MgO and AlO x.40–43What cannot be ignored is the remarkable breakthrough achieved in 2D materials synthesis and MTJs created based on them.44Due to their low dimensionality and quantum nature, 2D materials add many unique features to MTJs, such as flexi- bility and extremely high scaling.45,46 First-principles calculations based on density functional theory (DFT) have been the most widely used method in theoretical studies of 2D materials and their applications in devices such as MTJs. It can be said that the rational design of high-performing MTJs will always face challenges in experiments without proper theoretical guidance, such as in the development of (Co)Fe/MgO/(Co)Fe MTJs. Compared to other theoretical and computational methods, the first-principles method does not require empirical parameters and experimental inputs, which makes it an ideal method for studying new materials and their heterostructures. DFT calculations are also valuable in pre- dicting materials and device behaviors under extreme conditions that are difficult to achieve experimentally. DFT was proposed by Hohenberg and Kohn in 1964,47and in the very next year, Kohn and Sham48launched its primary fulfillment. DFT has become a convincing quantum simulation method in explor- ing the electronic structure of many systems through the use of the electron density as the fundamental variable instead of the electron wave function. The first MTJ based on Fe/MgO/Fe was proposed by DFT calculations49and was subsequently demonstrated experimen- tally.40,50Now, Fe/MgO/Fe-based MTJs are the main components in the reading head of the hard disk drive in personal computers. Besides, in searching for the magnetism of 2D materials and predicting new structures, first-principles calculations have also been used as a powerful tool for providing theoretical guides for experimental exploration. Even though DFT has been very successful in studying and pre- dicting new materials, it is still a computationally expensive method. Despite significant improvements in recent decades, it is still difficult to incorporate all the experimental “real world” subtlety. Concerning calculation of transport properties, many systems beyond MgO havebeen initially predicted by DFT to lead to high spin polarizations (with similar symmetry arguments), but experimentally, their performance so far failed to match that of MgO. As a result, direct comparison between computational prediction and experimental measurement for quantities such as MR ratio is nontrivial. Considering also the fact that there have been many computational studies on MTJs but a limited experimental realization of the predicted structures, we mainly focus on computational works in this review and discuss available related experimental works. We begin with a brief introduction to spintronics and conventional MTJs and then discuss the current status on MTJs, highlighting the problems encountered and challenges. This isApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-2 Published under license by AIP Publishingfollowed by the advantages of 2D materials in solving those MTJ chal- lenges. The rest of this review is organized as follows: In Sec. IV,w e review the recent progress of 2D materials–based MTJs. This is orga- nized into four subsections based on the key issues in 2D MTJs, i.e., targeting at spin polarization (Sec. IV A ), spin injection (Sec. IV B), spin manipulation (Sec. IV C), and stability (Sec. IV D ). We finally conclude and offer an outlook and perspectives in Sec. V. A. Spintronics Conventional electronic devices have one thing in common: They rely on electronic transport in semiconductor materials, such as silicon. With the size and function of silicon-based electronic devices reaching the limit, further downscaling of silicon-based electronic devices becomes impossible. New concepts are required for future electronic devices, which should also meet certain requirements such as low-power operation. In this respect, it is noted that the energy scale of spin dynamics is typically many orders of magnitude smaller than that of charge dynamics, and low-power electronics operation can thus be achieved in spintronic devices. Spintronics has become a rap- idly developing field on this background. Spintronics is based on the manipulation of spin of electrons to store, encode, and transmit data. In spintronics, information is first marked as up-spin or down- spin; the spin-carrying electrons are transported along a path; and at afinal point, the spin information is read. The conduction electrons’ spin orientation needs to be sustained for several nanoseconds in order for them to be used in an electrical circuit and chip. Transportingcurrent through ferromagnetic material and transmitting the spin- polarized electrons to the receiver is a common method to generatespin-polarized current. The successful implementation of spintronic devices and circuits ( Fig. 1 ) relies on the realization of six elementary functionalities: spin–orbital torque, spin detection, spin transport, spin manipulation, spin–optical interaction, and single spin device. Spin–orbital torque is induced through the spin–orbital interaction inFM/heavy metal bilayers by flowing an in-plane electrical current. 51 The spin detection includes detection of circularly polarized light;52 transient Kerr/Faraday linearly polarized light rotation;53spin Hall voltage;54,55electric resistance change;56and tunneling-induced lumi- nescence microscopy,57which has been the most convenient method from a device perspective and application to date. Spin transport isexpected to offer low-loss spin channels, which can provide long- distance propagation of spin signals and enable more operations of spin signals. 58,59Spin manipulation is required to achieve more func- tionality of spintronic devices like that in electronic devices.60–62 Spin–optical interactions include spin Hall effects in inhomogeneousmedia and at optical interfaces, spin-dependent effects in nonparaxial(focused or scattered) fields, spin-controlled shaping of light using anisotropic structured interfaces (metasurfaces), and robust spin- directional coupling via evanescent near fields. 63The single-spin, sin- glet, and polarized phases of a quantum dot allow different currents to flow through the dot. The spin state of the dot is controlled either by adding electrons or by tuning the magnetic field, and thus, a prototype single-spin transistor is produced.64,65 Spin quantum dot Spin LED Spin FETMTJSpin-orbital torque Spin manipulationSpin transportSingle spinSpin detection Spin-optical interactionSOT deviceSpin logic device M 1 0 FIG. 1. Overview of some spintronics devices. Key aspects, including physical effects, elementary functionality, and applications, are schematically il lustrated. The elementary functionalities include spin–orbital torque, spin detection, spin transport, spin manipulation, spin–optical interaction, and single spin. The corresponding applications of these functionalities include the SOT device, spin logic device, MTJ, spin FET, spin LED, and spin quantum dot device. In this review, we focus on the magneti c tunnel junction.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-3 Published under license by AIP PublishingAt present, a variety of spin electronic devices based on different mechanisms of spintronics have been studied and designed. Several applications are also highlighted in Fig. 1 . In a SOT-driven device, the heavy metal induces strong SOC and, thus, generates SOT-driven switching. In a spin logic device, by defining bistable magnetizations of electrodes along the easy axis as the input logic (“1” and “0”) and the as-detected current as the logic output , Boolean operations can be achieved. The spin electrons transport from one FM layer to anotherFM layer; by passing through a barrier, high- and low-spin current can be achieved by the magnetic alignment of the two electrodes. As such, this kind of device, called MTJ, can be used as a memory device to store information, even under the power-off state (nonvolatile). The spin field-effect transistor (FET) was first proposed by Datta andDas. 66It is based on manipulation of electron spin during transport driven by an electric field in semiconductors. This device works simi- larly to a charge-based transistor. A spin current is injected into thechannel material from an FM electrode (source) in which spin polari- zation is electrically manipulated by a gate voltage (or other means), and finally, spin polarization is detected at the drain. In a spin light- emitting diode (LED), when the spin-polarized electron is injected, it recombines with a hole and emits a circularly polarized photon that isused to assess the polarization of the injected spin. When a spin- polarized electron is injected into a quantum dot, the spin state of the quantum dot can be changed and can be controlled by a gate voltage. All these spintronic devices have lower power consumption, lower cost, and more stable and excellent performance in high-capacity stor-age than traditional electronic devices whose operation principle is only based on charge. Therefore, spintronic devices will play a great role in the next generation of electronic information science and technology. In the past several years, three important focus areas of spin- tronics research have been explored by scientists: (1) fabricating nano- scale structures, including new magnetic materials, hybrid heterostructures, and functional materials; (2) studying the spin effect,including spin injection, transport, and detection; and (3) improving the performance of MTJ-based devices. B. MTJs Using the well-developed knowledge on how to manipulate spins, 67–74one can generate state-of-the-art spintronics devices with desired properties. Thus, it is vital to explore the application possibili- ties of spintronic effects in order to achieve more promising spin- tronics devices. Such electronic devices have made a big impact oncomputer technology through achieving higher and higher informa- tion storage in hard disk drives as well as faster and faster reading speed of data in RAM. MTJ is one of the most important forms in spintronics applications, as mentioned in Fig. 1 . In this section, we will introduce MTJs. A basic MTJ consists of two ferromagnetic layers separated by a thin insulating layer, as schematically shown in Figs. 2(a) and 2(c). The tunneling conductance or resistance of such a device depends onwhether the magnetizations of the two electrodes are parallel or anti- parallel. If R PandRAPare the resistance in the parallel and antiparallel state, the TMR ratio is given by75 TMR ¼RAP/C0RP RP¼2P1P2 1/C0P1P2;where P1and P2are the spin polarization of the two electrodes. The origin of TMR arises from different density of states (DOS) for spin-up and spin-down electrons as shown in Figs. 2(c) and2(d).B e c a u s e electron spins are preserved during the transport, each type of spincan only tunnel into the subband of the same spin. Therefore, therewill be a high tunneling current (or low resistance) if the spindirections of two electrodes align parallel [ Fig. 2(a) ], which makes the two electrodes have symmetric spin density of states [Fig. 2(b) ]. Otherwise, the antiparallel alignment of the spin direc- tions of two electrodes [ Fig. 2(c) ] will generate asymmetric DOS [Fig. 2(d) ], resulting in low tunneling current (or high resistance). It is worth noting that although it is insignificant for small TMR,for large negative TMR, the resistance variation is sometimes nor-malized preferentially by R APso as to obtain the same value as pos- i t i v eT M Ra n d ,t h u s ,b ec o m p a r a b l ei na b s o l u t ev a l u e .76 TMR at room temperature was first demonstrated by Miyazaki and Tezuka77and Moodera et al.78Immediately after that, the TMR ratio was raised rapidly to 81% in a Co 0.4Fe0.4B0.2(3)/Al (0.6)–O x/ Co0.4Fe0.4B0.2(2.5) (thickness in nm) MTJ at room temperature.79 Subsequently, a TMR ratio as large as 604% was achieved in MgO-based MTJ Co 0.2Fe0.6B0.2(6)/MgO (2.1)/Co 0.2Fe0.6B0.2(4) (thickness in nm) at room temperature.80The dramatic increase in MR ratio, compared to that of its predecessor GMR devices, led to the domina-tion of MTJs in the magnetic data storage industry. The first successful application of MTJs was demonstrated in computer read head technology with Al 2O3barrier and MgO barrier MTJs. The magnetic recording density in the hard disk drive increasedconsiderably compared to traditional devices. 81–85Another MTJ appli- cation is to develop the MRAM, which exceeds the density of dynamicRAM, speed of static RAM, and nonvolatility of flash memory.Moreover, these nanoelectronics generate less heat and operate atlower power consumption. Because there is no or almost no interlayer coupling between the two ferromagnetic layers in MTJs, only a small external magnetic fieldis needed to reverse the magnetization direction of one ferromagnetic layer, thus realizing a huge change in tunneling resistance. Therefore, MTJs have much higher magnetic field sensitivity than metal multi-layer films. At the same time, MTJs have high resistivity, low energyconsumption, and stable performance. All in all, MTJs act as one ofthe most important spintronics applications, which can be used as akey component in many spintronics devices, such as the read head ofhard disk drives, microwave oscillators, and MRAM, as shown in Figs. 2(e)–2(g) . Recent studies have demonstrated that magnetization in the free layer of an MTJ can be switched by STT or SOC, even though in practice it is not so easy due to the highly spin-polarized current den- sity required. To fabricate an MTJ with a giant TMR is crucial for practical applications. With the development of nanotechnology, there are more and more ways to construct junction structures. For the prepara-tion methods in the laboratory, methods such as molecular beam epi-taxy, 86magnetron sputtering,87electron beam evaporation,88and chemical vapor deposition (CVD),89are often used. In industry, meth- ods used to prepare micrometer, submicrometer, and nanomagnetictunnel junction, magnetic tunnel junction array, TMR magnetic read-out head, and MRAM include lithography, electron beam exposure,ion beam etching, chemical reaction etching, focused ion beam etch-ing,etc. Among them, lithography combined with ion beam etching isApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-4 Published under license by AIP Publishingthe preferred process, with low cost and mass production achievable in the micromachining process. Generally, all MTJs consisted of FM layers and an insulator layer. The most common ways to fabricate theFM layer in past years is sputter deposition (magnetron sputtering and ion beam deposition). 90–92The magnetic alignment and thickness are the key parts of MTJ fabrication in the experiment. A better method to fabricate insulating layers always keeps forging ahead. For example, ion beam oxidation,93glow discharge,94,95plasma,96atomic oxygen expo- sure,97and ultraviolet-stimulated oxygen exposure98have been used as alternate ways for the insulator layer deposition. In terms of prepara- tion and processing, the issues about the control of the oxide barrier and the interfaces, the shielding tolerance, the thermal stability, and the robustness of the lifetime of the device need to be solved urgently. II. CURRENT STATUS AND ISSUES OF MTJs WITH 3D MATERIALS With the rapid development of MTJs, the TMR value of MTJs has increased rapidly in the past few years and has quickly approached the theoretical value. However, at present, despite extensive studies and much progress being made, many problems and challenges still exist that need to be understood and addressed to improve the effi-ciency, performance, and stability of MTJs. For example, one of the important issues is to control the quality of the interface between the ferromagnetic layer and barrier layer. 76T h ee f f e c to ft h ei n t e r f a c e bonding on magnetoresistive properties must be considered99,100 because it determines the effectiveness of transmission of electronswith different orbital properties (and/or symmetry) through the inter- face, and electrons with different orbital properties carry unequal spinpolarization. 101Therefore, the interface bonding has profound effects on conductance.102F o rt h i sa s p e c t ,l a y e r e d2 Dm a t e r i a l s ,w h i c ha r e connected by van der Waals (vdW) force without chemical bonding, can avoid these problems. Furthermore, it is known that a long spinlifetime in the NM materials and an efficient polarization of the injected spin are required. Fert and Jaffre `s100showed that introducing a spin-dependent interface resistance at the FM/NM interfaces cansolve the problem of the conductivity mismatch between FM and NMmaterials. They found that a significant magnetoresistance can beobtained if the junction resistance at the FM/NM and NM/FM interfa-ces is chosen in a relatively narrow range, depending on the resistivity,spin diffusion length, and thickness of NM. However, introducing a3D tunneling barrier, such as Al 2O3, indeed has an effect on improve spin injection efficiency (SIE),103,104but new issues, such as pinholes and clusters, emerge.105Using 2D materials like h-BN, which has a well-defined interface contact and fewer defects to act as a tunnelingbarrier, can avoid these new issues perfectly. 106In addition, the spin- dependent electronic structure of electrodes,107the symmetry selection rules known to control TMR in MTJs with electrodes and crystallinetunnel barriers, 50and the role of the tunnel barrier layer and its elec- tronic structure49,108are all important issues and deserve serious atten- tion. Moreover, the diffusion and oxidation process of elements in thebarrier layer is another puzzling issue in the growth process. TakingCoFeB/MgO/CoFeB MTJs as an example, Burton et al. 109found that B atoms in the crystalline CoFeB electrodes tend to migrate to the interface, which leads to a decrease in the TMR ratio due to a signifi- cant suppression of the majority-channel conductance through statesofD 1symmetry. Similarly, the diffusion and oxidation process also h a sa ni n fl u e n c eo nt u n n e l i n gp r o c e s s .110Another thing is the tunnel- ing mechanism of the barrier layer.111MgO systems show much larger TMR than that of traditional AlO xsystems.112–114However, the tunneling mechanism in single-crystalline or textured MgO barriers isquite different from traditional AlO xamorphous barrier materials.111 In addition, perpendicular magnetic anisotropy (PMA) of out-of-planemagnetized MTJs aroused a lot of attention because in-plane anisot-ropy only yields a typical anisotropy field in the 100–200 Oe due to (f) Microwave oscillator (e) Read head(c) (d)(a) (b) (g) SOT-MRAM FIG. 2. Two different magnetic configurations of MTJs in real and reciprocal spaces and three possible spintronics applications based on MTJs. (a) and (b) Sch ematic diagram of an MTJ in parallel configuration and its corresponding band diagram, which shows that both spin-up and spin-down electrons can pass through the barr ier from the left elec- trode to the right one. (c) and (d) Schematic diagram of an MTJ in antiparallel configuration and its corresponding band diagram. Both spin-up and spin- down channels are blocked due to the asymmetry of two states. (e) and (f) Examples of the read head, microwave oscillator, and MRAM of MTJs, which are represented by cylin ders. The orange arrows indicate the direction of magnetism. The black arrow shows the direction of charge current.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-5 Published under license by AIP Publishingshape anisotropy, while PMA can yield an effective anisotropy field of several kOe.115In order to realize the miniaturization of devices, it is urgent to develop processing technology on the atomic scale.However, controlling thickness of metal oxide tunnel barriers ischallenging. 116,1172D materials have the natural advantage of ultrathin to atomic scale, which can avoid some troubles in proc- essing, and can be used to develop devices working at room tem-perature. But most of the existing FM materials have a too-lowCurie temperature ( T C), causing the need to search for room- temperature FM. Last but not least, the origin and the influence of different layer thicknesses on transport properties are also impor- tant. Understanding and resolving these issues will greatly promotethe progress of MTJs in theoretical and practical applications. Thediscovery of 2D materials and their heterostructures provides anew playground for MTJs. Some of the 2D materials may offerpromising routes to resolve some of these issues with their unique properties, like sharp interfaces, a natural and tunable van de Waals insulating gap, layer-by-layer control of the thickness, highPMA, and the potential for a diffusion barrier (thermal stability),and even provide the possibility of new functionalities, such asspin filtering. On this background, this review discusses and sum-marizes the following four main problems, as shown in Fig. 3 :( 1 ) spin polarization, (2) spin injection, (3) spin manipulation, and (4) spin stability.III. 2D MATERIALS When the layered materials are thinned down to their physical limits, they exhibit novel properties which is different from their bulk counterpart. Thus, these materials are specifically referred to as “2D materials.” In other words, 2D materials refer to materials in which electrons can only move freely in two dimensions (in a plane) on the non-nanometer scale (1–100 nm). 2D materials emerged with the suc- cessful separation of graphene, a single atomic sheet of carbon atoms with a bonding length of 1.42 A ˚, by Geim’s team at Manchester University in 2004. 118Graphene is also regarded as the most widely studied 2D material. It has been well known that the pristine graphene is a unique 2D hexagonal structure with zero-bandgap and semimetal- lic property, which is an important allotrope of carbon. Due to 2D materials’ various crystal structures and physical properties, many other 2D materials beyond graphene are also undergoing a lot of research work, including semiconductors (e.g., transition metal dichal- cogenides such as MoS 2), insulators (e.g., h-BN), superconductors (e.g., NbSe 2), and magnets (e.g., Fe 3GeTe 2). These great advances have expanded 2D nanodevices. The research on MTJs has been committed to continuously improving the high TMR ratio. However, in the process of further improving the performance and reducing the size, more and more challenges emerge. Some of the 2D materials may offer promising routes to resolve some of these issues with their unique properties and Spin polarization Spin manipulation Stability Tc Hc HTPESpin injection (c) (d)(a) (b) FIG. 3. Schematics of spin polarization, spin injection, spin manipulation, and spin stability, which are the four main issues in current MTJ device design. (a) A charge current will be spin polarized when passing through a magnetic material. A high–spin-polarized current is desired in MTJs. (b) High-performance MTJs requir e a high efficiency of spin injection from a magnetic electrode into a nonmagnetic insulator through their interface. (c) Many methods have been proposed to manipulate the spin within MTJs effectively and efficiently, such as applied electrical field ( E), pressure ( P), magnetic field ( H), and magnetic torque ( T). (d) Both the thermal stability (high Curie temperature) and mag- netic stability (high coercivity [H C]) play a key role in the real application of MTJs.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-6 Published under license by AIP Publishingeven provide the possibility of new functionalities, such as spin filtering. A. Theoretical prediction and experimental synthesis Recently, more and more 2D materials have been predicted theo- retically, and some of them have been synthesized experimentally. Take the 2Dmatpedia database119as an example. This database contains 6351 2D materials at present, of which 1500 are magnetic based onspin-polarized DFT calculations (the total magnetic moment of thematerial is >0.5lB). The known 2D magnetic materials are summa- rized in Fig. 4 , 120and they are divided into different categories according to whether they are synthesized experimentally or predicted theoreticallyand their properties. By combining 2D materials with different proper-ties from this table as heterojunctions, rich 2D MTJs can be constructed. In order to make 2D magnetic devices, the long-range magnetic order in 2D magnetic materials is necessary. It is known that adjacentatomic moments or spins are coupled through an exchange interactionin a lattice, leading to the magnetic order of materials. 121The magne- tism depends on the lattice dimensionality or crystal structure as well as on the spin dimensionality of the system. Uniaxial anisotropy is able to sustain long-range magnetic ordering, which has been experi-mentally observed in several magnetic 2D materials recently. It isworth noting that the 2D Ising kind of behavior has been reported in neutron scattering experiments in layered materials much earlier. 122 The existence of magnetism down to monolayers in several magnetic 2D materials has been established very recently. Intrinsic magneticorder in 2D-layered Cr 2Ge2Te6was found at low temperatures in 2017.123,124Almost at the same time, the ferromagnetic order was alsofound in monolayer CrI 3up to 45 K. Subsequently, several magnetic materials, including VSe 2and Fe 3GeTe 2, have been found at room temperature. These findings offer a new opportunity to manipulate spin-based devices efficiently in the future.125,126 B. Advantages of 2D materials in solving current 3D MTJs problems In Sec. II, the potential problems of 3D MTJs when they are scal- ing down to the nano level are introduced. Various 2D materials with natural monolayers are now available through the large-scaled CVD growth. Thus, it is natural to seek high-performing, flexible, and stable MTJ tunnel barriers based on 2D materials and their heterojunctions. 2D materials provide a reliable solution to the problems in the manufacturing of high-performance MTJs through the layer-by-layer control of the thickness, sharp interfaces, and high PMA. In this direc-tion, low-resistance-area products, strong exchange couplings across the interface, and high TMR in MTJs were predicted and synthesized. IV. 2D MATERIALS-BASED MTJs As discussed in Sec. III, 2D materials are expected to offer solu- tions to some of the challenges when further improving the perfor- mance and reducing the size of MTJs. In this section, the currentdevelopment status of 2D MTJs around the current problems in the development of conventional MTJs will be reviewed. Following each problem and challenge, possible solutions and future development directions are discussed. A. Targeting high spin polarization The generation, transport, and detection of spin current in MTJs are three key parts to integrate spin into existing electronics success- fully. In this section, we focus on the current works targeted at improv- ing spin polarization in MTJs with 2D materials. The spin polarization is the most important factor for governing TMR performance because a high-spin-polarized current is essential for high magnetoresistance. The subtle offset between two spin channels is known to cause the net spin polarization, which is greatly affected by atomic, electronic, and magnetic structures of the system. A straightforward way to have high spin polarization is to use half-metallic magnetic materials in which the Fermi level only crosses one spin channel, resulting in 100% spin polarization. Thus, half-metallic materials become good candidates for MTJ devices. Using them as electrodes, 100% spin-polarized currents under a bias voltage may be generated in MTJs with high TMR. However, the MTJs with half-metallic materials do not show very high TMR, as expected from the materials point of view, 128because of the complicated geometry structure in MTJ devices made of several differ- ent materials and nonequilibrium electronic structure under bias. From the experimental aspect, with the fast development of CVD technology recently, MTJs using 2D materials, such as MoS 2,129gra- phene,130,131and boron nitride (BN)132as a nonmagnetic spacer, have been fabricated successfully. Their van de Waals interface is expected to overcome the disordered interface between two 3D bulk materials. However, their reported magnetoresistance is quite low, which is undesirable. This was attributed to the use of Permalloy electrodes,such as Fe, Ni, and Co, injecting current with a relatively low spin polarization. In addition, the inherent properties of these materials hinder the performance of MTJs. Taking the Co-Fe system as an Chalcogenides Halides OthersCr2Ge2Te6,F e 2P2S6, Fe 2P2Se6, Mn 2P2S6, Ni2P2S6, Ni 2P2Se6, CuCrP 2Se6∗, AgVP 2S6, AgCrP 2S6, CrSe 2, CrTe 3, Ni 3Cr2P2S9, MnBi 2Te4∗, CrCl 3, FeCl 2, FeBr 2, FeI 2,CuCrP 2S6 CuCl 2, CuBr 2, NiBr 2, NiI 2, CoI 2, SnO, GeS, GeSe, SnS, SnSe, GaTeCl, CrN, CrB 2MnI 2 α-RuCl 3MnBr 2, CoCl 2, CoBr 2, NiCI 2, VCl 2, VBr 2, VI 2, FeCl 3, FeBr 3, CrOCl, CrOBr, CrSBr, MnCl 2∗, VCl 3∗, VBr 3∗ MnX 3 (X = F, Cl, Br, I), FeX 2 (X = Cl, Br, I), MnSSe, TiCl 3, VCl 3MnBi 2Se4∗Cr2Si2Te6, Fe3GeTe 2, VSe 2∗, MnSe x∗ CrI3∗, CrBr 3, GdI 2 VS2, InP 3, GaSe, GaS FIG. 4. 2D magnetic materials library. In this diagram, the gray row below lists theo- retically predicted vdW ferromagnets (left), half metals (center), and multiferroics (right). Others above the line are 2D magnetic materials that have been experimen-tally confirmed. Among them are bulk ferromagnetic vdW crystals (green), bulk anti-ferromagnets (orange), bulk multiferroics (yellow), and RuCl 3[a proximate Kitaev quantum spin liquid (purple)].127Note (asterisks): The different phases of these materials have different properties. Reproduced with permission from Gong et al. , Science 363, 6428 (2019). Copyright 2019 The American Association for the Advancement of Science.120Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-7 Published under license by AIP Publishingexample, current-induced switching in FeCoB/MgO requires intense current densities to overcome the large Fe Gilbert damping.133Thus, it is important to search for different 2D materials stacks with other elec- trodes, which can offer better opportunities to implement such new technologies. High spin-polarized Heusler alloys, a large family of ter-nary compounds, 134appear as promising candidates of electrodes. For example, most early research on MoS 2-MTJs with Permalloy electro- des show relatively low TMR. Adopting the electrodes of Fe 3Si, a Heusler alloy with a lower Gilbert damping parameter and a higher saturation magnetization, Rotjanapittayakul et al.135reported a large TMR in Fe 3Si-MoS 2MJTs because of the similar lattice to MoS 2,s m a l l Gilbert damping, and high Curie temperature of Fe 3Si. A small Gilbert damping parameter leads to a potentially low critical cur- rent density for STT switching. The TCof Fe 3Si is large, >800 K, and the spin-polarization at low temperature ( /C2445%)136is larger than that of Co ( /C2434%) and Ni ( /C2411%).137In addition, Wu et al.138researched ferromagnetic Fe 3O4electrodes in Fe 3O4/ MoS 2/Fe3O4MTJs. A clear large TMR phenomenon appears at <200 K temperature. They also performed first-principles calcula- tions and found that Fe 3O4keeps a high-spin-polarized electron band at the interface of the MoS 2and Fe 3O4. This calculation pro- vides a clear and deep physical explanation on how the TMR phe- nomenon appeared in their experiment. By using the DFT combined with nonequilibrium Green function (NEGF) method, 80% SIE, and 300% magnetoresistance ratio are pre- dicted in multiple 2D barrier layers on the performance of MTJs. This is an effective means to improve spin polarization in such junctions. There are also some works in the same direction. For example, Zhang et al.139investigated the vertical transport across M/MoS 2/M (M ¼Co and Ni) MTJs with MoS 2layer numbers N ¼1, 3, and 5. Their results revealed that the thinner junctions are metallic because of the strong coupling between MoS 2and the ferromagnets, and the junctions with thicker MoS 2begin to show tunneling effects. A higher MR is achieved by increasing the number of interlayers. In their junctions, both posi-tive (63.86%) and negative MR (–70.85%) can be obtained. A similar model based on their prediction was later developed experimentally. Galbiati et al. 140reported the fabrication of NiFe/MoS 2/Co devices with mechanically exfoliated multilayer MoS 2using an in situ fabrica- tion protocol that allows high-quality nonoxidized interfaces to be maintained between the ferromagnetic electrodes and the 2D layer. Their devices displayed an MR ratio up to 94%. Beyond interfaces andmaterial quality, they suggested that spin-current depolarization could explain the limited MR. This points to a possible path toward the reali- zation of larger spin signals in MoS 2-based MTJs. Besides the number of layers of MoS 2, the length of scattering region can affect the MR ratio. By employing a three-band tight-bind-ing model combined with the NEGF method, Jin et al. , 141studied the spin-dependent electron transport in a zigzag monolayer MoS 2with ferromagnetic electrodes. Their results revealed that the conductance shows a quantized oscillating phenomenon in the P configuration, while the conductance exhibits a zero platform in a large-energyregion in the AP configuration. In addition, the length of the central part of the structure has a certain influence on the MR ratio. It is found that as the length of the middle region increased, the MR ratio decreased gradually. However, this prediction has not been approved by experiments yet. It is hoped that the experimental report will appear in the future.Beyond nonmagnetic MoS 2, 2D intrinsic magnetic materials play an important role in the spin polarization of MTJs. The reported molecular beam epitaxial growth of 2D magnetic materials for Fe3GeTe 2,V S e 2, MnSe x,a n dC r 2Ge2Te6opens new possibilities for MR devices. They are magnetic conductors or insulators, which pro- vide diverse application perspectives. For example, magnetic insulators are ideal for the central tunneling layer in MTJs, and CrI 3is a typical example of 2D magnetic insulating materials that have emerged in recent years. Atomically thin CrI 3flakes were fabricated recently by mechanical exfoliation of bulk crystals onto oxidized silicon sub- strates.124In CrI 3flakes, intrinsic ferromagnetism and out-of-plane magnetization are observed. Interestingly, it is FM in the monolayer but becomes AFM in bilayer and back to FM in both trilayer and bulk.142Also, n-layer CrI 3in the high-temperature phase exhibits interlayer AFM coupling, which provides a natural pinning layer for CrI 3.Y a n et al.143studied the electron transport properties of CrI 3/ BN/n-CrI 3(n¼1, 2, 3, 4) MTJs, as shown in Figs. 5(a)–5(d) ,a n d found that the n ¼3 MTJ shows a fully polarized spin current with /C243600% TMR ratio when at the equilibrium state. More interestingly, the odd–even effect appears due to the difference of the number of pinning layers. The usage of different numbers of CrI 3pinning layers greatly regulates the spin polarization. As CrI 3is a semiconductor that can serve as a spin-filter tunnel barrier when sandwiched between graphene electrodes, Song et al.144 performed an experimental work for improving spin polarization in MTJs by increasing the thickness of CrI 3layers [ Figs. 5(e)–5(g) ]. The TMR of these spin-filter MTJs can be drastically enhanced with the increase in the thickness of CrI 3layers, which corresponds to the for- mer theoretical results.143When the thickness of CrI 3increases to four layers, the TMR ratio can reach 19 000% at low temperature. The four-layer CrI 3MTJ points to the potential for using layered antiferro- magnets for engineering multiple magnetoresistance states in an indi- vidual multiple-spin-filter MTJ. The low TCof CrI 3(/C2450 K) limits its practical device application. It is urgent to find intrinsic 2D magnetic materials with high Curie or Neel temperature for room-temperature MTJ devices. A similar theoretical investigation of a 2D spin filter and spin- filter MTJs consisting of atomically thin Fe 3GeTe 2was also reported.145The models and the main data are shown in Figs. 6(a)–6(d) . By the DFT-NEGF method, the TMR effect is obtained in single/double-layer Fe 3GeTe 2-hBN-Fe 3GeTe 2heterostructures. For heterostructures consisting of single- and double-layer Fe 3GeTe 2,t h e calculated MR ratio is 183% and 252% at zero bias, respectively. The Fe3GeTe 2MTJ in the P state shows a spin polarization of >75%. Besides the use as a spin-filter barrier, the metallic nature of Fe3GeTe 2a l s oe n a b l e si tt ob eu s e da sam a g n e t i ce l e c t r o d ei nv d W MTJs to provide high spin polarization, which has advantages over insulating CrI 3that is used as a spin-filter barrier only. On the one hand, a large magnetic field up to 1 T is required in CrI 3-based MTJs to switch the antiferromagnetic ground state to ferromagnetic.146,147 On the other hand, CrI 3-based MTJs are volatile (i.e., the magnetic field needs to be maintained to preserve the ferromagnetic order), while Fe 3GeTe 2-based MTJs are nonvolatile due to two stable P and AP magnetization configurations that still appear under the absence of an applied field. For this 2D metal with room-temperature ferromag- netism,148Liet al.149found that the spin-dependent electron transport across vdW MTJs consisted of a graphene or an h-BN spacer layerApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-8 Published under license by AIP Publishing(a) (b) (c) (d) (f) (e) (g) 1000 500CrI3 0 –500 –1000 –400 –200 0 V (mV)0 T 0.9 T 9 TIt (nA) 200 400μ0H⊥ = 0 μ0H⊥ > μ0H⊥cμ0H⎥⎥ > μ0H⎥⎥chBN hBN SiO2Graphene Graphene1.0 0.5P AP0.0 1.0 0.5 0.0–1.04000% 3000% 2000% 1000% 0 –1.0 –0.5 0.0 0.5n=1 n=2 n=3 n=4 Voltage (V)Voltage (V)TMR SIE 1.0 –0.5 0.0 0.5 1.0Free layer Left Lead Right Lead Central RegionPinning layern=3y y x x zCu Cr I B Nz x ⎥⎥ ⊥ It It It FIG. 5. (a) Top and side views of bilayer CrI3in the high-temperature phase. (b) The model structures of Cu/CrI 3/BN/n- CrI3/Cu (n ¼3, and two of them are the AFM pinning layer). (c) TMR vs bias volt-age of Cu/CrI 3/BN/n-CrI 3/Cu. (d) SIE vs bias voltage of Cu/CrI 3/BN/n-CrI 3/Cu in P and AP configurations. Reproduced withpermission from Yan et al. , Phys. Chem. Chem. Phys. 22, 26 (2020). Copyright 2020 Royal Society of Chemistry.143(e) Schematic of magnetic states in bilayerCrI 3. Left panel: layered-antiferromagnetic state, which suppresses the tunneling cur- rent at zero magnetic field. Middle and right panels: fully spin-polarized stateswith out-of-plane and in-plane magnetiza-tions, which do not suppress it. (f) Schematic of a 2D sf-MTJ. (g) Tunneling current of a bilayer CrI 3sf-MTJ at selected magnetic fields. Top inset: optical micro-scope image of the device (scale bar ¼5lm). The red dashed line shows the position of the bilayer CrI 3. Bottom panel: schematic of the magnetic configurationfor each tunneling current ( I t)-V curve. Reproduced with permission from Song et al. , Science 360, 1214 (2018). Copyright 2018 The American Associationfor the Advancement of Science.144Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-9 Published under license by AIP Publishingwith Fe 3GeTe 2ferromagnetic electrodes [see Figs. 6(e) and6(f)]. The authors found that the resistance changes by thousands of a percent from the P to AP state for both (graphene and h-BN spacers). The two different electronic structures of conducting channels in Fe 3GeTe 2 arouse a remarkable TMR effect. The authors further argued that thestrain and interfacial distance are two main factors that can influencethis giant TMR ratio. Also using Fe 3GeTe 2as ferromagnetic electrodes, Zhang et al.150designed Fe 3GeTe 2jInSejFe3GeTe 2MTJs and foundthat TMR reached /C24700% in this kind of MTJs. By analyzing both complex band structure of the barrier and band structure of the elec- trode, the origin of the considerable TMR is disclosed, as shown in Figs. 6(h) and6(i). In studies of the influence of the number of layers on the MTJ performance, it turns out that the number of layers of the central bar-rier layer has a great influence on the value of TMR (see more details inTable I ). (a) (b) (d) (e) (g) (h) 2.0 1.51.0 0.5 0.0 –0.5–1.0 –1.5 –2.0–1.00 –0.75 –0.50 k(Å–1) kL/ΠεFεF –0.25 0.00 0.25 0.50 0.75 1.00K M Γ ΓFe Ge Te Se In Band structureY Z XFe3Ge3Te2 Fe3Ge3Te2vdW barrier (Graphene, h-BN)(f) 0.25TP↑ TAP0.00 K MTMR > 1000 % 2 1 0 0.0 0.2 0.4 bias voltage (V)current ( μA) 0.6 0.8 1.0 0.00100200 TMR (%) Energy (eV)2.0 1.51.0 0.5 0.0 –0.5–1.0 –1.5 –2.0Energy (eV)300 0100200 TMR (%)300 0.2 0.4 bias voltage (V)0.6 0.8 1.0 0.0 0.2 0.4 bias voltage (V)0.6 0.8 1.0IPIAP2 1 0 0.0 0.2 0.4 bias voltage (V)current ( μA) 0.6 0.8 1.0IPIAP(c)buffer region buffer regionscattering region left electrode right electrode FIG. 6. (a) Model of single-layer Fe3GeTe 2sandwiched between two Cu electrodes. (b) Spin-involved I-V curvesand TMR of current varying with bias volt- age of the model corresponding to (a). (c) The model of double-layer Fe 3GeTe 2 sandwiched between two Cu electrodes.(d) Spin-involved I-V curves and TMR of current varying with bias voltage of the model corresponding to (c). Reproducedwith permission from Lin et al. , Adv. Electron. Mater. 6, 1900968 (2020). Copyright 2020 Wiley-VCH. 145(e) The magnetic junction structures of Fe 3GeTe 2/ h-BN/Fe 3GeTe 2. (f) Electron tunneling probability in MTJ distribution in 2D Brillouin region. Reproduced with permis- sion from Li et al. , Nano Lett. 19, 5133 (2019). Copyright 2019 AmericanChemical Society.149(g) The structure of Fe3GeTe 2jInSejFe3GeTe 2 MTJ. (h) Complex band structure for the InSe part(left panel) and band structure ofFe 3GeTe 2(right panel). Reproduced with permission from Zhang et al. , J. Phys. Chem. C 124, 27429 (2020). Copyright 2020 American Chemical Society.150Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-10 Published under license by AIP PublishingB. Targeting effective spin injection Note that in conventional MTJs, the electrodes usually adopt a kind of metal material, while the central scattering region is composedof an insulator material. Thus, the electrodes and the central scatteringregion are usually composed of different materials. The contact between them plays a significant role in determining the transport properties. 152 Unfortunately, due to lattice mismatch and conductivity mismatch, the contact between metallic electrodes and the insulating barrier is usuallypoor, resulting in low spin injection , which weakens the deviceTABLE I. Central and electrodes materials and defined TMR and TMR ratio with the function of the number of spin-filter tunnel barrier layers. Central materialsElectrode materialsTMR max (%)Defined TMR Remarks Reference MnSe 2/1-layer h-BN/MnSe 2 Ir 222.12 TMR ¼GP/C0GAP GAPG is conductance 151 Ru 419.94 MnSe 2/2-layer h-BN/MnSe 2 725.07 VSe 2/1-layer h-BN/VSe 2 Ir 254.48 Ru 183.55 VSe 2/2-layer h-BN/VSe 2 199.15 1-layer MoS 2 Co(fcc) 63.86 TMR ¼GP/C0GAP GAPG is conductance 139 Co(hcp) 33.40 Ni(fcc) 5.30 2-layer MoS 2 Co(fcc) 58.80 Co(hcp) 55.60 Ni(fcc) /C013.79 3-layer MoS 2 Co(fcc) /C070.85 Co(hcp) 55.91 Ni(fcc) 55.91 1-layer MoS 2 Fe3Si (100) 109.44 TMR ¼GP/C0GAP GAPG is conductance 135 3-layer MoS 2 306.95 5-layer MoS 2 278.87 7-layer MoS 2 154.56 9-layer MoS 2 121.63 CrI 3/h-BN/1-layer CrI 3 Cu (111) 2900 TMR ¼IP/C0IAP IAPI is current 143 CrI 3/h-BN/2-layer CrI 3 1800 CrI 3/h-BN/3-layer CrI 3 3600 CrI 3/h-BN/4-layer CrI 3 /C240 1-layer Fe 3GeTe 2/ h-BN/1-layer Fe 3GeTe 2Cu 183 TMR ¼1=IAP/C01=IP 1=IPI is current 145 2-layer Fe 3GeTe 2/ h-BN/2-layer Fe 3GeTe 2289 1-layer Fe 3GeTe 2/ h-BN/1-layer Fe 3GeTe 2Cu 78 TMR Julliere ¼1=IAP/C01=IP 1=IP¼2P2 1/C0P2P¼(Dup/C0Ddown)/ (DupþDdown) is the spin polarization of electrode; Dupand D down are the spin-up and spin-down density of states in the Fermi level of electrode2-layer Fe 3GeTe 2/ h-BN/2-layer Fe 3GeTe 2520 1-layer Fe 3GeTe 2/h-BN/ 1-layer Fe 3GeTe 22-layer Fe 3GeTe 2160 TMR scattering ¼1=IAP/C01=IP 1=IP¼ðDup/C0DdownÞ2 2DupDdownT is transmission 2-layer Fe 3GeTe 2/h-BN/ 2-layer Fe 3GeTe 2215 h-BN/2-layer CrI 3/TaSe 2/h-BN Graphite 240 TMR ¼RAP/C0RP RPR is resistance 19 2-layer CrI 3/TaSe 2 40Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-11 Published under license by AIP Publishingperformance.153,154If these problems are not solved well, even in the off state, electrons from one side of the electrode have the possibility of tunneling through the central scattering region to the other side of the electrode, resulting in the leakage of current.155Accordingly, it is diffi- cult to fabricate high-performance devices based on the present configu- ration in experiments. As such, resolving the contact problem and/or spin injection into the central region is important for developing high- performance MTJs. On the other hand, the parcel on the BN would effectively reduce the spin mismatch problem. In this section, we reviewthe recently reported 2D MTJs that target improved spin injection. To resolve lattice mismatch between electrodes and the barrier layer for high spin injection, the most direct means is to adopt thesame material. One 2D material can achieve both metal and insulator with different phases, for example, MoS 2in H phase and T phase. By using this means, Zhou et al.156constructed a kind of graphene-based MTJ as shown in Figs. 7(a) and7(b). They systematically studied the transport properties of the zigzag graphene nanoribbon using DFT- NEGF method. Remarkably, a 100% SIE and a giant TMR of up to 107 is predicted in their designed graphene-based MTJ device, whichshows much better performance than that of traditional 3D MTJs. Besides choosing the same material, selecting two materials from the same family can also minimize the mismatch problem. Inspired by the experimental synthesis of the magnetic layered crystal of Mn 2GaC, its 2D counterpart of the half-metallic Mn 2CF2MXene layer can serve (a) (b) (c) (e)Current (nA) TMR(d) a Left electrode b b∞∞ ∞∞ ∞ ∞ ac cRight electrode Scattering region 1004.104 2.106 1.106 5.105 3.103 1.105 7.104 3.104 2.104 8.103 4.103 2.103 1.10390 80 70605040 30 2010 00 0.2 0.4 0.6 Voltage (Volt)0.8 1 0 0.2 0.4 0.6 Voltage (Volt)0.8 1Mn Ti C F O3100 5 6 7 8 9 10 1150 02 1 010 20 30 40PP Left leadTransmission η (%) TMR (%)Right lead Scattering region Left lead Right lead Scattering regionVg VgL L (Å)010 20 30 40 L (Å)0012×107 10 20 30 40 L (Å)PNVg VgL x zy FIG. 7. (a) The structures of the device where in-plane gate voltages are applied to the electrode regions in the same direc- tion (PP) and opposite direction (PN). Thelength of the central scattering region withno gate voltage is applied, as denoted by L. (b) Spin-down transmission coefficient (left panel), spin injection efficiency (mid-dle panel), and TMR (right panel) as afunction of L for the PP configuration. Reproduced with permission from Zhou et al. , Phys. Rev. Appl. 13, 044006 (2020). Copyright 2020 American PhysicalSociety. 156(c) Top view and side view of Mn2CF2unit cell. (d) The top and side views of an MXene-based MTJ (Mn 2CF2/ Ti2CO 2/Mn 2CF2). (e) I-V curve of P config- uration and TMR of the MTJ. Reproduced with permission from Balcı et al. , ACS Appl. Mater. Interfaces 11, 3609 (2018). Copyright 2019 American ChemicalSociety.160Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-12 Published under license by AIP Publishinga st h em a g n e t i ce l e c t r o d ef o rM T J s .T i 2CO 2MXene can be chosen as the tunneling barrier, which is one of the most studied MXenes in both experiments and theories.157–159Balcı et al.160designed an MXene-based MTJ, as shown in Figs. 7(c)–7(e) . The highlight of their work is that the electrodes and barrier layer materials they chose are from the same family, which avoided the lattice mismatch problem. Furthermore, the bandgap of the Ti 2CO 2barrier layer was almost the same as the half-metallic gap of Mn 2CF2electrodes. Both the barrier and the electrodes have a common carbon layer that con- tributes the most to the transmission. Based on this, the proposed MTJ is matched both in structure and electronic structure. Theproposed MTJ also exhibits TMR, with a peak value of up to 10 6 and average values >103within the bias of 61V . A k k u s ¸et al.161 similarly investigated characteristics of a Ti 2CT2(T¼Oo rF ) MXene-based device that consisted of semiconducting Ti 2CO 2and Ti2CF2metallic electrodes. Their DFT-NEGF transport calcula- tions suggested that the device, made of a Ti 2CT2CO 2semiconduc- tor and two Ti 2CF2metallic electrodes, shows field-effect transistor characteristics when the semiconducting part is >6n m . I t w a s found that devices with larger tunneling barrier width should have a much better response to the gate voltage. Another effective way to solve the conductivity mismatch prob- lem and improve spin injection in MTJs is the introduction of h-BN, as discussed in Sec. II. The insulating h-BN is proposed as an ideal covalent spacer for MTJs, which provides a higher MR ratio and stron- ger exchange coupling at the interface to remove the conductivity mis- match between the metal leads and the FM layer. Many theorical andexperimental works have proven that h-BN–based MTJs have a good TMR performance. Such a conclusion has great impact on the h-BN integration pathway. Beginning with theoretical works, Qiu et al., 162 proposed an effective method to control the spin current in a vertical MTJ by combining the strong spin-filtering effect of graphene/ferro-magnet interface with the resonant tunneling effect of graphene/h- BN/graphene vdW heterostructure, in which Ni(111) is used as elec- trodes. Their theoretical results revealed that when the electronic spec- tra of spin electrons in two graphene layers are aligned, the spin resonance would appear, which results in a negative differential resis-tance effect. By studying a similar structure with a periodic density functional method in conjunction with Julliere’s model, Sahoo et al. 163 constructed Ni/BN/graphene/BN/Ni and Ni/graphene/Ni tunnel devi- ces. It was found that the former, the graphene/h-BN multi-tunnel junction, has a much higher TMR than the latter. The underlying mechanism was, as explained by Wu et al. ,164that the minority-spin- transport channel of graphene can be strongly suppressed by the insu- lating h-BN barrier, which overcomes the spin–conductance mismatchbetween Ni and graphene, resulting in a high spin-injection efficiency. Another example of using h-BN as the tunnel barrier is MnSe 2/h-BN/ MnSe 2.151The schematic diagram of the proposed structure is shown inFig. 8 . The monolayer T-MnSe 2is selected as the ferromagnetic layer,165and Ir and Ru are employed as metal electrodes, which have a smaller lattice mismatch with XSe 2compared with conventionally well-used electrode metals (e.g., Au, Ag, and Cu). It was found that such a vertically vdW MTJ has a large TMR of 725%.151This is due to a large transmission in the majority channel in the P magnetic config- uration, while it is suppressed in the AP magnetic configuration, as shown in Figs. 8(b) and 8(e). The authors took two approaches for improving TMR: One was choosing suitable electrodes, and the otherwas finitely increasing the number of layers of the h-BN barrier. The former minimizes the lattice mismatch and the latter involving BN sol-ves the spin mismatch problem. Introducing an extra 2D BN layer has advantages in improving TMR than 3D tunneling barriers like Al 2O3, which has been supported in experiments successfully.106In addition, many efforts have been made in the actual manufacturing of BN-based MTJs to improveTMR further. For example, a TMR ratio of /C240.3–0.5% can be obtained by performing wet transfer on ferromagnet. 117While h-BN is exfoli- ated on perforated membranes, the TMR ratio is 1%,166and the TMR can reach up to 6% as grown by large-area CVD on Fe directly.132 Besides using calculations to predict the conclusions, using calcu- lations to explain experiments is also important and necessary becausecalculations can give some physical insight. Like in the research on dif- ferent ferromagnets to develop h-BN MTJs, it is particularly important to use calculations to understand the interfacial effect on the MR per-formance of MTJs. For example, Piquemal-Banci et al. 95fabricated two h-BN–based MTJs with different FM electrodes, Co/h-BN/Co and Co/h-BN/Fe MTJs. In these two MTJs, h-BN is grown directly byCVD on prepatterned Co and Fe stripes. The TMR ratio in these twoMTJs are 12% and 50%, respectively. By using calculations method, the expected strong dependence of the h-BN electronic properties on the coupling with the FM electrode is further investigated. This calcu-lations part explained how h-BN improves TMR in their experimentalconclusion. When h-BN acts as the tunneling barrier, the FM layers can be various beyond Fe and Co. Like CrI 3, which is introduced Sec. IA,i t s cousin CrBr 3monolayer also shows FM characteristics,142and CrX 3 heterojunctions have recently been reported as having large MR.144 Inspired by this, Pan et al.167systematically investigated the structural, magnetic, and spin-transport properties of CrX 3/h-BN/CrX 3(X¼Br, I) MTJs. The barrier layer is h-BN, and CrX 3is used as the ferromag- netic layer. The metals Au, Ag, Al, and Pt are chosen as electrodes. Taking the AgCrBr 3/h-BN/CrBr 3/Ag MTJ [ Fig. 8(f) ] as an example, the large TMR effect can be up to 1565% in this series of MTJs. Thediagram of the band alignment and the k-resolved transmission spec-tra of MTJ are shown in Figs. 8(g)–8(k) . C. Targeting spin manipulation Manipulating spin is another powerful means not only to improve TMR but also to realize different functionalities of spintronicdevices. There are many ways for spin manipulation, such as addingother vdW materials to induce local phase transition, introducing SOT, making use of interlayer interaction, doping FM, and so on. In this section, we mainly review recent works that targeted spin manipu-lation of MTJs. For the aspect of spin manipulation through phase transition induced by an adjacent material, Begunovich et al. 168proposed ultra- thin MTJs based on vanadium ditelluride monolayers with graphene as a tunnel barrier. Both trigonal prismatic (H phase) and octahedral(T phase) VTe 2were considered in their study. The authors found that the introduction of graphene makes the electronic characteristic of 2D T-VTe 2changeable from the metal to half-metal phase, making T- VTe 2a promising candidate for MTJ applications. Although several possible structures are considered, the one that follows the framework of the Julliere model shows the highest TMR ratio of up to 220%.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-13 Published under license by AIP PublishingInserting other vdW “heavy” materials may introduce SOT-driven operations. Combining NEGF with noncollinear density functional theory methods, Dolui et al.19constructed bilayer-CrI 3/ monolayer-TaSe 2vdW lateral heterostructure, as shown in Fig. 9(a) , in which bulk nonmagnetic metal electrodes are required in practicefor generating in-plane charge current. They found that the AFM–FMnonequilibrium phase transition can be induced by the SOT in these MTJs, where the unpolarized charge current is injected parallel to theinterface. The 1H phase monolayer of metallic TaSe 2is chosen because of a small lattice mismatch (0.1%) to CrI 3and inversion asymmetry. By introducing another heavy metal of WS 2,Z o l l n e r et al.36designed Cr2Ge2Te6/graphene/WS 2vdW MTJs [see Fig. 9(b) ], where SOC, valley-Zeeman and Rashba splitting, and exchange coupling can beobtained. As for enhancing TMR by making use of interlayer interaction, many efforts are also done. To understand the inside mechanism (a) (b)0.5 1.20 0.78 0.36 –0.0550.23 0.12 0.0060 –0.10 0.17 0.11 0.049 –0.0110.21 0.13 0.050 –0.0300 –0.5–0.5 0 kakb 0.5 (d) (f) (g) Vacuum level Ag Ag CrBr3 CrBr3h-BNe EFEc Ev0.5 0 –0.5–0.5 0 kakb 0.5 (h)Central region screening region 0.5 0.12 0.077 0.037 0.0032 0.0026 0.0018 0.0022 0.0012 1.9E–049.6E–04 1.6E–04–0.00250.0 –0.5 –0.5 0.0 kakb(i) 0.5 0.01.3E–04 9.0E–05 5.2E–05 1.4E–05–0.5kb 0.5 (j) (k) 0.5 0.0 –0.5 –0.5 0.0kb kb 0.5ka 0.5 0.0 –0.5 –0.5 0.0 0.5–0.5 0.0 0.5(e)0.5 0 –0.5–0.5 0 kakb 0.5(c)0.5 0 –0.5–0.5 0 kakb 0.5central regionscreening region ΦcFIG. 8. (a) The structure of Ru/MnSe 2/ h-BN/MnSe 2/Ru MTJs. (b–e) kjj¼(ka,kb)– dependent transmission spectra of MTJsbased on MnSe 2for majority-spin and minority-spin states in P and AP configura- tions, respectively. Reproduced with permis-sion from Pan et al. , Chin. Phys. B 28, 107504 (2019). Copyright 2019 CPB.151(f) The structure of CrBr 3/h-BN/CrBr 3MTJs. White-colored atoms on the left and rightsides are silver electrodes. (g) Diagram ofthe band alignment of this MTJ. (h–k) The k-resolved transmission spectra of MTJ for majority- and minority-spin states in P andAP configurations. Reproduced with permis-sion from Pan et al. , Nanoscale 10, 22196 (2018). Copyright 2018 Royal Society of Chemistry. 167Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-14 Published under license by AIP Publishingbetter, the calculations method is needed. For example, Heath et al.169 provided important insight from an atomistic viewpoint on the under- lying mechanism governing the spin transport in graphene/CrI 3spin- filter tunneling junctions by a combined first-principles and quantum ballistic transport calculation, as shown in Fig. 10 . The calculated elec- tronic structures revealed that tunneling is the dominant transportmechanism in these heterostructures. The tunneling effect boosts dif- ferentiated intermediate metamagnetic states presenting in the switch- ing process. This is manifested in an increase in TMR for energyabove the Fermi level due to enhancement of Bloch states near theedge of the conduction band of CrI 3. In order to manipulate spin in nonmagnetic 2D materials, one can dope them by charge transfer from FM metals or proximity-induced spin splitting in themselves. For example, Asshoff et al. 166fab- ricated vertical graphene-based devices where ultimately clean graphene–FM interfaces were obtained by depositing the FM metals (FM¼Co and FM0¼Ni0.8Fe0.2alloy) on the two sides of a suspended graphene membrane, thereby preventing oxidation, minimizing thenumber of fabrication steps, and limiting the exposure of the devices to solvents during preparation. Such treatment improves the perfor- mance of MTJs. Applying a finite external bias voltage has been proven to be an effective method to manipulate spin transport. For example, Chen et al. 170theoretically investigated the nonequilibrium spin-injection and spin-polarized transport in monolayer black phosphorus (MBP)with ferromagnetic Ni contacts. The top and side views of their model are shown in Fig. 11 . In this study, they explored the SIE, TMR ratio,spin-polarized currents, charge currents, and transmission coefficients as a function of bias voltage. Furthermore, they studied two different contact structures where MBP is contacted by Ni(111) and Ni(100). Both structures are predicted to have great spin-polarized transport performance. 170The Ni(100)/MBP/Ni(100) MTJ has the superior properties of the SIE ( /C2460%) and TMR ratio (40%), which maintains almost a constant value against the bias voltage. Hydrostatic pressure can be used for continuous control of inter- layer coupling by interlayer spacing in vdW crystals and then tuning the spin interaction and transport. For example, experimentally, Song et al.171demonstrated the changes of magnetic order by pressure in 2D magnet CrI 3. The MTJ structure is composed of bilayer/trilayer CrI 3sandwiched by top and bottom multilayer graphene contacts, and h-BN encapsulates the whole MTJ in order to avoid sample degrada- tion. Figure 12 shows the structure of a bilayer CrI 3MTJ. It is found that the interlayer magnetic coupling can be doubled by a hydrostatic pressure.171 D. Targeting stability Besides the target high performance of MTJs, one should con- sider the structural and magnetic stabilities in the practical application. These practical problems include whether the MTJs can operate at room temperature and whether the 2D materials used in MTJs can be successfully synthesized. Targeted at these problems, many efforts have been made in 2D materials–based MTJs. In this section, we review the recent work around efforts to target stability problems. CrI3 m2 m1SOT VSe2TaSe2 MoS2 VSe2(a) (b) (c) (d)1000 800 600 400 200 010020030040000.1 0.2 0.3 0.4 Bias (eV) E – EF (eV) –2 –1 0 1 2TMR (%) SHC [( h/e) S/cm]0.5TMR at 300 K SHCIread IwriteCr2Ge2Te6 Graphene WS2mTe Utg mC UbgRL z z y xyx VbVbmCr FIG. 9. (a) The geometric structure of the CrI3/TaSe 2SOT vdW MTJ. Reproduced with permission from Dolui et al. , Nano Lett. 20, 2288 (2020). Copyright 2020American Chemical Society.19(b) The structure of the Cr 2Ge2Te6/graphene/WS 2 vdW MTJ. Reproduced with permission from Zollner et al. , Phys. Rev. Res. 2,4 (2020). Copyright 2020 American Physical Society.36(c) The structure of the VSe 2/ MoS 2SOT vdW MTJ. (d) TMR and spin Hall conductivity (SHC) of VSe 2/MoS 2 SOT vdW MTJ. Reproduced with permis- sion from Zhou et al. , ACS Appl. Mater. Interfaces 11, 17647 (2019). Copyright 2019 American Chemical Society.20The key feature of these three SOT MTJs is that the write charge current horizontally flows a heavy nonmagnetic 2D material,while the read spin current vertically flowsa 2D vdW MTJ.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-15 Published under license by AIP PublishingIn order to achieve 2D materials–based MTJs, the first require- ment is that the 2D materials are stable at room temperature. Thus, thermodynamic stability, dynamic stability, and mechanical stabilityshould be assessed. In practical efforts, inert 2D materials like BN areoften used to wrap reactive 2D materials like black phosphorene. 172Room-temperature working devices also require magnetic stabil- ity. However, most 2D FM materials discovered to date suffer from lowTC. As a result, the MTJs constructed by these materials only work at low temperatures, such as CrI 3-based MTJs. High-temperature working MTJs can be achieved by using 2D FM materials that have (a) (b)(c) 102 101 100 10–1 –0.6 –0.4 –0.2 0 E (eV)E (eV) E (eV) E (eV) TMR 0.2 0.4 0.6graphene lead tunnel junction lead Layer 1 1 0 –1 1 0 –1 1 0 –1MK ΓΓ MK ΓΓ MK ΓΓ MK ΓΓ MK ΓΓ MK ΓΓ MK ΓΓ MK ΓΓ MK ΓΓLayer 2 Layer 3Crl3 FIG. 10. Atomic and electronic structures of graphene/CrI 3/graphene MTJs. (a) Atomic structures of graphene/CrI 3/graphene MTJs. (b) Band diagrams and corresponding band structures in trilayer graphene/trilayer CrI 3/trilayer graphene junctions for various states. (c) TMR as a function of Fermi level for both trilayer (TMR """ and TMR ""#) and bilayer (TMR "") systems. Reproduced with permission from Heath et al. , Phys. Rev. B. 101, 195439 (2020). Copyright 2020 American Physical Society.169Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-16 Published under license by AIP Publishing(a) (b) (c) (e) (f) (h) 40 30 20 10 0 30 20 10 0 2.0 1.6 1.2 0.8 0.4 0.0 0 2 04 06 000.00.30.60.91.2 PC APC 20 40 60 80100 80 100 Bias (mV)up, APC down, APC total, APCup, APC down, APC total, APCup, APC down, APC total, APCup, APC down, APC total, APCTMRCurrent (nA) Bias (mV)00.00.30.60.91.2 PC APC 2040 60 80100 Bias (mV)ηη 20 40 60 80 100(i)(g) c ab ac b(d)Left Lead Central Scattering Region Right Lead x zy yFIG. 11. [(a) and (b)] The top view of the structure of the Ni(111)/MBP/Ni(111) and Ni(100)/MBP/Ni(100) MTJ. The dashed rectangle is the central scattering region.The transport is along the y direction. [(c)and (d)] The side view of Ni(111)/MBP/ Ni(111) and Ni(100)/MBP/Ni(100) MTJ. Ni and P atoms are labeled in yellow andpurple, respectively. [(e)-(g)] The planview of MBP in the ab, ac, and bc planes, respectively. (h) I-V curves of PC configu- ration (top panel) and APC (middle panel)as well as the bias voltage dependence ofTMR (bottom panel) of the Ni(111)/MBP/ Ni(111) MTJ, respectively. The inset in the bottom panel is the bias dependence ofspin-injection efficiency, g. (i) I-V curves of PC configuration (top panel) and APC (middle panel) as well as the bias voltage dependence of tunnel magnetoresistance(bottom panel) of the Ni(100)/MBP/Ni(100)MTJ, respectively. The inset in the bottom panel is the bias dependence of SIE. Reproduced with permission from Chenet al. , Phys. Chem. Chem. Phys. 18, 1601 (2016). Copyright 2016 Royal Society of Chemistry. 170Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-17 Published under license by AIP Publishingbeen predicted/discovered to have high TC. Recently, monolayer VSe 2 is reported to be a room-temperature ferromagnetic 2D material experimentally.173Thus, a VSe 2/MoS 2vdW MTJ was theoretically designed by Zhou et al. ,20as shown in Figs. 9(c) and9(d). They pro- posed a concept of SOT vdW MTJs, which can achieve both reading and writing functions at room temperature. Their NEGF results show a TMR of up to 846%. These proposed SOT vdW MTJs based on VSe 2/MoS 2give 2D MTJs a new opportunity for many magnetic field–free device applications that can work at room temperature. Later on, more XSe 2(X¼Mn, V)-based MTJs151with a 300 K work- ing temperature are proposed. The search for room-temperature FM materials is always demanding for 2D MTJs. Recently, Yang et al.174 designed excellent ultrathin spin filters by using half-metal 2D Cr2NO 2(seeFig. 13 ), which has a TCof 566 K based on first-principles calculations. The half-metal feature with 100% spin polarization of Cr2NO 2g u a r a n t e e sag i a n tT M Ro fu pt o6 0 0 0 % . Perpendicular MTJs with out-of-plane interfacial magnetization have many advantages, such as high thermal stability, infinite endur- ance, and fast with low-power switching. They are the base on whichto construct an advanced nonvolatile memory device, which can be built as non–von Neumann–computing paradigms to overcome a power bottleneck. Some perpendicular MTJs have been proposed the-oretically 175a n dt h e na c h i e v e de x p e r i m e n t a l l y .41From the materials point of view, it is important and necessary to discover more 2D mag- netic materials with out-of-plane magnetic anisotropy, which can be used as building blocks for perpendicular MTJs. V. SUMMARY AND PERSPECTIVES Since the discovery of TMR in MTJs, the MTJ devices have shown a great and profound impact on spintronics applications,including hard disk drives, MRAM, radiofrequency sensors, micro-wave generators, and neuromorphic computing networks. In this review, we discussed the four main issues of conventional MTJs, the current progress of 2D materials-based MTJs and how they addressthe problems of conventional MTJs, and briefly, the new scientific problems and technical challenges in 2D MTJs. In the past decade, the emergence of 2D (magnetic) materials has brought fresh blood to the family of MTJs and its spintronics (a) (b) (c) Piston Oil environmentCell bodyhBN graphene0 GPa1.46 GPa1.82 GPa2.70 GPa 1.16 GPa graphene hBNCrl3 Epoxy StageMonoclinic(d) 8 6 4 5 4 4 3 4 3 1 0.5 –2 –1 0 μoH (T)It (μA) 1 2Rhombohedral aac aac cb b FIG. 12. Stacking order dependence of 2D magnetism and tunneling measurements of bilayer CrI 3under pressure. (a) Schematic of rhombohedral stacking with top (left panel) and side view (right panel), indicating the ferromagnetic interlayer coupling. The green (purple) atoms represent the Cr (I) atoms. (b) Schematic fo r monoclinic stacking, indicat- ing antiferromagnetic interlayer coupling. (c) Schematic of high-pressure experimental setup, under which CrI 3is compressed normal to the van der Waals interface. (d) Tunneling current ( It) under different magnetic fields ( H) and pressures. The insets are the changes of magnetic states under 0 GPa and 2.70 GPa. Reproduced with permission from Song et al. , Nat. Mater. 18, 1298 (2019). Copyright 2019 Springer Nature.171Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 8, 021308 (2021); doi: 10.1063/5.0032538 8, 021308-18 Published under license by AIP Publishingapplications. The good performance of electron transport properties presented by graphene, h-BN, MoS 2,C r I 3,a n dF e 3GeTe 2and their vdW heterostructures has led to the prediction and demonstration of perfect spin filtering and large TMR ratio, as elaborated in this review. A TMJ usually is a stack of two or more materials in terms of an FM- NM-FM configuration. The various ways of stacking 2D materials result in 2D MTJs with diverse structures, which are categorized as lat-eral and vertical vdW MTJs. The 2D materials can be the magnetic electrodes and/or the central insulating materials. Many factors have been reported to influence the performance of 2D MTJs, such as the thickness, strain, interface, and the number of layers of 2D materials, which have been discussed in this review. For the computational aspect, the combination of DFT with the NEGF method is one of the most common techniques to study MTJ transport properties. The main challenge of this method is the overes- timated TMR ratio because under the framework of DFT, the compli-cated interfacial configuration, such as disorder, and the temperature effect cannot be described properly due to the demand in computa- tional resources. Some advanced methods in calculations of transport properties of MTJs are needed. In 2008, Ke et al. 176developed a non- equilibrium vertex correction theory to handle the configurationalaverage of random disorder at the density matrix level so that disorder effects to nonlinear and nonequilibrium quantum transport can be cal- culated from atomic first principles in a self-consistent and efficient manner. Recently, Starikov et al. 177described a DFT-based two-termi- nal scattering formalism that includes SOC and spin noncollinearity.An implementation using tight-binding muffin-tin orbitals combined with extensive use of sparse matrix techniques allows a wide variety of inhomogeneous structures to be flexibly modeled with various types of disorder, including temperature-induced lattice and spin disorder. The low stability and T Cof 2D magnetic materials, such as CrI 3,h a m p e r the practical applications of MTJ devices.The future prospect in the development of 2D MTJs might be keeping pace closely with the emergence of novel stable 2D magneticmaterials with room-temperature T C. However, the conventional DFT method is not quick enough to discover more demanded 2D materials with high TCand out-of-plane magnetic anisotropy. Recently, high- throughput computations have been carried out to screen high- TC2D materials. For example, Jiang et al.178investigated the electronic and magnetic properties of 22 monolayer 2D materials with layered bulkphases. They showed that monolayer structures of CrSI, CrSCl, andCrSeBr have notably high T C(>500 K) and favorable formation ener- gies, making these ferromagnetic materials feasible for experimental synthesis. Besides the discovery of new 2D magnetic materials usinghigh-throughput calculations, theoretically rational design of novel systems, such as mixed dimensional heterostructures and mixed dimensional vdW heterostructures, certainly provides a stable geomet-rical and magnetic structure as well as gives the additional spin- injection and spin-state control tools through the spin–orbit combined with the Coulomb effect, improving MTJ device performance. In recent years, more and more research has combined experi- mental work with computational work. One should design a calcula-tion model that can better reflect the actual situation of the experiment instead of being too idealistic. The calculation should take full account of the actual experimental conditions as far as possible and not be toounconstrained. As for the experimental aspect, the key research direction to devel- oping a device mainly focuses on integration engineering and the under- standing of 2D ferromagnet growth processes. At present, the research for this aspect is still at an early stage, though some progress has alreadybeen made. In fact, some integration parameters are not understood clearly. Various methods are applied to growth 2D materials. Many parameters, such as 2D material quality, crystallinity, phase, surfacechemical state, thickness, and the interaction at the interface, would havegreat impact on the performance of MTJ devices. In addition to focusing on improving TMR ratio of 2D MTJs, the device lifetime, mechanical stress, thermal stability, and switching current are important and haveresearch value in the experiment. Finally, with the implementation of large-scale growth of advanced 2D materials through chemical vapor deposition methods, it is indubitable to expect that a real promise existsf o ran e wg e n e r a t i o no f2 DM T J so nal a r g es c a l e . ACKNOWLEDGMENTS The authors would like to acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 51671114 and U1806219), MOE Singapore (MOE2019-T2-2-030, R-144-000-413-114, R-265-000-651-114, and R-265-000-691-114).This work is also supported by the Special Funding in the Project of the Taishan Scholar Construction Engineering. The authors declare no competing financial interest. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1D. Akinwande et al. , “Two-dimensional flexible nanoelectronics,” Nat. Commun. 5, 5678 (2014). Au elecrode Au elecrode Au elecrodenA nAIEfE I Au elecrode FIG. 13. The structure of Au/bilay-Cr 2NO 2/Au. A half-metal 2D Cr 2NO 2, which has a Curie temperature of 566 K, is demonstrated to be a promising candidate for ultra- thin spin filters. 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1.4773210.pdf

Scaling the dynamic response and energy harvesting potential of piezoelectric beams Deniz Tolga Akcabay and Yin Lu Young Citation: Appl. Phys. Lett. 101, 264104 (2012); doi: 10.1063/1.4773210 View online: http://dx.doi.org/10.1063/1.4773210 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v101/i26 Published by the American Institute of Physics. Related Articles Effect of strong electron correlation on the efficiency of photosynthetic light harvesting JCP: BioChem. Phys. 6, 08B617 (2012) Effect of strong electron correlation on the efficiency of photosynthetic light harvesting J. Chem. Phys. 137, 074117 (2012) Power generation from conductive droplet sliding on electret film Appl. Phys. Lett. 100, 213905 (2012) Tuning of nonlinear vibration via topology variation and its application in energy harvesting Appl. Phys. Lett. 100, 031902 (2012) Size effect of flexible proof mass on the mechanical behavior of micron-scale cantilevers for energy harvesting applications Appl. Phys. Lett. 99, 243506 (2011) Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 26 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsScaling the dynamic response and energy harvesting potential of piezoelectric beams Deniz Tolga Akcabay and Yin Lu Y oung Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Michigan 48109-2145, USA (Received 28 August 2012; accepted 10 December 2012; published online 26 December 2012) This work presents dynamic scaling relations for the fluid-structure interaction (FSI) response and energy harvesting potential of fluttering piezoelectric beams. The results show that it is difficult to find a model-sized material to satisfy all the scaling relations using Reynolds scaling. Machscaling, which does not imply that flow compressibility is important, but simply requires the same inflow speed and operating fluid, allows the same material as full-scale to be used; this enables the model to emulate similar FSI response, energy harvesting potential, and material failuremechanisms, if both the model and full-scale operate at sufficiently high Reynolds numbers (/C211000). VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4773210 ] The specific focus of this paper is to derive and validate dynamic scaling relations for the fluid-structure interaction(FSI) response and energy harvesting potential of fluttering piezoelectric beams 1–10for designing and interpreting model-scale experiments. Two different scaling strategiesare examined: the Reynolds (Re) and the Mach (Ma) number scaling. Re scaling requires the Reynolds number to be the same between the model and the full-scale. Ma scaling, doesnot imply that fluid compressibility effects are important, but rather requires that the relative inflow speed and the fluid properties are the same for both the model and the full-scale.It should be noted that although different scaling laws have been derived and analyzed in the context of aerospace 11–14 and marine15–19structures, none concerned the FSI response and energy harvesting potential of fluttering piezoelectric beams. Consider an energy harvesting device as shown in Fig. ( 1). The device consists of a cantilev er beam with two outer piezo- electric layers, each of thickness q p0, that are separated by an inner substrate layer. The beam is exposed to steady, uniformflow. The governing parameters of this device are given in Table I, assuming two-dimensional response and q 0/C252qp0. Furthermore, the beam is assumed to be stiff enough to neglectany changes in beam length, so only the beam bending stiff- ness, K b0,i sc o n s i d e r e d .T h eb e a mt h i c k n e s si sa s s u m e dt ob e small (q0/C28L0), so that the Euler-Bernoulli beam theory is ap- plicable. There are 10 dimensional parameters in the units of mass [M], length [L], time [T], and electric current [I] units. Buckingham’s PI theorem yields 10 /C04¼6 independent dimensionless parameters, shown in Table II. These dimen- sionless parameters were derived using qf0L03,L0,L0/Uo0,a n d Uo02L0(e0qf0)1/2as characteristic mass, length, time, and electric current scales, respectively. The electromechanical equations are derived by assum- ing: (i) laminar, viscous, Newtonian, and incompressible flow,(ii) large, elastic beam deformation model with small thickness and negligible extension, (iii) zero structural damping, and (iv) linear piezoelectric constitutive material behavior. 20,21 The mathematical modeling of the device is described in Ref. 22. The model solves the full Navier-Stokes equations,the Euler-Bernoulli beam equation with consideration for large deformations and electromechanical coupling, and applies thePenalty Immersed Boundary method 23to solve for the FSI response and energy harvesting. Table IIIshows the scaling relations for the dimensional parameters as functions of the length scale ratio, kL0, which is the ratio between the model and full-size lengths, as well as the required piezoelectric coupling coefficient scale ratio,k t0(note that k/is the ratio of any quantity /between the model and full-scale). It assumes the same fluid medium between the model and the full-scale and keeping the samedimensionless b q,bU,tq, Rb, and q parameters. Table IIIreveals the challenges of the Re scale: Re scal- ing requires kUo0¼(kL0)/C01 ,which may be difficult to achieve ifkL0is low, due to the limitations to the maximum attain- able flow speed in wind or water tunnels. In addition, Kb0(/C24E0q03) scales with L0, where E0is Young’s modulus, which requires kE0¼(kL0)/C02and the same scaling applies to the yield stress. For most piezoelectric ceramics, E0is between 60 and 100 GPa, and E0/C243 GPa for the PVDF plas- tic. Hence, it will be very difficult to find a suitable model- scale material that will simultaneously satisfy the stiffness and density scaling requirements. Moreover, ke0¼(kt0kL0)2 brings yet another difficulty due to the limited range of t0 available for piezoelectric materials. On the other hand, Table IIIreveals that Mach scaling can circumvent many of the scaling difficulties. Since kUo0¼1, kqs0¼1 (because kqsq0¼kL0), and kE0¼1 (because kKb0¼kL03), it permits the use of the same material for both the model and FIG. 1. The energy harvesting device. Reprinted with permission from D. T. Akcabay and Y. L. Young, Phys. Fluids 24, 054106 (2012). Copyright 2012 American Institute of Physics. 0003-6951/2012/101(26)/264104/3/$30.00 VC2012 American Institute of Physics 101, 264104-1APPLIED PHYSICS LETTERS 101, 264104 (2012) Downloaded 26 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsthe full-scale, which will allow kt0¼1a n d ke0¼1. Ma scale will also lead to kR0¼1. On the other hand, the major defi- ciency of Ma scaling is that Re will be scaled with kL0. Using bq¼10, bU¼10, q ¼1/646, tq¼/C05.3, and Rb¼3.22/C210/C03—corresponding to a 3 m long beam with PZT-5 A layers in air flow of 24 m/s at full-scale, Fig. 2 shows the effect kL0while applying Ma scaling on the pre- dicted time histories of the normalized beam vertical tip dis-placement ( Dy tip/L0), non-dimensional generated voltage— scaled by L0Uo0(qf0/e0)1/2, and the maximum bending stress (rmax) normalized by the yield stress ( rY0). The computa- tional setup is similar to Sec. IV B in Ref. 22, but in this casethe top and bottom fluid-domain boundaries are modeled with symmetry conditions. For simplicity, material damping is neglected in the current computations. Readers should refer to Ref. 22on discussions about the influence of mate- rial and fluid damping. The Re is 2000, 1000, and 200 for kL0¼1, 0.5, and 0.1, respectively. The tip displacement, gen- erated voltage, and maximum stresses are smaller for thek L0¼0.1 case because of the greater contribution of viscous damping. On the other hand, the results are comparable for kL0¼0.5 and 1.0 because the Re was high enough; indeed, the rms voltage and maximum stress were only about 4% and 5% different between kL0¼1 and 0.5, respectively. The plots in Fig. 2demonstrate that Mach scaling could be used to produce similar displacements, stresses, energy har- vesting characteristics, and potential material failure mecha- nisms with the full-scale operating conditions, as long asRe>/C241000 s. Note that the ( r Y0) is exceeded in Fig. 3for kL0¼0.5 and 1.0 and the avoidance of this issue is discussed in Ref. 22. Figure 3shows the predicted vorticity shedding patterns and stroboscopic plots of the beam deflections for different kL0obtained using Mach scaling—see Ref. 22for detailed analyses of the vorticity shedding patterns. The stroboscopic images for kL0¼1 and 0.5 are very similar, while the case with kL0¼0.1 has smaller displacements due to the signifi- cance of viscous damping at low Re. As expected, the vortic- ity structures get thicker with decreasing Re, but the difference in vorticity structure is negligible on the FSIresponse for Re /C211000. The above findings show while it might be impractical to apply Reynolds scaling, especially due to material selec-tion issues, it is feasible to apply Mach scaling to simulate the FSI response, energy harvesting potential, and potential material mechanisms in model-scale experiments, as long asRe is high enough ( /C211000 s).TABLE I. Critical dimensional parameters. L0Beam length [L] qf0Fluid density [M/L3] q0Total beam thickness [L] lf0Fluid viscosity [M/(L /C1T)] qs0q0Total beam density times thickness [M/L2] t0Piezoelectric layer coupling coefficient [I /C1T/L2] Kb0Beam bending stiffness [M /C1L2/T2] e0Piezoelectric layer electric permittivity [I2/C1T4/(M/C1L3)] Uo0Inflow speed [L/T] R0b0Electric circuit resistance times beam width [M /C1L3/(I2/C1T3)] TABLE II. Critical dimensionless parameters. Definition Meaning Re qf0L0Uo0/lf0Ratio of fluid inertial to viscous force bq qs0q0/(qf0L0) Ratio of solid to fluid inertial force bU [(qf0Uo02L03)/Kb0]1/2Ratio of fluid inertial to solid bending stiffness force tq t0q0/(Uo02L02qf0e0)1/2Ratio of total electric to fluid inertial force Rb R0b0e0Uo0/L0Ratio of electric damping to fluid inertial force qq0/L0Ratio of beam thickness to length TABLE III. Scale ratios of dimensional parameters as a function of kL0and kt0for the Re and Ma scales. Re Ma Re Ma kL0 kL0 kL0 kqf0 11 kq0 kL0 kL0 klf0 11 kqs0q0 kL0 kL0 kt0 kt0 1 kKb0 kL0 kL03ke0 (kt0kL0)21 kUo0 kL0/C011 kR0b0 kt0/C02kL0 FIG. 2. The effect of kL0while applying Ma scaling on the normalized time histories of the beam vertical tip displacement, generated electric voltage, and maximum beam stresses.264104-2 D. T. Akcabay and Y . L. Y oung Appl. Phys. Lett. 101, 264104 (2012) Downloaded 26 Dec 2012 to 152.14.136.96. 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1.rcpa11406R37201.pdf

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Birch, R. S.; Gerges, S. N.; and Vergara, E. F. Design of a Pulse Generator and Shock Tube for Measuring Hearing ProtectorAttenuation of High-Amplitude Impulsive Noise. Appl. Acoust. ~UK!64~3!, 269–286 ~2003!. Phys. Abstr.: Iss. 09, Classif. Code 4385D,Abstr. Nr. 1 ~2003!. INSPEC A2003–09–4385D–001. Dehmel, S.; et al. Electrophysiological Characterization of the Superior Paraolivary Nucleus in the Mongolian Gerbil. Hear. Res. ~Netherlands !172~1–2!,1 8– 3 6 ~2002!. Phys. Abstr.: Iss. 03, Classif. Code 8734, Abstr. Nr. 2 ~2003!. INSPEC A2003–03–8734–002. Ferrara, M.; et al. Topographical Changes in N1-P2 Amplitude upon Awakening from Recovery Sleep After Slow-Wave SleepDeprivation. Clin. Neurophysiology ~Ireland !113~8!, 1183–1190 ~2002!. Phys. Abstr.: Iss. 24, Classif. Code 8770F, Abstr. Nr. 4 ~2002!. INSPEC A2002–24–8770F–004.43.60.P 43.64.À R398 R398 J. Acoust. Soc. Am., Vol. 114, No. 6, Pt. 2, Dec. 2003 References to Contemporary Papers on Acoustics Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.174.21.5 On: Tue, 23 Dec 2014 08:05:00

1.2815287.pdf

Physics News in 1986 Phillip F. Schewe Citation: Physics Today 40, 1, S1 (1987); doi: 10.1063/1.2815287 View online: http://dx.doi.org/10.1063/1.2815287 View Table of Contents: http://physicstoday.scitation.org/toc/pto/40/1 Published by the American Institute of PhysicsAn American Institute of Physics Special Report Phillip F. Schewe', EditorTHE AMERICAN INSTITUTE OF PHYSICS The American Institute of Physic s is a not-for-profit mem- bership corporation chartered in New York State in 1931 for the purpose of promoting the advancement and diffusion of the knowledge of physics and its application to human wel- fare. Leading societies in the field of physics and astronomy are its members. AIP's activities include providing services to its Member Societies in the publishing, fiscal, and educa- tional areas, as well as other services which can best be per- formed by one operating agency rather than dispersed among the constituent societies. Member Societies arrange for scientific meetings at which information on the latest advances in physics is exchanged. They also ensure that high standards are maintained in the publication of the results of scientific research. AIP has gen- eral responsibility for the publication and distribution of journals. The Institute is expected to stay in the forefront of publishing technology and to ensure that the services it per- forms for its Member Societies are efficient, reliable , and economical. The Institute publishes its own scientific journals as well as those of its Member Societies; provide s abstracting andindexin g services; serves the public by making available to the press and other channels of public information reliable communications on physics and astronomy; carries on ex- tensive manpower activities; encourages and assists in the documentation and study of the history and philosophy of physics; cooperates with local, national, and international organizations devote d to physics and related sciences; and fosters the relations of physics to other sciences and to the arts and industry. The scientists represented by the Institute through its Member Societies number more than 85,000. In addition, approximately 7000 students in over 500 colleges and uni- versities are members of the Institute's Society of Physics Students, which includes the honor society Sigma Pi Sigma. Industry is represented through some 115 Corporate Associ- ate members. The purpose of the AIP's Public Information Division is to promote a public understanding of the role of physics in society and to provide an appreciation of the developments in physics by assisting Member Societies in conveying infor- mation about research, education, and manpower. Preface Physics News in 1986, prepared by the Public Information Division of the American Institute of Physic s (AIP), is the 18th in a series of annual reviews of physics news. In past years Physics News was published in bookle t form and was distributed to reporters, students, libraries, teachers, scientists, and to the general public. More recently Physics News has been published as a supple- mentary report in the January issue of Physics Today, beginning with the January 1984 issue. The article s in Physics News in 1986 were selected and prepared by the AIP Member Societies. The following individuals helped to organize the chapters and in some cases to write articles: The American Physical Society: Division of Chemical Physics—Stephen R. Leone, Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards Division of Condensed Matter Physics—Phillip J. Stiles, Brown University Division of Atomic, Molecular, and Optical Physics— Joseph Callaway, Louisiana State University Division of Particles and Fields—Stephen Adler, Institute for Advanced Study Division of Fluid Dynamics—Elaine S. Oran, Naval Research Laboratory Division of Plasma Physics—James D. Callen , University of Wisconsin Division of High Polymer Physics—Matthew Tirrell, University of MinnesotaOptical Societ y of America: Jarus W. Quinn, OSA American Association of Physics Teachers: Jack M. Wilson, University of Maryland American Crystallographic Association: William L. Duax, Medical Foundation of Buffalo American Astronomical Society: Stephe n P. Maran, NASA-Goddard Space Flight Center American Association of Physicists in Medicine: Gary T. Barnes, University of Alabama, Birmingham American Vacuum Society: Paul H. Holloway, University of Florida American Geophysical Union: Jack B. Hartung, AGU AIP Corporate Associates: John R. Reitz, Ford Motor Company 1 December 1986 Phillip F. Schewe S-2 PHYSICS TODAY / JANUARY 1987AIP OFFICERS Hans Frauenfelder, Chair, Governing Board H. William Koch, Executiv e Director Roderick M. Grant, Secretary Gerald F. Gilbert, Treasurer Lewis Slack, Director of Educational Programs Robert H. Marks, Director of Publishing PUBLIC INFORMATION DIVISION David Kalson, Manager Phillip Schewe, Senior Science Writer James Berry, Science Broadcast Coordinator Audrey Likely, Consultant MEMBER SOCIETIES The American Physical Society Optical Society of America Acoustical Society of America The Society of Rheology American Association of Physics Teachers American Crystallographic Association American Astronomical Society American Association of Physicists in Medicine American Vacuum Society American Geophysical Union COVER: A composite sagittal (midline) nuclear magnetic resonance image of the head and spinal cord. Surface coils provide improved resolutio n and contrast of the cervical, lumbar , and thoracic spine. (Courtesy of Philip s Medical Systems.) © 1986, American Institute of Physics Permission is hereby granted to journalists to use the material in this special report at their discretion and without referencing its source. Pub. R-225.17, December 1986Physics News in 1986 An AIP Special Report ASTROPHYSICS 4 Voyager at Uranus/Encounters with Comet Halley/Afloat in the Clouds of Ve- nus/New Determinations of the Distance to the Galactic Center/Bubble Structure in the Universe/SomeRecord-Breaking White-Dwarf Stars/Quasi-Periodic Oscil- lations in Galactic X-ray Sources/Solutions to the Solar Neutrino Problem? CHEMICAL PHYSICS 11 Alignment and Orientation Effects in Collision Dynamics/Molecular Recogni- tion/Vibrational Energy Exchange in Molecular Collisions With Solid Surfaces/ High-Resolution Optical Spectra of Large Organic Molecules CONDENSED MATTER PHYSICS 14 Squeezed States/Universal Conductance Fluctuations/Convection in Fluid Mix- tures/Lattice Dynamics in Picosecond Time Domain/Scanning Tunnel Micros- copy/Conformal Invariance and Critical Exponents in Two Dimensions CRYSTALLOGRAPHY 21 Small-Angle Scattering/Fractals and Small-Angle Scattering /Time-Resolved X- Ray Scattering/Colloidal Dispersion Structures Analyzed by Small-Angle Neu- tron Scattering ELECTRON AND ATOMIC PHYSICS 23 Quantum Jumps in a Single Atom/Parity Nonconservation in Atoms/Uranium Lamb Shift/A toms in Strong Laser Fields/Discovery of the Soliton Self-Frequency Shift ELEMENTARY PARTICLE PHYSICS 27 Superstrings/Snowmass 1986/New Issues in Cosmology/Heavy Quark Physics/ Cosmic Strings/Advanced Accelerator Research and Development FLUID DYNAMICS 34 The Structure of a Propagating Detonation/Triangle-Based Grids in Computa- tional Fluid Dynamics/The Condensational Instability GEOPHYSICS 37 Geology of Venus/Geomagnetic Reversals/Geomagnetic Main Field and Secular Variation/Rock Magnetism/Marine Geology/Oceanography, El Nino, and Cha- os/Stochastic Analysis of the Transport of Contaminants in Groundwater MEDICAL PHYSICS 42 Nuclear Magnetic Resonance in Medicine/Dual-Energy Chest Radiography NUCLEAR PHYSICS 46 Sharp Positron Peaks/Double Beta Decay and Nuclear Structure Calculations OPTICS 47 Experimental Observations of Laser Cooling and Trapping of A toms/Scattering of Free Electrons by Light/Binary Optics: An Emerging Diffractive Optics Technolo- gy/Tunneling and Photoconductivity/Photon Localization PHYSICS APPLIED TO INDUSTRY 51 Research in Scanning Tunneling Microscopy/Direct Observation of Ballistic Elec- tron Transport in GaAs/Nuclear Magnetic Resonance Spin-Offs/Electronic Con- duction in Silicon Dioxide PHYSICS EDUCATION 56 The International Physics Olympiad/Conference on the Teaching of Modern Physics/Support for Undergraduate Physics Programs PLASM A AND FUSION PHYSICS 59 Progress Towards Breakeven on the Tokamak Fusion Test Reactor/Recent Ex- periments in Inertia! Confinement Fusion/Recent Free Electron Laser Experi- ments/Relativistic Plasma Waves and Particle Acceleration/Transport Near the Onset of Chaos POLYMER PHYSICS 64 Polymer Photophysics/Polymer Theory: Crossover and Criticality/Polymer Sur- face Forces VACUUM PHYSICS 66 GaAs on Si: Progress and Opportunities/Surface States in Real Space/Studies of Surface Phonons by Electron Energy Loss Spectroscopy/XPS and A uger Forward Scattering in Epitaxial Films 70 PHYSICS NOBEL PRIZE PHYSICS TODAY / JANUARY 1987 S-3ASTROPHYSICS Astrophysics, the investigatio n of the universe beyond the Earth, made great strides in 1986 despite the terrible loss of the Challenger and its crew. As the Challenger tragedy un- folded, Voyager 2 was charting the remarkable moons, dark rings, and tilted magnetic field of Uranus, the third giant planet to be explored by this remotely-controlled spacecraft, now en route to Neptune. Less than two months later, a fleet of five interplanetary probes , VEGA 1, VEGA 2, Sakigake, Suisei, and Giotto, flew near or through Comet Halley in a single week, and shortly thereafter the International Come- tary Explorer, veteran of an earlier comet mission, encoun- tered waves and particles from Halley in the solar wind, tens of millions of kilometers upstream of the comet. Also in 1986 the first detailed scientific results were pub- lished from the balloon payloads that VEGA 1 and VEGA 2 deposited at Venus as they flew past in June 1985 on their way to Halley. Tracked by twenty radio telescopes around the world, the VEGA balloons explored the hot, turbulent middle cloud layer of our twin planet where vertical drafts as strong as those in earthly storms are commonplace. The technique of Very Long Baseline Interferometry (VLBI) was critica l to precision determination of the balloons' tra- jectories, as it was to a new determination of the distance to the cente r of our galaxy, the Milky Way. The center of the Milky Way, close to the radio source Sagittarius B2, was found to be about 7.1 kiloparsecs from Earth, versus 8.5 kpc for the previous best estimate. Most distances in astrophysics beyond the solar system are expressed in parsecs (pc) and their multiples. One par- sec, the distance at which the angle subtended by the semi- major axis of the Earth's orbit would be one second of arc, is about 3.26 light years, or 3.09 XlO13 kilometers. Kilopar- secs—thousands of parsecs—ar e good enough to measure distances across the galaxy, but megaparsecs (Mpc—mil- lions of parsecs) are the units of choice when the vast dis- tances between external galaxies and clusters of galaxies are investigated. Although VLBI now offers the prospect of di- rect distance determinations for nearby external galaxies, the distance scale over megaparsecs is still uncertain by about a factor of two. Thus, the survey of a slice through the universe, which revealed that galaxies may be located pre- dominantly on the surfaces of thin-shelle d spherical bubble structures, extended to 100 Mpc for galaxies of typical brightness, assuming a particular value (100 km/s/Mpc) for the Hubble constant, a measure of the expansion of the universe. However, in fact, the survey objects may be twice as far away, if the Hubble constant is twice as small. New kinds of stars and star systems also made news in 1986. White dwarfs, typically as large as the Earth but with S-4 PHYSICS TODAY / JANUARY 1987about two-thirds the mass of the sun, were found to include some objects with immense magnetic fields, around 500 megagauss, some with exceptionally high temperatures, to 350 000 K by one estimate, and at least one with the lowest luminosity yet observed for a star. The European Space Agency's satellite EXOSAT made pioneering studies of a new type of stellar phenomenon, the QPOs or quasi-periodic oscillators. EXOSAT operated for over 1000 days, from May 1983 through May 1986 in a high orbit (initial apogee, 200 000 kilometers; period 99 hours), considered ideal for monitoring variabilit y in x-ray sources. An old, well-known star, the sun, still excites the greatest scientific interest. Astrophysicists still seek an explanation for the discrepancy between the observed flux of solar neu- trinos at the Earth and the predictions of solar interior mod- els. If the theory of the solar interior cannot withstand ex- perimental verification, what hope is there for the vast corpus of stellar interior and evolution models? Fortunately, two new explanations for the observed neutrino flux seem to deserve the most serious consideration. There may be a reso- nance in neutrino scattering within the sun, or weakly inter- acting massive particles may lower the temperature in the solar core. Either process would reduce the neutrino flux at the Earth. Stephen P. Maran, NASA-Goddard Space Flight Center Voyager at Uranus The durable and venerable Voyager 2 spacecraft (launched August 1977), swept by Uranus on January 24, 1986. Four days later, the space shuttle Challengerblew up. The reaction to the shuttle tragedy meant that almost everyone would overlook Voyager at Uranus, no matter what it found. But this planetary encounter was, if anything, more surprising than the earlier visits to Jupiter and Saturn. Planetary scien- tists expected something strange, because Uranus is remark- able in having its rotational axis tilted 98 degrees from the perpendicular to its orbital plane. The orbits of the Uranus satellites and rings share this unusual orientation, suggesting perhaps that some event tilted the Uranus system on its side just as the planet was forming. Uranus's rotational axis pointed almost directly towards the sun at the time of en- counter, and so half of the planet had been experiencing night for 42 Earth years. The variety of terrain on the coal- black satellites was unexpected, as were the bizarre tilt of Uranus's newly discovered magnetic field and the complex geometry of its rings.1'2PHYSICS NEWS IN 1986—ASTROPHYSICS The discovery of active volcanoe s on Jupiter's moon Io by Voyager 1 in March 1979 dispelled any notion that the satel- lites of the major planets would prove to have cratered sur- faces resembling that of the Earth's moon. Saturn too has strange satellites with geologically young, grooved surfaces. Voyager 2 discovered ten small moons of Uranus and ob- tained close-up images of the five large moons discovered many years ago. They are all considerably darker than the satellites of Saturn. The bright areas on their surfaces are probably water ice, but the dark areas have an albedo of 0.1 or less, as dark as carbon black, and among the darkest stuff in the solar system. The Uranian rings are even darker. The nature of the dark matter in the Uranus satellite system is a major mystery. Broadly speaking, the large Uranian satellites are divided into two major groups. The outer three have relatively old surfaces, with many impact craters. This indicates that little internal activity has altered these three moons since forma- tion, although the details of the formation process may have been affected by events in the formation of Uranus itself. The inner two large moons (Ariel and Miranda) are the strange ones. Tiny Miranda, innermost, has many deep, globe-gir- dling grooves and enormous scarps. Planetary scientists speculate that the event which tipped Uranus on its side dis- rupted Ariel and Miranda. Miranda, some suspect, may even have been disrupted and reaggregated five times. The small numbers of craters in parts of the Ariel and Miranda surfaces suggest that violent events continue to occur. Per- haps water in their interiors is spiked with an ammonia anti- freeze that keeps it liquid . Then, flows of the ammonia-water mixture might fill in craters on the surface and produce a young-looking terrain. Another possible explanation for the young surfaces of Ariel and Miranda is that these moons may contain more rocky matter than the outer Uranus satel- lites, making radioactive decay a more significant source of internal heat. The rings of Uranus were discovered in March 1977 from the Kuiper Airborne Observatory when they occulted the light from a background star. Voyager 2's photography of this ring system revealed new rings and ringlets. A carefully planned sequence of pointings imaged the rings that contain very small dust particles; these particles scatter light strong- ly in the forward direction but are much harder to see in the backscattered light visible from the Earth. The dusty rings are distinct from those compose d of larger particles. Any small particles near Uranus will spiral into the planet in about 2000 years, so these measurements suggest that the Uranian ring system is young and quite dynamic. Uranus itself is generally a featureless greenis h disk seen through ground-based telescopes , and the raw images from Voyager 2 were similarl y uninteresting. But sophisticated image processing brought out patterns in the atmosphere. Uranus proved to look surprisingly like Jupiter and Saturn, despite the unusual orientation of its polar axis. The planet is girdled by bright and dark zones aligned along parallels oflatitude. Although the present dark hemisphere has not ex- perienced sunlight in 42 years, it is as warm as the sunlit hemisphere. Another Uranus surprise was the strange magnetic field. The very existence of a field was surprising, since Uranus has no internal heat source (unlike Jupiter and Saturn). But stranger still was the field's orientation. The Earth and all other magnetized planets have magnetic axes that are rea- sonably close to parallelism with their rotation axes, with fields roughly like that of a dipole. In the other planets, then, it's much as though a giant bar magnet were buried in the interior. However, Uranus's magnetic field is tilted 60 de- grees from the rotation axis. An offset dipole, in which the center of the planet is not at the planet's center, is a good initial model, but there are probably complex components to the field. The nature and origin of Uranus's field raise ques- tions; Ness and co-worker s speculat e that Uranus may be undergoing a field reversa l like those which , geological evi- denc e suggests, have occured in the past on Earth.3 Harry L. Shipman, University of Delaware 1. E. C. Stone and E. D. Miner, Science 233, 39 (1986) and accompanying papers. 2. J. K. Beatty, Sky and Telescope 71, 333 (1986) and A. Chaikin, ibid., 338 (1986). 3. N.F. Ness ef a/., Science 233, 85 (1986). Encounters with Comet Halley 1986 will go down in astronomical history as the year of Comet Halley, and will be particularly remembered for the studies made by five remote-controlled spacecraft which en- countered the comet between March 6 and 14. An interna- tional cooperativ e effort of remarkable proportions yielded the first direc t images of a comet nucleus and its dust jets, as well as information on the parent molecules of the gases ob- served, the dust emissions, the solar wind and many other phenomena.1"3 Although Comet Halley was a visual disappointment to much of the public, who saw it only with difficulty or not at all, the success of the spacecraft encounters made it an excit- ing object to the many scientific teams. Planning for the mis- sions had begun as long ago as the late 1970's. The Japanese Institute of Space and Astronautical Sciences (ISAS) devel- oped two vehicles, Suisei and Sakigake, which were launched in January and August of 1985. These craft were designed primarily for studies at fairly large distances of the solar wind and the comet's hydrogen halo. The Soviet Un- ion's Space Research Institute also sent two missions, VEGA 1 and VEGA 2, in December of 1984; these studied the plane t Venus before being rerouted to Comet Halley. (See the following article.) Their intent was to observe the PHYSICS TODAY / JANUARY 1987 S-5PHYSICS NEWS IN 1 986—ASTROPHYSICS nucleus, if possible, and to study the processes going on in the coma and the compositio n of the gas and dust. The European Space Agency's Giotto spacecraft was launched by an Ariane rocket in July 1985. It was designed to fly closest to the comet (ideall y about 500 km away) and to obtai n images of the nucleus, as well as information on the inner coma, neutral gas and ion composition and distribu- tion, dust particles and dust jets, and magnetic fields. All five missions achieved their primary goals, despite the dangers from possible collisions with dust particles. The most spectacular results were the first direct observa- tions of a comet nucleus, obtained both by VEGAS 1 and 2 from distances of 8000-900 0 km, and by Giotto from about 600 km. These offered support for Whipple's icy conglomer- ate model of comet nuclei.4 Halley's nucleus is nonspherical, rather like a potato, about 15 km long by 10 km wide. Its surface is extraordinarily dark: the albedo is less than 4%, suggesting a thick layer of nonvolatile material coating the nucleus. The surface is irregular and rough, probably with pits and hills on the order of 100 meters in size. Two large dust jets and some smalle r ones projected from the sunward side of the nucleus , evidently from relatively localized areas of the surface, covering in all perhaps 10% of the total sur- face area. A large part of the surface must consist of a thick dark crust that insulates the ice within, since the observed rate of gas and dust release was lower than expected. This insulation is also suggested by the VEGA probes' measure- ments of the temperature of the nucleus as 300-400 K, hotter than that which would cause water ice to sublime. The irreg- ular shape of the nucleus further suggests that sublimation of ice below the crust does not occur uniformly, so the nucleus is not homogeneous. Both Giotto and the VEGA probes also looked at the gas emissions in the coma near the nucleus. Water seems to be the most abundant parent molecule , with CO2 next; among other neutral molecules detecte d were OH, C2, CH, CN, and NH. Ions observed included H+,C+,H2O+,O+,He+, and others; C "*" was unexpectedly abundant, and may have come from the dust grains or from carbon atoms released at the surface. The VEGAs first detected dust particles while still over 3X 105 km away from the comet, but the number density increased rapidly within 1.5 X 105 km. These grains ranged in mass from the smallest detectable (about 10" 16 g) to 10 "6 g, the largest encountered by the VEGAs. Giotto re- corded some 12 000 dust particle encounters, the largest be- ing 1.4 X 10 " 4 g. The dust particles appear to be rich in H, C, N, and O. The dust production rate was estimated by Giotto experimenters at 3.1 X 106 g/sec; dust activity is much high- er close to the nucleus, and heavie r particles are also more abundant there. Interactions of cometary ions with the solar wind were detected at distances of 107 km. The Japanese vehicles moni- tored the solar wind's behavior for about a year, especially near the closest approach times of the other spacecraft. Thecomet/solar wind interactions are fairly complex; Giotto de- tected a bow shock, a turbulent magnetic region, and various other regions including a zone near the nucleus with almost no magnetic field. Suisei observed the comet's hydrogen cor- ona in November 1985, at a distance of about 2.5 X 108 km. Its brightness changed regularly in a 2.2 day cycle, suggest- ing a rotation of the nucleus with that period. The hydrogen corona changed with time in non-periodic ways also, and brightened appreciably in early March of 1986. Further analysis of the many experiments is still in prog- ress; but it can already be said that our knowledge of comet nuclei and structure has been vastly enriched by these five encounters with Comet Halley. It is to be hoped that the success of this international adventure will lead to further cooperation in the future on other projects. Katherine Bracher, Whitman College 1. R. Z. Sagdeevera/., Nature 321, 259 (1986) and accompanying papers. 2. K. Hirao and T. Itoh. ibid.. 294 (1986) and accompanying papers. 3. R. Reinhard. ibid.. 313 (1986) and accompanying papers. 4. F. L. Whipple, Astrophys. J. Ill, 375 (1950). Afloa t in the Clouds of Venus The first balloon exploration of planetary meteorology be- yond the Earth was accomplished in June 1985.' The Soviet agency Intercosmos delivered two balloons to the middle cloud layer of Venus through release from the VEGA 1 and 2 spacecraft, which also deposited lander probes and flew on to encounter Comet Halley. French and U.S. investigators also played major roles in the mission . (For an article on the geology of Venus, see the chapter on geophysics.) The helium-inflated, 3.5-m diameter, superpressure Tef- lon balloons each supported a 6.9-kg instrumented gon- dola,2 coate d to withstand sulfuric acid in the hot (305 K) clouds. Released, respectively, at latitudes 7.3 deg N and 6.6 deg S near the midnight meridian, the balloons were caught up in the retrograde (east to west) zonal wind and tracked by six Soviet antennas and by a worldwide network coordi- nated by CNES, the French space agency. These included the three 64-m reflectors of NASA's Deep Space Network, and eleven radio astronomy observatories. Instrument data were telemetered , while each payload was tracked by differ- ential Very Long Baseline Interferometry on the signals from each balloon and the on-flying spacecraft that released it.3 Tracking of each balloon extended for over 46 hours, until the lithium batteries in the gondola were exhausted. The surface of Venus is hidden from view by thick clouds. However, this near-twin of the Earth (the two planets have similar size and mass) has been explored by many Soviet landers, by the U.S.'s Pioneer Venus descent probes and Or- S• 6 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1986—ASTROPHYSICS biter, and by radar astronomy from the Earth's surface. Con- ditions within the clouds have been measured during brief intervals as the probes and landers fell through them. The VEGA balloons represented, in contrast, the first extended survey of conditions within the cloud layer . Each gondola was equippe d to measure temperature, pressure, ambient light level, cloud particle backscatter, and frequency of lightning. Three-dimensional wind data, de- rived with the aid of the tracking network, support investiga- tions of atmospheric dynamics, including vertical and hori- zontal heat and momentum transport, and the presence and origin of large-scale eddies and waves. Some previous work has suggested, amidst dispute, that lightning is prevalent over putative volcanic mountains on Venus, a result that, if sustained, would support other indications, also controver- sial, that there is active volcanism on the planet. No light- ning flashes were observed from the VEGA gondolas, nor were obvious breaks in the clouds detected. The balloons were initially deployed at 50 km altitude (900 mbar) and rose to an equilibrium float altitude of 53.6 km (535 mbar). They were blown westward into the sunlit hemisphere by the prevailing 70 m/s horizontal winds, but were subjected to intermittent gusts of vertical wind larger in amplitude than many investigators had expected. The ob- served predominance of downward vertical winds may be a consequence of horizontal convergence that biases the loca- tion of free-floating balloons to regions where such wind s are present. The vertica l winds were frequently larger than 0.5 m/s and reached 3.5 m/s; sometimes one balloon or the oth- er was subject to such wind s for a few hours at a time.4 A strong and consistent vertical temperature and pressure gra- dient prevails in the cloud layer . Thus, the vertical gusts allowed the balloon instruments to explore a greater range in the atmosphere than anticipated, from about 535 to 880 mbar. The horizontal range of the observations was about 11 000 km, or roughly one-third of the way around Venus. A consistent 6.5 K temperature difference between read- ings taken at the same atmospheric pressure by the balloons to the north and south of the Venus equator was observed, perhaps an indication of long-lived, large-scale eddies that propagate in the zonal (east-west) direction. Tracking data suggest that some eddy motions are either fixed in relation to the sun direction or fixed with respect to the planetary sur- face. The VEGA-2 balloon experienced severe downdrafts near the end of its observed float, when it passed over the high mountainou s region Aphrodite. This may be attribut- able to a mountain or lee wave, familiar phenomena on Earth.5 The success of the VEGA balloon experiments suggests that balloons deployed from spacecraft may have further utility in the future exploration of Venus and perhaps other planets. Stephen P. Maran, NASA-Goddard Space Flight Center1. R. Z. Sagdeeve/ al, Science 231, 1407 (1986). 2. R. S. Kremnev etai, ibid., 1408 (1986). 3. R. A. Preston era/., ibid., 1414 (1986). 4. V. M. Linkin etai, ibid., 1420 (1986). 5. J. E. Blamont etai, ibid., 1422 (1986). New Determination of the Distance to the Galactic Center The distance to the center of our Milky Way Galaxy, R 0, when combined with measurements of the 21-cm wavelength radio emission line from neutral hydrogen atoms is an impor- tant parameter for understanding the mass distribution in the Galaxy. The most recent standard value, Ro =8.5 kpc, was adopted by the International Astronomical Union in order to insure that different investigators can intercompare results based on standard calibrated distance scales. Classically, the determination of R 0 has relied heavily on observations of the density maximum in the distribution of RR Lyrae stars in direc- tions close to the galactic center. Uncertainties in our knowl- edge of the intrinsic luminosity of these "standard candles" and in the amount of interstellar absorption, however, have histori- cally influenced the value of R 0. The 8.5-kpc value, for example, was a downward revision from the "round number" of 10 kpc previously adopted in 1964. Recently, a geometric technique making use of the precise sub-milliarcsecond accuracy of Very Long Baseline Interfero- metry has been employed to determine a new value of R 0. The method depends on the existence of moving H2 O maser spots in Sagittarius B2 (Sgr B2), a young star-forming region very close to the galactic center. In this technique VLBI images of the maser spots were repeatedly obtained over the course of about one year. Individual moving maser spots were identified and measured, and Doppler velocities were also obtained for each spot. For a large assemblage of emitting sources, statistically, the conversion between angular motion across the plane of the sky and velocity components toward the observer determines the distance to the emitting region. Previously, the power of the VLBI technique was demon- strated through monitoring the motions of many H2 O maser clumps associated with star-forming regions throughou t the Galaxy; the most prominent one is Orion A, less than 500 pc distant from the sun. For Sgr B2, however, independent consid- erations place it within 300 pc of the galactic center, and its distance was an important reference value to establish. An international team led by Mark J. Reid at the Harvard- Smithsonian Center for Astrophysics analyzed the VLBI data.' They estimated the distance to Sgr B3 at 7.1 + 1.2 kpc. This constitutes a new determination of the distance to the galactic center. At the same time, the VLBI technique will provide an additional check on the value of R 0, once the distance of an- other strategically located source, W49, is determined. Final improvements in R 0 are also dependent on a thorough under- standing of the systematic error in the distance to both sources. Beyond the immediate importance of geometrically deter- mining the distance scale of the Galaxy, the VLBI technique PHYSICS TODAY / JANUARY 1987 S-7PHYSICS NEWS IN 1 986—ASTROPHYSICS shows promise for determining extragalactic distances. H2O maser sources also exist in the young star-formation regions of nearby galaxies, where careful monitoring should provid e valu- able independent calibrations of the extragalactic distance scale. Robert J. Havlen, National Radio Astronomy Observatory 1. M. J. Reid et al., in Star Formation, edited by M. Peimbert and J. Jugaku (Reidel, in press). Bubble Structure in the Universe A recent survey of a slice through the universe indicates that galaxies are distributed on the surfaces of thin spherical shells.' This conclusion has strong potential implications for crucial issues in cosmology, including the formation of galaxies,2 the origin of clusterin g and superclustering,3 the nature and preva- lence of dark matter in intergalactic space,4 and the interpreta- tion5 of an equally recent discovery, that galaxies in a vast re- gion that includes the Milky Way and its Local Group of Galaxies share a common directed motion.6 The survey slice through the universe measures six degrees wide and 117 degrees long on the sky (see Fig. 1). It is centered near the north galacti c pole and passes through the Coma clus- ter of galaxies. Since the survey was magnitude limited, the dis- tances to which galaxies were observed depend on their intrinsic brightnesses; brighter galaxies are detected at greater ranges. Galaxies of average intrinsic brightness were observed to depths in the slice of abou t 100 Mpc (about 330 000 000 light years), assuming that the Hubble constant Ho equals 100 km/s/Mpc. For smaller values of Ho favored by some astrophysicists, the distance surveyed and the scales of structures identified may be as much as twice as great. As in all studies of large scale structure in the universe, dis- tances were not measured directly but were determined from the redshifts of the 1100 galaxies in the study, which were attri- buted exclusively to the expansion of the universe. Man y of the redshifts were measured with a 1.5-meter telescope at Mount Hopkins, near Amado, Arizona. Presumably each galaxy has a so-called peculiar velocity, due to its motio n with respect to nearby mass concentrations such as galaxy clusters. The effects of the peculiar velocity components were ignored, a circum- stance that may requir e further analysis. The galaxies in the slice surveyed appear to be located on the surfaces of shells, the "bubbles," with typical diameters of 25 Mpc; there is a pronounced dearth of detectabl e galaxies in the interiors of the bubbles, which range up to 50 Mpc in size. The data encourag e advocates of the explosive galaxy formatio n the- ory, since blast waves predicted by that theory might explain the thinness of the shell walls. However, existing versions of the theory cannot readily account for the great diameter of a typical bubble. The pancake theory for the formatio n of large-scale structure (such as superclustering) in the universe leads natu- rally to the prediction of filamentary structure in the universe, which some observations seem to reveal. However, provable filamentary structure is lackin g in the slice of the universe sur- S-8 PHYSICS TODAY / JANUARY 1987oistsncG in millions of light ysdm FIG. 1. This two-dimensional representation of the 3-D map prepared by Margaret Geller, John Huchra, and Valerie de Lapparent of the Harvard- Smithsonian Center for Astrophysics shows galaxies (represented by dots) distributed on the surfaces of giant bubble-lik e structures. The torso of the "human figure" at the center of the map is a cluster of galaxies in the constellation Coma. veyed in the recent work. Clusters of galaxies, from the recent survey, seem to be located at or near where adjacent bubbles meet. Velocity dispersions for galaxies on the outskirts of clus- ters may need to be reinterpreted, or interpreted with extreme care, in the light of this observation. Such velocity dispersions are often used to support claims that there is a very high mass- to-light ratio in clusters, presumably due to dark matter of un- certain nature. Stephen P. Maran, NASA-Goddard Space Flight Center 1. V. de Lapparent, M. J. Geller, and J. P. Huchra, Astrophys. J. Lett. 302, LI (1986). 2. Physics Today 39, 17 (May 1986). 3. J. Silk, Nature 320, 32 (1986). 4. N. Turok, Nature 322, 111 (1986). 5. J. Silk, Nature 322, 207 (1986). 6. D. Burstein et al, reported by J.R. Bond and S. van den Bergh, Nature 320,489 (1986). Some Record-Breaking White-Dwarf Stars White dwarf stars are tiny balls, the size of the Earth, in which several tons of starstuff are packed into a volume the size of a thimble. They are dying cinders, the final remnants of low mass stars like our sun. White dwarf matter, with its high density, is weird enough, but other extreme values of stellar properties are found amon g the white dwarf stars. During the past year, sever- al different groups have establishe d a number of extremes in the properties of these objects: the highest temperatures, the stron- gest magnetic fields, and possibly the lowest luminosities among all ordinary single stars.PHYSICS NEWS IN 1986—ASTROPHYSICS A few percent of white dwarf stars have extremely high mag- netic fields, which range from a few to a few hundred megagauss (for comparison, the magnetic field at the surface of the Earth is 0.3 gauss). In these stars the magnetic field disturbs the energy levels of the atoms in the stellar surface, altering the star's spec- trum and making interpretation difficult. On the basis of calcu- lations of the behavior of the common element hydrogen in high magnetic fields,1 two groups estimated that the first magnetic white dwarf star ever discovered (a star called Greenwich + 70° 8247) had a 300 megagauss field.2 Gary Schmidt and collaborators at the University of Arizona identified the object PG1031 + 234 as a magnetic white dwarf with a still higher field, exceeding 500 megagauss.3 The rotation periods of mag- netic white dwarfs can sometimes be determined, since the spec- trum and polarization pattern change as the star's rotation brings different regions with different field strengths into view. A curious difference between these two magnetic stars is that PG1031 + 234 rotates reasonably rapidly, turning once in 3 hours and 24 minutes, while Grw + 70° 8247 takes at least two centuries to spin once (its spectrum has remained unchanged for 50 years). These magnetic fields are far stronger than those known in any other type of more or less ordinary star. The only known stronger fields are those of the pulsars (1012 gauss). Two determinations of the highest well-established tempera- tures among ordinary single stars appeared during the year. The star called HI504 + 65 was first discovered as the seventh brightest source in the soft x-ray sky by the HEAO-1 all-sky x- ray survey, and follow-up work identified it as a very blue, faint star, with a very peculiar spectrum that rises steeply from the visible through the ultraviolet. For such a faint star to be so bright in x rays, it must have a temperature of 160 000 K, mak- ing it the hottest white dwarf star known.4 But its chief pecu- liarity may be its composition. Most stars contain hydrogen and helium, but this star contains no sign of either of these two astronomically common chemical elements. It could be a very hot helium-rich star, so hot that the helium lines don't show up. More likely, it is a unique carbon-oxygen star, the remnant of a red giant star which has shed its envelope, exposing its nuclear burning core. An even more extreme temperature was established for an object about to become a white dwarf star, the central star of the planetary nebula NGC 2440. In at least some, if not all cases, stars which are about to become white dwarf stars are surround- ed by a cloud of glowing gas, the last remains of the outer enve- lope of the dying star. This cloud of gas is called a "planetary nebula" because it can resemble a planet when seen through a telescope. The nebula itself can be used to determine how many high energy photons are emitted by the central star, because these photons cause the nebula to emit light by fluorescence. Atherton, Reay, and Pottasch were able, for the first time, to measure the visual brightness of the central star of NGC 2440 by using a charge-coupled device (CCD) camera at the focus of the Anglo-Australian Telescope.5 The central star is very faint, only six times brighter than the faintest objects visible on large scale sky surveys. If such a faint object is to produce a visible nebula, it must be very hot indeed: in this case about 350 000 K. To complement this set of record high temperatures, a white dwarf star which is one of the coolest and dimmest, if not thecoolest, known was discovered in the course of a supernova search program at the University of Chile. Ruiz, Maza, Wisch- nejwsky, and Gonzalez recognized this object, called ER 8, as peculiar because it moves very rapidly across the sky and is quite faint.6 Its spectrum is generally similar to the spectra of two other very cool white dwarfs, which have temperatures of around 4000 K. Its faintness and high proper motion suggest (but do not prove conclusively) that it may be cooler, which would make it the least luminous star known. Very cool white dwarfs like ER 8 are not the coolest stars, but because of their small size, they are the stars of lowest known luminosity. The discoveries of these weird inhabitants of what has been called the "astrophysical zoo" raise questions about the origin and evolution of the strangely different types of white dwarf stars. The extreme characteristics of some white dwarfs suggest that you don't need to discover fundamentally new classes of objects in order to face new puzzles. Harry Shipman, University of Delaware 1. H. Forsterefa/., J. Phys. B 17, 1301 (1984); R. J. W. Henry and R. F. O'Connell, Pub. Astron. Soc. Pacific 97, 333 (1985); G. Wunnerefa/., Astron. and Astrophys. 149, 102 (1985). 2. J. R. P. Angel, J. Liebert, and H.S. Stockman, Astrophys. J. 292, 260 (1985); J. L. Greenstein, R. J. W. Henry, and R. F. O'Connell, As- trophys. J. Lett. 289, L25 (1985). 3. G. D. Schmidt et al., Astrophys. J. (in press). 4. J. A. Nousekef a/., Astrophys. J. (in press). Quasi-Periodic Oscillations in Galactic X-ray Sources The brightest galactic x-ray sources were discovered over twen- ty years ago, but in many respects they are still mysterious. Astronomers have developed an acceptable general scenario in which a companion star dumps mass onto a neutron star (or in some cases a black hole), producing the x rays. But progressing beyond this general picture has been difficult because the neu- tron star itself is unobservable directly, hidden beneath a com- plex, turbulent, and possibly magnetized accretion disk. In the past year and a half a number of teams have discovered that the x-ray intensity of several galactic x-ray sources, as measured by the European EXOSAT, varies semiregularly. Reanalysis of earlier data from the EINSTEIN satellite provided confirma- tion. This phenomenon may prove to be a valuable observa- tional tool which will allow us to probe the complexities of ac- cretion disks surrounding neutron stars.1 The teams were looking for periodic x-ray emission pro- duced by the rotation of the neutron star, similar to the well- defined, regular pulses seen in pulsars. While such pulses were not seen, the persistent searches paid off with the discovery of quasi-periodic oscillations (or QPO's) in many sources. There are now ten known QPO's. These QPO's show up as a broad peak in the power spectrum, quite different from the sharp peak which would be expected from a spinning neutron star. The first PHYSICS TODAY / JANUARY 1987 S-9PHYSICS NEWS IN 1986—ASTROPHYSICS reported detection2 of a QPO was in the bright source GX5-1, and this was quickly followed by similar reports of a number of other sources.3'4 Any newly discovered phenomenon immediately stimulates efforts to interpret it. Models published so far suggest that the oscillations arise from the interaction of a rapidly spinning neu- tron star with material at the inner part of an accretion disk. Material binaries being dumped from one star to another in an interacting binary system forms a disk around the accreting star, and velocities in the disk increase as you go closer to the accreting star. The temperature of the disk becomes higher, and the magnetic field (if any) of the accreting star becomes stron- ger, producing complex interactions. Alpar and Shaham5 interpreted the periodicity of the QPO's as arising from the difference between the frequency at which material at the inner edge of the accretion disk orbits the neu- tron star and the frequency at which the neutron star spins on its axis—as a beat frequency. In a later elaboration of this model, Lamb, Shibazaki, Alpar, and Shaham6 introduced the idea that clumps of material at the inner edge of the disk could make the beat frequency observable. In this scenario, a clump of gas en- counters magnetic fields of different strength and geometries as it orbits the neutron star. Each time it reaches the same longi- tude, relative to the spinning neutron star, material is stripped off of the clump, falls to the neutron star surface, and emits a more intense burst of x rays. This model can predict a relation between the frequency of the quasi-periodic oscillations and the intensity of the x-ray source. The rate at which material is being dumped onto the neutron star varies in these sources; in general, a higher accre- tion rate produces a denser disk, which extends closer to the surface of the neutron star, and higher x-ray emission. A rea- sonable choice of model parameters fit the data for the first reported QPO, GX 5-1, very well, where higher frequency oscil- lations occurred when the source was more intense (see Fig. 2). However, the intensity-frequency relationship is quite differ- ent in different sources, and in some cases it can be quite com- plex. In the brightest galactic x-ray source (Sco X-l), the fre- quency goes down slightly, not up, with increasing source intensity, as long as the source is not too bright.3 But when conditions in the source change, the energy as well as the inten- sity of the x rays can change too. A drop in the x-ray intensity at a particular energy observed by some satellite instrument does not necessarily mean that the total x-ray flux has gone down, and so the interpretation of intensity-frequency relationships can be quite complex. Other models for the QPO's have been proposed. All models (including the one described above) are basically similar in that interaction between a spinning neutron star and the environ- ment immediately surrounding it, in the inner part of the disk, produces oscillations. In one scenario, x rays produced directly by the central neutron star are scattered from hot gas located above the accretion disk,7 and it is the periodic rotation of blobs of hot gas around the neutron star which produces the quasi- regular bursts of x rays. In a second scenario, the complex inter- action of magnetic fields and hot gas in the inner part of the disk channels material to the surface of the neutron star, producing40 60 Frequency (Hz) FIG. 2. Power density spectra of quasi-periodic oscillations in x-ray emis- sion from GX 5-1. When the source is at low intensity, the oscillations correspond to the broad peak near frequency 20 Hz, while at high intensity, the peak appears near 36 Hz. (Illustration from William C. Priedhorsky, Los Alamos National Laboratory.) bright spots which appear randomly on the neutron star surface and persist for a few rotation periods.8 Gas accreting onto a neutron star must produce a torque on the spinning neutron star or the disk of matter surrounding it, and Priedhorsky has ar- gued that changes in this torque could explain a number of observed phenomena including the QPO behavior of Sco X-l.9 Still more schemes may be published in the coming months. A common theme of many of these scenarios is that similar, much slower QPO's are seen in cataclysmic variables, binary systems in which a white dwarf rather than a neutron star is at the center of the accretion disk. Instabilities in the Earth's magnetosphere have also been used to guide theoretical interpretations . The QPO phenomenon is rather complex. Work in future years will show whether these (or other) models can explain the data. It may be that QPO's will be a very good observational probe of x-ray binaries, since the frequency with which an astro- nomical object oscillates is often fundamentally related to that object's structure. But it is also possible that the cause of this phenomenon may remain as an intriguing puzzle in high-energy galactic astronomy for some time. Harry Shipman, University of Delaware 1. N. E. White, Nature 316, 210 (1985); Physics Today 38, 17 (October 1985). 2. M. Van der Klis et al, Nature 318, 225 (1985). S-10 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1986—ASTROPHYSICS 3. G. Hasinger etal. Nature 319,469 (1985); L. Stella, A. Parmar, and N. White, IAU Circular No. 4102; L. Stella et al, IAU Circular No. 4110; W. Priedhorsk y <tf a/., Astrophys. J. Lett. 306, L91 (1986). 4. J. Middleditch and W. C. Priedhorsky, Astrophys. J. 306, 230 (1986). 5. M. A. Alparand J. Shaham, Nature 318, 239 (1985). 6. F.K. Lamb et at, Nature 317, 681 (1985). 7. C. B. Boyle, A. C. Fabian, and P. W. Guilbert, Nature 319, 648 (1986). 8. J.-M. Hameury, A. R. King, and J.-P. Lasota, Nature 317, 597 (1985). 9. W. Priedhorsky, Astrophys. J. Lett. 306, L97 (1986). Solutions to the Solar Neutrino Problem? Do we understand how the sun shines? Do we know how neu- trinos propagate? These questions have plagued astronomers and physicists for almost two decades. An experiment by R. Davis and his collaborators using 100 000 gallons of perchlor- ethylene in the bottom of the Homestake Gold Mine in South Dakota has yielded a puzzling result: observation differs mar- kedly from calculation. The measured value is about a factor of three lower than the theoretical expectation based upon the standard theory of how the sun shines.' This persistent discrep- ancy has spawned many imaginative solutions, some based upon the idea that conventional astronomical theory is wrong and some based upon speculative ideas in physics. Two new suggestions from the arena of fundamental physics have stimu- lated investigators around the world to reexamine the proper- ties of weakly interacting particles.2 Soviet cosmic-ray physicists Mikheyev and Smirnov,3 fol- lowing up earlier work of L. Wolfenstein,4 have shown that electron neutrinos may be converted to muon neutrinos by re- sonantly scattering off electrons that the neutrinos encounter on their way out from the core of the sun to the Earth. The asymmetry between electron and muon neutrinos is caused by the extra (charged-current) interaction that exists between electrons and electron-neutrinos. This process, now called theMSW effect, after its inventors, is the most conventional solu- tion yet proposed for the solar neutrino problem since it re- quires only that: (1) neutrinos produced in radioactive decays are a mixture of neutrino states; (2) the difference of neutrino masses be somewhere in the range between 10~2 and 10~4 eV; and (3) the electron neutrino be lighter in vacuum than the muon neutrino. Two groups of American astrophysicists5'6 have proposed, instead, the existence of weakly interacting massive particles (WIMPS) that would simultaneously solve two fundamental problems: the missing matter problem and the solar neutrino problem. The abundance of these massive (typically several GeV or heavier) particles produced in the Big Bang would be just sufficient to account for the unseen matter discovered astro- nomically, provided they also have the right cross sections to be captured by the sun and transport a significant amount of ener- gy from its core outward to larger radii. The captured WIMPS could lower the central temperature calculated for the sun and, in turn, the important neutrino flux, because they add to the efficiency with which photons transport energy. Parameters can be postulated for the WIMPS that are consistent with a single solution of both the missing-matter and the solar neu- trino problems. There remain important questions as to why the WIMPS do not annihilate efficiently or reveal themselves by their mutual destruction. John Bahcall, Institute for Advanced Study 1. J. N. Bahcall and R. Davis, Science 194, 264 (1976). 2. Physics Toda y 39, 17 (June 1986). 3. S. P. Mikheyev and A. Y. Smirnov, Yad. Fiz. 42, 1441 (1985), English translation in Sov. J. Nucl. Phys. 4. L. Wolfenstein, Phys. Rev. D20, 2634 (1979). 5. W. H. Press and D. N. Spergel, Astrophys. J. 296, 679 (1985). 6. J. Faulkner and R. L. Gilliland, Astrophys. J. 299, 994 (1985). CHEMICAL PHYSICS Alignment and Orientation Effects in Collision Dynamics Investigation of new aspect s of atomic and molecular align- ment and orientation effects on collision dynamics using po- larized laser radiation is emerging as a stimulating field of research.u Intriguing new experiments use the weak van der Waals interaction between atoms and molecules to orient molecules in clustered species for state-to-state dynamical studies. The early classic exampl e of the effect of molecular align- ment on reaction dynamics uses a hexapole magnetic field toorient molecules of CH3I in a crossed beam reaction with an atom such as Rb.3 The scattering is found to have a large reactive asymmetry and the reaction probability is signifi- cantly attenuated when the Rb approaches the CH3 end of the molecule. Thorough studies of orientation effects in the NO + O3 system demonstrated that the NO molecule can approach and react with either the central O atom or an O atom on the end of the ozone molecule.4 More recently, laser excitation schemes have been used to align electronically excited orbitals of Ca and Na atoms in studies of reactions and energy transfer of these atoms.5"7 PHYSICS TODAY / JANUARY 1987 S-11PHYSICS NEWS IN 1 986—CHEMICAL PHYSICS The reaction of a Ca P state with Cl2 shows a dramatic de- pendence on the yield of the CaCl(A) excited state with the direction of the orbital alignment in the collision. The reac- tion has a marked preference for perpendicular alignment of the p orbital with the incoming reaction partner. Energy transfer induce d by rare gas collisions from the Ca 5[P state to the near-resonant 53P states shows a 50% effect on the cross section with varying direction of the 5P orbital align- ment. The results have been successfully explained in terms of the differing potential energy surfaces for the perpendicu- lar and parallel approaches in the collision. Crossed beam studies with good angular resolution and intricate laser state preparation find the favored alignment for reaction of an Na(4c?) orbital with HC1 when the orbital points along the direction of approach. For molecula r reagents, a high degree of alignment can be achieved by selective photodissociation with a high-powered laser of effectively all the molecules except those along one axis. In this way, the plane of rotation of IBr can be selected, and one finds that the reaction of IBr with excited Xe atoms is enhanced when the molecular plane is parallel to the rea- gent approach.8 Information about the alignment of molecu - lar orbital s involved in the transition states of chemical reac- tions is obtained by analysis of the "lambda doublet" spectroscopic components of the products. In the reaction of translationally fast H atoms with O2, for example, one lamb- da doublet component, in which the lone/7-orbital electron is left in the plane of rotation of the product OH molecule, is strongly favored.9 Weak van der Waals cluster formation is used to prepare selective geometries for reaction with subsequent laser exci- tation or photofragmentation. In this way, the axial and per- pendicular electroni c configurations of the metal atom p or- bital are excited to produce an internal reaction of the Hg- H2 van der Waals complex; the resulting product states of HgH are detected and show dramatically different dynami- cal behaviors for the two alignments.10 A complex of CO2- HBr is formed and reacted internally by photodissociation of the HBr to produce OH products.'' The OH rotational pop- ulations differ considerably in this geometry-fixed case com- pared to the reaction in the bulk, in which all impact param- eters and orientations are sampled. These novel experiments are providing conceptually new ways to understand the dynamics of reactive and energy transfer collisions and the geometrica l effects of orbital alignment on electron transformations. Stephen R. Leone, Joint Institute for Laboratory Astrophysics, University of Colorado and the National Bureau of Standards 1. R. N. Zare and Ber Bunsenges, Phys. Chem. 86, 422 (1982). 2. S. Stolte and Ber Bunsenges, Phys. Chem. 86, 413 (1982). 3. D. H. Parker, K. K. Chakravorty, and R. B. Bernstein, J. Phys. Chem. 85,466 (1981). 4. D. van den Ende, S. Stolte, J. B. Cross, G. H. Kwei, and J. J. Valentini, J. Chem. Phys. 77, 2206 (1982).5. C.I. RettnerandR. N. Zare, J. Chem. Phys. 77, 2416 (1982). 6. M. O. Hale, I. V. Hertel, and S. R. Leone , Phys. Rev. Lett. 53, 2296 (1984). 7. P. S. Weiss et al, presented at the Conference on Recent Advances in Molecular Reaction Dynamics, Aussois, France (1985). 8. M. S. de Vries, V. I. Srdanov, C. P. Hanrahan, and R. M. Martin, J. Chem. Phys. 78, 5582 (1983). 9. K. Kleinermanns and J. Wolfrum, J. Chem. Phys. 80, 1446 (1984). 10.W. H. Breckenridge, C. Jouvet, and B. Soep, J. Chem. Phys. 84, 1443 (1986). 11. G. Radhakrishnan, S. Buelow, and C. Wittig, J. Chem. Phys. 84, 727 (1986). Molecular Recognition Much of chemistry, biochemistry, and pharmacology de- pends on the specific formation of noncovalent complexes between molecules in solution. '"3 The ability to bind certain molecules and not others is essential to the activity of many chelating agents, catalysts, enzymes, antibodies, drugs, and other types of molecules . Although the physical principles responsible for selectivity in binding have been fairly well understood for some time, it has only recently become possi- ble to predict and quantitatively interpret such recognition phenomena in a systematic fashion. These recent developments reflect advances both in fun- damental theory and in the power and availability of com- puters. The specificity of molecular activity can often be ex- pressed in terms of differences in free energy, which can be calculated using perturbation methods.2^1 Consider , for ex- ample, the binding of two different ligands L and M to a receptor molecule R in some solvent. The ligand that binds with the greatest decrease in the appropriate free energy will bind most strongly to R. In principle, the free energy of bind- ing for each ligand can be computed by forcing each ligand- receptor pair to associate (or dissociate) during a molecular dynamics or Monte Carlo computer simulation. In practice, it is usuall y much easier to simulate the "transmutation" or perturbation of L to M because this involves relatively local- ized changes in the system . By reference to a thermodynamic cycle, the difference in free energies of binding is seen to be equal to the difference in free energies of "transmutation" of the uncomplexed and complexed ligands.2"10 The thermodynamic cycle-perturbation method has been used successfully to compute the relative free energies of binding different halide anions to an organic receptor,3 dif- ferent inhibitors to an enzyme,4 and a given inhibitor to ge- netically engineere d enzymes.4 The method can be applied to other types of molecular activity as well. Reaction rates can be studied by computing relative free energies of activa- tion. The simulations can also be analyzed to provide phys- ical interpretations of the computed properties. For exam- ple, the relatively weak affinity of one inhibitor for an enzyme was shown to be due to the difficulty of desolvating the inhibitor rather than to a poor fit to the enzyme.4 S 12 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1986—CHEMICAL PHYSICS In the case of diffusion-controlled processes, molecular recognition may occur as a kinetic property and may reflect long range interactions. For example, the rates of diffusion- controlled reaction between charged or polar molecules will be increased if electrostatic interactions tend to steer the dif- fusing reactants toward productive collision geometries.2 The rate constants for such diffusion-controlled reactions can be computed by analyzing reactant trajectories genera- ted by Brownian dynamics simulations.11'12 Such calcula- tions have been used to show that the reaction of superoxide anion catalyzed by the enzyme superoxide dismutase is ac- celerated by electrostatic steering of the anions toward the active sites of the enzyme.12 J. A. McCammon, University of Houston-University Park 1. See, for example, the Physics News In 1985 contribution by W. Duax, Phys. Today 39, S-25 (January 1986). 2. J. A. McCammon and S. C. Harvey, Dynamics of Proteins and Nucleic Acids (Cambridge University, London, in press). 3. T. P. Lybrand, J. A. McCammon, and G. Wipff, Proc. Natl. Acad. Sci. USA83, 833 (1986). 4. C. F. Wong and J. A. McCammon, J. Am. Chem. Soc. 108, 3830 (1986). 5. M. Born, Z. Phys. 1, 45 (1920). 6. H. L. Friedman, A Course in Statistical Mechanics (Prentice-Hall, En- glewood Cliffs, NJ, 1985). 7. J. P. M. Postma, H. J. C. Berendsen, and J. R. Haak, Faraday Symp. Chem. Soc. 17, 55 (1982). 8. W. L. Jorgensen and C. Ravimohan, J. Chem. Phys. 83, 3050 (1985). 9. C. L. Brooks, J. Phys. Chem. (in press). 10. M. Mezei and D. L. Beveridge, Ann. N.Y. Acad. Sci (in press). 11. S. H. Northrup, M. S. Curvin, S. A. Allison, and J. A. McCammon, J. Chem. Phys. 84, 2196 (1986). 12. J. A. McCam mon, S. H. Northrup, and S. A. Allison, J. Phys. Chem. (in press). Vibrational Energy Exchange in Molecular Collisions with Solid Surfaces Although vibrational energy exchange between gas-phas e molecules and solid surfaces plays an important role in such practical areas as thermal transport, plasma confinement, and corrosion, it is only in recent years that our understand- ing of the microscopic dynamics of this process has made rapid advances. New molecular beam and laser techniques have provided detaile d measurements that awe the experi- mentalist and challenge the theorist. The process of vibrational excitation has been examine d as a function of translational energy, scattering angle, and sur- face temperature in collisions of NO with Ag( 111), where the (111) denotes the planar orientation of the Ag surface.1 NO was prepared with high translational energies by super - sonic expansion, while the vibrational excitation was moni- tored by laser-induced fluorescence. The fraction of NOmolecules vibrationally excited by their collision with the surface was found to increase weakly with the incident kinet- ic energy and strongly with the surface temperature, reach- ing 7% at 120 kJ/mole and 760 K. Angular distributions for the vibrationally excited molecules were quasispecular (that is, the angles of incidence and scattering were nearl y equal). Although other investigators have observed vibrational exci- tation following trapping at the surface,2"4 this is the first report of direct excitation during a single collisional encoun- ter. Vibrational deactivation has been investigated for colli- sions of the first vibrational state of NO, denoted as NO(y= 1), with Ag(lll), Ag(110), and LiF(100) sur- faces.5'6 A pulsed infrared laser was used to excite a fraction of the incident molecular beam to a single rotational level of v = 1, and the subsequent relaxation was monitored by re- sonantly enhanced multiphoton ionization using a second, ultraviolet laser. On all three surfaces studied , the surviving NO (v = 1) was found to be scattered roughly into the specula r direction with a broad rotational distribution and with a 30% loss of the original translational energy. Survival probabilities for the vibrational excitatio n were found to be about 0.9 for all three surfaces. Survival probabilities of 0.7, 0.8, and 0.3 have also been measured for CO(u = 2), CO2(001), and CO2(101) on poly crystalline silver using a laser-induced fluorescence technique.7 The detailed results from these experiments suggest that an electronic mechanism, probably one involving electron- hole pairs8 or molecular ion formation,9 is responsible for the vibrational excitatio n and deactivation. Calculations of vi- brational relaxation at metal surfaces seem to require such a mechanism, although the agreement for insulating surfaces is adequate without the inclusion of electronic processes.10 Paul L. Houston, Cornell University 1. C. T. Rettner, F. Fabre, J. Kimman, and D. J. Auerbach, Phys. Rev. Lett. 55, 1904 (1985). 2. M. Asscher, W. L. Guthrie, T. H. Lin, and G. A. Somorjai, J. Chem. Phys. 78, 6992 (1983); Phys. Rev. Lett. 49, 76 (1982). 3. M. Asscher, G. A. Somorjai, and Y. Zeiri, J. Chem. Phys. 81, 1507 (1984). 4. D. A. Mantell, Y. F. Maa, S. B. Ryali, G. L. Haller, and J. B. Fenn, J. Chem. Phys. 78, 6338 (1983). 5. J. Misewich and M. M. T. Loy, J. Chem. Phys. 84, 1939 (1986). 6. J. Misewich, H. Zacharias, and M. M. T. Loy, Phys. Rev. Lett. 55, 1919 (1985); J. Vac. Sci. Technol. B 3, 1474 (1985). 7. J. Misewich, C. N. Plum, G. Blyholder, P. L. Houston, and R. P. Mer- rill, J. Chem. Phys. 78, 4245^249 (1983); J. Misewich, P. L. Houston, and R. P. Merrill, J. Chem. Phys. 82, 1577-1584 (1985). 8. B. N. J. Persson and M. Persson, Surf. Sci. 97, 609 (1980). 9. J. W. Gadzuk, J. Chem. Phys. 79,6341 (1983). 10. R. R. Lucches e and J. C. Tully, J. Chem. Phys. 80, 3451 (1984). PHYSICS TODAY / JANUARY 1987 S 13PHYSICS NEWS IN 1986—CHEMICAL PHYSICS High-Resolution Optical Spectra of Large Organic Molecules It has traditionally not been possible to apply the techniques of rotationally resolved spectroscopy (that is, the study of the rotational quantum states of a molecule) to electronic transitions in large molecules with energy level densities suf- ficient to permit rapid occurrence of singlet-triplet intercon- version processes, intramolecular vibrational relaxation pro- cesses, isomerizations, and other chemically interesting phenomena. In fact, there are chemists who would state that if a molecule is small enough to permit a complete rotational analysis of its spectrum, then it is probably too uninteresting chemically to merit study. However, this historical point of view is no longer as true as it once was. The ground-breaking work of Levy, Wharton, Smalley and collaborators at the University of Chicago about a decade ago,1-2 as well as numerous later studies, have shown that the three techniques of high-resolution tunable dye laser excitation, fluorescence detection, and molecular cooling in supersonic beams can be utilize d simultaneously for remarkably successful high-resolution studies on large organic molecules containing a visible chromophore. A particularly important class of problems which can be studied with this technique concerns the behavior of a single, discrete rotation-vibration energy level of a given electronic state when this level is imbedded in a higher densit y mani- fold (or even a "quasi-continuum") of rotation-vibration levels belonging to one or more lower electronic states. One example of the tremendous potential of the latest technology applied to such problems is illustrated by a study3 of rotation vibration-electronic eigenstates in the vi- cinity of the o = 0 (vibrational) level of the lB3u electronic state of the pyrazine molecule (C4H4N2). Pyrazine exhibits a complicate d fluorescence decay4 from this 'B3u state. Al- though the densit y of states of the manifold into which the excited state decays is high, it is still sufficiently spars e that only a limited number of molecular eigenstates are contained within the coherence width of the exciting source. The ob-served decay then depends on the width of the exciting source, which accounts for the greatly varying reports on decay behavior and quantum beat patterns in the literature. A second exampl e of high-resolution information on states imbedded in a continuum is the work of Riedle and Neusser5 on the homogeneous linewidths of single rotational lines in the "channel three" region of benzene (C6H6). Us- ing Doppler-free, two-photon, dye-laser spectroscopy, with instrumental linewidths of about 5 MHz, they have accu- rately measured the rotational and vibrational dependence of molecular linewidths in the 10-150 MHz range. Such data leads directly to information on the extent of participation of rotational and vibrational motions in the intramolecular re- laxation processes. Theoretical model s for intramolecular relaxation and de- cay processes have often been forced in the past to rely rather heavily on a variety of partially tested assumptions. High- quality energy-domain measurements like those described above, especially when combined with more traditional time-domain decay measurements, can be expected to lead to significant quantitative tests and dramatic qualitative im- provements in our understanding of intramolecular energy flow. As a bonus result, we may also hope for a speedy reducv tion in some of the semantic (and perhaps even scientific) confusion which arises when workers in the time domain and energy domain attempt to discus s related observations on the same molecule. Jon T. Hougen, National Bureau of Standards 1. D. H. Levy, L. Wharton, and R. E. Smalley, "Laser Spectroscopy in Supersoni c Jets," Chapter 1 in Chemical and Biochemical Application of Lasers, Vol. II, edited by C. B. Moore (Academic, New York, 1977). 2. R. E. Smalley, L. Wharton, and D. H. Levy, Ace. Chem. Res. 10, 139 (1977). 3. B. J. van der Meer, H. T. Jonkman, J. Kommandeur, W. L. Meerts, and W. A. Majewski, Chem. Phys. Lett. 92, 565 (1982). 4. G. ter Horst, D. W. Pratt, and J. Kommandeur, J. Chem. Phys. 74, 3616 (1981). 5. E. Riedle and H. J. Neusser, J. Chem. Phys. 80, 4686 (1984). CONDENSED MATTER PHYSICS Condensed matter physics is a broad field with ill-defined boundaries, as one might expect. It encompasse s properties of matter and indeed, some phenomena that can be indepen- dent of material properties. Technological advances often follow from knowledge gained in the pursuit of fundamental physics in condensed matter research. Advances in this field result from theory, experiment and the development of new tools. In addition to the articles in this chapter, two other articles describing condensed matter research, those on bal- S 14 PHYSICS TODAY / JANUARY 1987listic transport and on electronic conduction in silicon diox- ide, appear in the chapter on physics applie d to industry. The discussion by Lee of the universal conductance fluc- tuations illustrates one aspect of small samples, namely the concept of an ensemble averaging that does not lead to a correct description as it does for large systems. There is a signature of a small system that is different from that for another "identical" small system . With relation to some pa- rameter, these systems have fluctuations that are universal inPHYSICS NEWS IN 1986—CONDENSED MATTER PHYSICS their behavior, as well as in how they differ from each other. That fluctuations produce noise in dynamical systems is well known. It is not surprising, therefore, that the observa- tion that it is possible to reduce the noise in the electromag- netic field has generated a great deal of interest. Kimble and Levenson discuss "squeezed states," a phenomenon in which the granularity of the quantum nature of light is ex- plored. Although the generation of squeezed states is expect- ed to be independent of the media in which it is observed, it should modify decay rates and hence become a new tool in the study of the properties of matter. The study of nonequilibrium phenomena is as important as those of equilibrium. Convectio n is of interest not only as a fundamental physica l phenomenon but from many techno- logical points of view as well. Hohenberg describes studies of convection in fluid mixtures. A new dimension is added, as diffusion may play an important role in studies of chaos and other such phenomena. Of particular interest is the study at the poly critical point. Friedan, Shenker, and Qiu discuss a fluid and a gas in their introduction to a description of phenomena near the critical point where there is no characteristic length. Such studies also seek universa l behavior, that is, the appearance of certain critical exponents for several classes of phenome- na. The scanning tunneling microscope (STM) continues to expand, at a fantastic rate, the horizons of what may be ex- plored. The ability to discover what the three-dimensional geometry of a surface is by studies in ultra-high vacuum con- tinues to be important in surface physics. Now the energy of surface states can be determined, and the directions of the wave functions in real space as well. The ability to carry on studies in air, liquid nitrogen, and water illustrates the power of this relatively new tool. (In addition to the description here by Hansma, articles on STM can also be found in the chapters on industrial physics and vacuum physics and in the article on the 1986 Nobel prize for physics.) Finally, Narayanamurti describes how the development of additional new tools allows studies of lattice dynamics in the picosecond time domain. One of the critical parameters in lattice dynamics is the lifetime of individual photons. The use of picosecond and femtosecond laser pulses has allowed realtime studies on very short time scales for the first time. Thus, the development of techniques hitherto unavailable, extends considerably the range in which experiment and the- ory can be compared. Phillip J. Stiles, Brown University Squeezed States In the past year a great deal of excitement has been generated in the community of optical physicists by the observation of squeezed states of light in several laboratories.1"5 Althoughthe prospect of "squeezing light" may conjure up a variety of images regarding the "elasticity" of the electromagnetic field, what has in fact been observed are quantum states that directly display the intrinsic granularity associated with the quantum nature of light. Any measurement of the amplitude or phase of a light wave must have some uncertainty, if only that required by quantum mechanics. Even the vacuum state—which has zero average amplitude—shows quantum-mechanical zero- point motion around its average value. Modern theories of quantum optics attribute the uncertainty in measurements of highly coherent laser beams to interferences between the laser and these quantum-mechanical fluctuations of the vacuum. Experiments in four laboratories have now shown that the vacuum fluctuations do not pose an intrinsic limit to the precision of optical measurements. Rather, it is possible to "squeeze" the fluctuations in such a way that those that interfere with the laser beam to produce noise have magni- tudes below the vacuum level, while those that are out of phase with the laser are above it.6 Such squeezed light may ultimately be useful for gravity wave detection, optical data storage, communications, and spectros copy . The squeezing of the vacuum fluctuations is accom- plished by nonlinear optical interactions. In a nonlinear me- dium, light waves shifted equally above and below a pump frequency are coupled together in such a way that changes in the amplitude or phase of one wave cause changes in phase or amplitude of the other. Correlations are thus created even among the vacuum fluctuations. When waves that have been subject to such a nonlinear interaction are incident on a de- tector along with a coherent beam of the correct phase, the correlated fluctuations interfere destructively with one an- other, resulting in a detecte d intensity more stabl e than for a coherent beam alone. The noise in such a squeezed state of light is said to be below the standard quantum limit. If the phase of the coherent beam were change d by 90 degrees, the enhanced fluctuations required by the uncertainty principle would appear. The first experimental demonstration of this squeezing phenomenon was in late 1985 at AT&T Bell Laboratories.1 A 7% reduction in quantum noise was observed when light that had interacted with a sodium atomic beam in an optical cavity was mixed with a laser beam on a detector. When the nonlinearly generate d light was blocked, the noise increased. Later experiments on sodium increased the noise reduction to 17%, still far short of the 90% reduction necessary to be technically useful. In 1986 a second program, at IBM, reported a noise level 13% below the standard quantum limit for light that had propagated through an optical fiber cooled below 4.2 K.2 While still small, the noise reduction appeared over two wide bands of frequency rather than only at the resonant modes of an optical cavity . Both of these experiments employed non- degenerate four wave mixing as the nonlinear optical inter- PHYSICS TODAY / JANUARY 1987 S-15PHYSICS NEWS IN 1 986—CONDENSED MATTER PHYSICS action. At MIT. meanwhile, a 4^ reduction for four wave mixing in sodium vapor was obtained." A group at the University of Texas. Austin, employed a completely different approach to the problem of squeezed state generation that proved enormously successful: they used subharmonic conversion: a photon at frequency a>2 splits into two photons at frequencies near al = az/2.i A net noise reduction of greater than 50% relative to the vacu- um level was observed for infrared radiaton at a wavelength of 1.06/im when a lithium niobate crystal (also in a cavity) was pumped at 0.53 /zm. The large noise reduction resulted from the immunity of the subharmonic conversion process to noise from scattered pump light. Further efforts in para- metric down conversion are expected to result in quantum noise reductions of 909c or more. In addition to the intrinsic interest in the nonclassica l nature of squeezed states of the electromagnetic field, there are as well a number of exciting applications in measurement science and in optical communication associated with sensi- tivity beyond the standard quantum limit, which is deter- mined by the vacuum fluctuations of the field. Squeezed state technology is especially important when small forces must be measured, as in gravity wave detection. Squeezed states of light injected into an interferometric gravity wave detector would perturb the system less than coherent light, permitting repeated measurements without adding uncer- tainty. Such effects do not violate the uncertainty principle as fluctuations are being added to the system, but in a way that the extra noise does not affect the quantity being mea- sured. Harn J. Kimble, University of Texas and Marc D. Levenson, IBMAlmaden Research Center 1. R.E. Slusher. L.W. Hollberg. B. Yurke. J.C. Mertz. and J.F. Valley. Phys. Rev. Lett. 55. 2409 (1985). 2. R.M. Shelby. M.D. Levenson, S.H. Perlmutter. R.G. DeVoe, and D.F. Walls. Phys. Rev. Lett. 57. 691 (1986). 3. M.W. Meade, P. Kumar, and J.H. Shapiro, postdeadline paper. 14th International Quantum Electronics Conference (June 9-13, 1986). 4. H.J. Kimble and J.L. Hall, 14th International Quantum Electronics Conference (June 9-13. 1986); L.A. Wu, H.J. Kimble. J.L. Hall, and Huifa Wu, Phys. Rev. Lett, (submitted 1986). 5. Physics Toda y 39. 17 (March 1986). 6. A special symposium on squeezed states will be held at the 1986 meeting of the Optical Society of America in October 1986. A summary should appear in the December issue of J. Op. Soc. Am. Universal Conductance Fluctuations In condensed matter physics, conventional wisdom says that physical measurements are well represented by ensemble averages. In the past few years, this belief has been chal- lenged by work on small samples at low temperatures. Oneof the most striking discoveries was that the conductance of a metalli c sample at sufficiently low temperature exhibits fluctuations as a function of magnetic field, chemical poten- tial, or impurity configuration with a root-m ean-s quare (rms) equal to the ratio e: /h. where e is the charge of the electron and h is Planck" s constant. This effect, called uni- versal conductance fluctuation, is due to the fact that the fluctuation amplitude is independent of sample size and the numerical coefficient in front of e: /h depends weakly on dimensionality and the amount of disorder, as long as the sample is metallic and the temperature sufficiently low. The original impetus for this discovery came from experi- ment. Along their way to observing Bohm-Aharonov effect in normal metalli c rings (see Physics News in 1985, p. 19, fora review of this interesting story, which will not be repeated here). Umbach et al. found that the conductance of a wire of approximate dimension (400 A): X 7000 A exhibited aperi- odic fluctuations as a function of a perpendicular magnetic field.1 The fluctuations are not time-dependent noise, but are reproducible for a given sample upon cycling through magnetic field or temperature. The magnetoconductance curve is, in effect, a signature of a given sample. The physical origin of this effect has to do with quantum interference of the electron wave function and the fact that the magnetic field changes the phase of the wave function. Stone per- formed numerical simulation on a two-dimensional lattice of tight binding model with random site energies, and pro- duced fluctuations in the conductance very similar to the experiment." However, the key question remained: how can numerical results on a few thousand sites be extrapolate d to a wire with 10s atoms? The answer was provided by Lee and Stone, who showed analytically that the rms conductance fluctuation should be approximately e2 Ax, and this result is consistent with both the numerical simulation and the exper- iment.3 Essentially the same result was obtained indepen- dentl y by Altshuler in the Soviet Union. The requirement for universal conductance fluctuation is that the temperature must be sufficiently low so that the inelastic length—that is, the length that an electron can dif- fuse in between inelastic scattering—exceed the sample size. In this case the conductance must be understood as quantum mechanical transmission through a disordered region. In a disordered medium, the motion of an electron is diffusive and its quantum mechanical description leads to conduc- tance fluctuations which are much larger than what is ex- pected classically. If the low temperature is not met, one can simply divide the sample into regions of size given by the inelastic lengt h and add their contributions as classical se- ries-paralle l resistance fluctuations. In practice, for metallic wires several thousand angstrom in length, the low tempera- ture limit is reached at a few tens of millikelvin. Owing to the much longer deBroglie wavelength of the electron in a sili- con inversio n layer , the low temperature limit can be reached at 4.2 K in a structure of a comparable size.5 The inversion layer has the additional advantage that it is possi- S-16 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1 986—CONDENSED MATTER PHYSICS ble to observe the conductance fluctuation by tuning the chemical potential as well as the magnetic field.5'6 By mak- ing very small structures, Skocpol et al. were able to test the predictions of the universa l conductance fluctuation in great detail.5 The unexpectedly large fluctuations which are specific to a given sample (or a given impurity configuration) leads naturally to the following question: how sensitive are the conductance fluctuations to a small change in the impurity configuration, and in the extreme limit, to the motion of a single atom? The result, obtained independently by Alt- shuler and Spivak7 and by Feng, Lee, and Stone,8 is that in one and two dimensions, moving a single atom by a distance on the order of the deBroglie wavelength, the conductance changes by an amount which is independent of sample size, and equals e2 /h when the disorder is strong. In three dimen- sions the effect is somewha t weakened, but still much larger than what is expected classically. This surprising result can be understood by recognizing that the Feynman paths of an electron in a disordered medium can be thought of as ran- dom walk. In two dimensions, a random walker which crosses the sample basically visits a finite fraction of all the sites, so that the alteration of a single site affects a finite fraction of all the Feynman paths. In the strong disorder limit, changing a single impurity is then the same as chang- ing the entire configuration and leads to a conductance change of e2 /h. It should be possible to observe this effect directly by experiments, and indeed discrete changes in the magnetoconductance curves are observed in the inversio n layer and are attributed to the change of a single scattering center.5 Very recently, time-dependent jumps in the conduc- tivity have been observed in small bismuth wires and films and interpreted in terms of defect motion.9 While the initial discovery of sample specific fluctuations was made at low temperatures, it has become clear that quantum mechanical coherence plays an important role as long as the inelastic scattering length greatly exceeds the mean free path, a condition which is satisfied even at room temperature if the metal is sufficiently disordered. For ex- ample, Feng et al.s have pointed out that the sensitivity to impurity configurations provides an estimate of the magnitude of ///noise due to defect motion in highly disordered metals. At the same time, it is likely that insights gained from the studies of electron wavefunctions will deepen our understanding of the propagation of classical waves such as electromagnetic or sound waves through a random medium. P. A. Lee, Massachusetts Institute of Technology 1. C. P. Umbach, S. Washburn, R. B. Laibowitz, and R. A. Webb, Phys. Rev. B 30, 4048 (1984). 2. A. D. Stone, Phys. Rev. Lett. 54, 2692 (1985). 3- P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985).4. B. L. Altshuler, Pis'ma Zh. Eksp. Teor. Fiz. 41, 530 (1985); [JETP Lett. 41,648 (1985)]. 5. W. J. Skocpol, P. M. Mankiewich, R. E. Howard, L. D. Jackel, D. M. Tennant, and A. D. Stone, Phys. Rev. Lett. 56, 2865 (1986). 6. J. C. Licini, D. J. Bishop, M. A. Kastner, and J. Melngailis, Phys. Rev. Lett. 55,2987 (1985). 7. B. L. Altshuler and B. Z. Spivak, Pis'ma Zh. Eksp. Teor. Fiz. 42, 363 (1985); [JETP Lett. 42, 447 (1986)]. 8. S. C. Feng, P. A. Lee, and A. D. Stone, Phys. Rev. Lett. 56, 1960 (1986); Erratum 56, 2772 (1986). 9. D. E. Bentler, T. L. Meisenheimer, and N. Giordano, Purdue University (preprint). Convection in Fluid Mixtures Many of the most striking examples of nonequifibrium pattern formation and chaos have been observed and studied in Ray- leigh-Benard convection. This is the flow of a thin fluid layer contained between horizontal, thermally conducting plates held at different temperatures. This research has been extended now from the study of pure fluids to fluid mixtures; these are particularly interesting because the additional diffusion mode allows for the possibility of two different types of convective threshold, an instability to steady convection as in pure fluids, and an oscillatory instability to a time-periodic state.1 This ad- ditional degree of freedom offers the hope of observing interest- ing dynamic phenomena (such as chaos, intermittency, etc.) immediately above threshold, and thus of developing a quanti- tative "first principles" theory of chaos based only on the hy- drodynamic equation. A phenomenon of particular interest is the polycritical point where the oscillatory and steady instability thresholds coincide. Interesting dynamic behavior was predicted by Brand, Hohen- berg, and Steinberg2 in the vicinity of this point, and some, but not all, of the qualitative predictions of their theory were con- firmed in experiments by Rehberg and Ahlers3 on cryogenic 3 He-4 He mixtures in a porous medium. An important advance came from flow visualization experi- ments by Walden, Kolodner, Passner, and Surko on alcohol- water mixtures, which showed that the oscillatory state above threshold was in the form of traveling waves.4 Indeed, it was argued theoretically5'6 that the standing waves previously as- sumed2 were unstable to traveling waves in this system. Sur- prisingly, however, even traveling waves did not occur as stable oscillations immediately above threshold,413'7 but were only visi- ble as transient growing waves. The propagation of these waves and their reflection from the sidewalls of the container were studied experimentally by the Bell Labs group,4b and under- stood theoretically on the basis of a linear theory by Cross (Cal- tech) .8 When the amplitude of the waves grows beyond a cer- tain range, the system makes a large transition to a nonlinear state of slowly traveling waves which has also been studied ex- perimentally by Moses and Steinberg,7 but whose natur e is at present only imperfectly understood.6'8 Although this system has complicated behavior and has shown many features unanticipated by theory,9 it is reasonable PHYSICS TODAY / JANUARY 1987 S-17PHYSICS NEWS IN 1 986—CONDENSED MATTER PHYSICS to hope that further fruitful interaction between experimenta- lists and theorists will help us to identify rich dynamics10 and interesting pattern forming behavior.4'7 PC. Hohenberg, AT&T Bell Laboratories 1. J. K. Platten and J. C. Legros, Convection in Liquids (Springer, New York, 1984) for a review. 2. H. Brand, P. C. Hohenberg, and V. Steinberg, Phys. Rev. A 30, 2584 (1984), and references therein. 3. I. Rehberg and G. Ahlers, Phys. Rev. Lett. 55, 500 (1985). 4. (a) R. W. Walden, P. Kolodner, A. Passner, and C. M. Surko , Phys. Rev. Lett. 55, 496 (1985); (b) P. Kolodner, A. Passner, C. M. Surko, and R. W. Walden, Phys. Rev. Lett. 56, 2621 (1986). 5. C. S. Bretherten and E. A. Spiegel, Phys. Lett. 96A, 152 (1983); P. Coullet, S. Fauve, and E. Tirapegui, J. Physique Lett. (Pans) 46, L-787 (1985). 6. E. Knobloch, A. Deane, J. Toomre, and D. R. Moore, "Multipara- meter Bifurcation Theory" to be published in Contemp. Math. 56 (1986). 7. E. Moses and V. Steinberg, Phys. Rev. A, preprint (to be published). 8. M. C. Cross (preprints). 9. G. Ahlers and I. Rehberg, Phys. Rev. Lett. 56, 1373 (1986); N. Gao and R. P. Behnnger, Phys. Rev. A 34, 697 (1986). 10. B. J. A. Zielinkska, D. Mukamel, V. Steinberg, and S. Fishman (pre- print). Lattice Dynamics in the Picosecond Time Domain A direct determination of the lifetimes of both acoustic and optic phonons as a function of frequency and temperature is of fundamental interest to a variety of problems in solid state phys- ics. The determination of acoustic phonon lifetimes, for exam- ple, is crucial to understanding the thermal transport properties of insulators, semiconductors, and amorphous materials. The kinetics of the decay of optic phonons, and their coupling to other excitations in the condensed phase, is of importance for such diverse problems as hot electron transport in semiconduc- tors and the generation of nonequilibrium high frequency acoustic phonons in insulators. The lifetimes of acoustic phonon excitations in short mean-free-path materials or at ele- vated temperatures are often in the picosecond time regime. Optic phonons have typical frequencies (5-10THz) which cor- respond to the femtosecond time domain. Using picosecond and femtosecond laser pulses, several researchers have made consid- erable progress in the last two years in studying (in real time) phonon excitations at very short time scales. In this article I shall discuss this recent progress. Acoustic Phonons. Over the last fifteen years, much work has been done using heat pulse techniques to study acoustic phonon propagation in solids at liquid helium temperatures.1 At low temperatures and moderately high frequencies (hundreds of gigahertz) acoustic phonon lifetimes in dielectric solids and semiconductors is often in the 10"7 to 10 ~ h sec range, and time-of-flight techniques can be readily applied using conven-tional electronics. As noted above, at higher temperatures one has to be prepared to study phonons with very short lifetimes and short mean free paths. In one experiment2 a "pump" light pulse (duration 0.2 ps) is focused on a suitable absorbing trans- ducer film (typically 100 A thick). The heating of this layer sets up a thermal stress which in turn results in the generation of a strain wave (phonon pulse) which propagates into the sample film behind. The transducer film also serves as a detector through measurements of the change in reflectivity due to the acoustic strain using a "probe" pulse whose time delay can be varied with respect to the pump. At phonon energies of about 2 eV, Mans et al. have found strong detector responses in a-As2Te3 InSb, and Al, among several materials. They have performed measurements at fre- quencies as high as 400 GHz and have studied the attenuation in o-SiO2 films of thickness 600 A at room temperature. The me- thod shows considerable promise to measure acoustic phonon attenuation over a wide frequency and temperature range and should help in elucidating, for example, the natur e of heat trans- port in amorphous materials, particularly in the region of the thermal conductivity "plateau." ' Optic Phonons. Even though linear spectroscopy in the far infrared and nonlinear optical and Raman spectroscopy have provided a wealth of information on optical phonon decay pro- cesses, until recently the time resolution was insufficient to mea- sure the individual oscillations of an optical phonon. In some beautiful recent experiments, Auston et al. have shown that the propagation of femtosecond optical pulses in electro-optic ma- terials is observed to produce a Cherenkov cone of pulsed far IR radiation in the terahert z spectral range.3'4 The above technique has led to the first4 coherent excitation and detection of traverse optic (TO) phonons in LiTaO,. Two femtosecond pulses were used: one to generate the radiation field and the other to detect it. The distance between generator and detector was as small as 7 ^m. This was essential to mini- mize the influence of the strong frequency-dependent absorp- tion near the lattice resonance. The damping time (380 fs) of the lowest TO phonon at 6 THz has been measured directly through the decay rate of the oscillations. The method should be applicable to all noncentrosymmetric crystals. Through the use of more intense optical pulses, the effects of nonlinear processes may possibly be observed directly for the first time. V. Narayanamurti, AT&T Bell Laboratories 1. See, e.g., Proceedings of Fifth International Conferen ce on Phonon Scat- tering in Condensed Matter, Urbana, Illinois, 1986 (Springer-Verlag, to be published). 2. H. J. Maris, C. Thomsen, and J. Tauc in Ref. 1: see also C. Thomsen, J. Strait, Z. Vardeny, H. J. Maris, J. Tauc, and J. J. Hauser, Phys. Rev. Lett. 53,989 (1984). 3. D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, Phys. Rev. Lett. 53, 1555 (1984). 4. K. P. Cheung and D. H. Auston, Phys. Rev. Lett. 55, 2152 (1985). S-18 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1986—CONDENSED MATTER PHYSICS Scanning Tunneling Microscopy There have been many exciting new developments in the study of scanning tunneling microscopy1 since it was first described in Physics News three years ago {Physics News in 1983, p. 16). Within the past 18 months atomic resolution images have been published for surfaces in air,2 in liquid nitrogen,3'4 and even in water.5 The ability to operate in new environments makes pos- sible opportunities for research on technologically importan t processes. For example, Sonnenfeld and Schardt6 were able to obtain atomic resolution images of a graphite surface immersed in a 0.05 M HgC104 solution before they electroplated the surface with silver, between islands of plated silver, and after stripping the silver from the surface. Their images of the plated surface were consistent with an island film growth mechanism. Their pioneering study demonstrates that atomic-scale studies of elec- trochemical processes occurring in electrodes in solution are now possible without ever removing the elctrode from the solu- tion. At an international conference in Spain7 last summer, other exciting work included a report of molecular vibration spectra obtained in liquid helium, images of a DNA-protein complex that clearly showed the protein spiraling around the DNA and many improved designs for scanning tunneling microscopes. Thus the future looks bright for scanning tunneling micros- copy.8 We can expect not only topographic, but also spatially resolved spectroscopic images for a variety of environments. These images may contribute to our understanding of techno- logically important devices and processes. Paul Hansma, University of California at Santa Barbara 1. G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel Phys. Rev. Lett. 49, 57 (1982); G. Binnig and H. Rohrer, Sci. Am. 253, 50 (August 1985). 2. S.-I. Park and C. F. Quate, Appl. Phys. Lett. 48, 112 (1982). 3. R. V. Coleman, B. Drake, P. K. Hansma, and G. Slough, Phys. Rev. Lett. 55, 394 (1985). 4. C. G. Slough, W. W. McNariy, R. V. Coleman, B. Drake, and P. K. Hansma, Phys. Rev. B 34, 994(1986). 5. R. Sonnenfeld and P.K. Hansma, Science 232, 211 (1986). 6. R. Sonnenfeld and B. Schardt, Appl. Phys. Lett, (to be published). 7. STM '86, organized by N. Garcia and R. Jaklevic, held in Santiago de Compostella, Spain, July 14-18, 1986. Proceedings scheduled to be pub- lished by Surface Science in March 1987. 8. C. Quate, Physics Toda y 39, 26 (August 1986). Conformal Invariance and Critical Exponents in Two Dimensions As water is heated through its boiling point it suddenly changes from liquid to gas. When the pressure is raised, this phase tran- sition becomes less and less abrupt, until at a certain critical pressure it becomes continuous. At this critical point the den- sity fluctuations in the fluid have a remarkable property. Under normal conditions the fluctuations are coherent only over lengths on the order of the space between water molecules, but at the critical point fluctuations occur at all length scales. The system has no characteristic length, so it is invariant underchange of scale. Because of these large fluctuations, thermody- namic quantities (e.g., the specific heat) develop power-law sin- gularities at critical points. The numerical powers are called critical exponents. Scale invariant critical points occur in a wide range of mate- rials, but physical systems which are quite different under nor- mal conditions have exactly the same critical exponents. This is the remarkable phenomenon of universality. Only the most ba- sic properties, like spatial dimensionality, determine the univer- sality class, making the problem of determining the possible universality classes a fundamental one. The renormalization group has provided a satisfying conceptual framework for un- derstanding the phenomenon of universality, but it provides no effective general procedure for determining the possible univer- sality classes. Most physical systems have the important property of local- ity—the interactions between degrees of freedom are short ranged. Local systems at their critical points should respond in a simple way to local scale transformations, called conformal transformations.2 Conformal transformations preserve angles but not scale lengths at different points. Two dimensions is an especially promising place to study conformal invariance be- cause the set of conformal transformations in two dimensions is enormous. Any analytic mapping of the complex plane is con- formal. This very large symmetry group is a powerful tool to study critical phenomena in two dimensions.3 In a certain do- main it allows us to determine all possible values of critical exponents.4 The algebra-of two-dimensional conformal transformations is called the "Virasoro algebra" and was first studied in the rather different context of string theory.5 This is a branch of high energy physics (see the article on strings in the chapter on elementary particle physics) originally developed to explain the behavior of the strong interactions and currently showing great promise as a unified, self-consistent description of all forces. Two-dimensional scale invariant systems are representa- tions of the Virasoro algebra. The mathematical properties of these representations are crucial to understanding the allowed forms of critical phenomena. The representations which can occur in the kinds of systems most often encountered—genuine thermal phase transitions with spatial isotropy—have the mathematical property of unitarity. As in quantum mechanics, unitarity means that the states of the representation cannot have a negative metric. Unitarity puts severe constraints on the possible values of critical exponents.4 The constraints are rath- er similar to those that arise from rotational symmetry in quan- tum mechanics. There the z component of angular momentum is constrained to have integer or half-integer values. In the case of conformal symmetry, the critical exponents are constrained to have certain rational values. This explains the occurrence of rational critical exponents in many known two-dimensional systems—the most famous case being the Ising model. To be more precise, the realization of conformal symmetry is described by a parameter c which measures the response of the system to distortion of the substrate6 and describes the leading finite size correction to scaling.' For c less than 1 only a certain infinite set of rational values of c is allowed, the discrete series. PHYSICS TODAY / JANUARY 1987 S 19PHYSICS NEWS IN 1 986—CONDENSED MATTER PHYSICS At each of these values a finite set of rational critical exponents are possible.45 For c greater than or equal to 1, all positive critical exponents are permitted. This still leaves unanswered the question of which combina- tion of these representations can occur in a critical system. The first constraint is that consistent correlation functions must ex- ist.' In the discrete series, correlation functions are miraculous- ly simple—they obey linear differential equations—so this con- dition can be checked.3" A second constraint is that a consistent partition function must exist. This condition has been effectively implemented for systems in the discrete series on a torus.10 Recently, an exhaustive list of solutions to this constraint has been conjectured.'' A complete classification of the university classes for c less than 1 is close at hand. Physical realizations of the discrete series are known.12 These include familiar systems such as the Ising model and cer- tain other generic multi-critical systems. Recently, models real- izing the additional members of the conjectured exhaustive list have been constructed.13 Conformal symmetry can be extended to include supersym- metry, a symmetry relating fermions and bosons. The two-di- mensional version of this, first studied in string theory, was the original example of supersymmetry. The possible physical rep- resentations of supersymmetric conformal algebras have been classified, with results similar to the ordinary conformal case.4-14 A laboratory system has been identified that displays this exotic structure414—helium adsorbed on krypton-plated graphite, a realization of the Ising model with annealed vacan- cies.15 This is the first realization of a supersymmetric field theory in nature. Recently an even more comprehensive and abstract descrip- tion of two-dimensional critical phenomena has been given, as analytic geometry on the space of all surfaces of arbitrary topo- logy.16 This will perhaps provide the framework for a complete classification of all two-dimensional critical phenomena. It also provides the basis for a new approach to string theory.17 Daniel Friedan and Stephen Shenker, University of Chicago and Zongan Qiu, Institute for Advanced Study1. For re\iews of all aspect s of phase transitions see Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green, vols. 1-6; C. Domb and J. Lebowitz, vols. 7-9 (Academic. New York, 1973-1977). 2. A. M. Polyakov, Pis'ma ZhETP 12, 538 (1970) [JETP Lett. 12, 381 (1970]; ZhETP 66, 23 (1974) [JETP Lett. 39, 10(1974)]. 3. A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, J. Stat. Phys. 34 (1984) 763; Nucl. Phys. B 241, 333 (1984). 4. D. Friedan, Z. Qiu, and S. H. Shenker. Vertex Operators in Mathemat- ics and Physics, Proceedings of a Conference .Xo. 10-17, 1983, edited by J. Lepowsky, S. Mandelstam, and I. M. Singer (Springer-Verlag, New York); Phys. Rev. Lett. 52. 1575 (1984). 5. For a review of early work in this field, see Dual Theory, edited by M. Jacob (North-Holland, Amsterdam. 1974). Important for develop- ments in statistical mechanics was A. M. Polyakov, Phys. Lett. 103B, 207, 211 (1981). For more recent developments see Proceedings of the Workshop on Unified String Theories, Institute for Theoretical Physics, Santa Barbara, July 29-August 16, 1985, edited by M. Green and D. Gross (World Scientific, 1986). 6. D. Friedan, 1982 Les Houches Summer School, Les Houches, Session 39—Recent Advances in Field Theory and Statistical Mechanics, edited by J.-B. Zuber and R. Stora (North-Holland, Amsterdam, 1984). 7. H. W. J. Blote, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986); I. Affleck, Phys. Rev. Lett. 56, 746 (1986). 8. That all the representations in the discrete series are, in fact, unitary has been proven by P. Goddard, A. Kent, and D. Olive, Commun. Math. Phys. 103, 105 (1986). 9. V. S. Dotsenko, J. Stat. Phys. 34, 781 (1984); Nucl. Phys. B 241. 54 (1984); V. S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240, 312 (1984); B. L. Feigin and D. B. Fuchs, Moscow preprint (1983). 10. J. L. Cardy. Nucl. Phys. B 270. 186 (1986). 11. A. Cappelli, C. Itzykson, abd J.-B. Zuber (submitted to Nucl. Phys. B). 12. G. F. Andrews, R. J. Baxter, and P. J. Forrester, J. Stat. Phys. 35, 193 (1984); D. A. Huse. Phys. Rev. B 30, 3908 (1984). 13. V. Pasquier, Saclay preprint PhT 86/124. 14. D. Friedan. Z. Qiu, and S.H. Shenker , Phys. Lett. 151B, 37 (1985). 15. M. J. Tejwani, O. Ferreira, and O. E. Vilches, Phys. Rev. Lett. 44, 152 (1980); W. Kinzel, M. Schick, and A. N. Berker in Ordering in Two Dimensions, edited by S. K. Sinha (Elsevier. North-Holland. Amster- dam, 1980). 16. D. Friedan and S. H. Shenker , "The Analytic Geometry of Conformal Field Theory'," Chicago preprint EFI 86-18A. Nucl. Phys. B (to ap- pear). 17. D. Friedan and S. H. Shenker , Phys. Lett. 175B, 287 (1986). S-20 PHYSICS TODAY / JANUARY 1987CRYSTALLOGRAPHY Small-Angle Scattering The scattering of x rays or neutrons at small angles has been used to address several key questions in the areas of materi- als and polymer science. Availability of intense sources, such as either synchrotron radiation facilities or rotating anode sources, has advanced the use of time-resolvin g scattering techniques to probe kinetic processes in these disciplines, providing key pieces of information on several outstanding questions. The enhanced flux from storage ring facilities has permitted substantial increases in the resolution limits of small-angle x-ray scattering, producing a bridge between light scattering and x-ray scattering methods. This has been of tantamount importance in the area of fractal structures. In addition, the brilliance of the beam from synchrotron sources has made possible the simultaneous measurement of x-ray scattering with other characterization techniques, such as, for example, differential scanning calorimetry. The combination of these methods yields information on phase transitions that was hitherto impossible to obtain. The fol- lowing articles examine recent progress in small-angle scat- tering research. William Duax, Medical Foundation of Buffalo Fractals and Small-Angle Scattering Although disordered materials are ubiquitous in nature, the characterization and modeling of such structures have elud- ed physicists for decades. The recent application of fractal geometry to the interpretation of scattering data, however , has substantially change d this situation. Fractal geometry is important in disordered systems be- cause most simple models of random growth lead to self- similar, or fractal, objects. Such objects look the same under different degrees of magnification and show power-la w spa- tial correlations. Typically, one measures the fractal dimen- sions D from the power-law slope of scattering profiles. Re- cent scattering experiments showed fractal behavior in branched polymers,1 colloidal aggregates,2"4 porous materi- als5"8 and rough colloidal particles.9 Fractal structures are typically observed on length scales exceeding 10 A, so small-angle x-ray and neutron scattering, as well as light scattering, are the techniques of choice for characterizing fractals. In fact, the desire to study fractal materials prompted an Exxon group4 to develop a synchro- tron-based instrument capable of resolving structures between 10 and 10 000 A. Substantial progress was made in 1986 in the understand- ing of the origin of random structures in terms of chemicaland physical growth processes. The polymerization of silica is a prime exampl e where, depending on catalytic conditions, either branched polymers, smooth colloidal particles, or rough colloidal particles can be prepared.''9 Based on simple rules, silica polymerization can be mapped on to computer- simulated fractal growth models.9'10 Catalytic conditions control growth by determining whether growth occurs by monomer addition or cluster-cluster growth. For suspended colloidal aggregates , the essential factors which control growth are less clear. Several groups find structures consistent with reaction-limited cluster-cluster growth.2'4 In the case of diffusion-limited aggregation , how- ever, the extreme rapidity of aggregation coupled with res- tructuring, !' precludes definitive experiments. Fractal aggregates can also be grown in the gas phase— fused silica12 and carbon black13 being two commercially important examples. Studies of commercial powders gener- ally reveal D 's which fall between the diffusion-limited and the reaction-limited regimes. Recent studies on the in-situ growth in flames, on the other hand, show very small D 's, suggesting that growth is modified by electrostatic interac- tions.14 Fractal structures have been found in numerous porous materials, both natural5'7'8 and synthetic.6 In the case of the synthetic porous silicates, the catalytic factors described above for solutio n growth, ultimately determine the struc- ture of the dry porous material.6'9 For natural materials, however, the origin of fractal porosity is not yet understood. Fractal geometry is not a necessary concept to interpret all power-law scattering curves. The Debye function, for ex- ample, represents a complete structure factor for an ideal linear polymer. The fact that such polymers are fractal with D = 2 represents no new information. The real value of fractal concepts is the ability to charac- terize complex structures and thereby clarify the essential chemistry and physics which control structure. Physic s of- ten advances on simple unifying concepts. The burst of activ- ity in 1986 on the structure of disordered materials suggests that fractal geometry falls in this category. Dale W. Schaefer, Sandia National Laboratory 1. D. W. Schaefer and K. D. Keefer, Phys. Rev. Lett. 53, 1383 (1984). 2. D. W. Schaefer, J. E. Martin, P. Wiltzius, and D.S. Carnell, Phys. Rev. Lett. 52, 2371 (1984). 3. J. K. Kjems, T. Freltoft, D. Richter, and S. K. Sinha, Physica 136B, 285 (1986). 4. P. Dimon et al., Phys. Rev. Lett, (in press). 5. H. D. Dale and P. W. Schmidt, Phys. Rev. Lett. 53, 596 (1984). 6. D. W. Schaefer and K. D. Keefer, Phys. Rev. Lett. 56, 2199 (1986). PHYSICS TODAY / JANUARY 1987 S 21PHYSICS NEWS IN 1 986—CRYSTALLOGRAPHY 7. P. Z. Wong, J. Howard, and J. S. Lin (to be published). 8. D. F. R. Mildner and P. L. Hall, J. Phys. D (in press). 9. K. D. Keefer and D. W. Schaefer, Phys. Rev. Lett. 56, 2376 (1986). 10. D. W. Schaefer and K. D. Keefer, in Better Ceramics through Chemis- try II, Mat. Res. Soc. Proc. edited by C. J. Brinker. D. R. Ulrich, and D. E. Clark (Elsevier. North-Holland, New York. 1986). 11. C. Aubert and D. S. Cannell . Phys. Rev. Lett. 56, 738 (1986). 12. J. E. Martin. D. W. Schaefer, and A. J. Hurd. Phys. Rev. A 33, 3540 (1986). 13. D. W. Schaefer. J. E. Martin. A. J. Hurd. and K. D. Keefer. in Physics of Finely Divided Matter, edited by M. Boccara and M. Daoud (Springer- Verlag, New York, 1985) , p.31. 14. A. J. and W. L. Flower (to be published). Time-Resolved X-Ray Scattering The measurement of real-time x-ray scattering has been used in the past to study deformation,1 crystal annealing,2 and phase separation3 in high molecular weight polymers. Re- cently , Keller and co-workers,4'5 using synchrotron radi- ation, performed an important set of measurements resolv- ing a discrepanc y between solution and bulk crystallization processes in polymers. Theoretical arguments predict that for both bulk and solution crystallization the primary crystal thickness should vary inversely with the difference between the melting and crystallization temperatures. This had been found to be true in solution crystallization but not in the bulk. Via a combination of Raman and time-resolved x-ray scattering measurements, these workers found that the pri- mary crystal thickness was nearl y a factor of two smalle r than that measured by static scattering methods, and varied in a manner similar to that of solution crystallized material, thereby resolving this outstanding discrepancy. The discrep- ancy between the time-resolved and static measurements was due to a rapid crystal thickening. The ability to measure the scattering at the very early stages of crystallization was made possible by the enhanced flux of the storage ring facili- ties. The combination of time-resolved, small-angle x-ray scattering (TRSAXS) measurements with differential scan- ning calorimetry (DSC) has also been developed recently.6 While the full potential of these combined measurements has not been realized, it is clearly evident that an understanding of complex thermal behavior in materials can be attained. Of particular note is a recen t study on the multiple endothermic response (a reaction in which heat is absorbed) exhibited by polyurethane block copolymers.7 Typically , these materials exhibit three endotherms when they are heated from am- bient temperatures to temperatures above the melting point. Combination of TRSAXS and DSC has clearly shown that these materials undergo a melting of shorter length crystal- lites followed by a mixing with the amorphous soft segment phase. At somewha t higher temperatures, a melting of the larger crystallites is observed. It is essentially impossible to unravel such complex thermal behavior without the use of S 22 PHYSICS TODAY / JANUARY 1987these combined techniques. Further advances are also in progress to couple these techniques with wide-angle diffrac- tion methods, thus providing structural information for a range of size scales from several to thousands of angstroms, along with the corresponding calorimetric data.8 Thomas P. Russell, IBMAlmaden Research Center 1. W. W. Wu, H. G. Zachmann, and C. Riekel , Polym. Commun. 25, 76 (1984). 2. D. T. Grubb, J. J. Liu, M. Caffrey, and D. H. Bilderback, J. Polym. Sci.; Polym. Phys. 22, 367 (1984). 3. T. P. Russell, G. Hadaiioannou, and W. Warburton, Macromolecules 18, 78 (1985). 4. P. J. Barham, R. A. Olivers, A. Keller, J. Martinez-Salazar, and S. J. Organ, J. Mat. Sci. 20, 1625 (1985). 5. J. Martinez-Salazar, P. J. Barham, and A. Keller, J. Mat. Sci. 20, 1616 (1985). 6. T. P. Russell and J. T. Koberstein, J. Polym. Sci.. Phys. Ed. 23, 1109 (1985). 7. J. T. Koberstein and T. P. Russell, Macromolecules 19, 714 (1986). 8. B. Fuller, H. R. Brown, and T. P. Russell (to be published). High-Resolution Small-Angle X-Ray Scattering A high-resolution small-angle x-ray scattering (SAXS) in- strument, designed to exploit the characteristics of synchro- tron radiation, has recently been developed by a group at the Exxon Research and Engineering Company and installed at the Stanford Synchrotron Radiation Laboratory. This in- strument extends the SAXS technique into areas and appli- cations hitherto accessible to conventional x-ray sources only with great difficulty. These research areas include: (a) measurements of SAXS diffraction patterns over a wave vec- tor (the invers e of the wavelength) range from 3 X 10 " 4 to 0.1 A"1; (b) resonant SAXS studies, that is, SAXS mea- surements with the incident radiation tuned alternately close to, and then away from an absorption edge of one atomic component in the system under study, thus providing an additional means of contrast variation; and (c) real-time SAXS measurements to study the relaxation and growth processes is subjected to strong initial perturbation. Charles Glinka, National Bureau of Standards Colloidal Dispersion Structures Analyzed by Small-Angle Neutron Scattering Colloidal particles, which are denned as having at least one spatial dimension between 1 and 1000 nm, exhibit a variety of fascinating phenomena when dispersed in aqueous or oil- based solvents. Typical examples are polymer-based thixo- tropic paints, which tend to spread when sheared by a brush and then immediately re-thicken to prevent dripping, or oilsPHYSICS NEWS IN 1986—CRYSTALLOGRAPHY which have been modified to become magnetic, viscoelastic, or micro-emulsified. A multitude of industrial processes rely upon at least a phenomenological understanding of such be- havior, but the complexity of many colloidal dispersions has until recently inhibited the development of a more detaile d description at the fundamental level. Dispersion structures may be classified in three broad ca- tegories: isotropic dispersions of particles which interact iso- tropically (e.g., spherical polymer lattices), isotropic disper- sions of particles which interact anisotropically (e.g., magnetic colloids or ferrofluids), and anisotropic structures which may be formed from particles which interact through either isotropic or anisotropic potentials (e.g., colloidal crystals). Understanding the way in which particles relate to each other at the microscopic level in such systems is compli - cated by the fact that many of the most interesting particles, especially bio-colloids, self-assemble from molecules in the dispersion and have no separate existence; we must simulta- neously and unambiguously determine the structures of both the dispersion and the particles from which it is formed. Small-angle neutron scattering (SANS) has proved to be an ideal tool for such studies for several reasons: light-atom structures may be studied in light-atom solvent s with good contrast, selection rules for magnetic scattering allow mod- el-free separation of magnetic contributions, and isotopic substitution permits wide variation of the refractive index of sub-assemblies to allow specific sites to be studied. In the first class of studies, on strongly interacting charged spheri- cal colloids, a major breakthrough was achieved by appro- priate use of modern liquid theory to develop a quantitative description of the scattering in fully analytic form.' Such studies are now routinely used, for example, to measure the size and charge of small colloids in situ in a concentrated dispersion.2 Ferrofluids provided a second class of colloids,a major practical as well as theoretical interest, in which a quantitative analysis of SANS studies was made possible by advances in magnetic liquid theory.3'4 The most recent breakthrough has been in the field of dispersions of anisotropic colloidal particles, such as, for ex- ample, the rodlike micelles which form in many amphiphilic molecular solutions. In these systems, analytic liquid theory is rendered intractable by its ability to predict the probability of the relative orientation of two such particles at a given distance. In this case, a new experimental technique has been developed in which all particles are given the same orienta- tion by subjecting the dispersion to a viscous shear flow, thus side-stepping the theoretical difficulty completely.5'6 A further advantage of this method is that the SANS scattering patterns show directly the symmetry of the colloidal parti- cle. As a result of these and other developments, SANS has become routinely used in colloidal studies in recen t years. An interesting spin-off is that many colloidal systems are now sufficiently well understood to be used as controllable model systems for the study of other physical processes, such as shear melting.7 John B. Hayter, Oak Ridge National Laboratory 1. J. B. Hayter and J. Penfold, J. Chem. Soc. Faraday Trans. 177, 1851 (1981). 2. J. B. Hayter, Faraday Discuss. Chem. Soc. 76, 7 (1983). 3. J. B. Hayter and R. Pynn, Phys. Rev. Lett. 49, 1103 (1982). 4. R. Pynn, J. B. Hayter, and S. W. Charles, Phys. Rev. Lett. 51, 710 (1983). 5. J. B. Hayter and J. Penfold, J. Phys. Chem. 88, 4589 (1984). 6. J. B. Hayter, Physica 136B, 269 (1986). 7. B. J. Ackerson, J. B. Hayter, N. A. Clark, and L. Cotter, J. Chem. Phys. 84,2344 (1986). ELECTRON AND ATOMIC PHYSICS Quantum Jumps in a Single Atom Quantum mechanics seeks to describe the dynamical process of spontaneous emission, whereby an excited atom emits a photon during de-excitation. Dramatic quantum effects, which are normally washed out in the emission from a col- lection of many atomic emitters, can be seen in the fluores- cence from a single atom. The sudden switching on and off of the fluorescence as the atom makes "quantum jumps" is a recently observed exampl e of such an effect. In 1985, Cook and Kimble1 realized that three atomic energy levels in the "V" configuration (two excited levels which decay to the same ground state), where one excited level decays in a short time and the other remains excited fora much longer time, is a system ideally suited for studying the photon emission and absorption process of atoms. Ori- ginally, this system was discussed by Dehmelt2 as a photon amplification scheme for the detection of a very weak optical transition. If a laser is used to couple the ground state to the "strong" (quickly decaying) upper level, the fluorescence from the strong level appears as a steady intensity when the averaging time is sufficiently long. Ordinarily, when observ- ing the fluorescence from a collection of many atoms, the effect of a second laser, which couples the ground state to the "weak" (slow decaying) upper level, is to reduce the average fluorescence intensity, since, on the average, the population of the strong level is diminished. However, the question of how this reduction in average intensity is manifested for a PHYSICS TODAY / JANUARY 1987 S 23PHYSICS NEWS IN 1 986—ELECTRON AND ATOMIC PHYSICS single atom has two possible answers: (. 1) it may diminish to a steady but reduced level as occurs with a collectio n of many atoms, or (2) it may switch abruptly from "on" to "off" if the atom "quantum jumps" between the ground state and the weakly coupled level. Many authors have examined the general three level sys- tem theoretically in the past year and have calculated the statistical properties of the emitted radiation.3 They con- clude that the quantum jump picture is the correct one. A recen t experiment has unambiguously shown that the flu- orescence intensity (from the strong transition) from a three-level "V" system is, indeed, two-valued rather than continuous when measured with a time resolution shorter than the lifetime of the weakly coupled level.4 In this experi- ment, a single positively charged mercury ion was confined to less than one micron in an electromagnetic ion trap, and simultaneousl y irradiated by two lasers. .Similar experi- ments, employing a single trapped barium ion, were per- formed by two other groups.5'6 A fourth group observed the fluorescence emitte d by single barium atoms in a very weak atomic beam, and by measuring correlations in the time in- terval between detecte d photons, were able to infer that quantum jump picture is correct.7 These experiments pro- vide insight into the statistical nature of the atomic emission and absorption process . Randall G. Hulet, National Bureau of Standards 1. R. J. Cook and H. J. Kimble. Phys. Rev. Lett. 54. 1023 (1985). 2. H. Dehmelt, Bull. Am. Phys. Soc. 20, 60 (1975). 3. T. Erber and S. Putterman, Nature 318.41 (1985); J. Javanainen, Phys. Rev. A 33, 2121 (1986); A. Schenzle, R. G. DeVoe, and R. G. Brewer, Phys. Rev. A 33, 2127 (1986); C. Cohen-Tannoudji and J. Dalibard, Europhys. Lett. 1.441 (1986); D. T. Pegg, R. Loudon, and P. L. Knight. Phys. Rev. A 33,4095 (1986); A. Schenzle and R. G. Brewer, Phys. Rev. A34, 3127 (1986); H. J. Kimble, R. J. Cook, and A. L. Wells. Phys. Rev. A34, 3190 (1986); P. Zoller, M. Marte, and D. F. Walls (to be published in Phys. Rev. A). 4. J. C. Bergquist, R. G. Hulet. W. M. Itano. and D. J. Wineland, Phys. Rev. Lett. 57, 1699 (1986). 5. W. Nagourney, J. Sandberg, and H. Dehmelt, Phys. Rev. Lett. 58, 2797 (1986). 6. T. Sauter , W. Neuhauser, R. Blatt, and P. E. Toschek (to be published). 7. M. A. Finn, G. W. Greenlees, and D. A. Lewis (to be published in Opt. Commun.). Parity Nonconservation in Atoms One of the great successes of theoretical physics has been the electroweak theory, which describes the weak and electro- magnetic forces in a single unified framework. One predic- tion of this theory is that a small force exists between the electron s and nucleons in atom s which does not conserve the spatial symmetry known as parity. If parity is not conserved, it means that the atom has a "handedness," and therefor e appears different from its mirror reflection. The measure- ment of the amount of this "handedness" or parity noncon- S-24 PHYSICS TODAY / JANUARY 1987servation(PNC) provides a unique test of the electroweak theory. Such measurements are extraordinarily difficult, however, because this "handedness" involves a distortion of the atomic shape which is only about one part in 10" . In spite of the many difficulties, PNC has now been ob- served in several different atoms, and the precision of these measurements has steadily improved over the past decade.1 Recently, PNC in an atom was measured with an accuracy better than 10%.2 In this experiment a tunable laser was used to excite a transition between two selected states of ce- sium atoms. Perpendicular electric and magnetic fields were applied to the excitation region. These fields, in combination with the polarization of the laser light, defined a coordinate system or handedness for the excitation process which was thereby sensitive to the handedness of the atom. A small but measurable change in the excitation rate for the atom oc- curred when the handedness of the excitation was reversed. A reversa l was accomplished by changing the sign of either field or the laser polarization. This change in the excitation rate is directly related to the amount of PNC in the atom. To relate the experimental measurements to the funda- mental electron-nucleon force, one must calculate the struc- ture of the atom, particularly in the vicinit y of the nucleus. This is a challenging problem in theoretical atomic physics, which several groups around the world are now attacking with a variety of new techniques. Improved calculations for a number of atoms are being carried out, and results ob- tained to date for cesium have an estimated uncertainty of a few percent. Combining the results of atomic theory and experiment, one obtains a value for the PNC interaction in cesium which agrees with the predictions of the electroweak theory to within the experimental uncertainty. Atomic PNC measure- ments complement high energy accelerator tests of this the- ory; they involve lower energies and are sensitive to different electron-quark couplings. Thus atomic PNC measurements are now providing important constraints on possible alterna- tives to the standard electroweak theory, such as superstring theories. Work is under way at a number of laboratories which should lead to substantially more precise measurements of atomic PNC in the near future . These small scale experi- ments will provide new information on the fundamental forces of nature. Carl E. Wieman, University of Colorado 1. M. A. Bouchiat and L. Pottier. Sci. Am. 250. 100 (June 1984). 2. S. L. Gilbert. M. C. Noecker. R. N. Watts, and C. E. Wieman, Phys. Rev. Lett. 55, 2680 (1985). Uranium Lamb Shift Quantum electrodynamics (QED), the quantum theory of the interaction of charged particles and light, is the funda-PHYSICS NEWS IN 1986—ELECTRON AND ATOMIC PHYSICS mental theory of the atom and the basis for theories which explain the interaction of quarks (quantum chromodyna- mics) and also the weak nuclear force. QED accurately de- scribes the hydrogen atom—a single electron bound to a sin- gle proton . But to be a complete theory, it must also describe the uranium atom where the electrons are bound to the much stronger electric field of a charge-92 nucleus. It is diffi- cult to test QED in neutral uranium because the QED effects are swamped by the ordinary interaction of any one electron with the other 91 electrons. Stripping off most of the electrons can make it easier to interpret the experiments, and an experiment has now been performed using uranium ions from which all but two of the normal 92 electron s were removed.1 The experiment mea- sured the average time for a spontaneous transition between two atomic levels. This time is very sensitive to the energy difference between these levels, and so to any QED contribu- tion to the energy difference. The shift in the energy levels of an atom due to quantum electrodynamic effects is referred to as the Lamb shift, for Willis Lamb Jr., who first measured the shift in hydrogen. For the energy levels chosen in two- electron uranium, calculations predict that QED effects con- tribute 20% of the energy difference, and the ordinary inter- action between the two electrons contributes most of the remainder. Because only two electrons remain, QED and non-QED effects can be calculated precisely. Uranium with two electrons is made by passing uranium ions traveling at half the speed of light through a paper-thin metal foil. Uranium and other atoms at relativistic velocities are produced by the Lawrence Berkeley Laboratory Beva- lac—the combinatio n of the Super-HILAC, a heavy-ion lin- ear accelerator, operating in tandem with the Bevatron, a synchrotron. The Bevalac is capable of producing uranium, or any other atom, with all of its electrons removed. The experiment uses foils both to strip electrons and to put two-electron uranium in the desired atomic level. The uranium passes through the foil and the atomic level decays in flight, emitting x rays whose intensity decreases exponen- tially with distance past the foil. The measured exponential decrease determines the average transition time and henc e the energy difference between the levels. The result is a value for the Lamb shift in uranium which is accurate to 12% and which agrees with the predictions of QED. Future experi- ments with more intense beams may be able to measure the Lamb shift in uranium to 0.1 %. QED provides a complete description of the interaction of charged particles and light. But Newton's Laws also once provided a complete description of gravity, the only unex- plained phenomenon being the anomalous advance of the perihelion of Mercury, the planet bound in the strongest gra- vitational field that could be studied. By studying an electronbound to the strongest electric field available, one may also find unexplained phenomena. Harvey Gould and Charles Munger, Lawrence Berkeley Laboratory 1. C. T. Munger and H. Gould (submitted to Phys. Rev. Lett.). . Atoms in Strong Laser Fields It has been known1 for more than 15 years now that any atom exposed to a laser of intensity above 1010 W/cm2 will undergo substantial ionization, losing one of its outer elec- trons. Around 1982, experiments at Saclay2 produced evi- denc e of significant multiple ionization (ejection of more than one electron per atom) in the rare gases. Since the wavelength of the lasers employed in those experiments were either in the infrared (1064 nm) or optical (532 nm) range, the absorption of a number of photons (a so-called multi- photon process) was require d for the ejection of the first electron; of an even larger number for the ejection of a sec- ond electron, and so on. Such multiphoton absorptions have been investigated since the mid-sixties, but the accumulated experience1 and wisdom appeared to suggest that processes requiring large numbers of photons, more than 10 or so, should be rather unlikely at those intensities. Thus it came as a surprise that processes requiring the absorption of 20 or 30 photons per atom were occurring with great ease at laser intensities of the order of 1012 to 1014 W/cm2. In 1983, reports3 of multiple ionization, with up to 9 electron s ejected, not only of rare gases but of a number of other atoms as well, extended the observation of these phenomena to regions of ultraviolet wavelengths (193 nm) and higher power, up to 1016 W/cm2. Moreover, radiation of wavelength shorter than that of the incident laser was in some cases emitted in the process . Several important question s with potentially far-reaching implications are posed by these experiments: (a) Why is it so easy to strip electron s from atoms and what is the underlying mechanism? (b) Are they stripped one by one or in groups? (c) If excitation in groups contributes significantly, can it be exploited towards the creation of short-wavelength and especially x-ray lasers? (d) What is the role of above-thresh- old ionization? (e) What role do the pulse duration and shape play in the overall process? These and a number of other related questions provoked widesprea d debate, speculation, and a flurry of activity in the theory of such strong field phenomena.4 In mid-1985, events took an ironic twist when it was demonstrated5 that some of the answers were lurking within the existing theoretical PHYSICS TODAY / JANUARY 1987 S-25PHYSICS NEWS IN 1 986—ELECTRON AND ATOMIC PHYSICS framework. Theoretical analysis and experimental evidence now conclude that under the conditions of the existing ex- periments, successive stripping of single-electrons is the dominant mechanism. The resulting understanding under- scores the importance of the laser pulse duration and shape, and appears to suggest the possibility of novel behavior if the duration and especially the rise time of the pulse become shorter, say 10 fs or less. At this point, little is known about the role of doubly and multiply excited states in these processes. It is nevertheless possible to speculate on certain possibilities based on the fun- damentals of atomic structure as well as some existing data. We can expect that photons of optical and near ultraviolet wavelength will interact mainly with electrons of the outer shell and possibly the subshell just below it. This is in con- trast to double excitation by extreme ultraviolet or x-ray photons where one photon excites (or ejects) an electron from an inner shell. It appears, therefore, that multiphoton excitations with powerful lasers of suitably chosen frequen- cy, intensity, and pulse properties may lead to novel atomic states that would be virtually impossible to create otherwise. The Is2 3p4 state of a carbon-lik e ion would be one such example which must, however, be understood as a state em- bedded in the strong field. Thus, in addition to possible ap- plications, the next few years are apt to reveal new vistas of atomic structure and its behavior under strong laser fields. P. Lambropoulos, University of Southern California 1. Recent review article s and background material can be found in Multi- photon Processes, edited by P. Lambropoulos and S. J. Smith (Springer- Verlag, Heidelberg, 1984) , as well as in Multiphoton Ionization of Atoms, edited by S. L. Chin and P. Lambropoulos (Academic, Toronto, 1984). 2. A. L'Huillier, L. A. Lompre, G. Mainfray, and C. Manus, Phys. Rev. Lett. 48, 1814 (1982); also Phys. Rev. A 27, 2503 (1983). 3. T. S. Luk, H. Pummer, K. Boyer, M. Shakidi, H. Egger, and C. K. Rhodes, Phys. Rev. Lett. 51, 110 (1983). 4. A. L. Robinson, Science 232, 1193 (1986). 5. P. Lambropoulos, Phys. Rev. Lett. 55, 214t (1985). Discovery of the Soliton Self-Frequency Shift In recen t years, the prediction1 that soliton pulses could exist and propagate stably in single mode optical fibers has been well verified experimentally.23 Several exciting uses have been found or propose d for these non-dispersiv e pulses. One is the soliton laser (see Physics News in 1984, S- 84), a mode- locked color center laser which produces pulses of any de- sired width, from psec down to a few tens of fsec, as con- trolled by a fiber in its feedback loop.4 Another is the possibility of creating an "all optical" communications sys-tem—one without electronic repeaters—in which a single fiber could transmit as much as 100 Gbits/s over thousands of kilometers. Such a system would use Raman gain to over- come loss, and the signal pulses would be transmitted as solitons.56 The development of such applications has created the in- centive to better understand the basic physics of fiber soli- tons. The nonlinear Schrodinger equation—the familiar equation of quantum mechanics, but with different coeffi- cients and the addition of a term to reflect the fiber's index nonlinearity—seems to describe well the propagation of soli- tons with widths of several picoseconds or greater. But we have recently discovered a new and unexpected effect, one that is not predicted from the nonlinear Schro- dinger equation as normally written.7'8 We observe a contin- uous red shift in the optical frequency of the soliton as it travels down the fiber. The effect is caused by a Raman self- pumping of the soliton, by which energy is transferred from the higher to the lower frequency parts of its spectrum. The effect becomes particularly dramatic in the subpico- second regime, as it scales inversely as the fourth power of the pulse width. For example, in 400 m of fiber, we see an 8 THz shift out of an optical frequency of 200 THz (at a wave- length of 1.5 /zm) for a 500 fs input pulse; similar shifts are obtained in just a few meters as the pulse widths are reduced to about 100 fs. Thus, it may be possible to use this phenome- non to derive femtosecond pulses of different optical fre- quenc y from the same laser source (even possibly outside the laser tuning range); this would be of interest for pump-probe experiments. The soliton self-frequency shift does not render the pro- posed scheme for soliton-based communications obsolete; for other reasons, the optimal scheme6 would involve pulses of width greater than 20 ps, but the scaling of the effect with the minus fourth power of the width should make the effect negligible for such pulse widths. Subpicosecond telecom- munications schemes would seem to be ruled out, however. EM. Mitschke, L.F. Mollenauer, andJ.P. Gordon, A T& T Bell Laboratories 1. A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973). 2. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980). 3. L. F. Mollenauer, R. H. Stolen, and M. N. Islam, Opt. Lett. 10, 229 (1985). 4. L. F. Mollenauer and R. H. Stolen, Opt. Lett. 9, 13 (1984). 5. A. Hasegawa and Y. Kodama, Proc. IEEE 69, 1145 (1981). 6. L. F. Mollenauer, J. P. Gordon, and M. N. Islam, IEEE J. Quantum Electron. 22, 157 (1986). 7. F. M. Mitschke and L. F. Mollenauer, Opt. Lett. 11, 659 (1986). 8. J. P. Gordon, Opt. Lett. 11, 662 (1986). S-26 PHYSICS TODAY / JANUARY 1987ELEMENTARY PARTICLE PHYSICS Superstrings One of the fundamental goals of theoretical particle physics is the formulation of a unified theory of the various forces of nature: gravity, electromagnetism, the weak nuclear force (which controls the burning of the sun), the strong nuclear force (which binds quarks into protons and neutrons and these into nuclei), and perhaps forces yet to be discovered. Superstring theories (sec Physics News in 1985, p. 31) are the latest, and by far the most promising, attempt to achieve this goal. Unification of the weak and electromagnetic forces ap- pears to have been achieved. A parameter in the electroweak theory which determines the relative strength of the electro- magnetic and weak forces is correctly predicted by certain "grand unified theories," which incorporate the strong in- teractions as well, although more direct experimental tests of these theories are so far lacking. Attempts to incorporate gravity in unified theories have met with less success, owing in part to the lack of a theory of gravity which is consistent with the principles of quantum mechanics. String theories, which are based on generalizin g thenotion of a point particle to that of a string-like object are rapidly changing this situation. In analogy to a string of pi- ano wire, the lowest note of the string corresponds to mass- less or very light particles, such as the photon, graviton, or electron; the harmonics or higher modes of the string corre- spond to very massive particles which modify the theory at high energies in a way that cures the divergences (infinitely large contributions to cross-section calculations) which have plagued quantum theories of gravity in the past. Su- perstrings also incorporate supersymmetry, a theory which predicts that each particle is accompanied by a superparticle with similar properties, except that the spin must differ by one half unit. Such superparticles are characteristically pre- dicted to be heavier than ordinary particles. Much of the recen t enthusiasm for string theory is due to a flurry of activity which occurred in the fall of 1984. Before that time it was thought that string theories that incorporat- ed forces other than gravity suffered from certain mathemat- ical inconsistencies. In a period of only a few months the absence of these inconsistencies in certain string theories was discovered,1 a new type of string theory that incorporated both gravity and other forces in a new and fundamental way was invented,2 and a plausible scenario for obtaining certain grand unified theories from this new string theory was pre- sented.3 It seemed that the goal of unification was just around the corner. In the last year and a half the difficulty of this undertaking has become more apparent, owing to the highly mathematical nature of string theory and to the fact that, in this case, one is attempting to reformulate the foun- dations of physics without experimental input. While enthu- siasm for string theory remains high and the number of prac-titioners is growing rapidly, significant new developments may be farther in the future than was originally hoped. One of the unsatisfactory aspects of string theory is the lack of a simple fundamental principle which would explain many of its marvelous properties. The search for the founda- tions of string theory has largely concentrated on formulat- ing a field theory of strings in analogy to field theories of point particles. This approach has been more successful for the older "open" strings; a satisfactory field theory for the more popular "closed" string theories (in which the strings close on themselves to form loops) does not yet exist. Another area of much effort has been the development of techniques for calculating string scattering amplitudes and, in particular, for proving the finiteness of the theory. While no universally accepted proof of finiteness yet exists, many feel that one is likely to appear in the next year or two. The third major area of string research involves attempts to relate the low-energy structure of string theory to more conventional grand unified theories. Superstring theory re- quires the existence of nine space dimensions and one time dimension . It is thus necessary to assume that six of the space dimensions form a very small space whose size must be com- parable to the Planck length (10 ~33 cm). The structure of this space is strongly constrained in string theory and pro- foundly influences the particle types and interactions which are observed in our four-dimensional space-time. This ap- proach to obtaining predictions from string theory suggests the possible existence of new weak forces which might one day be detected. Eventually the hope is not only to explain the properties of the known particles, but to predict the properties of particles which have not yet been detected, such as possible further generations of elementary particles or the superpartners of the known particles. Research in string theory involves many of the frontiers of mathematics, often in seemingly unrelated areas. Thus it is likely that string theory will eventually have a profound influence on mathematics. Given the extremely high ener- gies involved, direc t experimental tests will probably involve either dramatic changes in technolog y or discovery of some completely unexpecte d aspects of the theory. This fact has caused some theorists to question the wisdom of the particle physics community pursuing string theory en masse.4 Al- though not a direct test of string theory, the discovery of supersymmetric partners of known particles, one of the main goals of the Superconducting Supercollider , would be an im- portant indication that superstring theorists are on the right path. Jeffrey A. Harvey, Princeton University 1. M. Green and J. Schwarz, Phys. Lett. B 149, 117 (1984). 2. D. J. Gross, J. A. Harvey, E. Martinec, and R. Rohm, Phys. Rev. Lett. 54, 502 (1985). 3. P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten, Nucl. Phys. B 258, 461 (1985). 4. P. Ginsparg and S. L. Glashow, Phys. Today 39, 7 (May 1986). PHYSICS TODAY / JANUARY 1987 S 27PHYSICS NEWS IN 1 986—ELEMENTARY PARTICLE PHYSICS Snowmass 1986 A summer study devoted to the physics of the proposed Su- perconducting Supercollider (SSC) was held at Snowmass Village, Colorado during the summer of 1986. Successor to similar workshops held in 1982 and 1984, this year's study was organized by Lee Pondrom of the University of Wiscon- sin under the sponsorship of the Division of Particles and Fields of The American Physical Society, and was attended by over 250 physicists. The principal goal of the 1986 study was the critica l evaluation by the high energy physics com- munity of all aspects of the SSC in the light of the recentl y submitted conceptual design report (CDR)' and in view of progress in accelerator technology, collider physics, and in- strumentation. To achieve this goal, working groups were established to address four principal study topics: accelera- tor physics, physics and Monte Carlo studies, instrumenta- tion and detectors, and non-accelerator physics . The accelerator physics working group concluded that, overall, the SSC conceptual design rated high marks. The primary design goal2 of the SSC is to achieve 20 TeV proton- on-proton collisions (40 TeV in the center of mass frame) at a luminosit y of 1033 cm"2 s" 1. A particular concern has been whether the aperture of the conceptual design would in fact permit obtaining the design luminosity. A detaile d ex- amination concluded that a luminosity of 1033 is achievable, but that there is little room for error. Considerable effort was devoted to the analysis of methods for compensation of mag- net errors, with the aim of (economically) increasing the dynamic aperture. In particular, changes in the magnet lat- tice design to permit the incorporation of local corrections were suggested. The accelerator physics group studied the layout of test beams and the key issue of the design of the interaction re- gions . One important innovation to emerge from the Snow- mass study was the proposal of a beam bypas s around the interaction regions. Such a bypas s would require a small en- largement of the main ring, but would greatly add to the accelerator's flexibility. By permitting the assembly of criti- cal components of large detectors in place , it would make possible future innovation and growth without long inter- ruptions in the physics program. A bypas s could also pro- vide long straight sections, permitting the study of certain types of interactions requiring detectors very close to the beam lines. The other working groups addressed themselves to sever- al related questions. What are the best signatures and rel- evant background interactions for SSC experiments? What requirements do these physics considerations, and the high design luminosity, impose on detector design? How can non- accelerator physics experiments be expected to complement the SSC program? The physics and Monte Carlo discussions at Snowmass determined that the long sought-after Higgs boson should be observable at the SSC, unless perhaps its mass lies some-where between twice the mass of the top quark and twice the mass of the W boson, in which case top-quark decays would dominate in a study of collision events. Other subjects under discussion included heavy-quark physics (see the related ar- ticle in this chapter), CP violation, rare B meson decays, and possible experimental signatures of supersymmetry and su- perstring gauge theories. The instrumentation and detector working group report- ed encouraging progress towards the design of "4TT" detec- tors, detectors affording complete spherical coverage of the interaction area, capable of working at the SSC design lumi- nosity . The prospects for muon detection look very good, and the problems associated with silicon chip microvertex detectors, and with the needed electronics and triggering, look solvable. Electron identification is likely to achieve a pion-to-electron rejection of 10"3, and high quality calori- metry should be feasible using a uranium/lead-plus-liquid- argon combination. Central tracking still poses formidable problems because of high cell occupancy, and considerable further work will be needed to establish the feasibility of solving the pattern recognition problem and reconstructing tracks at design luminosity. In addition to considering detector components, layout studies were done for two overal l detector designs. These studies indicate that the muon system should be built in place , and that strong consideration should be given to building the central detector in the interaction hall, thus eliminating the need for an extensive assembly area under- ground (which would be possible if there were a bypass, as discussed above). On a cautionary note, nearly all of the advances in instrumentation and detectors reporte d at Snowmass '86 have come from research and development for detectors for other storage rings or accelerators. Since most of these outside detectors are now past the develop- ment phase, and since many of the unresolved problems are specific to TeV energies and high luminosity, funding of an SSC-oriented detector research and development program is essential if one is going to be able to construct proper SSC detectors in time. While the SSC will be the primary tool for the study of particle physics in the 1990s, nonaccelerator experiments can provid e complementary information on the same funda- mental questions, and also offer the possibility of studying particle physics beyond the TeV energy regime . The non- accelerator physics working group set itself the task of pre- dicting the contributions of non-accelerator experiments to particle physics in the 1990s in order to assess the needs for new facilities. The mounting evidence that objects like the Cygnu s X-3 are the sources of high energy cosmic rays (see Physics News in 1985, p. 10) suggests new cosmic-ray experi- ments. The recent idea that solar matter can amplify neu- trino oscillation effects promises to open up a whole new field of neutrino studies. Neutrino physics may also benefit from promising devel- opments in detectors which utilize quasiparticles and phon- S 28 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1986—ELEMENTARY PARTICLE PHYSICS ons to detect very low energy particles. Such detectors could revolutionize experiments to observe double beta decay, so- lar neutrinos, and dark matter. Experiments to detect proton decay are continuing to improve in sensitivity, and are now focusing on the kaonic modes, which may dominate in su- persymmetric unified theories. The first generation of large proton decay experiments have proved to be versatile instru- ments for cosmic-ray physics, magnetic monopole searches, and neutrino physics. A new generation of very large and powerful underground experiments, motivated in part by so- lar neutrino studies, would also provide unprecedented sen- sitivity to proton decay and other phenomena. European accelerator plans, described at the final sum- mary session, depend very much on American progress towards construction of the SSC. If the SSC is built on a schedule leading to completion in the mid-1990s, the Euro- peans will most likely move towards a complementary large electron-positron machine; if the SSC is not built, a favor- able European option would instead be a large proton col- lider (8.5 TeV on 8.5 TeV) in the LEP tunnel at CERN, targeted at essentially the same physics as the SSC. This op- tion would require an extensive development effort to achieve magnets capable of 10 T fields. Stephen L. Adler, The Institute for Advanced Study 1. Phys. Today 38, 58 (December 1985). 2. J. D. Jackson, M. Tigner, and S. Wojcicki, Sci. Am. 254, 66 (March 1986). New Issues in Cosmology The interface of particle physics and cosmology in the early universe has provided much excitemen t over the past few years. This comes in part from the possibility that observa- tions and experiments in different fields can be used to inves- tigate and constrain theoretical ideas. In this present survey of new developments, two are specifically of this type; (1) the beginning of experimental verification of the cosmologi- cal neutrino counting and (2) the new large scale structure observations of "foam," high velocities, and spatial correla- tions, and their implications for early phase transitions that might have produced the fluctuations which eventually led to the present structure of the universe. The other topic that I shall discuss is still at the theoretical stage but is beginnin g to lead to observational and experimental results. This is the present status of the inflationar y scenario. Almost ten years ago, it was shown that Big Bang Nucleo- synthesis constrains the number of generations of light (masses less than 10 MeV) neutrino types, TV, which in turn seems related to the number of quark flavors. Thus cosmol- ogy was setting a constraint on one of the most fundamental aspects of nature. Over the years these ideas were refined, until it now seems clear that cosmology requires that N be equal to 3, plus or minus 1. Since three v's are alreadyknown, this is a very powerful constraint. Particle physics theories might be just as happy with much larger values for N. Collider experiments are now beginning to check the cos- mological prediction using the decay width (the width of a plot of the mass of the Z as reconstructed from the properties of its decay products) of the intermediate vector boson Z° which decays to all v species. Current results3 from CERN and the PEP collider at Stanford show that N must be less than or equal to 5, a limit which is coming down towards the cosmological limit. New experiments at the Stanford Linear Collider, Fermilab's Tevatron, and CERN's LEP should re- fine the limit much further and provide the ultimate test of the theory. This is the first time a cosmological prediction is being checked with a collider experiment. In fact, since de- termining TV and the width of the Z° are the bread and butter experiments for SLC and LEP, one might even say these machines are new style telescopes for probing the early uni- verse. The early universe, where particle physics effects domi- nate, is opaque to photons and thus invisible to astronomers. Shortly after the universe becomes transparent, structures begin to form in the universe and these structures can be observed by astronomers. The conditions that produce these structures are what were left from the elementary particle epoch. Thus the formation of structure in the univers e has become a point of common ground for particle theorists working in cosmology, as well as more traditional cosmolo- gical observers using large telescopes, and everyone in between. Recentl y our ideas about the very largest structures in the universe have undergone a revolution. Observers have found that galaxies and clusters of galaxies are not randomly dis- tributed on the sky but in fact seem to lie in a foam-like pattern with lots of empty regions and the glowing galaxies lie on the walls of the foam bubbles.4'5 The scale of the bub- bles seems to be several tens of megaparsecs (1 parsec = 3.26 light years). In addition, other observers6 claim to have demonstrated that similarly large regions are moving coher- ently with velocities of about 600 km/s. It is also becoming increasingly apparent that clusters of galaxies correlate7 with each other more strongly than galaxies correlate. To summarize, there appears to be some very large scale struc- ture superimposed on the universe. In order to enable the universe to reach its critical density, cosmologists have postulated various forms of dark matter and many experimentalists have begun designing experi- ments to search for the stuff. These new large-scale structure observation s now force an alteration of the assumptions and criteria used to select various dark matter candidates. Dark matter candidates can be divided into two basic categories: (1) hot matter: low mass neutrinos or other particles mov- ing at high velocity just prior to matter dominating over radiation's contribution to the cosmological density, and (2) cold matter: axions, GeV-mass photinos or any other particle which was slow moving at matter domination. PHYSICS TODAY / JANUARY 1987 S 29PHYSICS NEWS IN 1 986—ELEMENTARY PARTICLE PHYSICS Before these new large scale structure observations oc- curred, cold matter appeared preferable because it could cluster rapidl y on galactic scales and thus enable galaxies to form fast. Cold matter had a problem, however, because it all clustered on these small scales so its total contribution would be measured in cluster dynamics which implied cosmologi- cal densities far below critical. Thus, to avoid this constraint, cold matter advocates postulated biased galaxy formation. saying that only the density peaks shine and that there are many clumps of cold matter and accompanying baryons which do not shine and yet which provide the dominant mass for the universe. While this scheme may work for the density problem , it is a disaster for the large scale velocity fields and the large cluster-cluster correlations. The more biased the galaxy formatio n is, the smaller the large scale velocities, since the glowing stuff is then in negligible isolated clumps. Hot, neutrino-like matter, seems to have the opposite problem: although it naturally yields large scale (40 Mpc) structures owing to its high velocities at the time of matter domination and although the collapsing 40 Mpc structures do produce large velocity fields, low mass neutrinos have fallen into disrepute because they can not rapidly make gal- axies. Now that the large scale structure predictions of the neutrinos are beginning to be confirmed, neutrinos are being resurrected to see if some alternative way to make galaxies can be devised. Two ideas for saving the neutrino picture and possibly even the cold matter scenario are: cosmic stringss and explo- sive galaxy formation.9 The cosmic string picture (see the article on this subject later in this chapter) has the advantage that it easily produces the cluster-cluster correlations,10 yielding a scale-free fractal structure from galaxies through rich clusters.11 The explosive scenario provide s small scale structure by explosively moving things about, but requires a sufficiently high density to percolate12 the explosions if it is to reach large scales. It also requires something to produce the seeds for the explosions, perhaps debris from the quark- hadron transition (strange matter nuggets or planetary mass black holes) or maybe clump s of cold matter or small loops of string. At present it is not clear if any combination of dark matter and galaxy formation scenarios can consistentl y satisfy all of the observational claims. However, it does seem clear that whateve r the scenario is, it involves some elementary-parti- cle initial conditions. The cosmic strings would be produced in an early phase transition, the explosion seeds may come from the quark-hadron transition, and the bulk of the matter itself may be some as yet undiscovered particle like the pho- tino or at the very least a neutrino would need a finite rest mass. The inflationary scenario first developed by Alan Guth (seePhysics News in 1982, ST1) has undergone some remark- able developments as well as demonstrated some glaring problems.13 If the early univers e undergoes a rapid expan- S 30 PHYSICS TODAY / JANUARY 1987sion prior to settling down to our current standard expan- sion, then most of the cosmological initial-condition prob- lems vanish. In particular, the horizon problem, of how the universe got to be so smooth on scales that appear causally disconnected is removed. Also the flatness problem or how did the universe managed to survive 15 billion years without going to zero or infinite density long ago, is solved by the rapid inflationar y expansion forcing the universe to be right at its critical density. The inflationary expansion can also remove any magnetic monopole density that might have been produced at the epoch of grand unification. The end of inflation also has the wonderful effect of producing fluctu- ations in the density which might be responsible for galaxy and structure formation. The good news is that new developments by A. Linde in the Soviet Union as well as several others show that inflation is not a special phenomenon, but that any scalar field that existed in the early universe will result in a rapidly expand- ing inflationar y epoch. Such an inflation will automatically solve the horizon and flatness problem. Thus we believe that something like inflation probably occurred even if we don't know the details. The bad news is that all known models tend to produce densit y fluctuations that are too large, unless some fine tun- ing is carried out. On the other hand, this is not so bad, since prior to inflation we had no idea how to produce any primor- dial fluctuations. Now at least we have a plausible mecha- nism. Unfortunately, they give fluctuations that would have been detected by the microwave anisotropy experiments. They also give a random phase spectrum which does not seem consistent with the large scale structure results (unless perhaps, strings or explosions form later). The major astronomical test of inflation, determining whether the universe is at the critical density, had, up until recently, appeared a serious failure. As mentioned above, the dynamics of galaxies and clusters imply densities far below critical. In addition, the ages of globular clusters and radio- active dating of the universe implies ages of about 15 billion years. A critical density universe would require a Hubble constant Ho of 45 km/s/Mpc. While Ho is traditionally list- ed as 50 to 100 km/s/Mpc, values as low as 40 are really still allowed, and, in fact, Sandage's and Tammann's best esti- mate14 from Type I supernovae gives a value of 42 km/s/Mpc. The first observational hints of a critical densit y universe have begun to appear. Loh and Spiller have found that at very large redshifts, the number density of galaxies is best fit by a flat geometry consistent with a critical density uni- verse. '5 In addition Rowan-Robinson has found that the In- frared Satellite (IRAS) catalogue of galaxies appears to have a high velocity distribution on scales of 200 Mpc, which is best fit if the univers e is at or near its critical density on these ultra-large scales. While both of these observations are very preliminary and numerous loopholes can be found, it is comfortin g that observational astronomy has finally pro-PHYSICS NEWS IN 1986—ELEMENTARY PARTICLE PHYSICS vided something positive for inflation proponents. Of course, the final results await many more observations, in- cluding the space telescope which should confirm or deny a low Ho. While there are many other developments at the particle- cosmology boundary, including direct probes of the quark- hadron transition through quark matter searches in heavy ion collisions, the three topics mentioned here clearly show that the new field of the boundary of particle physics and cosmology is rapidly developing, and that it is not free from real experimental and observational tests forcing theorists' feet to the hot coals of experimental reality. David N. Schramm, University of Chicago and Fermilab 1. G. Steigman, D. Schramm, and J. Gunn, Phys. Lett. B 66, 502 1979. 2. G. Steigman, M. Turner, D. Schramm, and K. Olive, Phys. Lett. B (in press). 3. D. Cline, review talk, Meeting on Anti-Proton-Interaction (Aachen, West Germany, 1986). 4. V. de Lapporent, M. Geller, and I. Huchra, Ap. J. Lett. 302, LI 1986. 5. D. Koo and R. Kon, talk presented at Aspen Center for Physics, Winter Physics Conference (1986). 6. A. Burstein, S. Faber, and D. Lynden-Bell, Proceeding of Hawaii Sym- posium on Cosmology (1986). 7. N. Bahcall and R. Souieva, Ap. J. 270, 20 1983. 8. J. Ostriker and L. Cowie, Ap. J. 243, L127 1981. 9. N. Turock, Phys. Rev. Lett. 55, 1801 1985. 10. A. Szalay and D. Schramm, Nature 314, 718 (1985). 11. J. Charlton and D. Schramm, Ap. J. (in press). 12. A. Guth, Phys. Rev. D 23, 347 (1981). 13. A. Sandage and G. Tammann, Ap. J. 256, 339 (1982). 14. E. Loh and E. Spillov, Ap. J. (submitted 1986). 15. M. Rowan-Robinson, Proc. ESO-CERN Meeting on Cosmology (1986). Heavy Quar k Physics In the standard model of elementary particles, there are three quark generations, each having one quark of charge 2/3 and one of charge — 1/3. The first generation consists of the up and down quarks (u and d), which are the constitu- ents of nucleons. The second generation consists of the charmed and strange quarks (c and s), and the third genera- tion includes the top and bottom quarks (t and b). Heavy quark physics, the study of the c, b, and t quarks, began in 1974 with the discovery of the charmed quark. The spectros- copy of particles containing heavy quarks has been used to measure the strong interaction between quarks at short dis- tances. The decays of mesons containing a heavy quark and a light antiquark have provided a new testing ground for the physics of weak decays. A great deal of experimental infor- mation on these weak decay s has been reported recently, providing more stringent tests for the theoretical models. The charmed quark is the lightest of the heavy quarks, and it is therefore the one about which we know most. The charmed particles (particles containing a charmed quark)have lifetimes of 10 u to 10 u s, which is characteristic of weak decays. Originally, it was expected that the charmed quark would decay with no effect from the antiquark, which would serve only as a spectator. This model (the simple spectator model) predicted equal lifetimes for all of the weakly decaying charmed mesons . (These are the D°, D + , and F + , in which the c quark is bound to a u, d, and s anti- quark respectively.) First measurement of the D° and D + lifetime showed that the lifetimes were not equal, thus con- tradicting the model. In the past year, precise measurements of charmed parti- cle lifetimes have been made. While earlier experiments had used samples of less than 50 D decays to measure lifetimes, present results are based on many more events.' One experi- ment at Fermilab is analyzing a sample of about 4000 events for each of the D° and D + ? The lifetime of the D + is measured to be about 1.0 ps, while the D° and F + live for approximately 0.4 ps. New measurements for the charmed baryon, the Ac, indicate a lifetime even shorter than those of the D° and F + . In some models, the differences in charmed meson lifetimes are explained by a relatively high rate for annihilation of the charmed quark and antiquark in the D° and F + , which reduces their lifetimes. In other models, the destructive interference which occurs only in the case of the D + increases its lifetime. The increasingly precise lifetimes for all of the weakly decaying charmed particles will help to constrain these models, and to obtai n a sharper picture of the underlying quark interactions. The other experimental data needed to describe charm decay are the branching ratios to various final states. The Mark III group, working at the SPEAR collider at Standard, has published a new analysis of all the major decay modes of the D° and D + .3 By collecting a large number of events in which both the charmed particle and antiparticle are recon- structed, they were able to make accurate measurements of the absolute branching ratios. The result was a shift upward of all D branching ratios, such that the known decays add up to almost 100%. In addition, three groups reported the ob- servation of a decay mode D° —0K°, which is effectively for- bidden in the simple spectator model.4 Intense effort, both experimental and theoretical, continues on the question of whether this is the first clear signal of quark-antiquark anni- hilation in the D° decay. The dominant weak interaction involves the conversion of one quark to the other quark in the same generation. There is, however, an important interaction with smaller coupling strength between quarks of different generations. Such a coupling is responsible for the weak decays of K me- sons, since they are the lightest particles containing second- generation quarks. A similar interaction between genera- tions is responsible for the weak decays of the B mesons, which are bound states of a bottom quark and a light anti- quark. In these decays, the bottom quark can be converted to a charmed quark, or less frequently to an up quark. The very weak coupling of the bottom quark to the up quark is of PHYSICS TODAY / JANUARY 1987 S 31PHYSICS NEWS IN 1 986—ELEMENTARY PARTICLE PHYSICS particular importance, because it determines the scale for CP violation in the model which explains CP violatio n in a natu- ral way. Groups at Cornell and the DESY accelerator in Hamburg have been analyzing their /?-decay events to search for final states with no charm, which would result from the b-to-u transition. Presently these groups quote up- per limits of around 10% on the fraction of bottom quarks decaying in this way.1 Improving the sensitivity of the search for B mesons decaying into noncharm states will con- tinue to be a central goal of experiments working in the area of bottom quark decay. Finally , the status of the sixth quark, the top quark, is still unclear. (The original evidence for this quark was discussed in Physics News in 1983, p. 31, and Physics News in 1984, p. 31.) Events compatible with the decay of a top quark of mass 30-50 GeV/c- were seen in the UA1 experiment at CERN in 1983 and 1984. With more data and more analysis, the pic- ture has not improved. More experimental work is required before the top quark can be considered to be well established. Michael Witherell', University of California at Santa Barbara 1. The experiments studying weak decays of heavy quarks were reviewed by M. G. D. Gilchries e at the International Conference on High Energy Physics, Berkeley (July 1986). 2. (A short article on this experimen t is to appear in the CERN Courier, probably in September 1986.) 3. R. M. Baltrusaiti s et a!., Phys. Rev. Lett. 56, 2140 (1986). 4. H. Albrecht etal, Phys. Lett. 158B, 525 (1985);C. Bebek et ai, Phys. Rev. Lett. 56. 1893 (1986); and R.M. Baltrusaitis etal, Phys. Rev. Lett. 56.2136 (1986). Cosmic Strings Modern theories of elementar y particles suggest that at ener- gies above 1015-10lb GeV all fundamental particle interac- tions (except gravity) have the same strength. Such energies are far beyond the scope of particle accelerators, and the main arena where these ideas can be tested is the early uni- verse shortly after the big bang. As the universe expands and cools down from extremely high temperatures, the symme- try between the particle interactions is spontaneously broken, typically in several steps , so that the strong, weak and electromagneti c forces become distinct. Cosmological symmetry breaking is similar to phase tran- sitions in familiar solids and liquids, such as boiling of water into vapor or its crystallization into ice. We know that a crystal formed by cooling a liquid is never perfect; it always has defects. Similarly, in cosmological phase transitions the symmetric high-energy phase can be trapped in various kinds of "defects," which can be in the form of surfaces, lines or points. The names for these defects are, respectively, do- main walls, strings and monopoles. (Strings, in this case, are not to be confused with the "superstrings" discussed earlier in this chapter.) The type of defect produced depends on the underlying theory of elementary particles.1 Observations S 32 PHYSICS TODAY / JANUARY 1987rule out the existence of domain walls in the visible universe and put a very tight upper bound on the number of mono- poles. Strings , on the other hand, can have interesting cos- mological implications and have attracted much attention recently because of their possible role in galaxy formation. Strings are forbidden to have ends; they either form closed loops or extend to infinity. Tension in curved strings makes them wiggle violentl y at a speed close to that of light. In the string scenario of galaxy formation, galaxies condense around oscillating loops of string, while the loops gradually lose their energy by gravitational radiation, shrink, and dis- appear.2 The lifetime of a loop is proportional to its mass. Loops responsible for the formation of galaxies have already decayed, but larger loops which served as seeds for rich clus- ters and superclusters may still be around. Numerical simu- lations based on this model produce a correlation between clusters of galaxies in agreement with observations.3 If light from a distant quasar encounters a string on its way to the Earth, the light rays would be slightly bent by the string's gravity, and we would see two distinct images of the same quasar, one to either side of the string. Other gravita- tional effects of strings include the gravitational radiation emitted by oscillating loops and a discontinuous change in the microwave background temperature across the string. In some elementary particle models strings can behave as su- perconducting wires.4 Such strings interact with magne- tized cosmic plasmas and can be detected as sources of radio waves.5 With so many observational side effects predicted by the string scenario, astronomers should be able to confirm it or to rule it out in the near future. Alexander Vilenkin, Tufts University 1. T. W. B. Kibble, J. Phys. A 9, 1387 (1976). 2. A. Vilenkin, Phys. Rep. 121, 263 (1985). 3. N. Turok, Phys. Rev. Lett. 55, 1801 (1985). 4. E. Witten, Nucl. Phys. B 249, 557 (1985). 5. E. M. Chudnovsky, G. B. Field, D. N. Spergel. and A. VUenkin, Phys. Rev. D 34 (1986). Advanced Accelerator Research and Development When one reaches for higher particle-collision energies, two effects become troublesome. First, at high energies the effec- tive center-of-mass collision energy of a high-speed particle incident on a stationary target increases only as the square root of the incident particle energy. Secondly , in circular orbits the energy loss due to synchrotron radiation becomes excessive. These effects dictate that for extremely high ener- gies, particles must be accelerated in linear accelerators and that high effective collision energies should be attained by colliding two high energy particle beams head-on. Such an arrangement is called a linear collider. The synchrotron radiation loss is much smaller for the heavie r protons compared to electrons. It is, therefore, stillPHYSICS NEWS IN 1 986—ELEMENTARY PARTICLE PHYSICS feasible to design a 20-TeV on 20-TeV circular proton-pro- ton collider, the Superconducting Supercollider (SSC), whereas the circular Large Electron Positron collider (LEP) under construction at CERN can reasonably go only up to 100 GeV on 100 GeV. For the SSC, furthermore, the high magnetic field strength and the low electric power con- sumption are facilitated by the use of superconducting mag- net technology. Nevertheless, these colliders are likely to be the highest energy machines of their kinds. LEP is schedule d to begin operation at 50 GeV on 50 GeV in 1989. The con- ceptual design of the SSC was completed early in 1986 and passed the review by a Department of Energy team with flying colors. With a price tag of about $3 to $4 billion, fund- ing for the SSC is still awaiting authorization by the U.S. Government. Beyond these machines the future lies in linear colliders. A modified linear collider, the SLC (Stanford Linear Col- lider), colliding electron and positron bunches accelerated simultaneously in the SLAC linac, is in an advanced stage of construction and is scheduled to go into operation in 1987. The e+ and e~ bunches that exit from the linac at 50 GeV are transported by magnets in two big arcs and then collide head-on. In its basic configuration a linear collider would have two linacs facing each other and the accelerated parti- cle beams colliding at the midpoint between the linacs. The requirements are clear: the accelerating field strength in a linac must be high so that its overall length is manageable, the power transfer efficiency from power supply to particle beam must be reasonably high, and the beam currents must be high and focusable to extremely small dimensions at the collision point in order to yield useful high luminosity (par- ticle collision rate). As energy goes higher, the collision cross-sections decrease as the square of the energy. In colli- sions of hadrons, the energy is shared among the constituent quarks and gluons, and the cross-section is, further, a sharp- ly decreasing function of the momentum-transfer relative to the collision energy. Thus, extremely high luminosity or over-energy is needed to obtain the minimum observable rates for interestin g violent quark or gluon collisions. Several rather bold and far-sighted R&D programs are being pursued to meet these requirements for future linear colliders, although only at minimal expenses of money and manpower. Most of the innovations consist of inventions of accelerating structures appropriate for novel power sup- plies. To obtain high power density—hence high accelerat- ing field—and high electric breakdown limit, one must go to high frequencies or short pulses. Frequencies from tens of GHz (cm and mm microwaves) up to 1014 Hz ( /xm laser waves) are being considered. As an exampl e of a cm-wave- length source, a Berkeley-Livermore group will use the free electron laser. The accelerating structure for this cm-wave is simply a miniature iris-loaded wave guide similar to the SLAC linear structure. The FEL microwave supply of this TBA (Two Beam Accelerator) has already been operated and the whole setup should be ready for study in 1987. Ac- celerating fields of hundreds of MV/m are expected.High-power glass lasers, gas lasers, and excimer lasers are all available. Here the principal development required is to obtain the high repetition rate necessary for high colliding beam luminosity. The propose d /zm-wavelength accelerat- ing structures are open low-g resonant structures excited simply by shinin g the laser beam in the proper orientation. These open structures can take on a variety of shapes: grat- ings, rows of micro-bumps, streams of micro-droplets, etc. If made of physica l material they will evaporate and turn into plasma when heated by the high-power laser, but will keep their integrity for a few transient nanoseconds to yield very high accelerating fields next to the surface. Theoretical esti- mates give fields on the order of 1-10 GV/m. Micrometer accelerating structures can also be formed in a plasma. In this case the strong fields in the plasma can be used to accelerate particles. The most promising scheme of this type is the Plasma Beat Wave accelerator being investi- gated experimentally at UCLA and at INRS in Canada. In this scheme, a plasma wave is resonantly excited by Raman scattering with two incident laser beams of slightly different frequencies such that their beat frequency equals the plasma oscillation frequency. The plasma wave will stay coherent for tens of oscillations before being destroyed by instabilities and non-linear effects. The field in a high density, strongly modulated plasma wave can be several GV/m. Fields as high as 1 GV/m were measured and electron s have been acceler- ated by about 1.5 MeV in 1.5 mm. Without any accelerating structure, a plane-wave laser beam can still be used to accelerate particles in a transverse plane while they are going through an undulator. This is an Inverse Free Electron Laser accelerator (IFEL). The accel- eration rate is limited, only a few hundred MeV/m, and the maximum energy attainable is a few hundred GeV being limited by the rapidly increasing synchrotron radiation loss of wiggling charged particles. Yet another source of accelerating field is the wakefield generated by an intense beam bunch propagating either in a wave guide or in a plasma. The arrangement must be so designed as to produce an amplification of the field (trans- former ratio) from the power-supplying beam (primary winding) to the accelerated beam (secondary winding). A straightforward theorem states that for a simple arrange- ment in which both beam bunches are short and travel along the same orbit, the most energy the accelerated beam can gain is only twice the energy of the power-supply beam. A complicated beam-wave guide structure giving a transform- er ratio of about ten is being tested by a DES Y-Los Alamos collaboration. These are all long shots, but they are the only hope for the long-range future of high-energ y physics. Lee C. Teng, Fermilab References: Details of all these works can be found in the proceeding s of the two workshops contained in AIP Conference Proceedings Numbers 91 (1982) and 130 (1985). The proceeding s of the third workshop held in August 1986 will be forthcoming shortly. PHYSICS TODAY / JANUARY 1987 S 33FLUID DYNAMICS Fluid dynamics is the study of the flow of gases and liquids. It underlies many areas of science, including astrophysics, geophysics, aeronautics, combustion, and plasm a physics . The equations governing fluid motion are derived from the conservation of mass, momentum, and energy that leave or enter a volume of space. The form of these equations is known, but they are notoriously difficult to solve. The recent availability of supercomputers and of new methods in ex- perimental flow visualization have led to a major increase in our ability to understand and predict complex fluid flows. The fluid dynamics article s presented below highlight the use of large-scale numerical computations to solve complex fluid problems. The Structure of a Propagating Detonation A detonation wave is supersonic and driven by rapid energy release behind it. It contains a shock wave, followed by an exothermi c chemical reaction zone, then a region of fully reacted hot material. Interactions between compressible , su- personic fluid dynamics and energy release lead to compli - cated shock interactions at the detonation front. The detonation front itself consists of interacting shock waves.1 Laboratory and field experiments have shown that a propagating detonation leaves a cellular pattern on the sidewalls of the confining chamber. These patterns also exist in detonations in free space. They are formed by "triple- points" created when three shock waves interact at the front of a detonation. These triple-points move through the sys- tems and trace out the cellular pattern. Detonation cells may be regular and repeating, but more often they are irregular, sometimes with a number of pre- ferred sizes. The size and regularity of the cell structure of a detonation are functions of the specific material, its dilution, and the nature of its confinement. Understanding this deton- ation structure and how it varies with geometry and material provides important information on the use and safety of ex- plosive materials. High-speed supercomputers with adequate computer memory and accurate numerical algorithms for time-depen- dent compressible flows perm it the numerical study of de- tonation structures.2 Numerical simulations have been used to study basic mechanisms of detonation cell formatio n and propagation, the structure of the interacting shock s at and behind the detonation front, detonation structure irregular- ity, and estimate the size of detonation cells. One unexpected and important phenomenon found in the experiments and simulations is the formatio n of unreacted pockets of material behind the detonation front.3 Experi- ments showed these first, but the pockets were not explained until they were observed in numerical simulations. Figure 1 is a series of "snapshots" taken during a calculation of a detonation propagating in a mixture of hydrogen and oxy- S-34 PHYSICS TODAY / JANUARY 1987gen diluted with argon. The upper figures show the extent of reaction. The detonation propagates to the right into the unreacted gas mixture. The fully reacted gas is on the left- hand side. The various shades in between represent different degrees of reactedness and together comprise the detonation front. In the third frame, we see a detached pocket of un- burnt gas behind the detonation front. The bottom figures show the temperature of the gases in the various regions. The figures show a cold, unreacted pocket cut off by interacting shock waves. The formation of the unburned pockets can be traced di- rectly to the curvature of the transverse shock waves. When two collide or one hits a wall, the interaction can cut off a portion of unreacted, cold material. If the material in the pockets burns slowly enough, the process effectively draws energy out of the detonation and can provide a mechanism for detonation extinction. If the pockets burn rapidly, they can generate new pressure pulses that perturb the system and cause new structures to form at the detonation front. This is one possible mechanism for generating the irregular cellular pattern observed in most gases and liquids. These findings are curious because they show how an ini- tially homogeneous material can develop an extremely inho- mogeneous structure as the shocks move through it. These findings are important because the existence of unreacted pockets and their properties can determine whether a deton- ation will live or die. Elaine S. Oran and K. Kailasanath, Naval Research Laboratory 1. See, for example, R. A. Strehlow, Combustion Fundamentals (McGraw- Hill, New York, 1984); W. Fickett and W. C. Davis. Detonation (Univer- sity of California Press, Berkeley. CA, 1979); J.H.S. Lee, Ann. Rev. Fluid Mech. 16, 311 (1984). 2. See, for example, K. Kailasanath, E.S. Oran, and J.P. Boris, Comb. Flame 61, 199 (1985); K. Hiramatsu. T. Fujiwara, and S. Taki, in Pro- ceedings of the Fourteenth International Symposium on Space Technol- ogy and Science, Tokyo, 549 (1984). 3. E.S. Oran, T.R. Young, J.P. Boris, J.M. Picone, and D.H. Edwards, Nineteenth International Symposiu m on Combustion, 573, The Com- bustion Institute, Pittsburgh (1982). Triangle-Based Grids in Computational Fluid Dynamics The last ten years have seen major advances in the numerical algorithms used to solve fluid dynamics problems. These ad- vances have been so substantial that the largest improve- ments in future calculations may now come from new ways to improve spatial resolution. The most straightforward way to fill space is with rectan- gular grids, which fill two-dimensional space with rectangles and three-dimensional space with parallelepipeds. The most innovative methods for improving spatial resolution, how- ever, now use triangular grids for fluid dynamics calcula-PHYSICS NEWS IN 1986—FLUID DYNAMICS DF.TONAT1ON MOVK S — INK) COM) INRKACTINC CAS FROM MOT . H l.l.Y RF.ACTF.D CAS. KXTENT OF REACTIONSHOCK RF.FLF.CTS FROM THK BOTTOM WALLAN CNBl RNF.D (;AS POCKF.T IS LF.FT BKHINI) THK FRONT -H TEMPERATURE FIG. 1. Extent of reaction and temperature contours showing the formation of an unburned gas pocke t during detonation propagation. The detonation is propagating from left to right. tions: triangles in two dimensions and tetrahedra in three. Figure 1 shows timestep s from calculations that use a trian- gular grid technique. Triangles are used to give fine resolu - tion throughout large parts of the grid. The triangles are smallest where there is the most structure in the flow. Figure 2 is a frame from the calculation of a shock wave passing over two irregularly shaped obstacles. The shock is initially planar and vertical. It is then disturbed as it hits the obstacles and a complicated evolving pattern of interacting shocks results. This calculation was done with an algorithm3 in which the fluid moves through the grid. In this kind of calculation, the grid does not naturally become distorted when the flow structure does, so there is no need to reconnect triangles sides. However, it is still important to change the resolution locally as the flow becomes more complicated. When structures develop in the flow, the resolution is auto- matically increased by dividin g triangles into smalle r trian- gles.4 As the flow field smoothes out, these small triangles are automatically removed to reproduce the larger triangles. At the time shown in the figure, the initially vertical shock wave has developed into an extremely complicated pattern of interacting shock s in the region of the obstacles. Triangular grids have been used for some time in finite- element calculations.5 What is new for computational fluid dynamics is the development of algorithms that combine these grids with accurate numerical methods for time-de- pendent fluid dynamics problems. The types of calculationsFIG. 2. Resolutions of the complicated pattern of shock interactions that develop as a shock passes two irregularly shaped obstacles. shown above would be extremely difficult and expensive, if not impossible, with other types of grid system. Elaine S. Oran, Naval Research Laboratory PHYSICS TODAY / JANUARY 1987 S-35PHYSICS NEWS IN 1986—FLUID DYNAMICS 1. M. J. Frittsand J. P. Boris, J. Comp. Phys. 31, 173 (1979); D. E. Fyfe, E. S. Oran, and M. J. Fritts, J. Comp. Phys. (to appear). 2. The Free-Lagrange Method edited by M. J. Fritts, W. P. Crowley, and H. Trease, (Springer-Verlag, New York, 1985). 3. R. Lohner, K. Morgan, M. Vahdati, J. P. Boris, and D. L. Book, J. Comp. Phys. J. (to appear). 4. L. Lohner, K. Morgan, and O.C. Zienkiewicz, Comp. Meth. Appl. Mech. Eng. 51, 441 (1985). 5. See, for example, O.C. Zienkiewicz, The Finite Element Method (McGraw-Hill, New York, 1977). The Condensational Instabilit y The condensational (or thermal) instability results from a delicate imbalance among convection , heating, cooling, and thermal conduction in a radiative, compressible fluid. In or- der to maintain pressure balance, the portion of the fluid that is cooling also becomes denser. If the radiation rate increases with increasing density, the cooler region radiates more ef- fectively, thus enhancing the instability. In this way, cool, dense regions are formed within a hotter, more rarified me- dium. Radiation-driven condensational instabilitie s can oc- cur in an ionized gas, for example, which is optically thin and which cools primarily by means of temperature-dependent and density-dependent radiative losses. The nature of the ambient heating mechanism and the presence or absence of a magnetic field also influence the behavior of this instability. This mechanism for forming filamentary structures in a complex fluid may operate throughout the solar atmo- sphere, which exhibits coexisting cool and hot regions over a wide range of spatial scales. Condensational instabilitie s also have been invoked as possible explanations for a variety of astrophysical phenomena, including interstellar clouds, condensations in planetary nebulae, and even galaxies.1 Therefore, we would like to know how condensational insta- bilities are initiated, how they grow in different environ- ments, and under what conditions the condensations stabi- lize or disappear. In addition, predictions of the spatial and temporal scales associated with the condensations provide a basis for comparison between theory and observation or ex- periment. Most theoretical research on this instability has ad- dressed the linear regime. '^ As with the well-known Kel- vin-Helmholtz and Rayleigh-Taylor instabilities, however, the nonlinear evolution is extremely important because the contributing physical processes are described by highly non- linear differential equations. At present, numerical simula- tions are the best method for studying the nonlinear evolu- tion of the condensational instability under a range of initial conditions. In recen t years, several nonlinear calculations have been performed which explore the condensational sta-bility of an ionized gas at temperatures and densities typical of the solar transition region and corona.5'6 Initial one-di- mensional simulations6 showed that the instability led to the development of a stable, bifurcated medium consisting of cool condensations imbedded in a hotter, more rarified envi- ronment. Solar phenomena such as spicules, prominences, and coronal mass ejections, all of which are cooler and denser than their surroundings, might thus be formed by means of the condensational instability. Most recently, multidimensional simulations have been performed to investigate the dynamics and energetics of the instability in greater detail.7"9 In one series of calculations, random velocity or density perturbations were used to trig- ger the instability in the gas.8'9 This method of initiation simulates turbulent effects which exist in many physical sys- tems of interest. As was found in the earlier one-dimensional work, stabl e cold structures are formed which oscillate only slightly around characteristic temperature and density val- ues. In addition, the multidimensional calculations indicate that the condensational instability can be responsible for the formatio n of filaments and associated turbulent flows, and that these structures can be stable over long timescales. It was also found that condensations will not form if a velocity perturbation of sufficiently large amplitude is used. More work is needed to explore the role of condensational instabilitie s in establishing stabl e structures in fluids whose energy balance is governed by comple x physical processes. For example, an analogous "runaway" situation might oc- cur in chemically reactive fluids through self-enhancing im- balances between endothermic reactions and the ambient en- ergy sources, thus leadin g to a similar type of instability. Detailed investigation of the applicability of the condensa- tional instability to the astrophysical phenomena mentioned above has barel y begun. To understand the behavio r of this instability in real physical systems, however , we must estab- lish its nonlinear development under less simplified condi- tions than have been modelled thus far: for example, in the presence of magnetic fields, continual forcing of the driving perturbations, or initially nonuniform media. Judith T. Karpen, Naval Research Laboratory 1. G. B. Field, Astrophys. J. 142, 531 (1965). 2. J. H. Hunter, Astrophys. J. 161, 451 (1970). 3. L. Sparks and G. Van Hoven, Solar Phys. 97, 283 (1985). 4. G. Van Hoven, L. Sparks, and T. Tachi, Astrophys. J. 300, 249 (1986). 5. E. Hildner, Solar Phys. 35, 123 (1974). 6. E. S. Oran, J. T. Mariska, and J. P. Boris, Astrophys. J. 254, 349 (1982). 7. L. Sparks and G. Van Hoven (in preparation). 8. R. B. Dahlburg, C. R. DeVore, J. M. Picone, J. T. Mariska, and J. T. Karpen, Astrophys. J. (in press). 9. J. T. Karpen, J. M. Picone, R. B. Dahlburg, and J. T. Mariska (in prep- aration). S 36 PHYSICS TODAY / JANUARY 1987GEOPHYSICS. In 1986 the American Geophysical Union became the 10th Member Society of the American Institute of Physics. Thus we include here for the first time a chapter on geophysics. Geology of Venus Because of the dense clouds that cover the entire planet, the surface of Venus has remained a mystery longer than that of any of the other inner planets. Venus is especially interesting to earth and planetary scientists, because it is our nearest planetary neighbor and is, in size and mass, almost the twin of Earth. The other terrestrial planets are much smaller, with diameters half that of the Earth or less.' They thus have smalle r internal "heat engines" (heat generated durin g planetary formation and from decay of ra- dioactive elements within the planet). Volcani c activity and plate recycling act to bring heat from the interior of a planet to its surface. The Moon, Mercury, and Mars are "one- plate" planets; they never developed plate tectonics as seen on the Earth, in which relatively rigid plates move horizon- tally, driven by thermal convection currents in the Earth's mantle.2 Much of our information about the tectonics of the other terrestrial planets has come from photographs of their sur- faces. Photogeologic data provide information on the basic features of a planet's surface (volcanoes, craters, faults , etc.); these data, when combined with information on the planet's size, mass, moments of inertia, and gravitational field, can be used to constrain model s for the internal struc- ture and geologic history of a planet. In the case of Venus, scientists must use radar imaging techniques, rather than normal photography, to "see" the surface of the planet through the dense atmosphere. Radar can provide informa - tion on topography , surface roughness, and the surface di- electric constant.3 The first planet-wide data on the surface topography of Venus came from the Pioneer Venus orbital spacecraft in late 1978.3 Altimetry data revealed that the surface of Venus has both very high and very low areas, including areas of high elevation comparable in size to Earth's continents. However, the horizontal resolution of the Pioneer Venus to- pographic data (100-200 km) was not good enough to settle the question of whether Venus is a one-plate or a multi-plate planet.1 In 1983 and 1984, the Soviet spacecraft Venera 15 and 16 surveyed the northern polar regions of Venus, producing radar images covering about 25% of the planet's surface with a horizonta l resolution as fine as 1-2 km.4"7 The recent- ly published Soviet images have revealed an astonishing var- iety and complexity of features on the surface of Venus. TheSoviets have also placed seven landers on Venus since 1975, which returned photographs of the surface, data on atmo- spheric conditions, and chemical analyses of rock sam- ples.8"11 Current research is focused on the analysis and in- terpretation of the Venera data. The Venera images reveal large numbers of circular fea- tures, ranging from 100-140 km in diameter down to about 10 km.7 Many of these features are morphologically similar to impact craters seen on the other terrestrial planets. The density of cratering can be used to estimate the age of a sur- face; age estimates for the cratered terrain on Venus are in the range 0.5-1.0 billion years.7 For comparison, plate recy- cling on our planet is so fast that 60% of the Earth's surface is less than 200 million years old, less than 5 % of the age of the planet.: Other circular features seen on the Venera images appear to be related to volcanic rather than impact processes, al- though the origin of some of these features is still uncer- tain.12 For example, Cleopatra Crater is a 100-km-diameter feature located just below the summit of Maxwell Montes, the highest area on Venus; if this feature is a volcanic cal- dera, then any model for the formatio n of Maxwell Montes must take into account this evidence for volcanism.13 In contrast to the one-plate planets, which are dominated by vertica l tectonics related to the emplacement of volcanic loads,2 Venus appears to have had lateral tectonics, includ- ing regions of both extension (rifting) and compression (ridges and mountains).4'514'15 Offsets of lineaments in the Maxwell Montes region may be the result of strike-slip fault- ing, horizontal motions in the same style as Earth's San An- dreas fault.16 Whether these features indicate true plate re- cycling is still unknown; one possibility is that the tectonic style of Venus falls somewhere between that of the smaller planets and the full plate recycling found on Earth.14 For the foreseeable future, American researchers will continue to depend on data provided by Soviet spacecraft. Launch of the Venus radar mapping mission ("Magellan"), originally scheduled for 1988, will be delayed at least 19 months owing to the Challenger disaster. Lynn Hall, Massachusetts Institute of Technology 1. J. W. Head, S. E. Yuter, and S. C. Solomon, Am. Sci. 69,614 (Nov.-Dec. 1981). 2. J. W. Head, and S. C. Solomon, Science 213, 62 (3 July 1981). 3. G. H. Pettengill, D. B. Campbell, and H. Masursky, Sci. Am. 243, 54 (August 1980). 4. V. L. Barsukove?a/.,J. Geophys. Res. 91, D378 (1986). 5. A. T. Basilevsky etal., J. Geophys. Res. 91, D399 (1986). 6. Yu. N. Alexandrov e/ al, Science 231, 1271 (14March 1986). PHYSICS TODAY / JANUARY 1987 S-37PHYSICS NEWS IN 1 986—GEOPHYSICS 7. B. A. Ivanovo a/., J. Geophys. Res. 91, D413 (1986). 8. V. L. Barsukov, V. P. Volkov. and I. L. Khodakovsky. J. Geophys. Res. 87. A3 (1982). 9. Yu. A. Surkov era/.. J. Geophys. Res. 88. A4S1 (1983). 10. K.P. Florenskiy etah. Venus, edited by D.M. Hunten. L. Colin, T.M. Donahue, and V.I. Moroz (University of Arizona Press, Tucson, 1983) , pp. 137-153. 11. R. G. Pnnn. Sci. Am. 252, 46 (March 1985). 12. O. V. Nikolaeva, L. B. Ronca, and A. T. Basilevsky. (abstract). Lunar and Planetary Science XVII (1986), pp. 617-618. 13. G. G. Schaber. R. C. Kozak. and H. Masursky, (abstract). Lunar and Planetary Science XVII (1986), pp. 762-763. 14. J. W. Head, (abstract). Lunar and Planetary ScienceXVII (1986), pp. 323-324. 15. J. W. Head, (abstract). Lunar and Planetary Science XVII (1986), pp. 325-326. 16. R. W. Vroder Bruegge and J. W. Head, (abstract), Lunar and Plan- etary ScienceXVII (1986), pp. 917-918. Geomagnetic Reversals The phenomenon of magnetic polarity reversa l has surged into the scientific limelight with the suggestion that Uranus may indeed be experiencing such an event just now (see the article on Uranus in the chapter on Astrophysics). For the earth, paleomagnetic observations from rocks indicate that changes in polarity have occurred throughout geologic time. How the dynamo in the Earth's fluid outer core manages to undergo reversal has become a hot topic of geophysical re- search and, with an influx of quality paleomagnetic record- ings spanning actual events, considerable new insight has recentl y been gained. What has emerged is a more defined picture of the core process, one which now appears to be neither simple nor continuous, yet sometimes surprisingly reproducible. For example, a recentl y reported, detaile d accounting of chrono- logical "snapshots" of transitional field behavior recorded in lavas shows the reversal mechanism to be highly sporadic with alternating periods of rapid directional changes and relative dynamo inactivity.1 The suggestion here is that a reversal may be accomplished through a staggered series of events by which magnetic flux is carried along, essentially frozen into the highly conductive core fluid as it undergoes rapid alterations in its convective flow.2 Also seen in this record is the occurrence of an initially unsuccessful attempt by the dynamo to secure the opposite polarity, during which time the field "rebounds" to an inter- mediate direction between the two polarity states before ulti- mately completing the transition. Data obtained from the southern hemisphere further show that, prior to a successful transition, the geodynamo may undergo a succession of at- tempts. Interestingly, each attempt appears to involve the same intermediate field geometry, like a steppingstone between the two polarity states.3 With regard to longer-term repeatability, sediments on Crete indicate that the spatial characteristics of the reversal process can remain virtually invariant over the time spanning a number of polarity inter- S 38 PHYSICS TODAY / JANUARY 1987vals.4 Over this time period, the magnetohydrodynami c mechanism in the core appears to be blind to the initial sign (polarity) of the reversing field. Paleomagnetic evidence consistently supports the hy- pothesis that transitional field geometries are not simply di- polar, but far more complex. This observation together with the above-cited studies suggest that, perhaps, only a portion of the core may revers e at any given time, with each such regional reversa l occurring over a time far shorter than the complete process. Results of an innovative statistical study of the long-term behavior in the rate of field reversal tend to support this suggestion.5 Specifically, the analysis suggests that reversals are actually triggered by some physical phe- nomenon in the core necessaril y associated with an energy source independent of that which powers the dynamo. The authors' preferred model5 assumes the geodynamo to be driven by chemical convection derive d from freezing at the base of the fluid core. The triggering source? Cold "blobs" produced far away at the outer boundary by way of heat loss into the mantle. Such material descending through the core might be expected to destabiliz e the local convection pat- tern, thus initiating a reversa l attempt. The number of such attempts and probability of success may then depend on such physica l properties as the size, number and free-fall velocity of the blobs. Kenneth A. Hoffman, California Polytechnic State University 1. M. Prevot, E. A. Mankinen, C. S. Gramme, and R. S. Coe. Nature 316, 230 (1985). 2. D. Gubbins and N. Roberts, Geophys, J. R. Astron. Soc. 73, 675 (1983). 3. K. A. Hoffman, Nature 320, 228 (1986). 4. J. P. Valet and C. Laj, Nature 311, 552 (1984). 5. P. L. McFadden and R. T. Merrill. Phys. Earth Planet Int. 43, 22 (1986). Geomagnetic Main Field and Secular Variation The ultimate aim of studies of the geomagnetic main field and secular variation (the gradual changes in the field with time) is to gain some understanding of the dynamo process in the Earth's core responsible for the regeneration of the field. In recent years the availability of satellite data, culmi- nating in the Magsat mission (1979-1980), has lead to a great upsurge of interest, the results of which are now com- ing to fruition. The problem of using measurements of the field, which of necessity are made on or just above the earth's surface, to infer the field at the core-mantle boundary some 3000 km below is formidable; however, recent advances in field modelling by two groups (D. Gubbins, K. Whaler, and J. Bloxham in Cambridge, England, and L. Shure, R. Park- er, and G. Backus in La Jolla, California) have led to much more detaile d maps of the field at the core-mantle boundary. Bloxham and Gubbins have been able, by using historicalPHYSICS NEWS IN 1986—GEOPHYSICS field observations, to extend these maps back in time to the beginning of the 18th century.1 The results have been sur- prising: the secular variation at the core-mantle boundary is confined almost entirel y to the hemisphere from 90°E to 90°W, and westward drift (the gradual westward movement of the field observed at the Earth's surface) is confined to an even smaller region. The maps also suggest the possibility of coupling between flow in the core and lateral heterogeneities in the lowermost mantle which have recently been mapped by seismologists. Other work, notably by C. Gire and J.-L. LeMouel (Paris, France),2 by Whaler (now in Leeds, Eng- land),3 and by C. Voorhies (Greenbelt, Maryland),4 has concentrated on using maps of the secular variation to derive models of the fluid motions at the surface of the core; the results depend strongly on the underlying assumptions, which are currently the subject of some controversy. The recognition of quite abrupt changes in the secular variation (so-called jerks) by V. Courtillot, J. Ducruix, and J.-L. LeMouel (Paris, France)5 and by S. Malin, B. Hodder, and D. Barraclough (Edinburgh, Scotland)6 continues to stimulate much interest and may also lead to new under- standing of processes in the Earth's core. The subject is cer- tainly in a period of rapid change and healthy controversy; indeed, it does seem possible that in the next few years we may finally begin to understand the basics of the dynamo, although more satellite data are an urgent requirement if the current pace of activity and rate of advance is to be main- tained. Jeremy Bloxham, Harvard University 1. J. Bloxham and D. Gubbins, Nature 317, 777 (1985). 2. C. Gire, J.-L. LeMouel, and T. Madden, Geophys. J. R. Astron. Soc. 84, 1 (1986). 3. K. A. Whaler, Geophys. J. R. Astron. Soc. 86, 563 (1986) 4. C. V. Voorhies, J. Geophys. Res. (to appear 1986). 5. V. Courtillot and J.-L. LeMouel, Nature 311, 709 (1984). 6. S. R. C. Malin and B. M. Hodder, Nature 296, 726 (1982). Rock Magnetism Two recent discoveries in rock magnetism have added new insights into the origins of stable natural remanent magneti- zation (NRM) recorded in rock and sediment. The first deals with direct observations of magnetic domain structure in titanium-substituted magnetite (a magnetic mineral found in basalti c lava flows) using the Bitter technique. It has been found that a significant proportion of small multi- domain (MD) grains (tens of microns) fail to nucleate do- mains after removal of a saturating magnetic field and thus remain in a metastable single-domain (SD) state.1 Domain nucleation will occur only after a reverse field of sufficient magnitude is applied. A model based on a random distribu- tion of nucleation sites was developed to explain the grain size dependence of saturation remanence.Recently, metastable SD grains were also observed after grains were given a weak-field thermoremanent magnetiza- tion.2 These observations are extremely important and may explain the SD-like intensities and stabilities found in small MD particles. Besides the importance of domain nucleation, the actual domain form in these materials has become a sub- ject of study and controversy.3 The surface domain struc- tures commonly observed in these materials are exceedingly complex, particularly in Ti-rich compositions, and can vary dramatically within single particles. Because magnetocrys- talline anisotropy is low and magnetostriction high in Ti- rich magnetites, stress could play an important role in con- trolling the domain structure. In this respect, the domain structures found in Ti-rich magnetites are very similar to those found in amorphous metals. Despite these similarities, however, the role of stress in controlling stable NRM re- mains obscure. The second discovery comes from the field of biomagne- tism. It has been known since 1975 that certain species of aquatic bacteria contain membrane-bound magnetite parti- cles, usuall y in chains of approximately 10-20 cuboidal parti- cles, each roughly 50 nm in size. The sizes of these particles are within the SD size range for magnetite. Since the initial discovery of magnetotactic bacteria, the question arose as to whether biogenic SD magnetite could contribute to stabl e NRM found in marine sediments. The answer now appears to be yes, based on two recen t reports.45 Biogenic magnetite has been identified in surface sediments at several localities off the coast of California and in deep-sea sediments from the South Atlantic. How general these findings are, however, must await further study. In particular, we do not know how far back in the geologic record magnetotactic bacteria may have existed, and hence it is uncertain whether older marine sediments contain biogenic magnetite. Bruce Moskowitz, Princeton University 1. S. Halgedahl and M. Fuller, J. Geophys. Res. 88, 6505 (1983). 2. M. Metcalf and M. Fuller, Nature 321, 847, (1986). 3. E. Appel and H. Soffel, J. Geophys. 56, 121 (1985). 4. J. F. Stolz, S. R. Chang, and J. L. Kirschvink, Nature 321, 849 (1986). 5. N. Petersen and T. von Dobeneck, Nature 320, 611 (1986). Marine Geology Two-thirds of the surface of the Earth is created along a vast underwater volcanic rift system known as the mid-ocean ridge. Aide d by recen t technological advances in sea floor mapping, navigation and instrumentation, marine geologists have made important advances over the past decade in un- derstanding the complex and interrelated volcanic , tectonic and hydrothermal processes involved in the formation of new ocean crust.l Among the most publicized discoveries to emerge from these studies are the spectacular sea floor hy- drothermal vent systems with their unique biological com- PHYSICS TODAY / JANUARY 1987 S 39PHYSICS NEWS IN 1 986—GEOPHYSICS munities and associated sulfide mineral deposits. This hy- drothermal activity is thermally driven by the presence of hot, molten rock at relatively shallow levels in the crust in reservoirs known as magma chambers. Although there is general agreement on the importance of crustal magma chambers in the formation of new ocean crust, the shape, longevity and variability of these magma chambers along the rise axis are still the subject of considerable controversy in the geological community. Over the past few years a general model of the volcanic processes at mid-ocean ridges has emerged from observa- tions of the variations in depth along the rise crest and from laboratory experiments with convecting fluids. The basic idea behind this model, which was initially proposed by Han Schouten and his colleagues at Woods Hole Oceanographic Institution,2 is that molten rock forming in the underlying mantle rises in localized blobs along the ridge, rather than in a continuous sheet. These upwellings feed melt into a crustal magma chamber , and this melt then flows laterally along the ridge, forming rift eruptions similar to those observed in Ice- land. The injection centers are associated with long-wave- lengt h topographic highs that occur with a spacing of about 50-100 km. This appealing model has been tested recently by two important experiments. The properties of magma chambers at oceanic spreading centers can be inferred indirectly from the compositio n of the lavas that are erupted at the sea floor. If the model out- lined above is correct, then the eruption temperature of the lavas should decrease systematicall y away from the bathy- metric high where the melt is initially injected into the magma chamber. Last year, Charles Langmuir of the La- mont-Doherty Geological Observatory led a team of investi- gators to the fast-spreading (6 cm/yr) East Pacific Rise off the western coast of Mexico to test this hypothesis.3 They systematically dredged nearl y 1000 km of ridge crest , ob- taining a much closer spatial sampling (8 km) than had ever been attempted over this lengt h of ridge. They found that, geochemically, the lavas are surprisingly diverse, with dis- tinct eruptive units occurring on a scale ten times less than the long-wavelength topographic variability along the rise axis. Their results appear to require either multiple regions of supply from below the crust or a magma chamber that is physically discontinuous on a scale much smaller than was previously suspected. The latter hypothesis was tested at nearly the same time by a novel two-ship, multichannel seismic experiment car- ried out along this same ridge segment by investigators from Lamont, Scripps Institution of Oceanography, the Universi- ty of Rhode Island, and the U.S. Geological Survey.4 Sound travels through the molten lava in a crustal magma chamber at a much lower velocity than in the surrounding rock. As a result, the roof of the magma chamber acts as a strong reflec- tor for seismic energy. Using the same technolog y developed in the petroleum industry to map subsurface geological structures in sedimentary basins,5 these investigatorsmapped the reflections off the top of a crustal magma chamber lying 1-2 km below the sea floor along the East Pacific Rise. They found that the magma chamber is surpris- ingly narrow (less than 3-4 km wide), but physically contin- uous along the rise axis over distances of at least 40-50 km, and in one case for more than 90 km. In order to reconcile these observations with the geochem ical diversity of the erupting lavas, the molten material within the chamber must be much more incompletely mixed and heterogeneous in its composition than previous magma chamber models had envisioned. The results from both of these experiments have led to a reexamination of current geological and geophysical models of mid-ocean ridges. Long-term, multidisciplinary investiga- tions of other portions of the mid-ocean ridge system in the central North Atlantic near the Kane Fracture zone, along the Juan de Fuca Ridge off the west coast of North America, and along a very fast spreading portion of the East Pacific Rise north of Easter Island are also under way, and promise, over the next few years, to significantly advance our under- standing of the geological processes involved in forming the oceanic crust. Roberts. Detrick, Jr., University of Rhode Island 1. K. C. Macdonald. Ann. Rev. Earth Planet Sci. 10, 155 (1982). 2. H. Schouten. J. Whitehead. and H. Dick. Nature (in press). 3. C. Langmuir, J. Bender, and R. Batiza, Nature (in press). 4. R. Detrick, P. Buhl, E. Vera, J. Mutter, J. Orcutt, J. Madsen and T. Brocher, Nature (in press). 5. J. Mutter, Sci. Am. 254, 66 (February 1986). Oceanography, El Nino, and Chaos Perhaps one of the most striking events in geophysics in re- cent years has been the reemergence of "chaos theory" in oceanography and meteorology. "Reemergence," because the modern (post-war) roots of the field are to be found in a paper by Lorenz' in the atmospheric science literature of the early 1960s, a paper largely neglected for several years. These ideas subsequently had a very large impact on the mathematical theory of dynamical systems and recently have filtered back into geophysics and fluid dynamics.2 It has been known for a long time that many problems are inherently unpredictable (even ignoring quantum effects). The weather is one example—a small error in our observa- tions of the state of the atmosphere at any one time will grow and cause our prediction to be wildly off in a matter of a few days. Chaos theory suggests that this may be an inherent, or generic, property of certain simpl e nonlinear systems, and that a high degree of complication is simply not needed to produce these effects. Indeed systems with as few as three degrees of freedom can produce such aperiodicity and un- predictability. S-40 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1986—GEOPHYSICS Now, many problems in geophysics and astrophysics do display aperiodicity, and seem very hard to predict. El Nino is one outstanding example. This is an event, occurring every few years, but not at regular intervals, in which the water off the coast of Peru becomes anomalously warm, devastating the fishing industry and possibly affecting climate world- wide. Chaos theory tells us that even though the events are aperiodic, we do not need a complicated explanation, or need to invoke some external random forcing. Of course it does not tell us that such explanations are not true, or that some random, small-scale noise is not important. Recently, rela- tively simple models have been proposed which seek to ex- plain El Nino.3'4 The simplest one is a conceptual model requiring only three variables. The model is a caricature of the real world, and should obviously not be used for predic- tive purposes, yet it explains many of the salien t features of El Nino in a very direc t way. Are those ideas correct? It's really too early to tell. Only observations and comparison with the theoretical predic- tions can tell. The observationists who must collect and ana- lyze long-time series of data, and the theoreticians, who must construct more meaningful models, must sit down together and interpret their results in the light of the new theories. Geoffrey K. Vallis, Scripps Institution of Oceanography 1. E. W. Lorenz, J. Atmos. Sci. 20, 130 (1963). 2. J. W. Miles, Adv. Appl. Mech. 24, 189 (1984). 3. M. Cane and S. Zebiak, Science 228, 1085 (1985). 4. G. K. Vallis, Science 232, 243 (1986). Stochastic Analysis of the Transport of Contaminants in Groundwater The occurrence of several widely reported incidents of groundwater contamination has led to the recognition that such incidents pose a significant threat to public health. The political response to this perception has been a recent strengthening of the regulatory climate through the RCRA (Resource Conservation and Recovery Act) and CERCLA (Comprehensive Environmental Response, Compensation and Liability Act) legislation . The requirements of these acts have led to vigorous research activity into the processes of contaminant transport in geologic media and the predic- tion of migration rates of contaminant plumes in ground- water flow systems. The most common sources of contami- nant plumes include hazardous-waste landfills, tailings ponds, waste-injection wells, and buried tanks. Such facili- ties represent point sources of pollution as opposed to the non-point pollution caused by agricultural use of fertilizers, herbicides, and pesticides. The contaminants of greatest con-cern include inorganic metals, organic solvents, and radioac- tive isotopes. The basic mechanisms of transport of chemical species by flowing groundwater have been recognized for many years.1 They include advection, diffusion, dispersion, and retarda- tion. Advective transport, the movement of contaminants with flowing groundwater, leads to the displacement of na- tive waters at background concentrations by contaminated waters at concentrations that may exceed acceptable stan- dards. Under conditions of steady-state, saturated ground- water flow, and with a source of constant strength, advective flow leads to a sharp plume front that moves at the velocity of the flowing groundwater. The rate of migration is con- trolled by the natural hydraulic gradient and by the porosity and hydraulic conductivity of the aquifer. Hydrodynamic dispersio n causes a spreading of the plume front under the influence of heterogeneities in the hy- draulic-conductivity field. Such heterogeneities are the rule rather than the exception in stratigraphically complex, un- consolidated geologic deposit s of glacial or alluvial origin, which commonly constitute the aquifers of concern. Retardation is a catch-all term that includes the in- fluences on migration rates of chemical reactions between waters and porous media, and between contaminated and native waters. The most common retardation mechanisms are ion exchange and adsorption on clays and organic mat- ter. These mechanisms of transport can be embodied in a boundary-value problem based on the advection-dispersion- retardation equation. Such models are usuall y solved with a finite-element method. They allow prediction of the rates of plume migration, contaminant travel times, and mass fluxes, which can then be used as input to risk analyses for proposed or existing waste-disposal facilities. Such analyses are used in the design of remediation at contaminated sites and in the siting , design, and regulation of new facilities. In the past, these boundary-value problems were solved in a deterministic framework. In recent years, however, it has become evident that the heterogeneity in the hydraulic- conductivity fields in real hydrogeologic applications breeds an uncertainty that favors a stochastic approach.2 Solutions can be obtained either through analytical solutions to the stochastic differential equations using perturbation tech- niques3 or through Monte Carlo simulation.4 The output from such simulations leads to probability densit y functions (or their first and second moments) for travel times and mass fluxes. Such output is well suited to risk analysis and to risk-based engineerin g design.5 The application of stochastic simulation at field sites is best carried out in a framework6 in which the estimates of the moments of the probability densit y function for hydrau- lic conductivity are updated,7 and predictive uncertainties in travel times are reduced,8 through the collectio n of addi- tional field measurements of hydraulic conductivity at the site. PHYSICS TODAY / JANUARY 1987 S-41PHYSICS NEWS IN 1 986—GEOPHYSICS A recen t review paper0 notes that "the current use of stochastic concepts in subsurface solute transport theory represents an initial attempt to capture field-scale variability in terms of a stochastic convection-dispersion equation." The authors caution, however, that "many of the approxi- mations remain both untested in detailed field experiments and underived from rigorous results in the theory of random processes." R. Allan Freeze, University of British Columbia1. J. Bear. Dynamics of Fluids in Porous Media (1972). 2. S. P. Neuman, Geol. Soc. Am. 189, 81 (1982), Special Paper. 3. L. W. Gelhar and C. L. Axness. Water Resources Res. 17. 1463 (1981). 4. L. Smith and F. W. Schwartz. Water Resources Res. 16, 303 (1980). 5. J. Massmann and R. A. Freeze. Water Resources Res. (in press). 6. P. K. Kiianidis. Water Resources Res. 22, 935 (1986). 7. E. Feinerman. G. Dagan, and E. Bresler. Water Resources Res. 22, 935 (1986). 8. W. Hachich and E. H. Vanmarcke, Am. Soc. Civ. Eng. J. Geotherm Eng. Div. 109. 373 (1983). 9. G. Sposito. W. A. Jnry. and V. K. Gupta. Water Resources Res. 22 77 (1986). MEDICAL PHYSICS Nuclea r Magnetic Resonance in Medicine In recen t years, there has been a dramatic acceleration in the application of nuclear magnetic resonance (NMR) tech- niques within the medical sciences. The potential utilization of NMR in clinical medicine was first demonstrated in 1971 by R. Damadian. who reported that the longer proton relax - ation times (7*! and Tz) exhibited by tumor tissue could pro- vide a basis for differentiation between malignant and normal tissues.1 In 1973, P.C. Lauterbur described a method using superimposed linear magnetic field gradients to spatially en- code the NMR data allowing formation of tomographic cross sectional images. By 1977, initial images of human anatomy were being made."'"" with some of the first refined head imaging results being published in 1980.6 These later advances were pio- neered in a large part by physicists at universities. By 1986 the intense academic and industrial research and development ef- forts had provided the technological capability for efficient, high resolution imaging and spectroscopic analysis of clinically important anatomical and pathological conditions. There are now more than 450 clinical NMR imaging units (referred to as magnetic resonance imaging, or MRI, by physi- cians) operating worldwide, including many in community hospitals and private clinics as well as in university medical centers. Depending upon the magnet specifications and siting configuration, the cost of these installations may range from under S500 000 to over S2 000 000. The major components of a clinical, whole-body NMR imaging system are: (a) a large bore magnet with fields on commercial units ranging from a low value of 0.02 T to a high of 2.0 T (for fields above about 0.3 T superconducting magnets are generally used), (b) coils to su- perimpose the magnetic field gradients required for spatial en- coding and slice selection, (c) radiofrequency transmitter and receiver coils, and (d) a computer to control the system and to reconstruct the image from the acquired data using Fourier transform techniques. Most clinical imaging to date is based on proton resonance. The images thus created can represent the mobile proton den- S-42 PHYSICS TODAY / JANUARY 1987sity in a thin slice of the body, or the image can be weighted with the proton relaxation times J", and T2 by using various rf pulse sequences including spin echoes. Figure 1 shows a 1-cm thick transverse slice through the eyes obtained using a 7\ -weighted, spin-echo technique at a field of 1.5 T. Note that the vitreous humor of the eye, which has a long 7\, appears bright, whereas the more crystalline lens, with a shorter 7\, appears dark. This image was obtained using a circumferential radiofrequency coil which surrounds the head. Recent advances in NMR imaging include the development of surface coils. A significant improve- ment in signal-to-noise ratio (S/N) can be achieved by using a receiver coil which is placed in close proximity to the region of interest. This increase in S/N can be translated into improved resolution by using smaller pixels. Surface coils have been used for imaging specific anatomical areas such as the eye, the ear, the spine, the knee, and the breast. FIG. 1. A transverse NMR image of the normal head at the level of the eyes at 1.5 T. For the 7\ -weighted image, the ventricular central spinal fluid appears bright. (Courtesy of Picker International, Inc.)PHYSICS NEWS IN 1986—MEDICAL PHYSICS FIG. 2. A gated image showing the chambers of the heart. (Courtesy of Picker International, Inc.) The image shown on the cover is a composite in which sever- al surface coil images of the spine have been photographically linked with the total head image. As illustrated, unlike x-ray- computed tomography (CT) which is constrained by the me- chanical gantry motion, NMR imaging may provide image planes at any orientation (i.e., sagittal, coronal, and oblique, as well as transverse) by means of electronic manipulation of the gradient fields. The images shown demonstrate the spatial reso- lution (less than 1 mm) and contrast presentation capabilities of clinical NMR imaging. Extension of these basic imaging techniques to small regions is now opening up the new field of NMR microscopy in which spatial resolution down to 10 fxm per pixel in the imaging plane has been demonstrated in vitro. This has allowed imaging of single cells (frog ova) with clear differentiation between the cell nucleus and various zones with- in the cytoplasma.7 Cardiac research and the associated topic of blood flow imaging have received considerable attention in the past year. EKG signal gating has been utilized as the control for NMR data acquisition. With this technique, it is possible to generate a "stop-action" image of the heart in any given part of the cardiac cycle (see Fig. 2) and to string these images together, thus pro- ducing a cine sequence of the beating heart. A number of meth- ods are available which allow visualization of flowing blood. One approach utilizes the spin state phase-dependence of the NMR phenomenon as an informational parameter to charac- terize the flow of a system. Velocity-dependent phase contrast in conjunction with electrocardiographic gating and imaging subtraction techniques have been used to produce non-invasive NMR angiographic studies.8'9 Other work involving fast data acquisition techniques has demonstrated the capability for visu- alizing dynamic flow patterns by real time NMR "movies."10 Originally, NMR imaging was considered a relatively slow modality in terms of data acquisition. Even with the conven- tional multislice NMR imaging today, a typical series of 16 slices requires about 12 minutes, owing to fundamental relaxa- tion time constraints which present a problem in body imaging since the patient moves and breathes. Special fast pulse se- quences have been proposed11 which would enable acquisition of a complete image within the time frame of the T2 relaxation(40 to 100 ms); however, this method gives coarse spatial reso- lution and low signal-to-noise. Development and refinement of new fast NMR imaging techniques has represented an area of intense activity over the past two years. A number of different pulse protocol strategies have evolved, generally involving lim- ited rf pulse flip angles and gradient recalled echoes.1213 Single slice images exhibiting relatively high S/N characteristics now may be obtained in 2 to 5 seconds while fast, 3-D data acquisi- tion techniques allow formation of a complete multi-slice series (more than 20 images) in four minutes.14 Finally, the role of nuclei other than hydrogen and the po- tential for obtaining diagnostic information through spectro- scopic analysis of various biomolecular species within tissue have been of emerging importance within medical NMR. NMR imaging of the natura l sodium-23 distribution within human heads has been reported.15 Fluorine-19, which is present endo- genously at extremely low levels, may be introduced in vivo in several biocompatible forms (including fluorinated glucose and the "blood substitute " materials) as a high contrast NMR agent. Applications for fluorine-19 NMR are being investigated extensively.16 Chemical shift phenomena resulting in complex multi-peak NMR spectra for a given nucleus provide a probe for the molecular environment and allow identification of various biochemical species and product metabolites. In fact, chemical shift imaging techniques have been developed which provide the capability for obtaining an image of a selected spectral com- ponent. For example, in the case of hydrogen, separate images may be generated for the protons bound in water and for those bound in fat (lipid).: 7 Stephen R. Thomas, University of Cincinnati and Robert L. Dixon, Wake Forest University 1. R. Damadian, Science 171, 1151 (1971). 2. P. C. Lauterbur, Nature 242, 190 (1973). 3. P. Mansfield and A. A. Maudsley , Br. J. Radiol. 50, 188 (1977). 4. R. Damadian, M. Goldsmith, and L. Minkoff, Physiol. Chem. Phys. 9, 97 (1977). 5. W. S. Hinshaw, P. A. Bottomley, and G. N. Holland, Nature 270, 722 (1977). 6. G. N. Holland, R. C. Hawkes, and W. S. Moore, J. Comput. Asst. Tomogr. 4, 429 (1980). 7. J. B. Aguayo, S. J. Blackband, J. Schoeniger et al, Nature 322, 190 (1986). 8. V. J. Wedeen, R. A. Meuli, R. R. Edelman et al, Science 230, 946 (1985). 9. R. A. Meuli, V. J. Weeden, S. C. Geller et al, Radiology 159, 411 (1986). 10. P. van Dijk et al, abstract Soc. of Mag. Res. in Med., 5th Annual Meeting (Montreal, Aug 19-22, 1986) , p. 625. 11. P. Mansfield, J. Phys. C: Solid State Phys. 10, L55 (1977). 12. P. Van der Meulen, J. P. Groen, and J. J. M. Cuppen, Mag. Res. Imag- ing 3, 297 (1985). 13. A. Oppelt et al. , Electromedica 54, 15 (1986). 14. J. Frahm, A. Haase, and D. Matthaei, J. Comput. Asst. Tomogr. 10, 363 (1986). 15. S. K. Hilalef al, J. Comput. Asst. Tomogr. 9, 1 (1985). 16. S. R. Thomas, SPIE Proc. 626, Medicin e XIV and PACS IV, edited by R. H. Schneider and S. J. Dwyer III, 7(1986). 17. W. T. Dixon, Radiology 153, 189 (1984). PHYSICS TODAY / JANUARY 1987 S-43PHYSICS NEWS IN 1986—MEDICAL PHYSICS Dual-Energy Chest Radiography The concept of utilizing two photon energies to obtain informa- tion on tissue characteristics was first suggested by Jacobson in 1953.' Dual-photon absorptiometry is currently in widespread use for bone mineral analysis and the idea has also been exploit- ed in a variety of x-ray imaging research projects during the last 15 years.2~* Most recently it has been applied to chest radiogra- phy, and a prototype unit (designated ESU, for energy subtrac- tion unit) has been under clinical trial at the University of Ala- bama at Birmingham Medical Center for the past two years. Dual-energy radiography utilizes both x-ray intensity and spectral information to decompose patient images into water (soft tissue) and bone (mineral) components. By contrast, con- ventional (single-energy) radiography utilizes only intensity information and is not capable of tissue decomposition. A help- ful analogy is to think of conventional radiography as black andwhite imaging and dual-energy radiography as color imaging. Just as the human visual response may be fully characterized in terms of three primary colors, x-ray attenuatio n (in biological tissue and in the 20 to 150 keV diagnostic energy range) may be fully characterized in terms of two attenuatio n processes— Compton scattering and photoelectric absorption or, alterna- tively, in terms of low atomic number (water) and high atomic number (bone) tissues.5 The first step in dual-energy imaging is to generate two si- multaneous images of the patient, identical in every respect ex- cept that one uses an x-ray spectrum with a relatively high mean energy (approximately 50 keV). These images are then decom- posed point by point into water and bone basis (component) images whose intensity values are proportiona l to the overlying thicknesses of water and bone. A bone-cancelled (soft tissue) image is then generated by adding a small fraction of the bone image to the water image to fill in the voids or "shadows " left by FIG. 3. Dual-energy chest radiography images: (a) soft tissue patient with calcined nodules in the left lung).-lung window, (b) soft tissue—mediastinum window, (c) and bone, (d) images of a S-44 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1986—MEDICAL PHYSICS the removal of bone. A final image, analogous to a conventional film and referred to as a normal or single energy image is gener- ated by adding the original low and high energy images togeth- er. In the ESU all these images are generated from a single pa- tient scan at a radiation level comparable to that employed in conventional chest radiography. The normal, bone cancelled, and soft tissue cancelled images (see Fig. 3) are presented to the radiologists simultaneously. Of practical interest is the fact that the bone-cancelled image shows only soft tissue structures; the ribs and spine are completely invisible. The bone image, on the other hand, shows the ribs and spine but on soft tissue struc- tures. Furthermore, the bone image clearly shows the mineral deposits occurring in the calcified lung nodules (see Fig 3d). The obvious advantage of dual-energy imaging in chest radi- ography is that it separates the two types of anatomical informa- tion present and provides the radiologist with simpler images to view. Thus, a nodule which may be hidden by the ribs of a conventional film is clearly visible on the soft tissue image. Re- cent results indicate that a dramatic improvement is obtained in nodule detection with the ESU compared with conventional chest films.6 To put this improvement in perspective, it should be noted that prior to this, nodule detection rates changed very little over the last 30 years, and as many as one in three nodules was missed which, retrospectively, could have been detected. Furthermore, lung cancer is the leading cause of cancer death in the United States, and its incidence is increasing, especially among women. Early detection and removal of cancerous lung tissue is the most effective treatment, and chest radiography is the primary tech- nique for early detection. Since a relatively large percentage of the U.S. population undergoes chest radiography, this capabili- ty offers great promise to decrease the mortality of the disease. In addition to detecting more nodules than conventional sys- tems, dual-energy chest radiography reveals nodule calcifica- tions with greater sensitivity and accuracy than other noninva- sive techniques.7 This is extremely important, since the presence of calcium is a primary radiographic criterion for ben- ignancy. In many cases, the ESU eliminates the need for nodule biopsy as well as the need to perform more expensive, higherradiation dose examinations. Not only is the calcium more evi- dent with the ESU unit, but it also can be quantified to a high degree of accuracy. However, at this time, the importance of this capability has not been clinically established. The remarkable performance of the ESU is a result of a syn- ergistic combination of x-ray imaging technologies which ma- tured over the last decade, including scanning fan-beam geome- try, linear array solid-state x-ray sensors, laser film writers, and dual-energy x-ray detection techniques.8 Prior to the dual-ener- gy detector development, the different x-ray photon energy in- formation was obtained by switching the x-ray tube voltage, typically between 85 and 135 kV. Since a time delay occurs between the low and high energy pulses, patient motion and misregistration artifacts can occur. These problems were eli- minated by employing a dual-energy x-ray detector and a con- stant (high) x-ray tube voltage. In addition to detecting pulmonary nodules and being able to differentiate between benign and malignant nodules, dual-ener- gy radiography holds considerable promise in several other areas of diagnosis. Studies are underway to evaluate its utility in mammography, measurement of bone mineral content (For de- tection and monitoring of osteoporosis), and evaluation of me- tastatic calcification of the lungs in kidney disease. R.A. Sones and M.M. Tesic, Picker International and G. T. Barnes, University of Alabama at Birmingham 1. B. Jacobson, Acta Radiol. 39, 436 (1953). 2. C. A. Mistretta, M. G. Ort, F. Kelcz, J. R. Cameron, M. P. Siedband, and A. B. Crummy, Invest. Radiol. 8, 402 (1973). 3. A. L. Hallef a/.,Proc. SPIE314, 155 (1981). 4. G. S. Keyes, S. J. Riederer, B. F. Belanger, W. R Brody, Proc. SPIE 347, 34(1982). 5. L. A. Lehmann, R. E. Alvarez, A. Macovski, and W. R. Brody, Med. Phys. 5, 659 (1981). 6. L. T. Niklason, N. M. Hickey, D. P. Chakraborty, E. A. Sabbagh, M. V. Yester, R. G. Fraser, and G. T. Barnes , Radiology 160, 589 (1986). 7. R. G. Fraser eta/., Radiology 160, 595 (1986). 8. G. T. Barnes, R. A. Sones, M. M. Tesic, D. R. Morgan, and J. N. Sanders, Radiology 156, 537 (1985). PHYSICS TODAY / JANUARY 1987 S-45NUCLEAR PHYSICS Sharp Positron Peaks In recent scattering experiments, collisions between such heavy nuclei as uranium, thorium, and curium have resulted in the production of positrons (antimatter counterparts of electrons). Such experiments (see Physics News in 1985, p. 50) were motivated in part by the desire to study "vacuum sparking," a phenomenon—predicted by quantum electro- dynamics, the theory which describes the electromagnetic force—in which the collision between two heavy nuclei creates for a fleeting moment (10~21 s) a nucleus with a charge Z of 173 or more. The electrostatic field of such a "supercritical" nucleus is so strong (1020 V/m), that an electron-positron pair ought to be spontaneously created out of the surrounding vacuum. The electron becomes bound to the nucleus, while the positron would presumably be emitted and recorded in detectors surrounding the collision area. Physicists at the Gesellschaft fur Schwerionenforschung (GSI) Laboratory in Darmstadt, West Germany did indeed observe the emission of positrons in the heavy-ion collisions, but were immediatel y puzzled by the narrowness of the peaks in plots of positron yield versus positron energy. '"3 That is, for a particular collision combination—say, between uranium and curium, with a combined Z of 188—a plot of the number of positrons observed at different energies exhib- ited a peak at an energy of 300 keV. The width of this peak was 75 keV, much narrower than one would expect if vacu- um sparking were the source of the positrons. Jack S. Greenberg of Yale University, a leade r of one of the GSI experiments, has reported that another surprise was the lack of any Z dependence: different combinations of heavy nuclei, with different composite Z 's, were tried, but each time a positron energy peak of 300 keV appeared in the plots. Still another surprise was the observation of positrons in collisions for which Z was actually below the critical value of 173. Also, in some scattering events, electrons with the same narrow energy peaks were observed.4 All of these facts sug- gested to Greenberg and to others that some phenomenon other than (or in addition to) vacuum sparking was at work. The most interesting speculation is that the positrons and electron s are the decay products of a previously unobserved neutral particle with a mass of about three times the electron mass. Such a particle would be of great interest to particle physicists since there would have been an obvious place for it in the generally accepted "standard model" of particle inter- actions. Phillip F. Scheme, American Institute of Physics 1. J.Schweppe<?ra/.,Phys. Rev. Lett. 51, 2261 (1983). 2. M. Clemente et al, Phys. Lett. 137B, 41 (1984). 3. T. Cowan eta/., Phys. Rev. Lett. 54, 1761 (1985). 4. T. Cowan et al., Phys. Rev. Lett. 56, 444 (1986). S-46 PHYSICS TODAY / JANUARY 1987Double Beta Decay and Nuclear Structure Calculations Properties of neutrinos, unlike those of other elementary particles, are often more conveniently studied at low ener- gies, using the techniques of nuclear physics, rather than at high energy accelerators. Double beta decay is an example of such an approach. Its study gives us indirect but very sensi- tive information about the mass and interactions of neu- trinos. Neutrinos are perhaps the most elusive of the elementary particles. Their existence was already postulated by Pauli in 1930, but was not experimentally verified until 25 years lat- er. Nowadays, observation of neutrinos is a rather routine matter, but many of their fundamental properties are still unknown. So, we do not know whether neutrinos are mas- sive or massless, or whether neutrinos and their antiparticles (antineutrinos) are distinct or identical. Other particles, such as electrons, are clearly distinct from their antiparti- cles: they are so-called Dirac particles. Neutrinos might be "ultimately" neutral, that is, identical with their antiparti - cles, and thus would represent the first realization of a so- called Majorana particle. If we could show that neutrinos are indeed massive, as a so far unverified Soviet experiment of 1980 suggests, then our basic ideas about elementary parti- cles and their interactions would have to be revised; the cos- mological consequences would be also profound. Beta decay is a process in which one of the neutrons bound in a nucleus spontaneously decay s into a proton (bound in the final nucleau), an electron, and an antineutri- no. Double beta decay is a simultaneous decay of a pair of neutrons into a pair of protons plus a pair of electrons and a pair of antineutrinos. Now, if neutrinos are Majorana parti- cles, another mode of double beta decay is possible. One can imagin e that the pair of antineutrinos is equivalent to a neu- trino plus its antiparticle which can, therefore, annihilate each other, leaving only a pair of electrons and no neutrinos as the decay products. Observations of such a "neutrinoless" double beta decay would be a proof that neutrinos are indeed Majorana particles. More detailed analysis shows that, moreover, the decay is possible only when neutrinos are mas- sive and that the decay rate is proportional to the square of the neutrino mass. Large experimental efforts are now de- voted to the study of both "modes" of the double beta de- cay.1 The "neutrinoless" double beta decay has not yet been observed, but one knows that the corresponding halflives are very long; for example, in the case of germanium-76 it is longer than 1023 years (for comparison, the universe is about 1.5 times 109 years old).2 In order to interpret correctly the results of experiments on double beta decay, one has to solve some important nu- clear structure problems. The decay lifetimes depend not only on the neutrino mass, but also on the details of thePHYSICS NEWS IN 1986—NUCLEAR PHYSICS structure of the initial and final nuclei (the so-called nuclear matrix elements). The reported work deals with calculation of these matrix elements. As a gauge of our ability to do the job correctly, we calculate the double beta decay mode with two electrons and two antineutrinos in the final state (a mode which has been observed in a few cases). The rather complex computer-assisted calculations lead to decay rates for both modes of the double beta decay for a number of nuclei.3 The calculated decay rates for the two- neutrino mode are faster than the observed ones. At the pres- ent time the discrepancy, noted also by other theorists, is not fully understood although an important mechanism which suppresses the nuclear matrix elements aand therefore pro- longs the calculate d lifetime has been found.3 In this work we attempt to relate the double beta decay to other processes which could be observed in a laboratory and which could give us a clue as to how to proceed further. At the same time the calculation has been extended to more nuclei, many of which are being studied experimentally now.Despite the noted discrepancy, the calculations allow us to put rather stringent upper limits on the mass of neutrinos, if they are indeed Majorana particles. That limit is substan- tially lower (approximately one tenth) than the mass value implied by the Soviet experiment. Thus, we have to conclude that either the experiment is incorrect or that the neutrino is not a Majorana particle. (Most theories which suggest that neutrinos are massive would prefer them to be Majorana particles.) Petr Vogel, California Institute of Technology 1. W. C. Haxton and G. J. Stephenson , Jr., Prog. Particle and Nucl. Phys. 12, 409 (1984); M. Doi, T. Kotani, and E. Takasugi, Prog. Theor. Phys. Suppl. 83 (1985). 2. D. O. Caldwell et a!., Phys. Nev. D33, 2737 (1986). 3. P. Vogel and P. Fisher, Phys. Rev. C32, 1362 (1985); P. Vogel and M. R. Zirnbauer (to be published). OPTICS Experimental Observations of Laser Cooling and Trapping of Atoms Since 1970 there has been great interest in the study of the forces exerted by laser light on atoms. Considerable atten- tion has been devoted to methods of using these forces to construct "optical traps" in which atoms could be confined for long periods of time. During 1986, a group of scientists at AT&T Bell Laboratories in Holmdel, New Jersey, demon- strated the optical trapping of atoms for the first time.1 Optical potential wells for atoms are very shallow and only extremely "cold" atoms can be confined in them. Be- fore an optical atom trap could be demonstrated it was nec- essary to learn how to create the collectio n of ultra-cold atoms needed to load it. In early 1985 the Bell Labs group demonstrated a technique for doing this using three sets of mutually orthogonal laser beams tuned half a linewidth be- low a resonance transition of the atom.2 In this configura- tion the laser beams exert strong viscous damping forces on the atoms and the atomic motion is cooled down to a tem- perature which is proportional to the natural linewidth of the transition. In the region where the laser beams intersect, the ultra-cold atoms execut e a random-walk motion that re- sembles the Brownia n motion of a particle in a highly viscous fluid. For this reason, the viscous "medium" of photons is called "optical molasses." Approximately 106 sodium atoms were cooled to 240 fiK (approximately the theoreticallimit for the transition used) and confined in a region of about 0.5 cm3 for a period of about 0.5 s. The optical trap demonstrated in early 1986 is comprised of a single, strongly-focused Gaussian laser beam tuned ATOMSINMOLASSES FOCUSEDTRAPBEAMATOMSINTRAP FIG. 1. Schematic diagram of the interaction region inside the vacuum chamber used for trapping atoms. The broad arrows represent the collimat- ed laser beams (about 1 cm in diameter) which intersect to form "optical molasses." The shaded spher e represents the fluorescence emitted by the collection of ultra-cold atoms contained and executing random-walk mo- tion withi n the optical molasses. The optical trap is formed just beyond the focus of the trap laser beam, which is also shown. The black dot represents the intense fluorescence emitted by the dense collection of atoms confined withi n the optical trap. PHYSICS TODAY / JANUARY 1987 S-47PHYSICS NEWS IN 1986—OPTICS about 104 natural linewidths below the atomic resonance.1 The trap relies on the "dipole force" of resonance radiation pressure, which is proportional to the gradient of the light intensity. For tuning below the resonance transition, the force is such that atoms seek out regions of greatest light intensity. This "single-beam, gradient-force trap" was first proposed by Ashkin in 1978.3 In the experiment , the trap laser beam was focused into a collection of atoms confined and cooled by optical molasses. The trapped atoms were observed as an intense, small spot of fluorescence within the much more diffuse light scattered from atoms in the molasses. The use of optical molasses was crucial to the success of the trapping experiment. First, ul- tra-cold atoms are needed to load the trap, which is only 5 mK deep. Second, the random walk motion and long storage time of atom s in optical molasses allow the atoms to continu- ously diffuse to the trap surface and be captured (see Fig. 1). Thus, considerably more atoms can be trapped than the number given by the molasses densit y times the trap volume. Finally, optical molasses was also used to provide cooling for trapped atoms. In the absence of this additional cooling mechanism, trapped atoms heat up and boil out of the trap in times on the order of 5 ms. Using 200 mW of trap laser power, about 500 atoms were trapped in a volume of about 10~9 cm3 for periods of several seconds. The atomic densit y in the trap was approximately 5x 1011 cm"3, about 106 times the molasses density. The trap lifetime appear s presently to be limited by the 2 X 10 "9 torr background pressure within the vacuum chamber; the innate trap lifetime was calculated to be about 104 s. The temperature of the trapped atoms was inferred to be about 350 fiK, only slightly hotter than atoms in optical molasses. The trapped atoms are confined transversely within 2 //m and the loaded trap can be rapidly moved without a signifi- cant loss of trapped atoms by simply redirecting the laser beam. The trap serves as "optical tweezers", giving one the ability to localize and manipulate ultra-cold atoms in a pre- cise way. The collection of trapped atoms comprise an interesting gas. The high densit y and ultra-low temperature represents a previously inaccessible regime for gaseous matter. Much higher densities and lower temperatures seem readily attain- able. It appears that these dense, ultra-cold gas samples will be ideal candidates for a variety of experiments in which fundamental atom-atom and atom-trap interactions are in- vestigated. /. E. Bjorkholm, AT&T Bell Laboratories 1. Steven Chu, J.E. Bjorkholm, A. Ashkin, and A. Cable, Phys. Rev. Lett. 57,314 (1986). 2. Steven Chu, L. Hollberg, J.E. Bjorkholm, A. Cable, and A. Ashkin, Phys. Rev. Lett. 55, 48 (1985). 3. A. Ashkin, Phys. Rev. Lett. 40, 729 (1978).Scattering of Free Electrons by Light Since the invention of the laser, a major focus of optical re- search has been the study of the interaction of light with single elementary particles. The effect of isolated particles on light in the form of scattering was observed in early experi- ments, but the complementary observation of the effect of the light on single particles has been very difficult to achieve and often ambiguous in interpretation. Recently researchers from AT&T Bell Laboratories and the University of Mary- land have demonstrated large angle scattering of free elec- trons from a focused laser beam.1 The experiment involves the production of low energy (0.5-10 eV) electrons by multiphoton ionization of xenon gas. The electrons were produced by high order processes (absorption of 11 or more photons) at very low densities so that there were no effects from collisions or space charge. Observations of angular distributions of the photoelectrons at low intensities established that the electrons had initial velocities which were highly collimated along the direction of the laser's polarization vector. As the intensity of the light was increased, large angle electron scattering resulted. The initial beam collimation (less than 15°) was destroyed and the distribution became nearly isotropic. Classically, the scattering is that of a charged particle off the electric and magnetic fields of the light, a phenomenon theoretically described by quantum electrodynamics in terms of stimulated Compton scattering. The first sugges- tion for experiments of this sort was made by Kapitza and Dirac in 19332 and a clear theoretical basis for an under- standing of the Bell experiments was provided by Kibble in 1966.3 Kibble showed that the behavior of a free electron in a radiation field can be described relativistically in terms of the motion of a particle with constant energy that depends on the local light intensity. The change in rest mass energy is exactl y given by the classical "ponderomotive" potential. This potential energy is the result of the oscillatory "quiver" energy of a free electron in the light field. For 1064 nm light focused to an intensity of 1013 Wcm"2, this amounts to a 1 eV potential. The electron moves under a force given by the gradient of the potential. The Bell-Maryland researchers studied the angular dis- tribution of electrons at fixed energies as a function of light intensity. The ponderomotive potential at the focus was var- ied from 0.5 to 5 eV and noncylindrical potentials were pro- duced using elliptical beams. The final trajectories depended on the distribution of the light intensity producing the con- servative force field as well as the initial momentum of the electrons, so that the angular distribution was a function of both the electron energy and the light intensity. A computer model of the ponderomotive force field was used to calculate the trajectories of the electrons as they left the region of focus and the model provided excellent predictions of the intensity S-48 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1986—OPTICS and energy behavior of the distributions as well as the de- pendence on the shape of the focused spot. With the production of plasmas using intense laser beams in order to produce coherent and noncoherent x-ray sources, the interaction of free electrons with light has become of interest beyond the theoretical regime. On a wider scale, the development of free electron lasers focuses attention on this interaction, which is at the heart of their operation, while optical accelerators are viewed by some as the only hope to reach the next domain of energy for elementary particle physics. These experiments on electron light scattering, for the first time, provide an easy and sensitive tool for studying the fundamental nature of this process with spatially com- plex, time varying radiation. R. R. Freeman, P. H. Bucksbaum, and M. Bashkansky, AT&T Bell Laboratories and T. J. Mcllrath, University of Maryland 1. R. R. Freeman etal, Phys. Rev. Lett, (to be published). 2. P. L. Kapitza and P. A. M. Dirac, Proc. Cambridge Phil. Soc. 29, 297 (1933). 3. T. W. B. Kibble, Phys. Rev. 150, 1060 (1966). Binary Optics: An Emerging Diffractive Optics Technology Sheet optics by the square meter, high-quality throw-away optical sensors and arrays of hundreds of small modular la- sers coherently added to form powerful laser beams—these are some of the promises of an emerging binary optics tech- nology. What's so binary about this optics? Binary optics refers to the two-level (high-low) nature of the phase relief patterns used to control the phase, the amplitude, and the polariza- tion of an optical wavefront. The benefit of two-level relief structures is access to the entire integrated circuit (VLSI/VHISIC) technology effort of the electronics indus- try. All lithographic and dry-etching technologies developed IIIIHIl IIIIMIII IIJ till II minimiiimiiii ilium minim iiimnii iiiiimni iiimnii iiimnii i i in i iinnii \\ MONOLITHIC DIODE USER ARRAYLENS-GRATING STRUCTUREEXTERNAL CAVITY MIRROR FIG. 2. A laser diode array with micro-turning mirrors etched in the surface produces a mesh of beams. A lens-binary-holographic-grating combination placed in an external cavity provides optical feedback and locks the laser sources in to appropriate phase states. The output is a coherent sum of the modular laser beams.by that industry apply directly to the fabricatio n of binary optical components. As in integrated circuit fabrication, with binary optics it is possible to produce many optical ele- ments from a single mask. These optical elements can be planar, low-cost, light-weigh t replacements of conventional optical elements such as lenses and scanners, or they can correspond to unique optical functions not feasible with con- ventional optics. The simple, elegant properties of binary optic s comple- ment conventional optics. Among these properties are the easy formation of multiplexers and phased-array structures and the simplicity of implementing spatial phase distribu- tions in the form of lithographically etche d relief patterns under computer control. Fundamental optical principles stay the same, but a shift from refractive to diffractive elements and the improved ac- curacy in the fabricatio n of planar phase relief structures resulting from the blending of high-resolution lithography and reactive ion-beam etching technologie s have made it fea- sible to generate two-dimensional optical transfer functions accurately and efficiently. Present high-voltage electron beams can write lithographic patterns with better than 300 A accuracy and resolution. High materials cost of convention- al infrared optic s and ease of binary mask fabricatio n make infrared optical elements particularly suited for replacement by binary optics. Infrared optical transfer functions such as beam shaping, beam multiplexing, beam steering, focusing, filtering, and scanning can now be accurately implemented on thin transmissive or reflective substrates and can exhibit very high (greater than 95%) diffraction efficiency. Researchers at the MIT/Lincoln Laboratory have used binary lenses in a CO2 laser radar telescope to image objects at ranges of a few kilometers. For monochromatic laser op- eration, far off-axis lenses are appropriate.1 The advantage of binary optics, however, is in infrared thermal imaging applications where on-axis lens operation is essential. In these applications multi-level lens structures can maintain the high diffraction efficiency. For non-imaging applica- tions, very large apertures can be assembled from planar- etched metal or kapton-embossed components. Present-day diffractive optics technology makes one-meter-diameter planar sub-apertures feasible. At the small end of the scale, arrays of micro-lenses, deep diffractive structures for agile beam steerin g and very high-spee d non-contiguous scanners on spinning thin substrates are recognize d applications. One application exploiting binary optics' unique diffrac- tive properties is in multi-branched laser cavities that coher- ently add one- and two-dimensional arrays of lasers. Re- searchers in the DARPA (Defense Advanced Research Projects Agency) sponsored binary optics program tested several techniques with GaAs, CO2 and HeNe laser arrays.2 They believe it is feasible to add coherently a few hundred modular lasers. The modular laser array technology permits high laser power from small sources at low cost and avoids catastrophic failures. PHYSICS TODAY / JANUARY 1987 S 49PHYSICS NEWS IN 1986—OPTICS Aperture filling with arrays of lasers already phase- locked by evanescent coupling is a related technique devel- oped at Lincoln (see Fig. 2). Here binary optical elements convert an array of laser sources (with associated strong far- field sidelobes) into a uniformly illuminated output aperture with near-perfect efficiency. Other novel applications of binary optic s at Lincoln are broadband anti-reflection coatings and the mixing of refrac- tive or reflective elements with diffractive components for more design flexibility in electro-optical sensors. Wilfrid Veldkamp, MIT/Lincoln Laboratory 1. G. J. Swanson, and W. B. Veldkamp, Opt. Eng. 24, 791 (1985). 2. J. R. Leger, G. J. Swanson, and W. B. Veldkamp, Appl. Phys. Lett. 48, 888 (1986). Tunneling and Photoconductivity Photoconductivity is a widesprea d phenomenon in semicon- ductors and has been known and understood for several dec- ades. ' If the lifetime of the photogenerated carriers exceeds the transit time of the majority carrier between the two oh- mic contacts, photoconductivity is accompanied by current gain. The latter is given by the ratio of the electron-hole pair lifetime to the transit time. The response time of the photo- conductor is primarily controlled by the lifetime, while the gain-bandwidth product is simply given by the reciprocal of the transit time. The performance of a "classical" photocon- ductive detector is therefor e determined by bulk properties such as mobilities and lifetimes. Recentl y a striking new type of photoconductivity based on quantum mechanical tunneling in superlattices has been discovered at AT&T Bell Laboratories.2 The underlying physical cause of this phenomenon is "effective mass filter- ing." Since the tunneling probability of carriers through the barrier layer s of a superlattice increases exponentially with decreasing effective mass , electrons are transported through a superlattice much more easily than the heavy holes. When light is shined on the superlattice, the photogenerated holes essentially remain localized in the wells, while electrons can tunnel through the barriers. Thus the superlattice acts as a filter for effective masses. This gives rise to a new photocon- ductive gain mechanism controlled by tunneling.23 Since the electron mobility perpendicular to the layer s depend s exponentially on the superlattice barrier thickness, it follows that the electron transit time, the photoconductive gain and the gain-bandwidth product can be artificially var- ied over a wide range. This offers great versatility in device design not available in standard photoconductors. High per- formance infrared photoconductors utilizing effective mass filtering have recentl y been demonstrated at AT&T Bell Laboratories.3 The devices grown by molecular beam epi- taxy consisted of 100 periods of AlInAs (35 A)/GaInAs (35 S 50 PHYSICS TODAY / JANUARY 1987A) between two contact layers and responded to wave- lengths in the 1.6-1.0 //m region. These detectors exhibit high current gain (up to 2 X 104 ) at very low voltage (less than 1 V) and response times in the 10 ~4 s range. The low bias operation cuts down on the device noise , which was only about 8 X 10"14 W/Hz~1/2. These are the highest gain and lowest noise photoconductors operating at such low voltage. Another important advance by the Bell Labs group is the first observation of sequential resonant tunneling through a superlattice.4 If the quantum wells are weakly coupled (rel- atively thick barriers) the states of a superlattice are well described by the quantum-mechanical quasi-eigenstates of the individual wells. Suppose now that a uniform electric field is applied to the superlattice. As the bias is increased, at some point the energy potential drop across the superlattice period equals the energy difference between the first two levels of the quantum well. At this point, resonant tunneling occurs between the ground state of the «th well and the first excited state of the (n + l)th well followed by intra-well energy relaxation by phonons. This process is repeated se- quentially through the superlattice. The Bell Labs workers observed two pronounced peaks in the photocurrent of a reverse-biased MBE grown p+ i n + diode. The low doped / region had 35 periods of AlInAs (135 A)/GainAs (135 A). The voltage position of the two peaks divided by the numbers of periods (35) gave exactly the calcu- lated energy differences between the first two excited states and the ground states of the quantum wells. This provides direct evidence of sequential resonant tunneling through the superlat- tice. There are some potentially very important device applica- tions of this effect. By appropriate realization of a population inversion between the first two excited states of the quantum wells, electrons, upon tunneling into the third energy level of a well, can be made to emit a laser photon, rather than relaxing the energy via phonons. Such a semiconductor laser, first pro- posed by Kazarinov and Suris,5 will emit in the infrared region of the spectrum between 5 to 10 fim (depending on the well thickness). Federico Capasso, AT&TBell Laboratories 1. A. Rose, Concepts in Photoconductivity and Allied Problems (Wiley, New York, 1983). 2. F. Capasso, K. Mohammed, A. Y. Cho, R. Hull, and A. L. Hutchinson, Phys. Rev. Lett. 55, 1152 (1986). 3. F. Capasso, K. Mohammed, A. Y. Cho, R. Hull, and A. L. Hutchinson, Appl. Phys. Lett. 47, 420 (1985). 4. F. Capasso, K. Mohammed, and A. Y. Cho, Appl. Phys Lett 48, 478 (1986). 5. R. F. Kazarinov and R. A. Suris, Sov. Phys. Semicond. 5, 707 (1971). Photon Localizatio n It has been appreciated for some time that the transport of elec- trons in a strongly disordered material will be governed by mul- tiple scattering of the electron wave function. Ultimately, if the scattering is sufficiently strong that the elastic mean free path isPHYSICS NEWS IN 1986—OPTICS comparable to the de Broglie wavelength (equal to Planck's constant divided by the electron's momentum), conduction will cease at a temperature of absolute zero.1 Before this strong lo- calization, or "Anderson localization," takes place, multiple scattering is thought to lead to "weak localization". Weak local- ization2 manifests itself by resistive corrections to the conduc- tivity at low temperatures and is caused by "coherent back- scattering" of the electron wave. Photon propagation in disordered dielectrics and electron localization in disordered conductors have both been shown to be governed by coherent backscattering. This enhanced scattering in the precise back- ward direction arises from the fact that the time-reversed scat- tering paths for the wave are in phase in this direction, and this results in a doubling of the scattering in a narrow range of angles about the backscattering direction. Researchers who have focused on electromagnetic wave propagation in disordered media have appreciated for some time that the cross section for multiple scattering of electromag- netic waves is enhanced in the precise backscattering direction. Ishimaru and collaborators at the University of Washington, who research photon propagation in disordered media, early reported the presence of coherent backscattering in media dominated by multiple scattering.3'4 But it was only recently that physicists in tune with the problems of electron localiza- tion, discovered the rather strict analogies between electron and photon coherent backscattering. Two groups of researchers, van Albada and Lagendijk5 at the University of Amsterdam, and Wolf and Maynard6 at Grenoble, independently reported experiments that revealed coherent backscattering peaks from dielectric spheres suspended in a fluid medium and pointed to the close analogy with electron weak localization. The most recent developments in electron localization have focused on so called universal fluctuations of the resistance ofsmall devices.7 Here it was pointed out that owing to the build- up of correlation through multiple scattering, the conductance of a microscopic sample will fluctuate in a universal fashion, that is, in a way which is independent of size and degree of disorder. Etemad and his co-workers at Bell Communications Research showed that photon "coherent backscattering" in rig- id media is dominated by fluctuations as well.8 The backscatter- ing peak was completely swamped by fluctuations in the scat- tered intensity, but they were able to recover the backscattering peak after ensemble averaging. There remains one glaring omission in the experimental studies of photon localization, especially as they mirror the properties of electrons. No one has reported a disordered media, without loss, in which scattering is sufficient enough that the mean free path for photons becomes comparable to the wave- length. Such a medium is required for the strong localization of photons. Nonetheless, the observation of the photon back- scattering peak, the appreciation of the critical role of ensemble averaging, and the rigorous analogy with electron transport in disordered conductors has both broadened and unified our view of wave phenomena in disordered materials. S. James Allen, Bell Communications Research 1. A. F. Ioffe and A. R. Regel, Prog. Semicond. 4, 237 (1960). 2. G. Bergman, Phys. Rep. 107, 1 (1984). 3. Y. Kuga and A. Ishimaru, J. Opt. Soc. Am. Al, 831 (1984). 4. L. Tsang and A. Ishimaru, J. Opt. Soc. Am. Al, 836 (1984). 5. M. P. van Albada and A. Lagendijk, Phys. Rev. Lett. 55, 2692 (1985). 6. P. E. Wolf and G. Maret, Phys. Rev. Lett. 55, 2696 (1985). 7. P. A. Lee and A. D. Stone, Phys. Rev. Lett. 55, 1622 (1985). 8. S. Etemad, R. A. Thompson, and M. Andrejco, Phys. Rev. Lett. 57, (1986). PHYSICS APPLIED TO INDUSTRY- For the past several years Physics News has had a chapter on physics in industry, thus highlighting the fact that a substan- tial number of physicist s are employed in industry and that their research is motivated at least in part by potential appli- cations in the area of new devices, new materials, or new diagnostic techniques. Physicists in industry do research in a broad range of physics subdivisions (such as condensed mat- ter physics, optics, polymer physics, and vacuum physics), and these overlap the conventional discipline-oriented sub- divisions which make up most of the other chapters in Phys- ics News. Thus, some "industrial research" is reported in those categories. In the present chapter we call attention to those develop- ments for which there is a clear industrial application or in which the major share of current research is being carried out in industrial laboratories. John R. Reitz, Ford Motor CompanyResearch in Scanning Tunneling Microscopy Scanning Tunneling Micros copy (STM) is a rapidly grow- ing field, with research groups at both universities and indus- trial research labs. STM is based on the pioneering efforts of Binnig, Rohrer, Gerber, and Weibel of the 1MB Zurich Re- search Laboratories.1'2 The STM instrument is based on electron tunneling (a quantum phenomenon) across an ad- justable vacuum gap maintained between a conducting tun- nel tip and a conducting surface. The atomic structure of the surface may be mapped by moving the tip over the surface using precision actuators. Atomic resolution imaging is pos- sible with STM since the electron tunneling phenomena, de- pending on overlap of tip and sample wave functions, probes a region of atomic dimensions. STM has already been ap- plied to a variety of different important systems. PHYSICS TODAY / JANUARY 1987 S 51PHYSICS NEWS IN 1 986—PHYSICS APPLIED TO INDUSTRY STM is expected to be an important contributor to indus- trial research. In particular, the development of semicon- ductor devices should be influenced by the capabilities of STM for investigation of device materials and structures. Considerable progress has been made in STM during 1986 by industrial investigators. Major advances have been made in the understanding of metal and semiconductor surfaces using STM imaging methods. Also, new STM devices with improved capabilities have been developed by several labo- ratories. Recently, at several industrial labs (IBM, AT&T, and Ford), STM spectroscopy techniques have been developed for the study of surface electronic structure.3"6 In these methods, a spectrum of tunnel current signal as a function of tunnel voltage is measured. The current-voltage spectrum reveals image potential states in the vacuum region at the surface3'4 and also intrinsic surface states.5'6 The surface electroni c structure imaging methods developed by Binnig7 have been extended to detaile d investigation s of several se- miconductor surfaces.8'9 These methods reveal the distribu- tion of surface states at the Si (111) and Si (100) surface for both vacuum-annealed crystals8 and cleaved samples.9 This new capability in STM allows, for the first time, the study of the contribution made by individual surface atoms, adsor- bates, and defects to surface electronic structure. The first high-resolution STM images of the Au (111) surface were obtained in 1986.10 This is a classic surface which has been studied for many years using standard sur- face physics techniques, including electron and ion scatter- ing methods. The STM images reveal the presence of an or- dered step reconstruction extending over a large (500 A by 1500 A) area. Arrays of large amplitude steps (7 atom lay- ers) are also seen. The observation of large amplitude steps has important implications for the mechanism of material transport at surfaces and for the epitaxy of materials. A collaboration of industrial and university researchers resulted in the development of a novel microscope based on STM. This new device, the Atomic Force Microscope (AFM), employs an STM apparatus for detection of very weak forces.'' This implementation of the STM is the most sensitive force detector available. The first version of the AFM measures forces between a sharp stylus and the surface under investigation. An image of an insulating Al2 O3 sur- face has been obtained with the AFM by mapping contours of constant force between a diamond stylus and the surface. The first ATM has a force sensitivity in the range of interato- mic forces. Also, the spatia l resolution obtained in the proto- type AFM images is comparable to STM images . AFM imaging is expected to reveal entirely new physica l proper- ties of surfaces relating to direct measurement of forces between surface and tip atoms. Progress in 1986 in STM and AFM imaging has a direct impact on industrial physics problems in the areas of surface technolog y and microelectronics. Since the engineerin g of catalysts and load-bearing surfaces relies on surface analyti- S-52 PHYSICS TODAY / JANUARY 1987cal tools, new STM electronic structure imaging methods should directly influence these important areas. In addition, many problems in industrial physics require the investiga- tion of insulating films for microelectronic and other appli- cations. The development of the AFM provides a means for high-resolution studies of these important insulators. The rapidly growin g capabilities of STM technology are expect- ed to provide novel methods for investigating a variety of industrial physics problems. W.J. Kaiser and R. C. Jaklevic, Ford Motor Company 1. G. Binnig, H. Rohrer, C. Gerber, and E. Weibel, Appl. Phys. Lett. 40, 178 (1981). 2. G. Binnig and H. Rohrer, Helv. Phys. Acta 55, 725 (1982). 3. G. Binnig, K. H. Frank, H. Fuchs, N. Garcia, B. Reihl, H. Rohrer, F. Salvan, and A. R. Williams, Phys. Rev. Lett. 55, 991 (1985). 4. R. S. Becker, J. A. Golovchenko, and B. S. Swartzentruber, Phys. Rev. Lett. 55,987 (1985). 5. R. S. Becker, J. A. Golovchenko, D. R. Hamann, and B. S. Swartzen- truber, Phys. Rev. Lett. 55, 2032 (1985). 6. W. J. Kaiser and R. C. Jaklevic, IBM J. Res. Develop. 30, 411 (1986). 7. A. Baratoff, G. Binnig, H. Fuchs, F. Salvan, and E. Stoll, Surf. Sci. 168, 734(1986). 8. R. M. Feenstra, W. A. Thompson, and A.P. Fein, Phys. Rev. Lett. 56, 608 (1986). 9. R.J. Hamers, R.M. Tromp, and J.E. Demuth, Phys. Rev. Lett. 56,1972 (1986). 10. W. J. Kaiser and R. C. Jaklevic, Bull. Am. Phys. Soc. 31, 227 (1986) and Proc. of Scanning Tunneling Microscopy '86, Santiago, Spain (1986). 11. G. Binnig, C.F. Quate, and C. H. Gerber, Phys. Rev. Lett. 56, 930 (1986). Direct Observation of Ballistic Electron Transport in GaAs Ballistic , or collisionless, transport of electrons has been a feature of technology for a long time. It occurs in TV picture tubes, for example, where the residual gas pressure is too low for the few remaining gas molecules to scatter many elec- trons: the electron beam is ballistic from the time it leaves the cathode until it impinges on the picture surface. In a solid, however , the motion of electrons is generally characterized by frequent scattering in all directions from, for example, thermal vibrations of the lattice (phonons), impurities and defects. Thus the electron drift velocity in a solid is signifi- cantly lower than the maximum value that would be achie- vable in the absence of scattering. Recently scientists at IBM made the first definitive observation of ballistic electron transport in semiconductors. The average distance travelled between collisions, or mean free path, is a characteristic of every solid and ranges widely. In pure metallic gallium, for example, the mean free path was found to be greater than 1000/zm at temperatures below 2 K;' on the other hand, in silicon it is estimated to be only about 100 A,2 smaller by a factor of 100 000. This largePHYSICS NEWS IN 1986—PHYSICS APPLIED TO INDUSTRY difference results mainly from the different energy-band structures of the materials. If the size of material along the path of electron travel can be made comparable to the mean free path, then ballistic transport would be expected to oc- cur. However, in semiconductors, ballistic transport had been only inferred from the results of several experiments on small samples.3'4 Recently, modern fabrication techniques (such as molec- ular-beam epitaxy, which allows the growth of high-purity, single-crystal semiconductors) and electron energy spec- troscopy techniques5'6 (which allow the electron energy to be measured) have been combined to fabricate spectrometer devices which can measure the transport of electron in GaAs in detail. In experiments that employ these devices (also called hot-electro n transistors) ,7 hot electrons of known en- ergies are injected into one side of a semiconductor layer . They are collected after emerging from the other side of the layer if they have enough energy to surmount the top of a potential barrier. By adjustment of this barrier height with an external bias voltage, the energies of the collected elec- trons can be determined and compared to the injected val- ues. Using such spectrometer devices, Yokoyama8 and Levi9 observed quasiballistic transport (electrons that suffered a few scattering events) for some of the electrons injected into n-type doped GaAs layer s between 650 and 1000 A thick. By developing a more refined injector and spectrometer, the IBM researchers observed true ballistic transport in 300-A- thick GaAs layers for up to 75% of the injected hot electrons moving at the limiting velocity of 108 cm/s.10'11 Narrow electron distributions, some 60 MeV wide, with peak ener- gies equal to the injection energies, have been measured. From this, and other measurements, we learn that scattering events remove the other 25% of the electrons completel y from the ballistic distribution by causing large energy loss or momentum changes. Ballistic electron s in high-resolution spectrometer de- vices are a powerful tool for studying electron transport. In addition to the direct determination of the electron energies, these structure s allow the study of many relevant parameters that influence electron transport.11'12 Since ballistic elec- trons in GaAs can travel as fast as 108 cm/ s, about five times faster than in present GaAs devices, such studies might lead to improved high-speed devices. Mordehai Heiblum and Marshall I. Nathan, IBM T. J. Watson Research Center 1. R. J. von Gutfeld and A. H. Nethercot, Jr., Phys. Rev. Lett. 18, 855 (1967). 2. J. G. Ruch, IEEE Trans. Electron Devices ED-19, 652 (1972). 3. F. Capasso, R. E. Nahory, and M. A. Pollack, Solid-Stat e Electron. 22, 977(1979). 4. L. F. Eastman, R. Stall, D. Woodard, N. Dandekar, C. E. C. Wood, M. S. Shur, and K. Board, Electron. Lett. 16, 525 (1980).5. D. J. Bartelink, J. L. Moll, and N. Y. Mayer, Phys. Rev. 136, 972 (1963). 6. P. Hesto, J-F. Pone, and R. Castagne, Appl. Phys. Lett. 40,405 (1982). 7. M. Heiblum, Solid-State Electron. 24, 343 (1981), 8. N. Yokoyama et al., in Technical Digest of the Int. Electron Dev. Meet- ing (San Francisco, 1984), p. 532. 9. A. F. J. Levi, J. R. Hayes, P. M. Platzman, and W. Wiegmann, Phys. Rev. Lett. 55, 2071 (1985). 10. M. Heiblum, M. I. Nathan, D. C. Thomas, and C. M. Knoedler, Phys. Rev. Lett. 55, 2200 (1985). 11. M. Heiblum, I. M. Anderson, and C. M. Knoedler, Appl. Phys. Lett. 49,207 (1986). 12. M. Heiblum, E. Calleja, I. M. Anderson, W. P. Dumke, C. M. Knoedler, and L. Osterling, Phys. Rev. Lett. 56, 2854 (1986). Nuclear Magnetic Resonance Spin-Offs That such a weak quantum-mechanical phenomenon as nu- clear magnetic resonance (NMR) can have such profound impact on chemistry, physics, industry, and medicine bears testimony to the long-term practical value of fundamental research. Now NMR imaging technology, whose develop- ment was spearheaded by the desire for improved medica l diagnostic images, is finding new applications in such di- verse disciplines as geology, agronomy , botany, materials science, and microscopy. The versatility of the technique de- rives from the sensitivity of the NMR signal to a large variety of molecular-level parameters associated with the chemical and motional environments of nuclear spins. Hydrogen (!H) is ubiquitous and the obvious choice for NMR imaging of water and oil in porous rocks and oil core samples.1 Figure l(a), for example, is a hydrogen image showing water distribution in a limestone core, revealing bedding planes and variations in porosity. Such images can be of use in evaluating effective oil extraction procedures which can be tailored to the local environment at the drill site. Since pressurized, dope d brine and hydrocarbons move relativel y slowly through rock, flow is measured by repeat imaging of cores. The study of water or fluid diffusion into polymers2 and other nonmagnetic solids3 is a similar appli- cation of NMR imaging, but to materials science. The fluid- absorption properties of solids are key considerations in the design and application of many man-made materials. Conversely, the ability to render the water in rocks and soil invisible makes hydrogen NMR imaging an ideal and currently unique tool for studying plant root systems undis- turbed in the soil medium in which they grow.4 Apparently, the water in roots is more mobile and has a longer NMR spin-spin relaxation time than water bound to soil particles, so that use of a spin-echo imaging sequence can discriminate against the soil water which often constitutes the dominant component. Figure 1 (b) is an NMR image of a bean plant in natural soil. Plant root imaging can be used for studying water transportation and response to light, water stress and disease, and ultimately may prove useful for evaluating the PHYSICS TODAY / JANUARY 1987 S-53PHYSICS NEWS IN 1 986—PHYSICS APPLIED TO INDUSTRY FIG. 1. Xuclear magnetic resonance images of (a) water distribution in a limestone core, (b) a bean plant in natural soil, (c) bloodvessels in a human head. effects of different COZ levels, pollutants, chemical s and her- bicides, as well as optimizing growth conditions in seed beds. All this does not mean that progress in medica l NMR imaging is at a standstill. Major advances in the past year have been (1) the development of rapid imaging pulse se- quences that reduce scan times to several seconds for high quality images," (2) the development of NMR angiography techniques that provide images of only the flowing blood in vessels,6V and (3) NMR microscopy at about 10 /zmX 10 /im resolution, enabling discrimination of structures within larger cells.* Rapid imaging is based on the use of many NMR excitation pulses spaced much shorter than the spin- lattice relaxation time. Such sequences bear some resem- blance to traditional driven equilibrium NMR pulse tech- niques.9 NMR angiography utilize s the fact that spins that move through a gradient change their phase relative to sta- tionary spins. This phase contrast can be used to select only the moving spins, or, in a living person, image the blood vessels. An example in the head is shown in Fig. l(c). Paul A. Bottomley, General Electric Research and Development Center 1. H. J. Vinegar, J. Petrol. Technol. 38, 257 (1986). 2. S. Blackban d and P. Mansfield, Proc. XXII Congress Ampere, edited by K. A. Muller. R. Kind, and J. Roos (Zurich Ampere Committee, Dept. Physics, U. Zurich, Zurich, Switzerland, 1984) , pp. 516-519. 3. G. C. Chingas, J. Milliken, H. A. Resing, and T. Tsang, Synth. Met. 12, 131 (1985). 4. P. A. Bottomley, H. H. Rogers, and T. H. Foster, Proc. Natl. Acad. Sci. USA 83, 87 (1986). 5. A. Haase, J. Frahm, D. Matthaei, W. Hanicke, K. D. Merboldt, J. Magn. Reson. 67, 258 (1986). 6. V. J. Wedeen, A. M. Reto, R. R. Edelman, S. C. Geller, L. R. Frank. T. J. Brady, and B. R. Rosen, Science 230, 946 (1985). 7. C. L. Dumoulin and H. R. Hart. Radiology (in press). 8. J. B. Aguayo , S. J. Blackband . J. Schoeniger, M. A. Mattingly, and M. Hintermann, Nature 322, 190 (1986). 9. H. Y. Carr, Phys. Rev. 112, 1693 (1958). S-54 PHYSICS TODAY / JANUARY 1987Electronic Conduction in Silicon Dioxide Despite its importance in silicon technology, thermally grown silicon dioxide remains a poorly known material. Thus, the recen t experimental1 and theoretical2 work per- formed at IBM on electronic conduction in SiO2 represents significant progress for both the microelectronics and the physics communities. The knowledge of the physica l mecha- nisms of electron transport in SiO2 may help us understand the mechanisms of degradation and breakdown of the thin insulators used in the sub-micron technology. In addition, the high fields and high rates of energy loss experienced by the electrons in insulators challenge our basic understanding of the physics of electron transport in solids. Previous to the IBM work, our understanding of electron transport in insulators still relied on the pioneering work of Frohh'ch.- In ionic materials, such as the alkali halides, an efficient mechanism by which electrons—accelerated by an external field—lose energy to the lattice is provided by the polar electron-phonon interaction. For slow electrons, the coupling to the dipol e field of the polar lattice is strong, since the lattice can easily follow the motion of the electrons and absorb their energy. But when the external field exceeds a critical value (historically called the "breakdown field"), the electrons acquire such a large velocity between succes- sive collisions that the lattice polarization cannot respond. The rate of energy loss decreases, the electrons reach even higher velocities, the rate decrease s even more, and so on. In this regime (normally called velocity runaway) nothing can now prevent the charge carriers from reaching the impact ionization threshold. Eventually, avalanche multiplication and the associated Joule-heating will lead to the breakdown of the dielectric. The extrapolation of these ideas to the less polar SiO2 seemed so obvious, that for many years the problem of the dielectric breakdown of SiO2 was considered almost set- tled.4 But there was little experimental confirmation of the theory. For this reason, in 1982 the IBM group started aPHYSICS NEWS IN 1986—PHYSICS APPLIED TO INDUSTRY series of experiments on the energy distribution of the elec- trons in SiO2.' The first results came as a surprise and suc- cessively refined experimental techniques confirmed that not all was going according to the extrapolated Frohlich pic- ture: the electron s were indeed running away from the polar interaction. But this was occurring at fields almost ten times smaller than the threshold field of 107 V/cm anticipated at that time.4'5 Already in 1975 Ridley had suggested that nonpolar elec- tron-phonon scattering could play a major role in the high- field behavior of insulators.6 Later, similar ideas were suc- cessfully applied to the alkali halides.7 Therefore, it seemed reasonable to identify the "missing" stabilization mecha- nism with the nonpolar electron-phonon scattering. This is negligible at low energy, but its effect grows larger above 2 eV or so, as the electrons will be scattered more effectively by short-wavelength phonons. Monte Carlo simulations confirmed the correctness of these ideas. Below 1.5 X 106 V/cm the polar scattering behaves as expected and keeps the electrons in steady state at a low average energy. Above this threshold field the electrons try to escape to higher energies, but they are soon affected by large-angle, quasi-elasti c non- polar collisons.8 Thus the nonpolar scattering behaves as a catalyst, the more frequent nonpolar collisions resulting in higher energy losses via polar processes. This allows steady state transport to occur even at the largest pre-breakdown fields, about 1.6X 107 V/cm. These results are encouraging for the device engineer, be- cause the ultimate dielectric strength of SiO2 does not seem to be limited by intrinsic electronic runaway processes but, rather, by technological limitations which we can hope to overcome: it is enough to recall that the highest breakdown fields have increased from 7 X 106 V/cm in the early 1970s to almost 2 X 107 V/cm. But also for the theoretical physicist, SiO2 becomes a unique prototype material. Despite the amorphous structure of thermally grown layers, short-range order renders them electronically similar to their crystalline counterpart, alpha-quartz, with the advantage of a higher purity. No other insulator can claim mobilities in excess of 20 cm2/V s.9 The enormous fields sustained by these layer s and the evidence of steady-state transport at these fields re- quires scattering and energy-loss rates which render the fa-miliar Boltzmann transport equation highly suspect. This has stimulated further research in the high-energ y "quan- tum" direction which complements the effort made in the low-energy area of heterostructures and quantum wells.10'11 Finally the growth of the high-quality films as thin as 5 or 6 monolayers has recently allowed the experimental observa- tion of ballistic electrons, of phonon replicas, and the study of current and energy oscillation owing to quantum reflec- tions at the SiO2-electrode interface.12 Silicon dioxide has given us the chance to analyz e directly the mechanism of electron transport at high fields. For the first time, we begin to grasp the principles of charge trans- port in wide-bandgap materials with the blessing of experi- mental confirmation. Perhaps further work on ultra-thin films will let us look directly at the physics of electron trans- port on the microscopic and quantum levels. M.V. Fischetti and D.J. DiMaria, IBM T. J. Watson Research Center 1. T. N. Theis, D. J. DiMaria, J. R. Kirtley, and D. W. Dong, Phys. Rev. Lett. 50, 750 (1983) and 52, 1445 (1984); D. J. DiMariaT. N. Theis, J. R. Kirtley, F. L. Pesavento, and D. W. Dong, J. Appl. Phys. 57, 1214 (1985); S. D. Brorson et al, i. Appl. Phys. 58, 1302 (1985). 2. M. V. Fischetti, Phys. Rev. Lett. 53, 1775 (1984); M. V. Fischetti, D. J. DiMaria, S. D. Brorson, T. N. Theis, and J. R. Kirtley, Phys. Rev. B 31, 8124 (1984). 3. H. Frohlich, Proc. R. Soc. London, Ser. A 160, 230 (1937). 4. See, for example, T. H. DiStefano and M. Shatzkes , Appl. Phys. Lett. 25, 685 (1974); P. Solomon and N. Klein, Solid State Commun. 17, 1397 (1975). 5. W. T. Lynch, J. Appl. Phys. 43, 3274 (1972); D. K. Ferry, Appl. Phys. Lett. 27, 689 (1975); J. Appl. Phys. 50, 1422 (1979). 6. B. K. Ridley, J. Appl. Phys. 46, 998 (1975). 7. M. Sparks etal, Phys. Rev. B 24, 3519 (1981). 8. D. J. DiMaria, M. V. Fischetti, E. Tierney , and S. D. Brorson, Phys. Rev. Lett. 56,1284 (1986); D. J. DiMaria, M. V. Fischetti, M. Arienzo, and E. Tierney , J. Appl. Phys. 60,1719 (1986); H.-H. Fitting and J.-U. Frieman, Phys. Status Solidi A 69, 349 (1982). 9. R. C. Hughes, Phys. Rev. Lett. 26, 1333 (1973); Solid-State Electron. 21,251 (1978). 10. K. K. Thornber and Richard P. Feynman, Phys. Rev. B 1, 4099 (1972). 11. M. V. Fischetti and D. J. DiMaria, Phys. Rev. Lett. 55, 2475 (1985). 12. D. J. DiMaria, M. V. Fischetti, J. Batey, L. Dori, E. Tierney, and J. Stasiak (submitted to Phys. Rev. Lett.). PHYSICS TODAY / JANUARY 1987 S 55PHYSICS EDUCATION In 1986 public attention, which for the last few years had focused on high school physics teaching, shifted to the prob- lems of undergraduate physics education. This was accom- plished without a corresponding decrease in attention to the high school situation. Publication of the Proceedings of the Physics Department Chair Conference on the Education for Professional Work in Physics, the testimony of A. P. French and R.R. Wilson to the National Science Board Task Com- mittee on Undergraduate Programs, and the Report of that Committee set the tone for the debate. The June 1986 issue of Physics Today was devoted to a detaile d look at some of these issues. As of this time, it appears that the NSF appropriation for the coming fiscal year will include a substantial sum for support of undergraduate programs. These programs are likely to be administered by the research directorates, which have not been known in the past for their sensitivity to edu- cational issues. The increase in support to the undergraduate science, mathematics, and engineering programs will never - theless be welcomed in the college and university communi- ty- Pre-college physics education has not suffered from lack of attention in 1986. The Physics Teaching Resource Agents (PTRA) program, sponsore d by the NSF and the American Association of Physics Teachers (AAPT), entered its sec- ond year with over 200 outstanding high schoo l teachers trained to work with other teachers in their regions. By mid- 1986, the PTRA had offered over 400 regional workshops with over 8000 participants. Among the workshop topics were: the microcomputer as a laboratory instrument, lecture demonstrations, a resource kit for new physics teachers, to- pics in modern physics, astronomy in the high schoo l curri- culum, developing student confidence in physics, and many others. In addition to building teachers' competencies, the workshops help to generate an enthusiasm for physics teach- ing that has attracted some new teachers and encouraged others to remain in teaching. Other national and regional workshops and programs have contributed to the overall improve d situation in the high schools. Example s of these projects include: The Wood- row Wilson Workshops, the VMI workshops on lecture demonstrations, the Mechanical Universe Projec t (Cal. Tech.), the Harvard Smithsonian Projec t STAR (Science Teaching from its Astronomical Roots), the Department of Energy's program of work at the National Laboratories, the Fermilab Conference on the Teaching of Modern Physics, the APS/AAPT College High School Interaction Commit- tee, the AIP/AAPT survey of high school physics teachers, the AIP physics poster distribution, the first participation by the U.S. in the International Physics Olympiad, and many local workshops for teachers. In spite of our best efforts, severe problems continue to face high school physics teaching. Low salaries, lack of pro- S-56 PHYSICS TODAY / JANUARY 1987fessional recognition, reductions in force to match declining enrollments, use of unqualified teachers, lack of enrollment in physics courses, lack of course offerings, lack of teachers, inadequate teacher preparation programs, poorly designed curricula, and poor working conditions remain problems, but progress has been made on each of these fronts. The relentless decay of the 1970s has been replaced by a moderate improvement in the 1980s, but we still have a long way to go. Participation in two international activities this year illus- trated just how inadequate our efforts are in comparison to the rest of the world. The U.S. team at the Physics Olympiad did quite well in spite of the great disparity between their exposure to physics and that of the students from other countries. In many ways the United State s could be charac- terized as an "underdeveloped" country when it comes to physics education. The National Science Foundation (through AAPT) also sponsored a delegation of 14 high schoo l teachers to the International Commission on Physics Education meeting in Tokyo in August of 1986. Each mem- ber of the delegation was to research some aspect of physics education, comparing the situation in the U.S. to that in other countries. Preliminary reports indicate that the teach- ers discovered significant differences in the quality and quantity of programs, teachers, and students. Computers in physics education and research in how stu- dents learn physics continue to be two of the hottest issues in physics education. Both issues were heavily explored in the two AAPT meetings and in the ICPE meeting in Tokyo. Computers in the laboratory, courseware evaluation, inter- active videodisk production and use, the teaching of chaos and other "new" topics on the computer, misconceptions, and artificial intelligence seemed to be among those topics raising the most interest. The spiraling numbers of comput- ers in the schools, the homes, the universities, and the work place will continue to make this an area of prime interest in the coming years. AIP is responding to that need by develop- ing a new journal on computers in physics. JackM. Wilson, AAPT The International Physics Olympiad The 17th International Physics Olympiad was held in July 1986 at Harrow School in London, England.12 The IPO has now truly become international. In 1985 Canada participat- ed for the first time, and China and the United State s joined this year. Altogether 21 countries and 103 students partici- pated. The U.S. team was organized by Jack Wilson (AAPT and the University of Maryland) while Arthur Eisenkraft (Fox Lane H.S., New York) and Ronald Edge (University of South Carolina) served as Academic Directors. SponsorsPHYSICS NEWS IN 1986—PHYSICS EDUCATION for this year's U.S. Olympiad team included: the American Association of Physics Teachers, The American Physical Society, the American Institute of Physics, the American Astronomical Society, the Optical Society of America, the American Association of Physicists in Medicine, the Univer- sity of Maryland, Duracell, Ford Motor Company, Interna- tional Business Machines, Exxon Corporation, John Wiley and Sons, Inc., Worth Publishers, Inc., and the Bond Manu- facturing Company. The selection of the team was a four-step process involv- ing nominations from each region, two qualifying exams, and final selection of the five team representatives from a team of twenty of the nation's best high schoo l physics stu- dents who were invite d to attend a training camp for the International Physic s Olympiad at the University of Mary- land from June 30 to July 11, 1986. The Olympiad itself was made up of two days of examina- tion during a five day period. The first examination day was devoted to a typical paper and pencil theoretical exam con- sisting of three problems. A practical exam in laboratory and computer work, was given on the second exam day. The five finalists making up the team were Howard Fu- kuda, Hawaii; Paul Graham, Colorado; Philip Mauskopf, North Carolina; Srinivasan Seshan, Virginia; and Joshua Zucker, California. Graham, Mauskopf, and Zucker came away with bronze medals in the competition. The USSR won three of the four gold medals with Romania taking the fourth. The International Physics Olympiad is designed to offer encouragement for the study of physics by publicizing the efforts of team members while involving thousands of stu- dents in the preliminary portions of the selection process. The first year was an encouraging start. Arthur Eisenkraft, Fox Lane High School, New York and Ronald Edge, University of South Carolina [ 1. Phys. Toda y 39, 51 (September 1986). 2. Ibid., 120 (Septembe r 1986). Conference on the Teaching of Modern Physics A Conference on the Teaching of Modern Physics was held at Fermilab in Batavia, Illinois, in April 1986. The aim was to promote the use of results of current research in high school physics and introductory level undergraduate courses. This conference had its roots in a similar meeting held at the CERN laboratory in Geneva, Switzerland in Sep- tember 1984. Leon M. Lederman, the Director of Fermilab, who has involved Fermilab heavily in science education, and Jack Wilson of A APT approached the National Science Founda-tion, which provided support for fifty high school teachers to attend the conference. Drasko Jovanovic, Director of Pro- gram Planning at Fermilab, and Gordon Aubrecht, Visiting Fellow at AAPT, were designated as conference coordina- tors/directors on behalf of the respective organizations. The specific purpose of the Fermilab Conference on the Teaching of Modern Physic s was to promote the use of cur- rent topics in physics research, especially particle physics and cosmology, in introductory level physics courses . It was decided to focus especially on particle physics and cosmol- ogy, partly because of the location (Fermilab) and partly because a short meeting would not allow a full exploration of all of the areas of modern physics. The steering committee agreed that the introduction of aspect s of modern physics into physics courses could serve to enliven physics teaching and would allow the teacher to maintain or even enhance the interest in science that students bring with them to school. This interest in many cases is lost by reliance on convention- al texts and lesson plans. As a result many students do not acquire even the minimum skills needed to attend college or university. Among the speakers were: Howard Georgi, one of the originators of grand unified theories, speaking on GUTs and elementary particles; Christopher Hill on symmetry and group theory; Chris Quigg, author of a recent book on gauge theories and coauthor of an important recent paper on the theoretical basis of the proposed Superconducting Supercol- lider, on discoveries, tools, and insights from particle phys- ics; Davi d Schramm on the Big Bang and the creation of the universe; Victor Weisskopf on the place of qualitative esti- mates in physics education; and Clifford Will describin g ob- servational tests of general relativity. The participants were assigned to interest groups on the basis of their responses to the acceptance letter. This ap- proach represented an attempt to assist the production of varied classroom materials on the five topics represented at the conference. The interest groups were further subdivided into six subgroups during the first discussion session: lecture presentations; demonstrations; software and audiovisuals; evaluation instruments; experiment and laboratory activi- ties; and homework problems. The subgroups discussed the choice of topic s as well as effective strategie s for bringing the materials into high school and beginnin g undergraduate courses. The final summary of the written reports from the sub- groups was published and distributed to the participants be- fore they left Fermilab. We expect that the ideas in this re- port will serve as the basis of a concerted effort by the conference participants to develop materials for use in the classroom. Production and testing of teacher-generated ma- terials is taking place between the April meeting at Fermilab and a follow-up meeting to be held at the AAPT/APS joint meeting in San Francisco in January 1987. This follow-up will focus on coordination of the materials produced and should stimulate further work. PHYSICS TODAY / JANUARY 1987 S• 57PHYSICS NEWS IN 1986—PHYSICS EDUCATION The plenary lectures at the Fermilab conference will be published in early 1987 and, along with videotape s of the entire conference, will be available from AAPT. The Lathi American physicist s at the Fermilab Confer- ence made gTeat progress in planning the next conference in the series, which will probably be held in Latin America (possibly hi Brazil) in 1988. There was also a discussion of the possibility of holding such a conference in Mexico in the near future. The topics for these conferences on teaching modern physics will most likely be condensed matter phys- ics. The International Commission on Physic s Education has scheduled a follow-up meeting on condensed matter physics hi Munich hi the Fall of 1988. Gordon J. Aubrecht II, Ohio State University Support for Undergraduate Physics Programs The problems in undergraduate physics education resulting from the withdrawal of federal support for undergraduate programs during the early 1980s have received much atten- tion lately (see Physics News in 1985, pp. 26-28). Several reports, including "Education for Professional Work hi Physics,"1 which arose from ajoint AAPT/APS conference held in 1985, and "The Impact of Undergraduate Physic s Programs on the Quality and Quantity of Physics Majors,"2 a survey organized by the APS Committee on Education, have formed the basis for AAPT and APS publi c positions on the state of undergraduate science and on defining those areas most in need of support. They have been used in a briefing to the actin g White House Science Advisor on the views of the physics community about the health of under- graduate and graduate physics programs and students, and they were undoubtedly a factor in helping to prompt the National Science Board to form a special task committee to look at the state of undergraduate science and engineering education. On November 20, 1985, A.P. French, President of AAPT, and Robert R. Wilson, President of APS, appeared jointly before the Task Committee. They made a strong case for a renewed NSF undergraduate program.3 They present- ed data from the Topical Conference of Physic s Department Chairs and from the Survey of Quality and Quantity of Un- dergraduate Majors and Programs. They pointed out that the United States relies upon the undergraduate science pro- grams to make up for the paucity of science taught at the pre- college level. French noted that "our undergraduate pro- grams must carry the burden of trying to bring our students, in the short space of four years, up to the level of the gradu- ates from universities in other technologicall y advanced countries." Wilson emphasize d the "intimate and vital rela- tionship between research and teaching which insures the vigor of science." In addition to their personal remarks, French and Wilson presented a detaile d series of recommen- dations that had been prepared by H. Lustig (APS) and S-58 PHYSICS TODAY / JANUARY 1987J.M. Wilson (AAPT). The recommendations were based on the two AAPT/APS conferences2-4 and the APS survey re- ferred to earlier. The Task Committee submitted its report in March, 1986.5 It begins: "The nation's undergraduate programs in science, mathematics, and technology have declined in qual- ity and scope to such an extent that they are no longer meet- ing national needs. A unique American resource has been eroded." The report highlighted the historical situation in which support for science education programs has declined from a high of 46.1 % of the NSF budget in 1960 to 5.6%to- day. Most of the existing support is being funneled into the pre-colleg e areas. The once healthy NSF program in under- graduate science education has dwindled to nearly nothing. As Robert R. Wilson observed: "I believe that the time has come for the resumption of programs and for new initiatives hi undergraduate science education." Sidney Drell, current President of APS, wrote strong letters to Erich Bloch (NSF Director) and Roland Schmitt (Chairman of the National Science Board) commending the report and encouraging di- rect action by the NSF. On 21 March, the full National Science Board accepted the report and passed the following resolution: "The Nation- al Science Board requests that the Director, hi close consul- tation with the NSB Committee on Education and Human Resources, prepare a plan of action to respond to the report focusing on new and innovative program approaches that will elicit creative proposals from universities and colleges, and submit such plan to the Board as part of the National Science Foundation FY 1988 budget process." The report called for expenditures of 100 million dollars by fiscal year 1989, alloted to laboratory development (S20M), instructional equipment ($30M), faculty develop- ment (S13M), curriculum development (S13M), compre- hensive improvement projects (S10M), undergraduate re- search participation (S8M), minority institutions programs ($5M), and long-range planning (SIM). The report esti- mated the total need for laboratory instrumentation alone to be $2^ billion, and recognizes that the NSF role must be one of providing seed money and stimulating improvements. Ro- land W. Schmitt, Chairman of the NSB, Senior Vice Presi- dent for Corporate Research and Development at General Electric, and former AIP Governing Board member, com- mented that "We cannot afford to allow the deterioration of our undergraduate science and engineering education sys- tem to go unchecked." A.P. French, Massachusetts Institute of Technology, and Jack M. Wilson, University of Maryland 1. "Education for Professional Work in Physics," AAPT (1986). 2. Phys. Toda y 39, special issue on physics education (June 1986). 3. Bull. Am. Phys. Soc. 31, (February 1986). 4. Proceedings of the Physics Department Chairs Conferen ce on the Edu- cation of the Physicist, AAPT (1984). 5. Undergraduate Science, Mathematics, and Engineering Education, a re- port of National Science Board Task Committee, National Science Foundation, March 1986.PLASMA AND FUSION PHYSICS' Progress Towards Breakeven on the Tokamak Fusion Test Reactor Much of the experimental program during the last year on the Tokamak Fusion Test Reactor (TFTR) at Princeton University has been focused on searching for enhanced ener- gy confinement regimes to improve the prospects for obtain- ing approximate energy breakeven in TFTR and thermonu- clear ignition in a future Compact Ignition Tokamak. A number of smaller tokamaks have demonstrated techniques for improving energy confinement, but none of these had yet been successfully applied to the current generation of large tokamaks: TFTR, the Joint European Torus (JET), and the Japanese JT-60. During 1986 two different approaches achieved success in different plasma regimes in TFTR. In the first approach, a high-speed pellet injector devel- oped at the Oak Ridge National Laboratory (ORNL) was used on TFTR to produce very high density plasmas with strongly peaked density profiles. Multiple 3-4 mm pellets of frozen deuterium were injected at speeds of 1250 m/s. These pellets penetrate ohmically heated TFTR plasmas deeply, depositing their deuterium fuel near the center of the dis- charge, effectively diminishing the effect of particle recy- cling at the plasm a edge. At central ion and electron tem- peratures of 1.3 keV, record nr (central densit y times global energy confinement time) products of 1.5xlO20 s/m3 were obtained,' nearly doubly the previous best result obtained in the Alcator C tokamak at MIT. The second successful approach employed on TFTR also involved the reduction of edge particle recycling, but used a different set of techniques. The first material surface a toka- mak plasma encounters is usually a system of graphite tiles, constructed to handle the high power densities in the plasma scrape-off without damage, and without the introduction of deleterious amounts of impurity ions into the plasma. The surface of this graphite "limiter" typically becomes filled with deuterium, and recycles lost particles back to the plas- ma with nearly 100% efficiency. The TFTR team pursued experiments in which long-pulse helium and low densit y deuterium discharges were used to deplete the limiters of trapped gas. When neutral beam injection was employed to heat plasmas with outgassed limiters, very favorable results were obtained. The 95-keV neutral deuterium atoms inject- ed in these experiments penetrated to the core of initially low density TFTR plasmas, resulting in very peaked density pro- files, and excellent confinement properties. In the best discharges, at a plasma current of about 1 MA, the total store d energy exceeded the prediction made using previous scaling laws by a factor of up to 3. The d-d neutron production rate reached a peak of 9X 1015/s, with a total injected power of 17 MW. Since the thermal electron compo-nent in tokamak plasmas usually has the poorest thermal insulation, it is very satisfactory that these discharges also showed an improvement by up to a factor of 2 over scaling- law prediction in regard to electron store d energy. The most spectacular feature of these plasmas is the central ion tem- perature, which reached 20 keV. The nr product was slightly over 1019 s/m3. For comparison, the record ion temperature achieved2 in the PLT tokamak at Princeton in 1980 was 7 keV, with an nr of 1018 s/m3. These results imply favorable prospects for future prog- ress on TFTR towards the goal of approximate energy breakeven. One of the original goals of TFTR, to achieve «r = 1019 s/m3 at an ion temperature of 10 keV, has already been exceeded. An ORNL pellet injector with increased ca- pabilit y was recentl y installed, and is being commissioned. This injector will allow investigation of pellet injection into higher-power TFTR discharges . The use of rf power in the ion cyclotron range of frequencies is planned to supplement neutral beam injection, to heat the center of the very high densit y discharges obtained with pellet injection. An intrigu- ing feature of the high-ion-temperature discharges is that the ion temperature, neutron emission, and total store d energy are all still rising at the end of the half second beam pulse currently available. The Lawrence Berkeley Laboratory, which built the original neutral beam system, is providing TFTR with a set of long-pulse ion sources which will pro- duce 2-s, 27-MW pulses at 120 keV. Furthermore, experi- ments in which the plasm a current was ramped at 1 MA/s have shown that global confinement improve d with plasma current above the levels which could be obtained at 1 MA. The ultimate current capability of TFTR (3MA), may thus be accessible in this regime through the use of the long-pulse beams, with an associated further improvement in confine- ment. R.J. Goldston, Plasma Physics Laboratory, Princeton University Recent Experiments in Inertial Confinement Fusion The inertial confinement approach to fusion relies on com- pressing and heating a mixture of deuterium and tritium (DT) contained in a small capsule to ignition conditions. The driving energy can be provided by focused laser or parti- cle beams, but most present experiments are being conduct- ed with short-wavelength high-power lasers. The full realization of the potential benefits of inertial confinement fusion (ICF) requires the attainment of high gain, which means obtaining a yield of 500-1000 MJ using a multi-megajoule driver. The attainment of ignition and breakeven will be an important scientific achievement, but PHYSICS TODAY / JANUARY 1987 S 59PHYSICS NEWS IN 1 986—PLASMA AND FUSION PHYSICS high gain is required to provide the large fluxes, x rays, and neutrons that can have many military and Chilian applica- tions as well as to experim entall y establish the scientific fea- sibility of economic ICF power generation. To achieve high gain with 5 to 10 MJ of driver energy, the requirements on the capsule implosion are stringent. Several milligram s of DT fuel must be compressed to a densit y of about 200 g/cm3 and a density-radius product (a parameter analogous to the confinement parameter nr used in magneti- cally confined fusion) of about 3 g/cm:. The thermonuclear burn must be initiated in a high temperature (3-5 keV) cen- tral hot spot (constituting a few percent of the fuel mass): the alpha particles generate d there will deposit their energy locally and initiate burn in the cold, surrounding dense main part of the fuel. The implosion velocity of the spherical cap- sule must be uniform to within about 19c to pro\ide symmet- rical compression and to minimize contributions to fluid in- stability break-up of the ablator. To allow the lowest driver energy, the driver-target coupling should be very efficient and the fuel preheat low. Preheat from early shocks, x rays, and hot electron s must be minimized to keep the fuel cold during compression. A combination of fluid instability and pre-heat issues makes it necessary to gradually shape the driver pulse temporally. There are two basic approaches to imploding ICF pellets; direct and indirect drive. In the direct drive approach, the driver energy is focused directly onto the fuel pellet to heat and ablate the outer regions and drive the implosion. In the indirect drive approach, the driver energy is converted to a flux of soft x rays which in turn drives the fuel capsule. One thrust of worldwide ICF research is in the area of laser-plasma interaction physics: absorption, energy trans- port and partitioning, and the conversion of driver energy to x rays. In particular, there has been progress in understand- ing the scaling of these processes to plasma sizes relevant to high-gain ICF targets. The requirements and techniques needed to provide symmetrical implosions that are not fatal- ly degraded by the growth of fluid-dynamic instabilities is also a major area of study. Diagnostic techniques for accu- rately measuring the density and temperature achieved in compressed fuel are being developed. The 100-kJ, 100-TW class Nova neodymium-glass laser facility at the Lawrenc e Livermore National Laboratory has been brought into full operation for experiments in two tar- get chambers with frequency-converted 0.53 and 0.35 fxm wavelength light. It has allowed the extension of short-wave- length laser-target interaction and implosion experiments to scales more relevant to ultimate high gain targets. Recent basic laser-target interaction results include quantificatio n of x-ray conversion efficiency as a function of laser intensity. Nova has produced a record ICF fusion yield of 1013 neu- trons (corresponding to a fusion gain of 0.18%) with sim- ple, directly driven DT filled glass microshells. These targets are intended as sources of 14-MeV neutrons for advanced neutron diagnosti c development. S-60 PHYSICS TODAY / JANUARY 1987Results of the first indirectly (x-ray) driven implosions conducted on Nova are consistent with a calculated confine- ment density-time in the range of 1.5-3 X 10u s/cm3 at a measured fuel ion temperature of 1.5—1.7 keV. This was achieved with 18 kJ of 0.35 fjm light. The results of both of these experiments are in close agreement with pre-experi- ment computer simulations. Substantial improvement in target performance is expected in the future using the full (50-80 kJ. 0.35 fim, 1.5—3 ns pulses) capability of Nova and optimized targets. A major comprehensive review of the U.S. ICF program was completed by a technical review committee convened under the auspices of the National Academy of Sciences.1 The previous review committee (1979) had cited several re- search goals as being important for progress in ICF: (1) the need to go to shorter wavelength lasers; (2) a vigorous labo- ratory program to understand issues of coupling of laser en- ergy to capsules; (3) \igorous prosecution of a program to investigate design characteristics of efficient ICF targets, and more attention to target fabrication; and (4) research into ion drivers. In its review, the (current) Committee found that the ICF program has made substantial progress in response to all these recommendations. The following are a few of the excerpts of the executive summary of the March 1986 Final Report. " Since the 1979 Review, it has become clear that the large numbers of hot electrons generated by long-wavelength 10-/im CO, laser radiation preheat the targets too much to permit high gain implosions . The CO; laser work at Los Alamos National Laboratory (LANL) has therefore been terminated. Hopes that target coupling would be improved at shorter laser wavelength s have been confirmed by experiments at the University of Rochester (UR) and Lawrence Livermore National Laboratory (LLNL) with doubled (0.53/im), tri- pled (0.35//m). and quadrupled (0.25 /urn) Nd:glass laser radiation. The plasma physics of the improved absorption is now understood quantitatively, as shown by the agreement between predictions and observations. The Nova laser has been completed and is able to deliver 60 kJ of \.0/um radiation, and 25 kJ of 0.35/im radiation. A KrF laser that produces 10 kJ of 0.25 ,um radiation directly has been constructed at Los Alamos. Techniques must be developed to shorten the natural 500 ns pulse length of KrF to the few ns needed for driving a target. A promising new method to reduce plasma instabilities and improve symmetry, induced spatial incoherence (ISI), applicabl e with broadband lasers, has been demonstrated at the Naval Research Laboratory (NRL). The 24-beam Ome- ga laser has been completed at UR and will serve United States' main direct drive facility in the immediate future. Fusion, Inc., the only private company in the ICF program, has made and continues to make important contributions to target fabricatio n and to methods for investigating target plasma physics.PHYSICS NEWS IN 1986—PLASMA AND FUSION PHYSICS The PBFA II light ion accelerator at Sandia National Laboratories (SNL) is potentially capable of depositing 1 to 2 MJ on targe t in 10 ns. The major construction of this facili- ty is complete and initial testing of the electrical system is under way. To focus an ion beam on target with suitable pulse shaping will require a considerable amount of addi- tional work." Although the NAS Committee did not address heavy-ion accelerators for ICF, recent experiments and analysis indi- cate that greater beam currents can probably be transported and focused than previously thought. This has significantly increased the possibility that a heavy ion driver in the 5 to 10 MJ range can be built for a cost significantly less than pre- viously believed.2 The prospects for the eventual success of ICF have never been brighter. L. W. Coleman and E. Storm, Lawrence Livermore National Laboratory 1. "Review of the Department of Energy's Inertial Confinement Fusion Program," Commission on Physical Sciences, Mathematics and Re- sources, National Research Council, National Academy Press (March 1986); Phys. Toda y 39, 19 (August 1986). 2. Proceedings of the 1986 International Symposium on Heavy Ion Fusion (to be published by American Institute of Physics). Recent Free Electron Laser Experiments There are several areas within the field of plasma physics that can benefit from a high-power, efficient source of tuna- ble radiation. Magnetic mirror machines, for example, re- quire high power sources of microwaves to create thermal barriers and end plugs which limit axial ion loss. Recently, there have been suggestions that electron cyclotron reso- nance heating of tokamaks could have a number of advan- tages as a heating technique, and may also be able to produce a current drive enabling steady-state tokamak operation. In both of these applications, tunability of the microwave source would be an asset. At much shorter wavelengths (250-1000 nm), laser- driven inertial confinement fusion also requires a high power source of coherent radiation. All of these sources share two requirements: they must be efficient and they must be inex- pensive. Recent experiments hold out promise that the free electron laser (FEL) can meet these diverse and demanding needs.1 The FEL directly converts the kinetic energy of a relativ- istic electron beam into coherent radiation. This conversion is accomplished by passing the electron beam through a spa- tially periodically reversing magnetic field (called an undu- lator) which allows energy transfer between the electron beam and a co-propagating radiation field. The successful operation of an FEL requires the maintenance of a very pre- cise relationship between the phase of the radiation field andthe position of the electron relative to the periodic magnetic field of the wiggler. The wavelengt h of peak optical/microwave gain is deter- mined by the electron beam's energy, the strength of the magnetic field, and the period over which the magnetic field reverses. These parameters are design variables, so that in principle any operating wavelengt h may be chosen, henc e the FEL has a broad range of applications. High efficiency is possible because the thermal loss channels of conventional lasers not present in the FEL. It was demonstrated long ago that the basic FEL princi- ple was valid, but it was only within the past year that very high conversion efficiencies at microwave frequencies were demonstrated at Lawrence Livermore National Laboratory. The LLNL experiment, called ELF, was configured as an amplifier: the output of a conventional 20 kW (peak), 34.6 GHz magnetron was directed through a 3-meter-long undu- lator along with an 800 A, 3.5 MeV electron beam produced by the Experimental Test Accelerator to yield an amplified microwave signal.2 Initial experiments produced 40 dB of gain and a peak output power of 180 MW for a net 6% extraction efficiency. This performance was achieved with only 1.4mofundulator. Long amplifiers did not increase the output power because the 6% energy extraction was enough to destroy the necessary phase relationships between the electrons and the microwav e field. The ELF undulator can, however, be tapered; that is, the undulator's magnetic field can be reduce d so that, in princi- ple, phase synchronism can be preserved even when substan- tial amounts of the electron beam's kinetic energy is extract- ed. When the Livermore undulator was tapered, the output power increased from 180 MW to over 1 GW with a net conversion efficiency of over 35%. Experiments at Los Alamos National Laboratory3 (10.6 /urn wavelength), the University of California at Santa Bar- bara4 (400 fim), Stanford University5 (2.6 /nm), TRW/Stanford6 (l^^m), andOrsay7 (0.6/urn) have dem- onstrated that FELs can work at optical wavelengths, albeit thus far at considerably reduced efficiencies. All of these experiments were configured as oscillators. The LANL ex- periment was able to demonstrate a considerable tuning range, demonstrating the versatility of the FEL. The success of all of these experiments is very encourag- ing, but there are many areas of both FEL physics (in terms of wavelength and power scaling) and accelerator physics which must be pursued over the next few years in order to demonstrate that the FEL is truly an inexpensive, efficient, and reliable source of coherent radiation. D. Prosnitz, Lawrence Livermore National Laboratory 1. P. Sprangle and T. Coffey, Phys. Today 37, 44 (March 1984). 2. T.J. Orzechowsk i et al., Phys. Rev. Lett, (to be published). 3. B. E. Newnam etal, IEEE J. Quant. Electron. QE21 , 867 (1985). 4. L.R. Elias et al, Phys. Rev. Lett. 57, 424 (1986). PHYSICS TODAY / JANUARY 1987 S-61PHYSICS NEWS IN 1 986—PLASMA AND FUSION PHYSICS 5. S. V. Benson et ah. Nuclear Instruments and Methods in Physics A250, 39 (1986) in the Proceedings of the 7th International FEL Conference, edited by E.T. Scharleman and D. Prosnitz. 6. J.A. Edignoffer etal., Phys. Rev. Lett. 52, 344 (1984). 7. M. Billardonera/.. IEEE J. Quant. Electron. QE21 , 805 (1985). Relativistic Plasma Waves and Particle Acceleration In the continuing search for the fundamental building blocks of matter, particle accelerators have become indispensable to physicists since the invention of the cyclotron in the 1930s. In contrast to their earliest table-top ancestor, today's synchrotrons are some of the largest machine s ever built, with circumferences measured in kilometers rather than in meters. It is natural to ask, therefore, if it is possible to achieve far higher accelerating gradients than the current typical gradient of 20 MeV/m, thereby permitting machines of ever-increasing energies, but reasonable size and cost, to be built. In any particle accelerator scheme, the basic requirement for obtaining particles with ultrahigh energies is an intense longitudinal electric field that interacts with particles for a long time. Since highly relativistic particles move at nearl y the speed of light (c), the energy gained by the particles is maximum if the accelerating field is made to propagate with the particles. Extremely large electric fields propagating with phase velocities close to c can be produced by space charge waves in a plasma. The maximum electric field of such a so-called relativistic plasm a wave is approximately equal to the square root of the plasma electron densit y per cm3. For instance, the longitudinal electric field of a relativ- istic plasm a wave with a background densit y of 1016/cm3 can be as high as 1010 V/m. Such waves can be either laser- driven, as in the Plasma Beat Wav e Accelerator,1 or excited by a short bunch of relativistic electrons, as in the Plasma Wak e Field Accelerator.2 In both cases the plasma acts as a single-mode, slow-wave cavity, in which the wavelengt h of the accelerating wave is typically several hundred /zm, as compared to 10 cm in linacs. This hitherto unexplored re- gime of parameter space may hold the key to the possible miniaturization of particle accelerators. In the Plasma Beat Wave Accelerator, two laser beams of slightly different frequencies resonantly beat in a plasma, in such a way that their frequency and wavenumber differences correspond to the plasma wave frequency and wavenumber. The amplitude modulated beat wave exert s a periodic pon- deromotive force on the plasm a electrons, causing them to bunch. The resulting space charge wave has a phase velocity that is equal to the group velocity of the beating waves. If the laser frequencies are much higher than the plasma frequency, the group velocity is nearl y c. If an electron is now injected with a velocity close to this, it can be trapped and accelerated by S-62 PHYSICS TODAY / JANUARY 1987the plasma wave much in the same way as a surfer riding an ocean wave . In the plasma wake field accelerator, a high-current but low-voltag e electron bunch is used to excite the plasma wave. The phase velocity of this plasma wave (like the wake of a boat) is tied to the velocity of the driving bunch, which is close to c. This wave then accelerates a trailing low-density bunch to high voltage or energy. The plasma thus acts as a transformer, increasing the voltage at the expense of current. The key to obtaining a high transformer ratio is to use a slowly ramped but sharply truncated driving bunch. In both the beat wave and the wake field cases, the trick to inhibiting most of the usual laser or beam plasma instabili- ties is to use a driver pulse that is only a few picoseconds long. To simulate the plasma wave excitation by a finite cross-section driver pulse and to optimize the energy extrac- tion by the accelerating beam, extensive two-dimensional particle simulations have been carried out.3'4 Theory pre- dicts, and the simulations confirm, that the maximum ener- gy that the particles get is limited by either the particles. Eventually outrunning the wave (dephasing) or by the pump depletio n of the driver. Experiments are underway at UCLA, Rutherford Labo- ratory (U.K.), ILE (Japan), INRS (Canada) and else- where to demonstrate the excitation of the relativistic plas- ma wave by the laser beat wave in a reproducible fashion and to demonstrate controlled acceleration of injected test parti- cles. In a recen t UCLA experiment, the relativistic plasma wave was excited by beating the 9.6/zm and 10.6,um lines of a CO2 laser, with a modest intensity of 2 X 1013 W/cm2 in a 1017/cm3 densit y plasma.5 The plasma wave electric field was inferred from Thomson scattering of a probe laser beam to be 103 Me V/m, a substantial improvement over the cur- rent benchmark gradient for accelerators. A new mecha- nism which saturates the beat-excited plasma wave in this parameter regime was discovered.6 The relativistic plasma wave saturates, on the time scale of a few picoseconds, by coupling to other plasma modes which have a much lower phase velocity, via an ion ripple due to stimulated Brillouin scattering of the laser beams. A scaled-up experiment which will demonstrate controlled acceleration of injected elec- trons is currently underway at UCLA. Experiments on the wake field concept are planned at UCLA and at Wisconsin. Finally, it is worth mentioning some of the other applica- tions of this new research area in plasma physics. The beat- excited plasma wave may be used as an intense submillimeter wave undulator for generating tunable, short-wavelength ra- diation using only a modest energy electron beam. Radial electric fields of a relativistic plasma wave with a transverse dimension on the order of a wavelength can be very intense and may be useful for focusing high-energy particles in a linear collider. A beat-excited plasma may also prove to be an ideal system for studying plasma evolution from a deter- ministic state into turbulence. ChanJoshi, University of California, Los AngelesPHYSICS NEWS IN 1986—PLASMA AND FUSION PHYSICS 1. T. Tajima and J. M. Dawson, Phys. Rev. Lett. 43, 267 (1979). 2. P, Chen, J. M. Dawson, R. W. Huff, and T. Katsouleas, Phys. Rev. Lett. 54,693 (1985). 3. C. Joshi, W. B. Mori, T. Katsouleas, J. M. Dawson, J. M. Kindel, and D. W. Forslund, Nature 311, 525 (1984); D. W. Forslund, J. M. Kindel, W. B. Mori, C. Joshi, and J. M. Dawson, Phys. Rev. Lett. 54, 588 (1985). 4. T. Katsouleas, Phys. Rev. A 33, 2056 (1986). 5. C. E. Clayton, C. Joshi, C. Darrow, and D. Umstadter, Phys. Rev. Lett. 54,2343 (1985). 6. C. Darrow, D. Umstadter, T. Katsouleas, W. B. Mori, C. E. Clayton, and C. Joshi, Phys. Rev. Lett. 56, 2629 (1986). Transport Near The Onset of Chaos Recent research has shown that systems with only a few dynamical variables can behave in surprisingly complicated ways.1 A set of three coupled first-order differential equa- tions, such as would describe a frictional pendulum with time-periodic forcing, is sufficient to give motion which is essentially as unpredictable as the proverbial toss coin. Simi- larly, an energy-conserving, or Hamiltonian, system with dynamical variables (two degrees of freedom ) is typically chaotic.2 Hamiltonian systems with completely regular or integrable motion can be devised, but virtually any perturba- tion of such a system gives a complicated mixture of regular and chaotic trajectories . The understanding of these systems is fundamental to de- signing fusion devices such as tokamaks and stellerators, building efficient accelerator storage rings, determining the stability of the solar system, estimating chemical reaction rates, and many other problems.2 An important example is the confinement of charged par- ticles by a magnetic field. When the field is strong, the parti- cle dynamics reduce to two degrees-of-freedom: gyration about the field line can be averaged out and, basically, parti- cles follow field lines. Since field lines never end, the confinement of particles require a toroidal configuration; if the torus is perfectly axi- symmetric there is a constant of motion, associated with the symmetry, which restricts the lines to two-dimensional to- roidal surfaces. Such configuration s are never realizable, partly because it is impossible to build perfectly axisymmet- ric field coils, but more generally because of symmetry- breaking collective motions of the plasma. These imperfec- tions cause some of the field lines to wander through three- dimensional regions of space in an extremely complicated, irregular or stochastic way. If these regions extend to the walls of the confinement device, particles will be rapidly lost. If the chaotic regions filled the entire confinement vessel, a diffusion coefficient could be obtained from a reasonable statistical hypothesis.3 However, the imperfections may be small enough that many field lines remain confined; on the other hand, a significant fraction of the orbits are often chao- tic.In this transition stage the notion of smoothly diffusive motion must be abandoned; chaotic trajectories linger for long periods in the neighborhood of invariant tori, and are impeded by the remnants of barely destroyed tori.4 These remnants are called "cantori" because they are invariant Cantor sets (a torus minus an infinitely long ribbon which winds around with irrational rotation number). The flux of trajectories through a cantorus is a well defined quantity and can often be very small even though the cantorus itself occu- pies zero area.5 Between the cantori, there are periodic orbits which re- sult from resonances between frequencies of each degree-of- freedom. These come in stable-unstable pairs. Near stable orbits there are encircling invariant surfaces which ensure local stability. By contrast, the unstable orbits have two- dimensional stabl e and unstable manifolds which form a "se- paratrix." The separatrix encloses the stable orbit, and the whole structure is called a "resonance." The volume of the resonance and the flux of trajectories entering and leaving it through the separatrices are well defined quantities.5 Numerical evidence indicates that resonances fill all of phase space, except that portion filled with invariant tori.5 This implies that phase space is divided into states, which are the resonances, separated by "fences," the separatrices and cantori, with gates or "turnstiles" with sizes determined by the flux. Transitions from state to state can be treated statis- ticall y because a chaotic orbit diverges from its neighbors exponentially in time: initially close trajectorie s have wildly different futures. The divergence rate is much faster than the transition rates between resonances; thus successive transi- tions are nearly statistically independent. Transition times from states near an invariant torus are arbitrarily long. This leads to the prediction that correlation functions decay algebraically with time.6'7 This theory of transport successfully predicts the escape times near onset of chaos in perturbed tokamaks. It also has been applied to the calculation of unimolecular chemical re- action rates and other situations. Numerical experiments to confirm the algebraic decay of correlations are difficult and time consuming, but so far confirm the theory. James D. Meiss, University of Texas, Austin 1. L. Kadanoff, Phys. Today 46 (Dec. 1983); J. Ford, Phys. Today 40 (April 1983). 2. M. V. Berry, in Topic in Nonlinear Dynamics, edited by S. Jorna, Ameri- can Inst. of Physics Conf. Proc. No. 46, 16 (1978); A.N. Lichtenberg and M.V. Lieberman, Regular and Irregular Motion (Springer-Verlag, New York, 1982). 3. B. V. Chirikov, Phys. Reports 52, 265 (1979). 4. R. S. MacKay, J. D. Meiss, and I. C. Percival, Phys. Rev. Lett. 52, 697 (1984);Physical3D, 55 (1984). 5. R. S. MacKay, J. D. Meiss, and I. C. Percival, "Resonance in Area Preserving Maps," Physica D (1986). 6. J. D. Hanson, J. R. Cary, and J. D. Meiss, J. Stat. Phys. 39, 327 (1985). 7. J. D. Meiss and E. Ott, Phys. Rev. Lett. 55, 2741 (1985); Physica 20D, 387 (1986). PHYSICS TODAY / JANUARY 1987 S 63POLYMER PHYSICS Polymer physics now encompasses research in condensed matter physics, physical chemistry, and materials science on substances composed of very large molecules. While synthe- sis of new polymeric materials falls outside of what is tradi- tionall y termed polyme r physics, progress in polymer phys- ics research has depended crucially , in many instances, on the availability or synthesis of macromolecules of specially- tailored molecular weight, topology, stereoregularity or some other structural characteristic. The following articles discuss recen t developments giving examples where this is the case: emission spectroscopy, theoretical polymer phys- ics, and polyme r surface forces. The choice is illustrative, not comprehensive. Matthew Tirrell, University of Minnesota Polymer Photophysics Emission spectroscopy of polymers—the observation of flu- orescence or phosphorescence after excitation of electronic transitions by visible or uv light—has received much atten- tion recently. Polymer photophysics has been the subject of two books1'2 and several major symposia. This apparent surge in interest is well-founded, since this technique may have a major impact on the understanding of the static and dynamic properties of polymer systems. This is possible be- cause of the wide applicability of the technique which re- quires only that some chromophore (an entity capable of absorbing and emitting ultraviolet or visible light) be pres- ent in the system. This may be a naturally occurring chromo- phore, as with the phenyl unit in polystyrene, or it may be a label attached to a polyme r lackin g chromophores. Analysis may involve one or more types of measurement: monomer or excimer fluorescence, fluorescence depolarization, fluores- cence recovery after photobleaching, energy transfer or mi- gration, phosphorescence, luminescence quenching, and transient luminescence measurements. Emission spectroscopy has proven particularly useful in studying phase separation in multicomponent polymer sys- tems. Morawetz pioneered the use of energy transfer and excimer emission to study phase separation.3 Recently, Monnerie's group has shown that reduction (quenching) of fluorescence of anthracene-labeled polystyrene (PS) by po- ly (vinylmethylether) [PVME] is at least as sensitive as ex- cimer fluorescence to phase separation.4 They have also shown that fluorescence emission is as well-suited as small- angle neutron scattering (SANS) in detecting the earliest stages and kinetics of phase separation. Furthermore, flu- orescence emission has an advantage over SANS when iso- tope effects are present. Emission spectroscopy is also being used to study phase behavior in more complex systems, in- cluding colloidal polymer particles5 and block copolymers and block copolymer/homopolymer blends.6 S-64 PHYSICS TODAY / JANUARY 1987Solution structure and interpenetration of polymer coils has been studied by Morawetz3 and more recently by Tor- kelson.7 Measurements of diffusional processes are possible with emission spectroscopy. Fluorescence recovery after photobleaching an area can be used to measure self-diffusion of polymers in solutions and melts; results obtained in entan- gled polymer systems are in good agreement with predic- tions of scaling concepts.8 Horie and Mita,9 andTorkelson et al.7 have used phosphorescence quenching in measuring in- teractions in polymer solutions which may be diffusion-limited. The technique is particularly well-suited for simulating (and isolating) interactions occurring in polymerization processes when polymer entanglement is important. Mobility and dynamics of polymer solutions and melts have also received considerable attention. By employing excimer flu- orescence10 Winnik and co-workers have done an extensive study of the rate at which the ends of a long chain come into contact with one another (cyclization) for polymers in dilute solution. Fluorescence anisotropy decay has been employed in Monnerie's group1! to determine orientation dynamics in poly- mer melts. For anthracene-labeled polybutadiene, the tempera- ture evolution of the orientation of the chain backbone is the same as for the mechanical properties, providing evidence for the connection between the glass transition and local molecular processes. J.M. Torkelson, Northwestern University 1. J. Guillet, Polymer Photophysics and Photochemistry (Cambridge Uni- versity Press, Cambridge, 1985). 2. Polymer Photophysics: Luminescence, Energy Migration and Molecular Motion in Synthetic Polymers, edited by D. Phillips (Methuen, 1985). 3. H. Morawetz, Polym. Prepr., Am. Chem. Soc. 27, 62 (1986). 4. J. M. Ubrich, F. Ben Cheikh Larbi, J. L. Halary, L. Monnerie, B. J. Bauer, and C. C. Han, Macromolecules 19, 810 (1986). 5. L. S. Egan, M. A. Winnik, and M. D. Croucher, Polym. Eng. Sci. 26,15 (1986). 6. J. M. Torkelson, E. G. Gordon, and M. D. Major, Polym. Prepr., Am. Chem. Soc. 27 (September 1986). 7. J. M. Torkelson, S. R. Gilbert, H.-S. Yu, and W. H. Schwimmer, Po- lym. Prepr., Am. Chem. Soc. 27 (September 1986). 8. B. A. Smith, E. T. Samulski , L.-P. Yu, and M. A. Winnik, Macromole- cules 18, 1901 (1985). 9. I. Mita, K. Horie, and M. Takeda, Macromolecules 14, 1428 (1981). 10. M. A. Winnik, A. M. Sinclair, and G. Beinert, Macromolecules 18, 1517 (1985). 11. J.-L. Viovy, L. Monnerie, and F. Merola, Macromolecules 18, 1130 (1985). Polymer Theory : Crossover and Criticality Most observed polymer solution properties actually are ob- served in regimes short of the infinite chain limit, since the chains in practice are of finite length. The description of such crossover regimes presents a challenge in theoretica l polymerPHYSICS NEWS IN 1986—POLYMER PHYSICS physics. In addition, the critical properties of phase separation of polymer-solvent systems and blends have so far been inter- preted only within the mean field formalism of Flory and Hug- gins. Experimental data on these systems clearly suggest that the polymer density fluctuations play a significant role in their critical behavior. The simplest problem of long-standing interest in polymer science is the calculation of the mean square end-to-end dis- tance, R2, of an isolated polymer chain of finite length with excluded volume interaction of arbitrary strength. The archty- pical model for this problem is the two-parameter model where the polymer is a random-flight chain with the short-ranged in- teraction between the segments. The dimensionless interaction parameter of this model, called z, equals w, the strength of the excluded volume effect, times the square root of L, the contour length of the chain. In the absence of excluded volume effect, R2 is approximate- ly equal to L. There have been many methods proposed to cal- culate R2 for any arbitrary value of z.1 The results from these schemes differ from each other depending on the severity of the intrinsic approximation of the method employed. Worse yet, because the approximations are uncontrolled, there is no way of estimating the uncertainties. It is therefore desirable to perform an accurate calculation of R 2 in order to (a) learn how to de- scribe a finite chain length in a systematic way, (b) assess the extent of the deviations of the various approximate schemes, and (c) find how close the two-parameter model is to reality. Towards this goal, Muthukumar and Nickel have derived a rel- atively long perturbation series for R2 in powers of z by evaluat- ing the leading six coefficients.2 They have obtained R2 numeri- cally for the full range of z.3 In addition, they have performed a systematic error analysis and found that their result is accurate within 0.5%. An entirely different analysis4 of their sixth-order series is in agreement well within the estimated uncertainty. The various previously known formulas deviate from these re- sults significantly. In order to compare With experimental data, similar lengthy calculations need to be done for several quanti- ties such as virial coefficients, radius of gyration, etc. Although much progress has been made in the theory of polymer physics, polymer science continues to be a rich mine of diverse and challenging theoretical problems. Experiments de- signed to test these developments clearly will rely on polymers of very well-defined structure. M. Muthukumar, University of Massachusetts 1. J. C. LeGuillou and J. Zinn-Justin, Phys. Rev. B 21, 3976 (1980). 2. M. Muthukumar and B. G. Nickel, J. Chem. Phys. 80, 5839 (1984). 3. M. Muthukumar and B. G. Nickel, J. Chem. Phys. 85, 4722 (1986). 4. J. des Cloizeaux, R. Conte, and G. Jannink, J. Physique Lett. 46, L-595 (1985).opment by Israelachvili of a version of the apparatus adaptable to studies of colloid, surface and polymer physics.2 This tool can provide information on surfaces in a liquid, or other desired ambient state in contrast to ultrahigh vacuum techniques. Smooth, basal, cleavage planes of muscovite mica are the sub- strate surfaces upon which can be deposited other substances between which one would like to measure surface forces. More than half a dozen groups around the world are now pursuing work with this device. Introductio n of this device to polymer studies was made first by Israelachvili and co-workers,3 though the first polymer ex- periments readily amenable to theoretical interpretation were made by Klein.4 We shall mention here only recent develop- ments in the study of forces between absorbed layers of poly- mers on mica immersed in solvent. These results are directly applicable to understanding the phenomena of polymer stabili- zation of colloidal dispersions described in Napper' s book;5 an introduction to some other lines being pursued with this device are described in Israelachvili's book.' Klein and co-workers have explored two systems in some detail: polystyrene (PS) in cyclohexane near the theta point4-6'7 and polyethylene oxide in good solvents, aqueous8 and or- ganic.9 In the PS system it has become clear6 that the attractive forces observed near the theta point consist of two contribu- tions: interactions of polymer segments in the solvent (osmot- ic) and individual macromolecules spanning the gap between the surfaces (bridging). Within the last year, Tirrell and co-workers have published two experimental efforts in part designed to clarify this situa- tion. Using block copolymers of 2-vinyl pyridine, which ad- sorbs more strongly than PS to mica and styrene, they have been able to affix PS cilia to the mica surface and eliminate bridg- ing.10 They have shown that there are no attractive forces between these PS layers at or above the theta points. They have also discovered some interesting configurational properties of adsorbed block copolymers. At the other extreme, they have examined PS only on one mica surface brought into contact with a bare mica surface.11 This enables a direct mechanical estimate of the sticking energy of polymer to mica. Other important developments in this area have been on the side of theoretical interpretation. Scheutjens and Fleer12 have published results of their lattice model calculations of the force profiles, while Pincus and co-workers13 have examined the problem from the point of view of gradient theory. Gast and Leibler have used a self-consistent field calculation to examine polymers in solution between the adsorbed layers.14 Klein and Luckham have recently taken a closer look at the interpretation of the range of the forces measured in these experiments.15 Matthew Tirrell, University of Minnesota Polymer Surface Forces Direct measurements of intermolecular forces by the technique of bringing two macroscopic surfaces into close proximity and measuring the force exerted between them have a long history.] Rapid progress has been made in the last decade since the devel-1. J. N. Israelachvili, Intermolecular and Surface Forces (Academic Press, Orlando, 1985). 2. J. N. Israelachvili and G. E. Adams, J. Chem. Soc. Faraday Trans. 174, 975 (1978). 3. J. N. Israelachvili, R. K. Tandon, and L. R. White, Nature 277, 120 (1979). 4. J. Klein, J. Chem. Soc. Faraday Trans. I 79, 99 (1983). PHYSICS TODAY / JANUARY 1987 S-65PHYSICS NEWS IN 1986—POLYMER PHYSICS 5. D. H. Napper, Polymeric Stabilization of Colloidal Dispersions (Aca- demic , New York, 1983). 6. J. N. Israelachvili, M. Tirrell, J. Klein, and Y. Almog, Macromolecules 17,204(1984). 7. Y. Almo g and J. Klein, J. Colloid Interface Sci. 106, 33 (1985). 8. J. Klein and P. F. Luckham, Macromolecules 17, 1048 (1984). 9. P. F. Luckham and J. Klein, Macromolecules 18, 721 (1985). 10. G. Hadziioannou, S. Patel, S. Granick, and M. Tirrell, J. Am. Chem. Soc. 108, 2869 (1986).U.S. Granick, S. Patel, and M. Tirrell, J. Chem. Phys. 85, 5370 (Novem- ber 1, 1986). 12. J. M. H. M. Scheutjens and G. J. Fleer, Macromolecules 18, 1882 (1985). 13. K. Ingersent, J. Klein, and P. Pincus, Macromolecules 19, 1374 (1986). 14. A. P. Gast and L. Leibler , Macromolecules 19, 686 (1986). 15. J. Klein and P. F. Luckham, Macromolecules 19, 2007 (1986). VACUUM PHYSICS GaAs on Si: Progress and Opportunities In the area of semiconductor technology, a relatively new material, gallium arsenide (GaAs), is emerging. Gallium arsenide promises much higher performance than the most common and well established material, silicon (Si). The growth of high-quality GaAs on a Si base (or substrate) by a process known as molecular beam epitaxy (MBE) promises to revolutionize telecommunications and computer chips. The fusion of these two material systems yields a hybrid which utilizes the advantages of both GaAs and Si. The high- ly developed Si-based chip technolog y and the superior qual- ity, thermal conductivity, strength, and size of Si substrates provid e an ideal foundation for semiconductor devices. On the other hand, electrical signals travel through GaAs much faster than through Si, and GaAs emits light when stimulat- ed by electrical signals. This combination of GaAs and Si generate s some exciting possibilities. Monolithic integration of GaAs and Si circuits will permit chip-to-chip communication via light signals. The viability of this technolog y has recently been confirmed with reports of room temperature operation of GaAs lasers on Si.123 High current GaAs devices can be used as out- put/interconnect drivers for low current Si NMOS (TV-chan- nel metal oxide semiconductor) transistor logic circuits to increase the overall speed of computers. Demonstration of this hybridization4 has aroused the interest of industry. The surge of activity in this technolog y has resulted from recen t solution s to two serious problems. The first of these obstacles is that the interatomic spacing of GaAs is 4% larg- er than that of Si. Because the growth of the thin film of semiconductor (epitaxial layer) mimics the size and struc- ture of the substrate, this mismatch in spacing causes strain which generate s imperfections. These structural flaws, called misfit dislocations, degrade the electrica l and optical performance of devices such as transistors and lasers. The solution to this problem lies in the proper choice of substrate orientation (the way in which the substrate is cut from the S-66 PHYSICS TODAY / JANUARY 1987bulk crystal or boule) and the growth of dislocation bar- riers.5 Confirmation of the defect control offered by these techniques has come from excellent results obtained from heterojunction bipolar transistors grown on Si.6 These tran- sistors, which have unique electrical transport characteris- tics, are very sensitive to defects. The second obstacle is en- countered in obtaining a coherent arrangement of atoms of a polar material (in this case GaAs) on a nonpolar substrate (Si). By MBE it is possible to deposit a single layer of As or Ga atoms on the substrate. This crystal growth process uti- lizes the precision-controlled evaporation of materials onto a substrate in ultra high vacuum. This technique provides an elegant solution to this problem by ensuring proper positions for Ga and As atoms. The potential for GaAs-on-Si technology is vast. Prelimi- nary results have shown that another material system, mer- cury cadmium telluride, which has specialized applications as a light detector, can be grown on a GaAs-coated Si sub- strate. Since the signal processing circuitry could be put on the Si substrate, improved optical detectors would result.7 Another fascinating possibility is the use of calcium fluoride, to serve as an insulator between layers of GaAs allowing three dimensional circuits to be realized.8 Utilizing the high speed of GaAs to perform time critical function s while using Si to perform the less demanding ones promises to alleviate bottlenecks which now limit overal l data processing and computer system speeds . Indeed GaAs on Si offers the best of both these material systems and promises to surpass them in high speed electronic and opto-electronic applications. G. Munns and H. Morkoc, University of Illinois 1. R. Fischer etal, Appl. Phys. Lett. 48, 1360 (1986). 2. J. P. van derZiel, R. D. Dupuis, and J. C. Bean , Appl. Phys. Lett. 48,1713 (1986). 3. T. H. Windhorn and G. M. Metze, Appl. Phys. Lett. 47, 1031 (1985). 4. R. Fischer et a/., Appl. Phys. Lett. 47, 983 (1985). 5. N. Otsuka, C. Choi, Y. Nakamura, S. Nagakura, R. Fischer, C. K. Peng, and H. Morkoc, Appl. Phys. Lett. 49, 277 (1986).PHYSICS NEWS IN 1986—VACUUM PHYSICS 6. R. Fischer, J. Klem, J. S. Gedymin, T. Henderson, W. Kopp, and H. Morkoc, Dig. of Int. Electron Dev. Meeting, 332 (1986). 7. K. Zanio, R. Bean, K. Hay, R. Fischer, and H. Morkoc, presented at the 1986 Materials Research Society Symposium on Compound Semicon- ductor Materials (Palo Alto, CA, April 15-18, 1986). 8. T. Asano, H. Ishiwara, H. C. Lee, K. Tsutsui, and S. Furukawa, extended abstract of the 17th Conference on Solid State Device and Materials (To- kyo, 1985), p.217. Surface States in Real Space Surface scientists have long tried to understand the interplay between surface geometric and surface electronic structure. Traditionally, these two have been studied in separate ex- periments with the link being provided by theory. In such studies experimental information is averaged over the probe area of the experiment and a quantum mechanical theory is employed to provid e an understanding at the atomic level. In a series of experiments on silicon surfaces, Scanning Tunnel- ing Microscopy (STM) has been used for the first time to bridge the gap between geometric and electronic structure with atomic resolution. STM images obtained with different voltages between the tip and the sample are generally different because only the wave functions with energies between the Fermilevels of the tip and the sample contribute to the tunneling process. As the voltage difference increases, the tunneling conductance will experience a stepwise increase each time a new quantum state starts to contribute to the tunneling process (provided that the sample-tip distance is held constant). Thus, the en- ergies at which surface states occur can be determined by measuring current-versus-voltage (J-V) curves. In order to create 2-D images of surface states we have developed a new method, Current Imaging Tunneling Spectroscopy (CITS), to separate spectroscopic information from other contribu- tions, in particular the surface geometry.' This new method circumvents several of the problems encountered by pre- vious workers attempting to image surface electronic struc- ture (see Fig. 1). The basic problem is to separate the geometric from the electronic contributions to the STM images . For a given model a purely geometric image can easily be calculated by superposition of atomic charge densities in which surface states are not considered. Surfaces of constant charge den- sity at the position of the tip reveal what the STM image should look like in the absence of surface states. Experimen- tally it was found that under suitable bias conditions the tip very closely follows such purely geometric contours. The CITS technique enables us to measure complete I-Vcurves m each pixel of an STM image while the tip independently follows the geometric contours. In this way the variations of tunneling current with lateral position directly revea l differ- ences in surface electronic structure. With this technique we nave succeeded in obtaining the first spatially- and energy- resolved images of the filled and empty surface states of theFIG. 1. A scanning tunneling microscopy image of the (a) silicon (111) 7x7surface, (b) adatom dangling bond state with an energy 0.35 eV below the Fermi energy, (c) restatom danglin g bond state at 0.9 eV below the Fermi energy, and (d) backbond state at 1.8 eV below the Fermi energy. One 7x7 unit cell is outlined in each frame. Si(lll)-(7X7) and Si(001)-(2x 1) surfaces with atomic resolution. The observed surface states agree very well with results known from photoemission and inverse photoemission ex- periments. In addition we were able to directly correlate these surface states with specific atomic features in the sur- face structure. Thus, we observed a filled surface state close to the Fermilevel associated with the dangling bond orbitals of 12 adatoms (atoms adsorbe d on a surface) in the Si (111) - (7X7) unit cell. In addition we found 7 orbitals 0.8 eV be- low the Fermilevel associated with broken bonds on the threefold coordinated silicon atoms (called restatoms) in the underlying surface layers. The location of these states further correspond with atomic features of the model pro- posed by Takayanagi etal.1 Recent theoretical work by North- rup on small subunits of the (7x7) cell is in good agreement PHYSICS TODAY / JANUARY 1987 S-67PHYSICS NEWS IN 1986—VACUUM PHYSICS with these results.3 The location of the restatoms states has provided direct, real space evidence for the presence of a stack- ing fault in the double layer directly underneath the adatoms in one half of the (7 X 7) unit cell. The Si(001)-(2x 1) surface reconstructs by forming dimer bonds between adjacent surface atoms, thus reducing the num- ber of broken bonds by a factor of 2. We have observed both the bonding state, located on and between the dimer atoms, and the antibonding orbitals, located on the outer ends of the dimers. Some of the dimers are buckled. On these we find that the bond- ing state is strongly localized on the "up" side of the dimer, whereas the antibonding state is found on the "down" side. This remarkable spatial separation of bonding and antibonding orbi- tals is strong evidence for charge transfer from the down to the up atom. The possibility of such a charge transfer in conjunc- tion with buckling has been a controversial subject for many years and explains the pronounced surface state bandgap. Our experimental findings are in good agreement with the theoreti- cal work by Ihm et al* In addition to the electronic features associated with the re- gular surface structur e we have also observed a variety of local- ized electron states introduced by atomic scale structura l de- fects. The study of such defects will be of importance in efforts to improve our understanding of many defect-dominated phys- ical phenomena, such as Schottky barrier formation, surface chemistry (steps, kinks) and oxide dielectric breakdown. Most importantly, surface geometric and electronic structur e can now be studied simultaneously at the atomic level. R. M. Tromp, IBM T. J. Watson Research Center 1. R. J. Hamers, R. M. Tromp, and J.E. Demuth, Phys. Rev. Lett. 56, 1572 (1986). 2. K. Takayanagi, Y. Tanishiro, M. Takahashi, and S. Takahashi, J. Vac. Sci. Technol. A3,1502 (1985). 3. J. E. Northrup, Phys. Rev. Lett. 57, 154 (1986). 4. J. Ihm, M. L. Cohen and D. J. Chadi, Phys. Rev. B 21, 4592 (1980). Studies of Surface Phonons by Electron Energy Loss Spectroscopy In both molecular and solid state physics, the study of vibra- tional modes provides crucial information on the natur e of the chemical bonding and subtle aspects of structura l geometry. Vibrational spectroscopies employ the absorption and inelastic scattering of photons, or of particle probes such as neutrons. For the same reasons, one wishes to study vibrational mo- tions of atoms in, or molecules on, crystal surfaces. Convention- al spectroscopies function in only limited circumstances be- cause it is very difficult to extract a signal from a monolayer of material against the large background from the crystal upon which the monolayer resides. Electrons of suitable energy (1-500 eV) are a powerful sur- face probe, since their mean free path in matter is only three orfour interatomic spacings. Electrons backscattered from sur- faces thus contain information on only the outermost atomic layers. As the electron backscatters, it may create a vibrational quantum of frequency &>v, and lose energy ha> v in the process, where h is Planck's constant. Measurement of this small energy loss, which requires the development of highly monoenergetic electron beams, thus provides access to vibrational frequencies. In the last two decades, electron energy loss spectroscopy has evolved into a powerful probe of surface vibrations, notable for its wide spectral range.1 Recent new experimental and theoreti- cal developments have greatly expanded the information one may obtain from such data. Traditionally, one measures the energy loss of only those electrons which suffer very small angular deflections in the vi- brational excitation process; such events are particularly in- tense by virtue of the long-ranged coulomb fields generated by surface vibrations. Such electrons emerge very close to the specular direction after reflection off the surface, and these losses are most intense if rather low primary energies—from one to ten electron volts—are employed. Selection rules allow one access to only a subset of the vibrational modes of the sur- face, in this mode of operation. A new generation of experiments2"6 study electrons deflected through large angles in the loss event. They then emerge far from either the specular or Bragg beam directions. The near- specular selection rule breaks down, and one has access to the complete spectrum of vibrational modes, in principle. Electron momentum transfer is a new variable; the dependence of the frequency of a surface vibrational mode (surface phonon) on wavelength may be extracted from the shift of the correspond- ing loss peak with momentum transfer. The dependence of fre- quency on wavelength is called the dispersion curve, and we now have access to surface phonon dispersion curves continu- ously from long wavelengths, out to the shortest allowed, which equals the interatomic spacing. High frequency surface phon- ons associated with adsorbate layers have now been studied for several systems, along with the influence of adsorbates on the substrate surface phonons. The new spectroscopy is a surface analogue of the inelastic neutro n scattering experiments which have proved a powerful probe of bulk matter. Essential to the success of the new method is use of high beam energies in the 100-300 eV range; it is an experimental tour de force to produce such beams so monoenergetic that the tiny energy losses from vibrational excitating can be detected. In this energy region, it has proved possible4 to calculate theoretically, in a remarkably quantitative manner, the energy and angle vari- ation of surface phonon excitation cross sections. These have guided the choice of scattering geometry, in a successful search for a surface phonon predicted by theory, but absent in the early data. The systematics of the energy and angle variation of the cross sections are also found to be sensitive to surface geometry, so we have a new means of probing the two closely related topics of surface structure and surface dynamics, simultaneously. In recent years, highly monoenergetic neutral helium beams have been developed, and employed to study surface phonons by energy loss spectroscopy.6 We now have two complemen- S-68 PHYSICS TODAY / JANUARY 1987PHYSICS NEWS IN 1986—VACUUM PHYSICS tary spectroscopies of surface phonons, and for the first time we have direct access to these modes, which have been studied for many years by theorists.7 D. L. Mills, University of California at Irvine 1. Electron Energy Loss Spectroscopy and Surface Vibrations, H. Ibach and D. L. Mills (Academic, San Francisco, 1982). 2. S. Lehwald, J. Szeftel, H. Ibach, T. S. Rahman, and D. L. Mills, Phys. Rev. Lett. 50, 518 (1983). 3. J. Szeftel, S. Lehwald, H. Ibach, T. S. Rahman, J. E. Black, and D. L. Mills, Phys. Rev. Lett. 51, 268 (1983). 4. The first quantitative comparison between theory and experiment has been given by M. L. Xu, B. M. Hall, S. Y. Tong, M. Rocca, H. Ibach, S. Lehwald, and J. E. Black, Phys. Rev. Lett. 54, 1171 (1985). Further results are found in L. L. Kesmodel, M. L. Xu, and S. Y. Tong (to be published), and B. M. Hall and D. L. Mills (to be published). 5. S. Lehwald, M. Rocca, H. Ibach, and T. S. Rahman, Phys. Rev. B 31, 3477 (1985), and for an erratum see Phys. Rev. B 32, 1354 (1985). 6. For a review of early work in this area, see J. P. Toennies, J. Vac. Sci. Technol. A2, 1055 (1984). 7. R. F. Wallis, Prog. Surf. Sci. 4, 233 (1973). XPS and Auger Forward Scattering in Epitaxial Films In angular-dependent x-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES) of single crystals, an effect has long been known in which XPS and AES peaks exhib- it enhanced intensities along major crystal axes.12 This effect was thought to be due to electrons channeling among planes of atoms in the lattice.1'3 However, recent studies on the develop- ment of these enhanced intensities during layer-by-layer epitax- ial growth in metals such as Co, Ni, and Cu on Ni( 100) and Cu( 100) substrates demonstrate that the enhancements are al- ready present when the metal under study is only two or three atomic layers thick.2 This is too thin to be the result of channel- ing, which would require much thicker films.2 The actual physical basis of the effect is forward scattering of the outgoing electron by overlying lattice atoms. The initial out- going electron wave interferes constructively with the scattered wave from an overlying atom to produce an enhanced intensity along the axis connecting the emitting and scattering atom.4'5 Such an interference was first identified in XPS of molecular CO.6This phenomenon has an interesting classical analogy in that the scattering atom is acting much like a lens, focussing the intensity of the emitting atom along the forward direction.4 This means the XPS and AES peaks act like "searchlights" pointing out the internuclear axes present in the top several atomic layers. This effect is a rather short-range one, applying primarily to nearest-neighbor and next-nearest-neighbor scattering atoms, since only the first scattering event tends to be forward focus- sing with subsequent scattering events (multiple scattering) tending to be defocussing.4 As a probe of local or short-range order this effect fills impor- tant gaps left by other techniques. In recent work, XPS and AES forward scattering have been found to provide a wealth of new information about such phenomena as the temperature de- pendence of surface alloying,2 the effect of surface contamina- tion on epitaxy,7 misfit dislocations and abrupt junctions in epitaxy,8 metal-semiconductor interfaces,9 and surface core- level binding energy shifts.2 W. F. Egelhojf, Jr., National Bureau of Standards 1. References to early work are given in Ref. 2. 2. W. F. Egelhoff, Jr., J. Vac. Sci. Technol. A2, 350 (1984); Phys. Rev. B 30 1052 (1984); J. Vac. Sci. Technol. A3, 1511 (1985); in The Structure of Surfaces, edited by M. A. Van Hove and S. Y. Tong (Springer, Berlin, 1985) , p. 199; and R. A. Armstrong and W. F. Egelhoff, Jr., Surf. Sci. 154, L255 (1985). 3. Early suggestions that forward scattering could be involved as well are found in: S. Kono, C. S. Fadley, N. F. T. Hall, and Z. Hussain, Phys. Rev. B 22, 6085 (1980); S. A. Chambers and L.W. Swanson, Surf. Sci. 131, 385 (1983); S. Takahashi, S. Kono, H. Sakurai, and T. Sagawa, J. Phys. Soc. Jpn. 51, 3296(1982). 4. H. C. Poon and S.Y. Tong, Phys. Rev. B 30, 6211 (1984); S.Y. Tong, H.C. Poon , and D.R. Snider, Phys. Rev. B 32, 2096 (1985); H. C. Poon, D. Snider, and S. Y. Tong, Phys. Rev. B 33, 2198 (1986). 5. E. L. Bullock and C. S. Fadley, Phys. Rev. B 31, 1212 (1985). 6. L. G. Peterson, S. Kono, N.F.T. Hall, C. S. Fadley, and J.B. Pendry, Phys. Rev. Lett. 42, 1545 (1979). 7. W. F. Egelhoff,Jr., in Layered Structures Epitaxy and Interfaces, edited by J. M. Gibson and L. R. Dawson (North-Holland, Amsterdam, 1985) , 443. 8. S. A. Chambers, T. R. Greenlee , C. P. Smith, and J. H. Weaver, Phys. Rev. B 32, 4245 (1985); S. A. Chambers, H. W. Chen, I. M. Vitomiron, S. B. Anderson, and J. H. Weaver, Phys. Rev. B 33, 8810 (1986). 9. S. A. Chambers, S. B. Anderson, and J. H. Weaver, Phys. Rev. B 32 581 (1985). PHYSICS TODAY / JANUARY 1987 S 69PHYSICS NOBEL PRIZE The 1986 Nobel Prize in physics was awarded for work de- voted to the development of microscopes. Ernst Ruska of the Fritz Haber Institute of the Max Planck Gesellschaft in West Berlin received half of the prize money for his inven- tion in 1931 of the electron microscope. The other half of the prize was shared by Gerd Binnig and Heinrich Rohrer, who work at the IBM Zurich Research Laboratory in Switzer- land, for their development in the 1980s of the scanning tun- neling microscope. The IBM microscop e employs a phenomenon known as quantum tunneling, in which subatomic particles can some- times pass through "forbidden" regions of space {see Physics News in 1983, p. 16). Imagine, for the moment, an experi- ment in which tenni s balls are rolled toward a concrete wall. Imagine also, on the other side of the wall, a detector which lights up every time a ball passes through the solid wall. In the macroscopic world such balls never pass through; the inside of the wall is a forbidden region, and the detector on the other side never lights up. Imagine now a microscopic version of this setup, with balls the size of electrons. Such tiny balls could, according to quantum mechanics, occasion- ally pass through the wall—the detector would now begin to light up. The likelihood of the quantum mechanical balls negotiating the forbidden passage through the wall is in- versely proportional to the thickness of the wall. The thicker the wall, the fewer the number of balls observed on the other side. By workin g backwards, one could determine the thick- ness of the wall at any point by observing how frequently balls were able to tunnel through. In Binnig's and Rohrer's device,1 electrons are used to probe the surface layer of atoms in a material sample. The forbidden region in this case is a layer of vacuum (only sever- al angstroms thick) separating two electrodes. One elec- trode, a very sharp needle (whose point is only a few atoms across), is equivalent to the source of tennis balls in the imaginary experiment. The other electrode is the sample whose surface properties are to be studied. By applyin g a tiny voltage difference between the needle tip and the sample sur- face, a tiny tunneling current of electrons will flow between them; the size of the current depends on the distance between the tip and the sample surface. By moving the needle (which is poised vertically above the sample) from place to placelike a phonograph stylus over a record, a series of readings measuring the tip-sample separation at many points can be compiled. This scanning data can be converted into a three dimensional map of the surface layer of atoms in the sam- ple.2 The whole setup acts, in effect, as a microscope for imaging the sample surface. In recent years this "scanning tunneling microscope" has been improved and now possesses a lateral resolution of 1A (the size of an atom) and an equivalent vertical resolution of about 0.01 A.3 The STM has generally been used to study the surface topography and chemical bonds between atoms in semiconductors and metals but has also been used to im- age biological subjects such as viruses. The STM has also been used as a device for measuring forces as small as 10 ~18 Newtons.4 (For further information on scanning tunneling microscopy, refer to the articles in this issue in the chapters on condensed matter physics, vacuum physics, and on phys- ics applied to industry.) In the electron microscope, developed by Ruska and M; Knoll in Germany in 1931-32, electrons take the place of the light waves used in conventional optical microscopy.5 In Ruska's device, which would now be called a transmission electron microscope, a beam of electrons strikes a sample. The scattered electrons pass through an aperture and are focused by a magnetic "lens" onto a screen or a photograph- ic plate. The advantage of using electrons over lightwaves is in their much smalle r wavelengths: 0.1 A for electrons in a typical device compared to thousands of angstroms for visi- ble light. The resolution of the final image is inversely pro- portional to the wavelengt h of the illuminating beam. Typi- cal electron microscope resolutions are about 2 A.6 Phillip F. Schewe, American Institute of Physics 1. G. Binnig, H. Rohrer, C. Gerber, and E Weibel, Phys, Rev. Lett. 49, 57 (1982). 2. G. Binnig and H. Rohrer, Sci. Am. 253, 50 (August, 1985). 3. C. Quate, Phys. Today 39, 26 (August 1986). 4. G. Binnig, C. F. Quate, and C. Gerber, Phys. Rev. Lett. 56, 930 (1986). 5. A. V. Crewe, Sci. Am. 234, 26 (April 1971). 6. J. M. Cowley and S. Ilijima, Phys. Today 30, 32 (1977). S-70 PHYSICS TODAY / JANUARY 1987

1.4971961.pdf

Defect-induced magnon scattering mechanisms in exchange-coupled bilayers R. A. Gallardo , R. L. Rodríguez-Suárez , and P. Landeros Citation: J. Appl. Phys. 120, 223904 (2016); doi: 10.1063/1.4971961 View online: http://dx.doi.org/10.1063/1.4971961 View Table of Contents: http://aip.scitation.org/toc/jap/120/22 Published by the American Institute of Physics Articles you may be interested in Band gap formation and control in coupled periodic ferromagnetic structures J. Appl. Phys. 120, 223901223901 (2016); 10.1063/1.4971410 Route toward high-speed nano-magnonics provided by pure spin currents J. Appl. Phys. 109, 252401252401 (2016); 10.1063/1.4972244 Low operational current spin Hall nano-oscillators based on NiFe/W bilayers J. Appl. Phys. 109, 242402242402 (2016); 10.1063/1.4971828 Performance degradation of superlattice MOSFETs due to scattering in the contacts J. Appl. Phys. 120, 224501224501 (2016); 10.1063/1.4971341 Defect-induced magnon scattering mechanisms in exchange-coupled bilayers R. A. Gallardo,1,a)R. L. Rodr /C19ıguez-Su /C19arez,2and P . Landeros1 1Departamento de F /C19ısica, Universidad T /C19ecnica Federico Santa Mar /C19ıa, Avenida Espa ~na 1680, 2390123 Valpara /C19ıso, Chile 2Facultad de F /C19ısica, Pontificia Universidad Cat /C19olica de Chile, Casilla 306, Santiago, Chile (Received 6 September 2016; accepted 26 November 2016; published online 15 December 2016) The influence of two-magnon scattering mechanisms, which may be activated by different sorts of defects, is theoretically studied in ferromagnetic/antiferromagnetic exchange-biased bilayers. Thespin-wave based model considers the influence of geometrical defects in the ferromagnetic (FM) layer as well as small domains in the antiferromagnetic (AFM) sub-lattice of the FM/AFM interface in such a way that both kinds of defects are randomly distributed over their respective surfaces.The in-plane angular dependence of the ferromagnetic resonance (FMR) linewidth allows detection of the relevant influence of such defects in the relaxation mechanisms, where the role of the exchange-bias field is clearly identified. Typical experimental findings, such as quadratic depen-dence of the linewidth with the exchange-bias field and the in-plane angular dependence, are well explained within the proposed model. This lends confidence in the model’s utility and leads to a better understanding of the role of the magnon-magnon scattering in the magnetization dynamicsof exchange-coupled antiferromagnetic/ferromagnetic bilayers. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4971961 ] I. INTRODUCTION When a ferromagnetic (FM) material is in contact with an antiferromagnetic (AFM) material, a subtle interfacial coupling arises and a notable hysteresis loop shift is observed.1–7This phenomenon, called exchange-bias (EB), was discovered more than 50 years ago1and has been very useful in technological applications like spintronic devices, where pinned ferromagnetic layers are frequently needed.8,9 The shift of the hysteresis loops is perhaps the iconic effect that appears when biasing an FM layer with an AFM layer. There are also other important signatures of EB such as the coercivity increment, the different reversal mechanisms for increasing and decreasing fields giving rise to asymmetrichysteresis loops, 10,11the positive EB,12,13training,14,15and memory effects.16,17Despite the tremendous technological impact of EB, a complete physical understanding has not yet been developed. This is perhaps because it is very difficult to examine the properties of antiferromagnetic layers, while theferromagnetic ones can be better understood with conventional magnetic characterization techniques. Knowledge of the interface between the FM and AFM layers is even more lacking, given its minimal volume and the cur- rent inability to properly assess it directly. The latter is key to getting a reasonably good physical picture of the EB phe- nomenon. 5–7Currently, it is well accepted that the AFM order is responsible for EB. Nevertheless, the behavior of the AFM spins during FM magnetization reversal is not completely clear.18 Given the importance for spintronics applica- tions,8,9,19–22EB has been widely studied using either static or dynamic characterization techniques.23–34A deepunderstanding of the relaxation mechanisms is fundamental, since they play a key role in the spin dynamics and magneti- zation reversal in ferromagnetic nanostructures.35In EB bilayers, additional dissipation mechanisms are present alongside the intrinsic (Gilbert) relaxation process, arising from the strong influence of the interfacial exchange cou- pling on the magnetization dynamics. As has been reported for these systems, the increase of the effective damping parameter23–28or the measured by ferromagnetic resonance (FMR) linewidth broadening,29–31depends on the magnitude of the exchange-bias field HE. In some cases, this is two orders of magnitude larger than the Gilbert contribution. In all of these works, the broadening and the angular depen- dence of the FMR linewidth have been attributed to the two- magnon scattering (TMS) mechanism35–40activated by the interface roughness of the films, and to the variation of prop- erties within the sample. These variations were phenomeno- logically introduced29–31by including fluctuations in the uniaxial anisotropy, effective magnetization, and exchange field among others, complicating the understanding of the relaxation mechanisms. For example, there has been no dis- cussion of the AFM grains41coupled to the FM film, influencing the linewidth broadening. In this paper, we apply the TMS formalism developed by Arias and Mills36to inves- tigate the contribution of small domains in the AFM sub- lattice adjacent to the FM film. As we show, this is important for having a complete description of the relaxation mecha- nisms in FM/AFM systems, and this may provide insights to the nature of the defects in a variety of systems. The paper isorganized as follows. In Section II, a theoretical description of the TMS process activated by both geometrical defects and small domains in the uncompensated AFM sub-lattice adjacent to the FM layer is presented. In Section III, the a)Electronic mail: rodolfo.gallardo@usm.cl 0021-8979/2016/120(22)/223904/6/$30.00 Published by AIP Publishing. 120, 223904-1JOURNAL OF APPLIED PHYSICS 120, 223904 (2016) results and the corresponding discussions are shown, and in Section IV, the main conclusions are summarized. II. THEORETICAL DESCRIPTION In this paper, the FMR response of a thin film with EB coupling is discussed analytically. The theoretical descrip-tion employs the two-magnon scattering picture and consid-ers both the influence of rectangular defects in FM film and small misaligned domains in the adjacent AFM sub-lattice. Both types of defects are randomly distributed over theirrespective surfaces (see Fig. 1). By means of the in-plane angular variation of the linewidth, the contribution of bothdefects is distinctively detected. As explained in theAppendix , expressions for the FMR frequency and the FMR field follow from Refs. 30and42. Useful information about magnetic relaxation mecha- nisms is usually extracted through the FMR linewidth DH, where both intrinsic and extrinsic processes must be consid-ered to describe it both qualitatively and quantitatively. TheFMR linewidth is generally expressed as DH¼DH G þDH2M, where the first term is associated to the phenome- nological Gilbert damping, given by DHG¼2ffiffiffi 3pa cN2pfR; (1)where ais the Gilbert damping parameter, fRis the resonance frequency, and cis the gyromagnetic ratio. Nis the so-called dragging function,40which accounts for the possible drag between the magnetization and the applied field and conse- quently depends on the external applied field, the anisotro-pies, and the equilibrium angles. Nevertheless, for the sakeof simplicity, it is assumed that the FM magnetization fol- lows the field ( u/C25u H), since the resonance fields are much larger than the in-plane anisotropy fields, which gives N/C251. In addition, note that in Eq. (1), the 2 =ffiffiffi 3p factor is intro- duced to obtain the peak-to-peak linewidth of the FMR sig- nal. The second term DH2Mcorresponds to the TMS contribution, whose fundamental idea is that magnons withzero wavevector k¼0 can be transferred to the state k 0as long as they have the same energy or frequency.35,36,39This magnon scattering may be activated by impurities or defects, allowing extraction of useful information about the magneticsample. Two contributions to the TMS linewidth originating with two kinds of defects will be considered. The first is alinewidth term DH 2M Iproduced by pits or bumps defects with an approximately rectangular shape that are randomly distributed over the FM film’s surface. The second contribu-tion,DH 2M II, is activated by small domains in the uncompen- sated AFM sub-lattice adjacent to the FM layer. Hence, the total extrinsic contribution to the FMR linewidth can be writ- ten asDH2M¼DH2M IþDH2M II, with DH2M I;II¼2ffiffiffi 3pC2MðÞ I;II c2HXXþHYY ðÞ; (2) where Cð2MÞ I;IIis the two-magnon damping rate [see Eq. (55) in Ref. 39], and HXXandHYYare given in Eqs. (A2) and(A3) in the Appendix . For rectangular defects, the two-magnon damping rate becomes30,43 C2MðÞ I¼K0 I HXXþHYY( H2 XX/C0Wp=2ðÞ /C0 uðÞ/C28c a/C29 /C01 ! þW0ðÞ þuðÞ/C28a c/C29 /C01 !) sin/C01ffiffiffiffiffiffiffiffi HXX HYYr ; (3) with K0 I¼8c2pIK2 sb2 pDM2 sd2: (4) Ksis the surface anisotropy constant, Msis the saturation magnetization of the FM layer, D¼2A/Msis the exchange stiffness, and dis the FM layer thickness. The quantity pIis the fraction of the surface covered by geometrical defects with a topology characterized by a side of length aparallel to the xaxis, a side of length cparallel to the zaxis, with b represents the height or depth of the defects [see Fig. 1(b)]. The brackets hiindicate average values over the randomly distributed population of defects. Also, the function WðdÞ 6ðuÞ has been defined as WðdÞ 6ðuÞ¼½ HYYcosð2uÞ6HXX cos2ðuþdÞ/C1382, which mainly describes the angular depen- dence of DH2M I. From Eq. (3),dcan be either 0 or p/2. FIG. 1. (a) Geometry of the system. The vector quantities ^uK;^uAK;MA, and Meqare arbitrarily oriented in the plane of the film, making angles uK;uAK;uA, and uwith the zaxis, respectively. The equilibrium magneti- zation Meqallows the definition of the XYZ reference system, where Zpoints along the MeqandXlies in the film’s plane. Schematic representations of randomly distributed defect structures include (b) the geometrical defects onthe FM surface, and (c) nucleated domains over the AFM sub-lattice adja- cent to the FM layer. The geometrical defects are assumed to be rectangular in shape, where bis the thickness, ais the side along the x-axis, and cis the z-axis side. Also, inside each domain, the magnetic moments of the AFM sub-lattice, represented by M d A, have an arbitrary orientation defined by ud A, while the rest of the AFM magnetic moments are oriented along MA.223904-2 Gallardo, Rodr /C19ıguez-Su /C19arez, and Landeros J. Appl. Phys. 120, 223904 (2016) To obtain an explicit expression for Cð2MÞ II, the energy change when small domains are introduced in the AFM sub- lattice adjacent to the FM layer [see Fig. 1(c)] must be con- sidered. Thus, defining ^Mdj Aas the unit vector oriented along the magnetic moments inside such misaligned AFMdomains, the energy change when the jth defect is introduced can be written as D/C15 j E¼JE Msð Sj dMj/C1^MAdSj d/C0JE Msð Sj dMj/C1^Mdj AdSj d;(5) Sj dis the surface of the j-th AFM domain and JEis the inter- facial exchange coupling constant. We define udj Aas the angle between the magnetic moments inside the jth domain’s defect and the zaxis. According to Eq. (5), when the mag- netic moments inside the AFM domains point in the samedirection as the rest of the sub-lattice, u dj A¼uAand the per- fect sub-lattice is recovered [see Fig. 1]. Then the magnetiza- tion can be written as Mj¼Mj Z^Zþmj X^Xþmj Y^Yand up to second order in spin deviation, the Z-component becomes Mj Z¼Ms/C0½ ðmj XÞ2þðmj YÞ2/C138=ð2MsÞ. Letting aside terms of zero and first order in mj X;Y, the energy change is given by D/C15j E¼JE 2M2 sð Sj dmj X/C16/C172 þmj Y/C16/C172/C20/C21 vuðÞdSj d; (6) where vðuÞ¼cosðu/C0udj AÞ/C0cosðu/C0uAÞ. Assuming that inside the defect’s zone, the magnetization does not vary sig-nificantly, therefore Eq. (6)becomes D/C15 j E¼Sj dJE 2M2 smj X/C16/C172 þmj Y/C16/C172/C20/C21 vuðÞ: (7) According to the model, uAdepends on the equilibrium con- ditions and therefore it is not fixed when the in-plane exter-nal field is swept. Likewise, a similar behavior for u dj Acan be argued. Summing over all domains and following the Arias and Mills theory for random defect distributions,35,36,39it is easy to write the following TMS damping rate: C2MðÞ II¼K0 IIvuðÞ2 HXXþHYYjHXXþHYYj2sin/C01ffiffiffiffiffiffiffiffi HXX HYYr ; (8) where K0 II¼2c2pIIJ2 ESd pDM2 sd2; (9) andpIIis the fraction of the surface covered by the AFM domain-like defects. Notice that in Eq. (9),Sdis written instead of Sj d, where now Sdrepresents an average value, since the domain surface is a fluctuating parameter. Thesame criteria apply to u dj A. Finally, the TMS linewidth is given by DH2M¼2ffiffiffi 3pC2MðÞ IþC2MðÞ II c2HXXþHYY ðÞ: (10) Some general conclusions about the TMS linewidth can be quickly stated. First, due to the surface nature of the defects,DH2Mscales as d/C02. Second, as HE¼JE/(Msd), one can state thatDH2M/H2 E. Third, in the in-plane angular variation of the linewidth, there is an angle u0at which vðu0Þ¼0 (or DH2M II¼0), which therefore infers that the main cause of the FMR linewidth broadening at any angle is due to geometricaldefects. Nonetheless, both DH 2M IandDH2M IIare relevant to explain the in-plane angular dependence, as will be devel- oped in the Sec. III. III. RESULTS AND DISCUSSION In this section, typical values of parameters for NiFe/ IrMn bilayers are considered to describe the measurable quantities such as the resonance field, various contributions to the FMR linewidth, and its dependencies on angular andgeometrical degrees of freedom. These are c¼17.6 GHz/ kOe, a¼0.005, H s¼0, and Ms¼800 emu/cm3. The uniaxial and unidirectional axes are parallel to each other with uK¼uAK¼0, i.e., both along the z-axis. Also, the uniaxial anisotropy field is HK¼5.1 Oe, hc=ai/C251:04, while K0 I=c2¼15 Oe and K0 II=c2¼1 Oe [see Eq. (2)]. The fre- quency is 8.61 GHz. In Fig. 2, the resonant field HRas a function of EB field HEand angle uare shown. The resonant field has a clear enhancement at u¼180/C14asHEincreases. At HE¼0, a two-fold symmetry is observed resulting from the uniaxial anisotropy field HK, which produces an easy orientation along u¼0a n d u¼p. In the same way, the unidirectional nature of the EB field produces a minimum only at u¼0o r FIG. 2. (a) Resonant field as a function of HEanduforHRA¼0. In (b), the resonant field as a function of uis shown for the cases HE¼0,HE¼50 Oe, andHE¼100 Oe.223904-3 Gallardo, Rodr /C19ıguez-Su /C19arez, and Landeros J. Appl. Phys. 120, 223904 (2016) 2p.29–31,34Further, the role of the rotatable anisotropy field is associated with the shift of the resonant fields, since the FM magnetization interacts with independent AFM grains, which follow the external field.41Thus, while HRAincreases, a decrease of the resonant field is produced (not shown). Experimentally, the FMR linewidth tends to mimic the angular variation of the resonance field,29–31and since the Gilbert contribution DHGis isotropic under the approach uH¼u(no dragging), extrinsic relaxation mechanisms are necessary to describe the angular behavior of the linewidth.Fig.3(a)shows the behavior of the linewidth as a function of uandu d AforHE¼25 Oe, when both geometrical and AFM domain defects are considered. Regarding DH2M II, the param- eter ud Adescribes the average orientation of the magnetic moments inside the domain defects; hence, ud Acan be assumed a fixed quantity or it can be influenced by the exter-nal field. Overall, an appreciable influence of u d Ain the angu- lar variation of the linewidth is detected. Figures 3(b),3(d), and 3(e) show the particular cases ud A¼180/C14;ud A¼90/C14, andud A¼0/C14, while Fig. 3(c)illustrates the case ud A¼u. The angular dependence of the FMR linewidth has a clear contri-bution from both DH 2M IandDH2M II. In the ( DH2M I) contribu- tion, geometrical defects produce an enhancement of thelinewidth, that is, larger than the Gilbert contribution. Thiscreates a clear two-fold symmetry with maxima at u¼90 /C14 and 270/C14, provided the defect side parallel to the easy axis (^uKin Fig. 1) is larger than the perpendicular one [ c>ain Eq.(3)]. The DH2M IIcontribution has a strong dependence on the magnetic moment orientation ud Ainside the domain zones. In the case ud A¼0/C14, we obtain DH2M II¼0 as shown in Fig. 3(e), since the AFM sub-lattice near the FM layer is fully uncompensated and thus the interaction in Eq. (5) becomes zero. For the case ud A¼180/C14shown in 3( b), it is understood that the small domains in the AFM sub-latticeare compensated with the FM layer. Here, the partial com-pensation of the moments is present in the FM/AFM inter-face, as has been previously reported. 44–46In the case ud A¼90/C14depicted in 3( d),DH2M IIhas a clear angular depen- dence with an asymmetric behavior around u¼180/C14. In thespecial case ud A¼u, shown in Fig. 3(c), a clear maximum at u¼180/C14results, since in such an orientation, the interaction in Eq. (7)becomes a maximum. Given the complex nature of the FM-AFM interfaces5,6 and the lack of experimental tools capable of extracting detailed information about such zones,47it is rather difficult to estimate a clear behavior of ud A. Nevertheless, based on the experimental data,29–31the EB coupling produces a clear enhancement of the FMR linewidth during the in-planeangular variation at u¼180 /C14, and therefore, it is inferred that conditions ud A¼uandud A¼180/C14(see Fig. 3) are the most reasonable ones. The case ud A¼uallows for aspects of the rotatable field, which appears since magnetic moments inthe AFM layer can follow the external field and therefore aneasy axis along the magnetization must be included. 41 Alternatively, the case ud A¼180/C14considers the partial com- pensation of the moments in the FM/AFM interface, as wasmentioned above. Overall, DH 2M IIdepends on the EB field HEthrough sin/C01ðHXX=HYYÞandK0 II, which is proportional to H2 Eas shown in Eq. (9). Fig. 4depicts the linewidth DH2M IIas a function of HEfor the case ud A¼180/C14and Sd¼225 nm2. The main dependence on the EB field comes from K0 II, con- sequently originating from the AFM domain defects. FIG. 3. (a) FMR linewidth as a function of uandud AforHE¼25 Oe. (b)–(e) Contributions DH2M I(dot-dashed line), DH2M II(dashed line), and DHG(solid line) separately, while the dots correspond to the full linewidth DH. In (b), (d), and (e), the cases ud A¼180/C14;ud A¼90/C14, and ud A¼0/C14are shown, respectively. The caseud A¼uis shown in (c), which assumes that magnetic moments inside the domain defects are able to follow the external field. FIG. 4. Linewidth due to domain defects DH2M IIas a function of HEfor u¼0/C14;ud A¼180/C14, and Sd¼225 nm2.223904-4 Gallardo, Rodr /C19ıguez-Su /C19arez, and Landeros J. Appl. Phys. 120, 223904 (2016) Furthermore, at higher concentrations of both types of defects, the linewidth significantly increases. This theoretical description explains some aspects of experimental observations. First, both DH2M IandDH2M II explain the in-plane angular dependence of the FMR line- width. Second, the modeling of geometrical defects gives aninterpretation of the cause of the typically large DH 29–31(at least two times larger than the Gilbert contribution). Third, the TMS scales as d/C02, which is experimentally con- firmed.24–26,29Fourth, the FMR linewidth DH2M IImainly depends on the square of exchange-bias field,28,48and this dependence could disappear for strong enough external fields. In this case, the magnetic moments inside the AFMdomains become parallel to M A(ud A¼uA) and the effective damping should be independent of the EB field HE. This has recently been measured in IrMn/Cu/CoFe trilayer films.48 IV. CONCLUSIONS An additional relaxation channel associated with the magnon-magnon scattering is included in a theoretical modelfor exchange-coupled bilayers. The theory considers both geometrical defects and small domains in the AFM sub- lattice adjacent to the FM layer. The role of two kinds ofdefects is clearly indicated through the in-plane angular vari-ation of the FMR linewidth. In particular, the influence ofthe domain defects turns out crucial in explaining both the angular variation and the EB field dependence of the line- width. These results are consistent with the previous experi-mental work. The calculations presented here provide abetter understanding of the relaxation mechanisms in EB bilayers and offer useful information for the manufacturing of devices based on the EB phenomenon. ACKNOWLEDGMENTS R. A. Gallardo acknowledges the financial support from CONICYT PAI/ACADEMIA, under Contract No. 79140033. This work was also partially supported byFONDECYT under Grant Nos. 1130705 and 1161403, andCenters of excellence with BASAL/CONICYT financing,Grant No. FB0807, CEDENNA. APPENDIX: FERROMAGNETIC RESONANCE CALCULATIONS A modified domain-wall formation model42is used in the theoretical description. Here the AFM spins develop a domain wall parallel to the FM/AFM interface, since it isenergetically favorable to deform the magnetization structureof the AFM instead of considering it a rigid system. 49In addition, non-collinearity between the magnetization of the AFM sub-lattice ( MA) in contact with the FM and the FM equilibrium magnetization ( Meq) is allowed (see Ref. 42and references therein). With such considerations, the resonancefrequency can be readily expressed by 30 fR¼c 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HXXHYYp ; (A1) where cis the gyromagnetic ratioHXX¼H0þHKcos2ðu/C0uKÞþ4pMeþHRAþH1(A2) and HYY¼H0þHKcos 2 ðu/C0uKÞþHRAþH2: (A3) Here H0is the external magnetic field, HKis the uniaxial in- plane anisotropy field, and 4 pMe¼4pMs/C0Hsis the effec- tive magnetization, where Hsis the surface anisotropy field, andMsis the saturation magnetization of the FM layer. The angles uanduKare measured from the z-axis to the equilib- rium magnetization Meqand to the uniaxial in-plane easy axis, respectively [see Fig. 1(a)]. It is assumed u/C25uH, where uHis the angle between the external field and z-axis. HRAis the rotatable anisotropy field,41,42which represents an additional uniaxial anisotropy with its axis oriented alongthe external field. The exchange-bias contribution is includedinto the fields H 1andH2, which are given by30,50 H1¼HEHWcosu/C0uA ðÞ /C0HEsin2u/C0uA ðÞ HWcosuA/C0uAK ðÞ þHEcosu/C0uA ðÞ(A4) and H2¼HEHWcosuA/C0uAK ðÞ cosu/C0uA ðÞ HWcosuA/C0uAK ðÞ þHEcosu/C0uA ðÞ:(A5) HEis the exchange coupling field and HWis the domain wall effective field. Also, uAanduAKare the angles that MAand the AFM pinning direction form with the z-axis. MAdepicts the magnetic moments at the AFM sub-lattice of the FM/AFM interface. From Eq. (A1), it is straightforward to obtain the FMR field, which is H R¼1 2/C26 HK1/C03 cos2u/C0uK ðÞhi /C0Fþ e/C02HRA þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi HKsin2u/C0uK ðÞ þF/C0 ehi2 þ4pfR=c ðÞ2r /C27 ;(A6) where F6 e¼4pMeþH16H2. 1W. H. Meiklejohn and C. P. Bean, Phys. Rev. 102, 1413 (1956). 2J. Nogu /C19es and I. K. Schuller, J. Magn. Magn. Mater. 192, 203 (1999). 3A. E. Berkowitz and K. Takano, J. Magn. Magn. Mater. 200, 552 (1999). 4R. L. Stamps, J. Phys. D: Appl. Phys. 33, R247 (2000). 5M. Kiwi, J. Magn. Magn. Mater. 234, 584 (2001). 6J. Nogu /C19es, J. Sort, V. Langlais, V. Skumryev, S. Suri ~nach, J. S. Mu ~noz, and M. D. Bar /C19o,Phys. Rep. 422, 65 (2005). 7F. Radu and H. Zabel, “Magnetic heterostructures,” in Springer Tracts in Modern Physics , edited by H. Zabel and S. 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1.1863420.pdf

Laser ablation of AlN films grown on sapphire substrate Mona R. Safadi, Jagdish S. Thakur, and Gregory W. Auner Citation: Journal of Applied Physics 97, 084901 (2005); doi: 10.1063/1.1863420 View online: http://dx.doi.org/10.1063/1.1863420 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Homoepitaxial AlN thin films deposited on m-plane ( 1 1 ¯ 00 ) AlN substrates by metalorganic chemical vapor deposition J. Appl. Phys. 116, 133517 (2014); 10.1063/1.4897233 Surface preparation and homoepitaxial deposition of AlN on (0001)-oriented AlN substrates by metalorganic chemical vapor deposition J. Appl. Phys. 108, 043510 (2010); 10.1063/1.3467522 Deep level defects in Si-doped Al x Ga 1 − x N films grown by molecular-beam epitaxy Appl. Phys. Lett. 86, 152109 (2005); 10.1063/1.1887817 Exciton spectra of an AlN epitaxial film on (0001) sapphire substrate grown by low-pressure metalorganic vapor phase epitaxy Appl. Phys. Lett. 81, 652 (2002); 10.1063/1.1493666 Molecular beam epitaxial growth of AlN single crystalline films on Si (111) using radio-frequency plasma assisted nitrogen radical source J. Vac. Sci. Technol. A 16, 2140 (1998); 10.1116/1.581321 [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: 141.209.144.159 On: Thu, 18 Dec 2014 13:36:04Laser ablation of AlN films grown on sapphire substrate Mona R. Safadi Department of Biomedical Engineering, Ligon Center of Vision, Wayne State University, Detroit, Michigan 48202 Jagdish S. Thakura! Department of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202 Gregory W. Auner Department of Electrical and Computer Engineering, Department of Biomedical Engineering, Ligon Center of Vision, Wayne State University, Detroit, Michigan 48202 sReceived 15 October 2004; accepted 11 January 2005; published online 31 March 2005 d Ablation threshold for single-crystal AlN semiconductor films grown epitaxially on sapphire substrateusingindigenouslybuilthollowcathodeplasmadepositionsourcemolecular-beamepitaxytechnique is investigated for a number of pulses by varying the fluence value of each pulse. Usinga KrF excimer laser sl=248 nm and t=25ns das a radiation source, we found that ablation ofAlN thin films is a discontinuous process and its onset requires a minimum threshold fluence<1.59 J/cm 2. The ablation depth is analyzed for different numbers of pulses and for each number asafunctionofincreasingfluencevalues.Theresultsshowthattheablationdepthincreaseslinearlywith increasing pulse fluence. It is found that the use of a single pulse for ablation at a given valueof fluence is more efficient than a large number of pulses at the same value of fluence/pulse. Inaddition,weinvestigatedthelowestpulse-fluencelimitthatcansustainablationonadisorderedAlNfilm surface. We present a theoretical discussion about the laser energy absorption mechanism andalsotherateofenergytransferfromtheconduction-bandelectronstolatticephononswhichcanleadto the ablation of AlN film. It is found that the rate of energy transfer increases linearly withincreasing temperature of the electron gas. © 2005 American Institute of Physics . fDOI: 10.1063/1.1863420 g I. INTRODUCTION Aluminum nitride sAlNdalong with other nitrides of group III is gaining wide interest due to its applications inhigh-power, high-temperature, and optelectronic devices op-erating in ultraviolet spectral region. 1–3Crystallized AlN semiconductor forms a hexagonal wurtzitelike structure witha high bond energy 1of 2.28 eV, a band-gap value of ,6.2 eV, an extremely hard and chemically stable material with a very high melting temperature in excess of 2275 K,and a high thermal conductivity s,3.2 Wcm −1K−1dsRef. 4 d at room temperature. Low chemical reactivity of AlN mate- rials in biological environment makes these materials usefulfor making biocompatible devices. Because of relativelylarge value of refractive index sn=2.1 d, AlN thin films be- come ideal candidates for broad-spectrum sdeep UV to infra- red region dwaveguide structures when bonded with sapphire sn=1.6 d. In particular, these films can be used to fabricate a waveguide array which can be employed in microfluidic drug delivery, miniaturized Raman spectroscopy devices anda variety of other applications. 2Due to their wide transmis- sion spectral range, high bond strength, and inertness, thesefilms are also useful in the applications of optical coatingswhich are particularly suitable for high-temperature environ-ments. Laser interaction with matter can produce various de- fects, cracks, and light-absorptive macroinclusions and mi-croinclusions which could in size be of the order of the laser wavelength. If a laser fluence value is close to the meltingthreshold of a material, then energy absorption from the laserleads to heating of the material followed by its melting.When the melted region solidifies, the electrical and opticalproperties of this solidified region can be a dramatic modi-fied. For example, the laser-ablated regions of an AlN filmswhich is a large band-gap semiconductor material dbecome metallic and as a result its resistivity value drops. Due to ahigher value of electron density in the metallic region, a qualitative change occurs in its optical properties, e.g., opti-cal reflectance and absorption. The ablation threshold studyonAlN thin films can provide data about the damage thresh-old which is important for industrial environments wherethere is a possibility of a laser interaction withAlN thin-filmmaterials. For fabrication of smooth and well-defined surface boundaries of an optical waveguide and other microstruc-tures, it is essential to determine the ablation threshold andablation depth as a function of laser fluence for these films.In spite of a wide range of applications of this group IIInitride, 1the ablation threshold and ablation rates sdepth/ fluence dhave not yet been discussed for AlN thin films de- posited on sapphire substrates. Threshold value of a thin filmcan be very different from that of the bulk material 5due to surface and other disorder effects which are more pro-nounced in thin films as compared to the bulk materials. It iswidely believed that roughness on the film surface can ini- adElectronic mail: jagdish@wayne.eduJOURNAL OF APPLIED PHYSICS 97, 084901 s2005 d 0021-8979/2005/97 ~8!/084901/6/$22.50 © 2005 American Institute of Physics 97, 084901-1 [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: 141.209.144.159 On: Thu, 18 Dec 2014 13:36:04tiate ablation at a much lower fluence value compared to a defect-free surface. In this paper, we focus on the determina-tion of ablation threshold and ablation depth for single andmultiple pulses. We also investigate the influence of the sur-face roughness on the threshold value of the film. The sur-face roughness in our films is created intentionally by laserirradiation with fluence value equal to the threshold value ofthe film. Irradiation dose at threshold alters the smoothnessof the film surface which we measure with atomic force mi-croscopy sAFM d. We address the question whether it is pos- sible to ablate such films at fluence values smaller than thethreshold value and we thus determine the lower fluencelimit which can initiate ablation in disordered films. II. THIN-FILM FABRICATION TECHNIQUE The fabrication of AlN/sapphire thin-film structures2,6,7 is done with a plasma source molecular-beam epitaxysPSMBE ddeveloped at the Wayne State University. The PSMBE system uses a hollow cathode plasma depositionsource, which employs a magnetically enhanced hollowcathode lined with the target material, or in this case, alumi-num. Nitrogen or nitrogen/argon plasma formed within thehollow cathode by an applied rf or pulsed dc power to thecathode dissociates the diatomic nitrogen into radicals andradical ions, as well as other products that accelerate into thewall and thus induce sputtering of atoms from the cathodesurface. Samples were loaded onto a substrate holder andheated to 800 °C for 1 h as a final surface cleaning proce-dure. AlN was deposited on double side epipolished s0001 d sapphire at 650 °C with a source power of 200 rf watts, at abias voltage of −12 V, a base pressure of 5 310 −10Torr, and a deposition pressure of 1 310−3Torr. The flow of nitrogen was kept at 10 SCCM sStandard cubic centimeter per minute d. III. ALN THIN-FILM CHARACTERIZATION It is well known that crystalline quality and surface dis- order of a material can influence the ablation process, whichthen can affect the ablation threshold value. Therefore, it isimportant to determine the crystal structure and its qualitybefore one investigates ablation threshold and other proper-ties. We determine these crystal properties using x-ray dif-fraction sXRD dexperiments and found that theAlN films are single crystalline but with small grain boundaries. The sur-face topology was measured using AFM and we found thatthe root-mean-square roughness value for our films is,4.7 nm. Determination of the other physical parameters, e.g., the band-gap value and optical-absorption coefficient, is also im-portant for the understanding of the energy absorptionmechanism during the laser ablation process. The band-gapenergy of AlN estimated from the optical-absorption spectrashows a strong absorption edge around 200 nm <6.2 eV sin the deep-UV range dconsistent with the known band-gap en- ergy value 8of AlN semiconductor. From the ellipsometry measurements we found the value of refractive index n =2.1 which agrees with other experimental results.9,10IV. EXCIMER LASER SETUP FOR THE ABLATION Micromachining of the AlN thin films into waveguide structures was performed using a Lambda Physik 200i exci-mer laser system which generates KrF laser pulses of 25-nsduration 11at a wavelength of 248 nm with a maximal output power of 600 mJ. A stainless-steel stencil was employed todefine and project the geometry of the ablation spot. Theprojected beam was condensed down by a factor of 8.9through the use of an UV optical objective onto an AlN thinfilm where it forms a square spot of size 281 3281 mm2.The AlN/sapphire substrate is secured down onto a microcon-trolled computer interfaced stage with micron resolution x-, y-, andz-axis controls. The laser is fired by means of a computer-operated trigger control and the parameters arevaried accordingly. We performed postablation analysis of our samples using a Dektak 3030 profilometer which can measure vertical fea-tures from approximately 130 mm to 50 Å. The profiler scans the ablated region and reports a surface-profile scanacross the region of interest. Using this characterization tech-nique we studied the ablation depth of the films as a functionof laser fluence and pulse number. V. RESULTS AND DISCUSSION A. Ablation threshold of a clean surface In Fig. 1, we discuss the variation in the ablation depth for a single pulse by increasing its fluence value from1.510 to 2.13 J/cm 2. No ablation is observed for the fluence values up to 1.566 J/cm2. However, when fluence is in- creased to next higher step equal to 1.61 J/cm2, we observed ablation about 0.02 mm in depth in our films. Once the ab- lation is initiated, its depth increases linearly with pulse en-ergy. For a pulse width of t=25 ns used in our experiments, the ablation does not start as soon as the pulse fluence be-comes greater than zero but it starts discontinuously whenthe pulse fluence is increased from 1.566 to 1.610 J/cm 2. The discontinuous laser ablation has been ascribed12–14to an FIG. 1. Determination of the ablation threshold for AlN thin films using a single laser pulse of KrF with l=248 nm and pulse width t=25 ns. The pulse fluence is varied from 1.510 to 2.13 J/cm2. The ablation occurs when the fluence is increased from 1.566 to 1.610 J/cm2.084901-2 Safadi, Thakur, and Auner J. Appl. Phys. 97, 084901 ~2005 ! [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: 141.209.144.159 On: Thu, 18 Dec 2014 13:36:04explosive boiling which leads to the formation of homog- enous nucleation in the material due to its superheating. Theactual threshold in our films will occur for a fluence value inbetween 1.566 and 1.610 J/cm 2, therefore, we estimate its value ,1.59 J/cm2from their average. A similar well- defined ablation threshold has also been observed in manyother experiments 15,16and computer simulations.17 In Fig. 2, we show the variation in the ablation depth with increasing numbers of pulses for a fixed value of flu-ence per pulse. In this figure, we also studied the ablationbehavior with increasing pulse fluence for a fixed number ofpulses. It is interesting to contrast the ablation behavior of asingle pulse sFig. 1 dwith higher number of pulses sFig. 2 d— ablation for higher number of pulses occurs even at fluencevalues smaller than 1.51 J/cm 2. However, the ablation still occurs discontinuously even for pulse number as large as ten.As one notices that when the pulse number is equal to ten,there is no ablation for the fluence values less than1.51 J/cm 2and ablation occurs when the fluence value be- comes larger than 1.51 J/cm2. Interestingly, for a given value of the pulse fluence, the ablation depth increases lin-early with number of pulses 18and this linear behavior per- sists even at higher values of fluence. B. Ablation versus pulse fluence In Fig. 3, we show the ablation depth with increasing values of fluence for different number of pulses. It is inter-esting to note that the ablation rate sdepth/fluence dincreases faster for higher number of pulses. This is more evidentwhen we compare data of a single pulse with that of tenpulses. However, the material removal depth per pulse athigher numbers of pulses is small compared to the singlepulse value. For example, the average removal rate per pulsefor ten pulses at fluence=2.1 J/cm 2is 0.037 mm while the corresponding value for a single pulse is 0.046 mm. So fromenergy consideration it is more energy efficient, for ablation purposes, to use a single pulse rather than a large number ofpulses at the same value of fluence/pulse. C. Ablation threshold of a disordered surface The laser energy absorption seems to be more efficient for surfaces which are less smooth. Here we discuss surfaceroughness effects on the ablation threshold value. We artifi-cially created surface defects by irradiating the film with la-ser fluence equal to its threshold value.The presence of thesedefects was measured using AFM. We then irradiated thefilm with fluence values smaller than the threshold value andfound that ablation continues even for smaller values of flu-ence, as shown in Fig. 4. For comparison, the ablation depthsfor both disordered and clean surfaces are shown in this fig-ure. For a single pulse, the ablation depth does not change FIG. 2. Variation in the ablation depth smmdofAlN thin films as a function of number of pulses for KrF s248 nm dlaser. The pulse number is varied as 1, 2, 5, and 10 pulses. For each pulse number, the laser fluence is variedfrom 1.566 to 2.13 J/cm 2. FIG. 3. Ablation depth smmdof AlN thin films vs that of laser fluence sJ/cm2dfor 1, 2, 5, and 10 pulses. The laser fluence for each pulse number is varied from 1.566 to 2.13 J/cm2. FIG. 4. Determination of the ablation threshold for disordered AlN filmsurface using KrF laser with l=248 nm. The pulse fluence is varied from 1.510 to 2.13 J/cm 2. For a disordered AlN film surface, the ablation begins when the laser fluence is ,1.10 J/cm2, a value much smaller than for a clean surface. The number of pulses used is marked by a label on thecorresponding curve.084901-3 Safadi, Thakur, and Auner J. Appl. Phys. 97, 084901 ~2005 ! [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: 141.209.144.159 On: Thu, 18 Dec 2014 13:36:04with the increasing fluence values, in contrast with the clean surface behavior. However, even for a highly disordered film,at some lower critical value of fluence the ablation processstops. This value provides the lower ablation threshold limitfor a disordered AlN film surface and its value is,1.10 J/cm 2which is about 31% less than the threshold value of a clean surface. Note that, like clean surfaces, theablation remains discontinuous for disorder films. It is inter-esting to note that surface morphology and quality can playan important role in the determination of the minimum dam-age threshold of a material. VI. ABLATION MECHANISM OF ALN FILMS The ablation threshold is a unique property of the mate- rial and three basic mechanisms—photochemical, photother-mal, and photomechanical—have been identified to initiatelaser ablation in materials. The ablation process starts withradiation interaction with the material and if this interactionoccurs then it leads to the laser energy transfer into the ma-terial, which then gets heated up and ablated.Various optical,electron-phonon interaction, thermodynamical, and othersparameters of the material strongly influence the ablationprocess. During the ablation, the energy scales associatedwith the optical se.g., band-gap and plasmon energies, etc. d and thermodynamical smelting temperature, specific heat, etc.dproperties of the material compete with the laser photon energy and thus determine the dynamics of the energy flowinto the material. In our ablation experiments, the energy of aKrF laser photon, " n=248 nm >5.0 eV, is much smaller than the binding energy,19EbAlN<11.4 eV, ofAlN, therefore, the ablation by a photochemical process with these low-energy photons is less likely to occur in the AlN film. For alaser pulse of short duration, the energy transfer by the pho-tomechanical process is a dominant mechanism for the abla-tion. However, in our case, the possibility of the photome-chanical ablation 17or “cold” laser ablation is also ruled out due to the fact that the pulse duration tis much larger than the mechanical equilibration time tsof the energy absorbing volume. tscan be estimated from the characteristic length L of the energy deposition area and sound velocity cfor the value of AlN using a relation20ts<L/c. Using the param- eters of our laser pulse and of the AlN sRef. 21 dfilm, we found that ts<2 ns, which is much smaller than the pulse duration, t=25 ns. This implies that, the laser-induced stress relaxes more quickly and is not limited by the pulse durationto initiate the photomechanical ablation process in the film. Defect energy states can also participate in the laser- material interaction and can lead to laser energy dissipation.However, the energy absorption by these defects dependsdirectly on the number of available energy states and energyof these states. For theAlN thin films, there are a number ofdifferent types of defect energy states distributed along itsband gap 22,23which can act as energy absorption centers for single-photon absorption. A donor level of Al at the N sitewith energy of about 5.0 eV could be quite efficient for thesingle-photon absorption process. Cation vacancies also cre-ate several acceptor states ranging from 0.2 to 1.9 eV abovethe valence-band maximum sVBM dwhich are less likely tobe filled by the electrons excited from the top of the valence band, but can be filled by the deep excitations from the va-lence band. The same argument applies for the Al and nitro-gen vacancies which are 0.5 and 1.7 eV, respectively, abovethe VBM. 23 Given the parameters of the laser pulse, it seems that photothermal ablation24,25is a more appropriate process for energy transfer26to the AlN films. The optical-absorption spectrum shows a strong energy absorption around the inci-dent energy " n=6.2 eV, due to electron excitation across the band gap. The energy of the laser photons used for the abla-tion purposes in our experiments is much less than the band-gap energy of theAlN film, therefore, the probability of elec-trons excitation by the single laser photons from the valanceto conduction band is very small. Two photons can excite anelectron across the band gap with maximum excess energy of2" n−Eg<3.8 eV. However, from probability considerations, it is much more favorable for a given laser photon to interact with thefree conduction-band electrons rather than with the boundvalance-band electrons. Therefore, the energy absorptionfrom laser seems to be done mainly by the free conduction-band electrons, rather than by the defect states.After absorb-ing laser photons, the electron’s occupation number for thoseelectrons excited from the vicinity of the Fermi level in-creases to an energy " n=248 nm >5.0 eV above the Fermi level. The electronic subsystem, as a result of these higher-energy excitations, goes out of the equilibrium and becomeshighly unstable. The excited electrons with excess energyequilibrate mainly through electron-electron scattering. Theelectron thermalization time is much shorter 27,28than other time scales, e.g., LO-phonon relaxation and pulse durationtime. After self-thermalization, the heated electron gas be-gins to transfer its energy to the lattice and externally ther-malizes with the lattice through electron-phonon smainly LO phonons dinteractions. There are other weaker scattering mechanisms which lead to the relaxation of electron momen-tum and energy. However, scattering with fixed ionized im-purities or crystal imperfections will only result in thechange of momentum of the electron. Thermalization takesmany collisions between the electrons and phonons due tothe large energy difference between the excess energy ofelectrons and phonons. The lattice acts like a sink for theinput laser energy, absorbs the energy in a very short periodof time, and becomes superheated. This superheated state iseventually ablated. Although during laser ablation the whole electron-lattice system is out of equilibrium, it is worthwhile to investigatethe rate of energy transfer from the conduction-band elec-trons to the lattice via LO phonons. Due to the polar natureof theAlN lattice cell, there is a large amount of electric fieldthat spills out of the unit lattice cell and interacts stronglywith the electrons. The link between the electrons and LOphonons becomes a most active channel of energy transferfrom the laser to the lattice. Since we have neglected electronexcitation across the band gap, we assume that the carrierdensity stays constant during the experiment. According tothe mechanism described above, the incident laser photons084901-4 Safadi, Thakur, and Auner J. Appl. Phys. 97, 084901 ~2005 ! [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: 141.209.144.159 On: Thu, 18 Dec 2014 13:36:04heat the conduction-band electrons, so the rate of the energy loss by the hot electrons through the emission of LOphonons can be calculated by 29 dEs«,Td dt=2˛2m"vLO "2pE −11 dfcossudg˛« 3f1−fs«+"vLOdgNphs"vLOd 2«+"vLO−2cos sud˛«s«+"vLOd 3Im1 «sq,"vLOd, s1d wherefsk,Tdis temperature-dependent Fermi–Dirac distri- bution function, Nphs"vLOdis the phonon distribution func- tion evaluated at the LO-phonon energy "vLO. The total di- electric function «sq,vdwithin the random-phase approximation for the electron-LO- phonon system is given by «sq,vd=«‘+svLO2−vTO2d svTO2−v2−ivgd+Vsqdxsq,v+iGd.s2d Here, «‘is the high-frequency dielectric and the second and third terms are the phonon and electron contributions to thedielectric function, respectively. Vsqd=4 pe2/q2is the Fou- rier transform of the Coulomb interaction potential between the electrons. From momentum conservation, the value of q used in the dielectric function are given by q=˛2m/"f2« +"vLO−2cos sud˛«s«+"vLOdg1/2. In the expression of «sq,vd,vLO/gand vTOare the unscreened LO-phonon resonant-frequency/damping parameters and TO-phonon fre- quency, respectively. For the AlN film parameters, we use vLO=893 cm−1andvTO=618 cm−1. The values of plasmon sGdand phonon damping parameters are taken about 1% of the plasmon and phonon frequency, respectively.30The temperature-dependent Lindhard function31is given by xsq,v+iGd=o k,sfsk,TdS1 Ek+q−Ek+"v+iG +1 Ek+q−Ek−"v−iGD, s3d whereEk="2k2/2mis the electron dispersion for a parabolic band at wave vector kandsis the electron-spin index. In Fig. 5, we plot the average energy loss by the electrons as afunction of temperature. Due to much shorter equilibrationtime of the electrons, we assume that they are described bythe equilibrium Fermi-distribution function whose tempera-ture increases with increasing laser fluence. Increasing laserfluence excites larger numbers of electrons which then un-dergo a self-equilibration process and as a result turn into ahot-electron gas. These hot electrons eventually dump theirenergy into the lattice via electron-phonon interactions. Therate of energy transfer to the lattice increases monotonicallywith increasing temperature of the electron gas.As the latticebegins to absorb energy from the electron gas, its tempera-ture increases and at some elevated lattice temperature it ab-lates. At higher laser fluence one expects a higher rate ofablation due to increased energy absorption by the phonons,as observed in Fig. 3 where the ablation depth increases lin- early with the laser fluence. VII. CONCLUSION In this paper, we discuss the ablation threshold and variation of ablation depth for clean and disordered surfaceAlN films by s1dvarying the energy of a single pulse and s2d varying the number of pulses at a fixed pulse energy. Wefound that in both cases sclean and disordered dthe ablation of AlN semiconductor films requires a minimum thresholdvalue, but for a disordered surface film the threshold valuebecomes smaller. The ablation depth increases linearly withpulse energy for energies greater than the ablation thresholdvalue.We also discuss theoretically the rate of energy loss byheated electrons due to laser energy absorption through theemission of phonons and found that the rate increases lin-early with increasing temperature of the electron gas. ACKNOWLEDGMENTS We are thankful to Yuri Danylyuk for the fabrication of AlN thin films and to Daad Haddad for assisting in the char-acterization of the films. This work has been supported byNational Science Foundation sNSFdGrant No. 98770720 and Wayne State University Center for Smart Sensors and Inte-grated Microsystems sSSIM d. 1J. W. Orton and C. T. Foxon, Rep. Prog. Phys. 61,1s1998 d. 2G. W. Auner, T. Lenane, F. Ahmad, R. Naik, and P. K. Kuo, Wide Band- gap Electronic Materials , edited by Z. L. Wu and M. A. Prelas sKluwer Academic, Dordrecht, 1995 d, p. 329. 3M. R. Safadi, G. W. Auner, R. Iezzi, P. McAllister, and G. Abrams, Proc. SPIE4982, 330 s2003 d. 4J. H. Edgar, Properties of Group III Nitrides sINSPEC, London, 1994 d, p. 11. 5J. K. Lumpp and S. D. Allen, J. Mater. Res. 12, 218 s1997 d. 6R. Krupitskaya and G. W. Auner, J. Appl. Phys. 84, 286 s1998 d. 7M. P. Thompson, G. W. Auner, and A. R. Drews, Mater. Res. Soc. Symp. Proc.537, G3.57.1 s1998 d. 8W. M. Yim, E. J. Stotko, P. J. Zanzucchi, J. I. Pankove, M. Ettenberg, and S. L. Gilbert, J. Appl. Phys. 44, 292 s1973 d; P. B. Perry and R. F. Rutz, Appl. Phys. Lett. 33, 319 s1978 d. 9J. Pastrnak and L. Roskovcova, Phys. Status Solidi 14,K 5 s1966 d. FIG. 5. Rate of energy transfer seV/sdby hot electrons to the lattice phonons as a function of electron temperature TsKd. The electron density, Ne=1 31013cm3, and phonon temperature are taken equal to 300 K.084901-5 Safadi, Thakur, and Auner J. Appl. Phys. 97, 084901 ~2005 ! [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: 141.209.144.159 On: Thu, 18 Dec 2014 13:36:0410D. J. Jones, R. H. French, H. Mullejans, S. Loughin, A. D. Dorneich, and P. F. Carcia, J. Mater. Res. 14,4 3 3 7 s1999 d. 11Q. Zhao, M. Lukitsch, J. Xu, and G. W. Auner, Mater. Res. Soc. Symp. Proc.595, W11.69.1 s1999 d. 12B. J. Garrison, T. E. Itina, and L. V. Zhigilei, Phys. Rev. E 68, 041501 s2003 d. 13A. Miotello and R. Kelly, Appl. Phys. A: Mater. Sci. Process. 69,S 6 7 s1999 d; A. Mele, A. G. Guidoni, R. Kelly, A. Miotello, S. Orlando, and R. Teghill, Appl. Surf. Sci. 96,1 0 2 s1996 d. 14M. M. Martynyuk, Sov. Phys. Tech. Phys. 19, 793 s1974 d. 15K. H. Song and X. Xu, Appl. Surf. Sci. 111, 127 s1998 d. 16J. H. Yoo, S. H. Jeong, X. L. Mao, R. Grief, and R. E. Russo, Appl. Phys. Lett.76,7 8 3 s2000 d. 17L. V. Zhigilei and B. J. Garrison, J. Appl. Phys. 88, 1281 s2000 d. 18J. Wei, N. Hoogen, T. Lippert, Ch. Hahn, O. Nuken, and A. Wokaun, Appl. Phys. A: Mater. Sci. Process. 69, S849 s1999 d. 19E. Ruiz, S. Elvarez, and P. Alemany, Phys. Rev. B 49,7 1 1 5 s1994 d;S .H . Kim, D. S. Lee, J. W. Lee, T. I. Kim, and G. Y. Yeom, Vacuum 56,4 5 s2000 d. 20C. Schafer, H. M. Urbassek, and L. V. Zhigilei, Phys. Rev. B 66, 115404 s2002 d.21D. Gerlich, S. L. Dole, and G. A. Slack, J. Phys. Chem. Solids 47,4 3 7 s1986 d. 22P. Boguslawski, E. L. Briggs, and J. Bernholc, Phys. Rev. B 51, 17255 s1995 d. 23T. L. Tansley and R. J. Egan, Phys. Rev. B 45, 10942 s1992 d; I. Gorczyca and N. E. Christensen, Physica B 185,4 1 0 s1993 d; I. Gorczyca, A. Svane, and N. E. Christensen, MRS Internet J. Nitride Semicond. Res. 2,1 8 s1997 d. 24P. L. Lui, R. Yen, N. Bloemberg, and R. T. Hodgson, Appl. Phys. Lett. 34, 864s1979 d. 25N. Bityurin and A. Malyshev, J. Appl. Phys. 92,6 0 5 s2002 d. 26A. Cavalleri, T. K. Sokolowski, J. Biaklkowski, M. Schreiner, and D. Von Der Linde, J. Appl. Phys. 85, 3301 s1999 d. 27J. J. Quinn, Phys. Rev. 126, 1453 s1963 d; T. Elsaesser, R. J. Bauerle, and W. Kaiser, Phys. Rev. B 40, 2976 s1989 d. 28T. Elsaesser, J. Shah, L. Rota, and P. Lugli, Phys. Rev. Lett. 66, 1757 s1991 d. 29M. P. Hasselbeck and P. M. Enders, Phys. Rev. B 16, 9674 s1998 d. 30M. Kuball, J. M. Hayes, Y. Shi, and J. H. Edger, Appl. Phys. Lett. 77, 1958 s2000 d. 31N. D. Mermin, Phys. Rev. B 1, 2362 s1970 d.084901-6 Safadi, Thakur, and Auner J. Appl. Phys. 97, 084901 ~2005 ! [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: 141.209.144.159 On: Thu, 18 Dec 2014 13:36:04

5.0004844.pdf

J. Chem. Phys. 152, 204104 (2020); https://doi.org/10.1063/5.0004844 152, 204104 © 2020 Author(s).The DIRAC code for relativistic molecular calculations Cite as: J. Chem. Phys. 152, 204104 (2020); https://doi.org/10.1063/5.0004844 Submitted: 18 February 2020 . Accepted: 22 April 2020 . Published Online: 26 May 2020 Trond Saue , Radovan Bast , André Severo Pereira Gomes , Hans Jørgen Aa. Jensen , Lucas Visscher , Ignacio Agustín Aucar , Roberto Di Remigio , Kenneth G. Dyall , Ephraim Eliav , Elke Fasshauer , Timo Fleig , Loïc Halbert , Erik Donovan Hedegård , Benjamin Helmich- Paris , Miroslav Iliaš , Christoph R. Jacob , Stefan Knecht , Jon K. Laerdahl , Marta L. Vidal , Malaya K. Nayak , Małgorzata Olejniczak , Jógvan Magnus Haugaard Olsen , Markus Pernpointner , Bruno Senjean , Avijit Shee , Ayaki Sunaga , and Joost N. P. van Stralen The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The DIRAC code for relativistic molecular calculations Cite as: J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 Submitted: 18 February 2020 •Accepted: 22 April 2020 • Published Online: 26 May 2020 Trond Saue,1,a) Radovan Bast,2,b) André Severo Pereira Gomes,3,c) Hans Jørgen Aa. Jensen,4,d) Lucas Visscher,5,e) Ignacio Agustín Aucar,6,f) Roberto Di Remigio,7 Kenneth G. Dyall,8,g) Ephraim Eliav,9,h) Elke Fasshauer,10,i) Timo Fleig,1,j) Loïc Halbert,3 Erik Donovan Hedegård,11 Benjamin Helmich-Paris,12 Miroslav Iliaš,13,k) Christoph R. Jacob,14,l) Stefan Knecht,15,m) Jon K. Laerdahl,16 Marta L. Vidal,17 Malaya K. Nayak,18,n) Małgorzata Olejniczak,19,o) Jógvan Magnus Haugaard Olsen,7 Markus Pernpointner,20,p) Bruno Senjean,5,21,q) Avijit Shee,22,r) Ayaki Sunaga,23,s) and Joost N. P. van Stralen5,t) AFFILIATIONS 1Laboratoire de Chimie et Physique Quantique, UMR 5626 CNRS—Université Toulouse III-Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse, France 2Department of Information Technology, UiT The Arctic University of Norway, N-9037 Tromsø, Norway 3Université de Lille, CNRS, UMR 8523—PhLAM—Physique des Lasers, Atomes et Molécules, F-59000 Lille, France 4Department of Physics, Chemistry and Pharmacy, University of Southern Denmark, DK-5230 Odense M, Denmark 5Department of Chemistry and Pharmaceutical Sciences, Vrije Universiteit Amsterdam, NL-1081HV Amsterdam, The Netherlands 6Instituto de Modelado e Innovación Tecnológica, CONICET, and Departamento de Física—Facultad de Ciencias Exactas y Naturales, UNNE, Avda. Libertad 5460, W3404AAS Corrientes, Argentina 7Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, UiT The Arctic University of Norway, N-9037 Tromsø, Norway 8Dirac Solutions, 10527 NW Lost Park Drive, Portland, Oregon 97229, USA 9School of Chemistry, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel 10Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus, Denmark 11Division of Theoretical Chemistry, Lund University, Chemical Centre, P.O. Box 124, SE-221 00 Lund, Sweden 12Max-Planck-Institut für Kohlenforschung, Kaiser-Wilhelm-Platz 1, 45470 Mülheim an der Ruhr, Germany 13Department of Chemistry, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica, Slovakia 14Technische Universität Braunschweig, Institute of Physical and Theoretical Chemistry, Gaußstr. 17, 38106 Braunschweig, Germany 15ETH Zürich, Laboratorium für Physikalische Chemie, Vladimir-Prelog-Weg 2, 8093 Zürich, Switzerland 16Department of Microbiology, Oslo University Hospital, Oslo, Norway 17Department of Chemistry, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark 18Theoretical Chemistry Section, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India 19Centre of New Technologies, University of Warsaw, S. Banacha 2c, 02-097 Warsaw, Poland 20Kybeidos GmbH, Heinrich-Fuchs-Str. 94, 69126 Heidelberg, Germany 21Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 22Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA 23Department of Chemistry, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji-city, Tokyo 192-0397, Japan Note: This article is part of the JCP Special Topic on Electronic Structure Software. a)Author to whom correspondence should be addressed: trond.saue@irsamc.ups-tlse.fr. URL: http://dirac.ups-tlse.fr/saue b)Electronic mail: radovan.bast@uit.no. URL: https://bast.fr J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp c)Electronic mail: andre.gomes@univ-lille.fr d)Electronic mail: hjj@sdu.dk e)Electronic mail: l.visscher@vu.nl f)Electronic mail: agustin.aucar@conicet.gov.ar g)Electronic mail: diracsolutions@gmail.com h)Electronic mail: ephraim@tau.ac.il i)Electronic mail: elke.fasshauer@gmail.com j)Electronic mail: timo.fleig@irsamc.ups-tlse.fr. URL: http://dirac.ups-tlse.fr/fleig k)Electronic mail: Miroslav.Ilias@umb.sk l)Electronic mail: c.jacob@tu-braunschweig.de m)Electronic mail: stefan.knecht@phys.chem.ethz.ch n)Electronic addresses: mknayak@barc.gov.in and mk.nayak72@gmail.com o)Electronic mail: malgorzata.olejniczak@cent.uw.edu.pl p)Electronic mail: markpp@gmx.de q)Electronic mail: bsenjean@gmail.com r)Electronic mail: ashee@umich.edu s)Current address: Institute for Integrated Radiation and Nuclear Science, Kyoto University, 2 Asashiro-Nishi, Kumatori-cho, Sennan-gun, Osaka 590-0494, Japan. Electronic mail: sunagaayaki@gmail.com t)Current address: TNO, Energy Transition Studies, Radarweg 60, NL-1043NT Amsterdam, The Netherlands. Electronic mail: joost.vanstralen@tno.nl ABSTRACT DIRAC is a freely distributed general-purpose program system for one-, two-, and four-component relativistic molecular calculations at the level of Hartree–Fock, Kohn–Sham (including range-separated theory), multiconfigurational self-consistent-field, multireference con- figuration interaction, electron propagator, and various flavors of coupled cluster theory. At the self-consistent-field level, a highly original scheme, based on quaternion algebra, is implemented for the treatment of both spatial and time reversal symmetry. DIRAC features a very general module for the calculation of molecular properties that to a large extent may be defined by the user and further analyzed through a powerful visualization module. It allows for the inclusion of environmental effects through three different classes of increasingly sophisticated embedding approaches: the implicit solvation polarizable continuum model, the explicit polarizable embedding model, and the frozen density embedding model. Published under license by AIP Publishing. https://doi.org/10.1063/5.0004844 .,s I. INTRODUCTION DIRAC is a general-purpose program system for relativistic molecular calculations and is named in honor of P. A. M. Dirac (Program for Atomic and Molecular Direct Iterative Relativistic All–electron Calculations), who formulated1his celebrated relativis- tic wave equation for the electron in 1928. The beginnings of the DIRAC code can be traced back to the four-component relativistic Hartree–Fock (HF) code written by Trond Saue during his mas- ter’s thesis, defended at the University of Oslo, Norway, in 1991. The original code stored all integrals, provided by the HERMIT code,2on disk, but during Saue’s Ph.D. thesis, defended in 1996, the code was extended to direct Self-Consistent Field (SCF) with integral screening3and a highly original symmetry scheme based on quaternion algebra.4A postdoctoral stay in 1996–1997 with Hans Jørgen Aagaard Jensen at the University of Southern Denmark focused on molecular properties, with the implementation of the calculation of expectation values and linear response functions5at the SCF level. Lucas Visscher, who had written a four-component direct Restricted Active Space (RAS) Configuration Interaction (CI) code for the MOLFDIR program system6during his Ph.D. thesis, defended at the University of Groningen in 1993, did a postdoc- toral stay with Jens Oddershede in Odense during the years 1996– 1997 and joined forces and code with Jensen and Saue to createthe DIRAC program system. Since then, the main author team has been joined by Radovan Bast and Andre Severo Pereira Gomes in addition to almost 50 contributors, and Odense has since 1997 hosted an annual “family” meeting for DIRAC developers. In addi- tion to the above authors, we would like to highlight the contri- butions to the code infrastructure by Jørn Thyssen and Miroslav Iliaš. The latest version of the code, DIRAC19,7was released on December 12, 2019. In Sec. II, we give a brief overview of the DIRAC program. Then, in Sec. III, we provide some implementation details, with focus on features that are little documented elsewhere and/or may be a source of confusion for DIRAC users. Throughout this paper, we are using SI units. II. PROGRAM OVERVIEW A. Hamiltonians Within the Born–Oppenheimer approximation, the electronic Hamiltonian may be expressed as ˆH=VNN+∑ iˆh(i)+1 2∑ i≠jˆg(i,j),VNN=1 2∑ A≠BZAZBe2 4πε0RAB, (1) J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where VNNrepresents the repulsion energy arising from point nuclei fixed in space. Notwithstanding the challenges associated with specific choices of the one- and two-electron operators ˆh(i) and ˆg(i,j), most quantum chemical methods can be formulated just from this generic form. This becomes perhaps even more evident by considering the electronic Hamiltonian in the second quantized form ˆH=VNN+∑ pqhp qa† paq+1 4∑ pqrsVpr qsa† pa† rasaq,hp q=⟨p∣ˆh∣q⟩ Vpr qs=⟨pr∣ˆg∣qs⟩−⟨pr∣ˆg∣sq⟩. (2) In this form, practical for actual implementations, the Hamiltonian is given by strings of creation and annihilation operators combined with one- and two-electron integrals. In relativistic calculations, the integrals are generally complex, in contrast to the nonrelativistic domain, and contain fewer zero elements, since spin symmetry is lost. The DIRAC code features several electronic Hamiltonians, allowing for molecular electronic structure calculations at the four-, two-, and one-component levels. Four-component relativistic cal- culations are sometimes referred to as “fully relativistic” in con- trast to “quasirelativistic” two-component calculations. However, a fully relativistic two-electron interaction, which would containmagnetic interactions and effects of retardation in addition to elec- trostatics, is not readily available in the closed form, rendering this terminology somewhat misleading. An overview over the most com- mon choices for relativistic molecular Hamiltonians can be found in Ref. 8. The default Hamiltonian of DIRAC is the four-component Dirac–Coulomb Hamiltonian, using the simple Coulombic correc- tion,9which replaces the expensive calculation of two-electron inte- grals over small component basis functions by an energy correc- tion. The one-electron part is the Hamiltonian ˆhDof the time- independent Dirac equation in the molecular field, that is, the field of nuclei fixed in space, ˆhDψ=[VeN c(σ⋅p) c(σ⋅p)VeN−2mec2][ψL ψS]=[ψL ψS]E,VeN(r)=∑ A−e 4πε0∫R3ρA(r′) ∣r′−r∣d3r′ ∫R3ρA(r)d3r=ZAe, (3) where cis the speed of light, σis the vector of Pauli spin matri- ces,pis the momentum operator, and VeNis the electron–nucleus interaction. The default model of the nuclear charge distribution is the Gaussian approximation,10but a point nucleus model is also available. The default two-electron operator of DIRAC is the instantaneous Coulomb interaction ˆgC(1, 2)=e2 4πε0r12, (4) which constitutes the zeroth-order term and hence the nonrelativis- tic limit11of an expansion in c−2of the fully relativistic two-electron interaction in the Coulomb gauge. It should be noted, though, that the presence of ˆgC(1, 2) induces the spin–same orbit (SSO) interac- tion, just as the presence of VeNinduces the spin–orbit interaction associated with the relative motion of the nuclei with respect to the electrons.8The spin–other-orbit (SOO) interaction may be included by adding the Gaunt term, which is available at the SCF level. Spin– orbit interaction effects may be eliminated by transforming to the modified Dirac equation and removing spin-dependent terms.12In the quaternion formulation of SCF calculations in DIRAC, this cor- responds to removing the three quaternion imaginary parts of the Fock matrix.13 It is also possible to carry out four-component nonrelativis- ticcalculations using the Lévy–Leblond Hamiltonian,13,14which is equivalent to the Schrödinger Hamiltonian within kinetic balance. The Schrödinger Hamiltonian is also available in DIRAC, but due toits four-component form, the Lévy–Leblond Hamiltonian is better integrated in the code, in particular for the calculation of molecu- lar properties. Contrary to the Schrödinger Hamiltonian, the four- component Hamiltonians are linear in vector potentials. In the rel- ativistic case, the calculation of 3 ×3 second-order magnetic prop- erty matrices thus requires the solution of just three linear response equations, contrary to the scalar non-relativistic case that requires the evaluation of at least six diamagnetic expectation values and often solving more than three linear response equations; for exam- ple, linear response equations become ten for paramagnetic spin– orbit, Fermi-contact, and spin-dipole (PSO, FC, and SD) contribu- tions to indirect spin–spin coupling between two nuclei.15,16On the other hand, in order to get the diamagnetic contributions converged, special basis set considerations are necessary, usually expressed through (simple) magnetic balance.17,18In the case of the Lévy– Leblond Hamiltonian, the diamagnetic contribution is calculated as an expectation value over the scalar non-relativistic operator.11 A troublesome aspect of the Dirac Hamiltonian is the presence of solutions of negative energy. Over the years, there has been exten- sive work on eliminating the positronic degrees of freedom of the Dirac Hamiltonian, leading to approximate two-component Hamil- tonians, such as the Douglas–Kroll–Hess (DKH)19–21and zeroth- order regular approximation (ZORA)22–24Hamiltonians. Various flavors of the ZORA Hamiltonian have been implemented in DIRAC,13but only for energy calculations. DIRAC features the very first implementation of a two-component relativistic J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Hamiltonian for molecular calculations where the one-electron part reproduces exactly the positive-energy spectrum of the par- ent four-component Dirac Hamiltonian.25,26This implementation, presented as the BSS Hamiltonian by Jensen and Iliaš,25referring to the previous work by Barysz, Sadlej, and Snijders,27carried out a free-particle Foldy–Wouthuysen transformation28on the Dirac Hamiltonian, followed by an exact decoupling of positive- and negative-energy solutions. This two-step approach allows for the construction of finite-order two-component relativistic Hamiltoni- ans such as the first- and second-order Douglas–Kroll–Hess Hamil- tonians but is unnecessary for exact decoupling. The code was there- fore superseded by a simple one-step approach, reported as the Infinite-Order Two-Component (IOTC) Hamiltonian by Iliaš and Saue.29Due to the equivalence with the exact quasirelativistic (XQR) Hamiltonian reported by Kutzelnigg and Liu,30it was later agreed31 to name such Hamiltonians eXact 2-Component (X2C) Hamiltoni- ans. The X2C decoupling transformation is available in matrix form and is used to transform the matrix of any one-electron operator to the two-component form, hence avoiding picture change errors. For the two-electron integrals, DIRAC employs the uncorrected two-electron operator supplemented with the atomic mean-field approach for including two-electron spin–orbit interactions.32,33 Recently, the X2C code in DIRAC was rewritten in a more modular and modern form by Knecht.34 For wave function-based correlation methods, the electronic Hamiltonian is conveniently written in the normal-ordered form ˆH=EHF+∑ pqFp q{a† paq}+1 4∑ pqrsVpr qs{a† pa† rasaq}, (5) where EHFis the Hartree–Fock energy, Fp qare elements of the Fock matrix, and curly brackets refer to normal ordering with respect to the Fermi vacuum, given by the HF determinant. For such calcu- lations, DIRAC features the X2C molecular mean-field approach:35 After a four-component relativistic HF calculation, the X2C decou- pling is carried out on the Fock matrix, rather than the Dirac Hamiltonian matrix, whereas the two-electron operator is left untransformed. In combination with the usual approximation of neglecting core electron correlation, this limits the effect of picture change errors to valence–valence electron interactions only: core– core and core–valence electron interactions are treated with the same accuracy as in the 4C approach. Last but not least, DIRAC features one-component scalar rela- tivistic effective core potentials (AREPs) as well as two-component spin–orbit relativistic effective core potentials (SOREPs).36 B. Electronic structure models 1. Self-consistent field (SCF) calculations At the core of DIRAC is an SCF module allowing for both Hartree–Fock (HF)3and Kohn–Sham (KS)37calculations. These cal- culations are Kramers-restricted and use a symmetry scheme based on quaternion algebra that automatically provides maximum point group and time-reversal symmetry reduction of the computational effort.4In nonrelativistic quantum chemistry codes, spin-restricted open-shell SCF calculations employ Configuration State Functions (CSFs) | S,MS⟩of well-defined spin symmetry. However, in the rela- tivistic domain, spin symmetry is lost, and so the use of CSFs wouldrequire linear combinations of Slater determinants adapted to com- bined spin and spatial symmetry, which is a challenge for a general molecular code. We have therefore instead opted for the average-of- configuration Hartree–Fock method38for open-shell systems. Indi- vidual electronic states may subsequently be resolved by a complete open-shell CI calculation.39Open-shell Kohn–Sham calculations use fractional occupation. SCF calculations are based on the traditional iterative Roothaan–Hall diagonalization method with direct-inversion-in- the-iterative-subspace (DIIS) convergence acceleration. By default, the start guess is provided by a sum of atomic local-density approx- imation (LDA) potentials, which have been prepared using the GRASP atomic code40and are fitted to an analytical expression.41 Other options include (i) bare nucleus potentials corrected with screening factors based on Slater’s rules,42(ii) atomic start based on densities from atomic SCF runs for the individual centers,43 and (iii) an extended Hückel start based on atomic fragments.44 In each SCF iteration, orbitals are by default ordered according to energy, and orbital classes are assigned by simple counting in the following order: (secondary) negative-energy orbitals, inactive (fully occupied) orbitals, active (if any) orbitals, and virtual orbitals. The implicit assumption of relative ordering of orbital energies according to orbital classes may cause convergence problems, for instance, for f-elements where closed-shell ns-orbitals typically have higher energies than open-shell ( n−2)fones. Such convergence problems may be avoided by reordering of orbitals combined with overlap selection, pioneered by Bagus in the 1970s45and nowa- days marketed as the maximum overlap method.46Overlap selection also provides robust convergence to core-ionized/excited states.47 Negative-energy orbitals are treated as an orthogonal complement, corresponding to the implicit use of a projection operator.48 An extensive selection of exchange-correlation energy func- tionals, as well as their derivatives to high order, needed for prop- erty calculations, is available for Kohn–Sham calculations. These XC functional derivatives are provided either by a module written by Sałek49using symbolic differentiation or by the XCFun library written by Ekström50,51using the automatic differentiation tech- nique. By default, Kohn–Sham calculations employ Becke partition- ing52of the molecular volume into overlapping atomic volumes, where the numerical integration within each atomic volume is car- ried out using the basis-set adaptive radial grid proposed by Lindh, Malmqvist, and Gagliardi53combined with angular Lebedev quadra- ture. The XC contributions to energy derivatives or Fock matrix elements are evaluated for a batch of points at a time, which allows us to screen entire batches based on values of basis functions at these grid points and enables us to express one summation loop of the numerical integration as a matrix–matrix multiplication step. 2. Correlation methods a. Four-index transformations. While the AO-to-MO index transformations are subordinate to the correlation approaches described below, some features are worth describing in a sepa- rate section. Irrespective of the Hamiltonian that is used in the orbital generation step, the approach always assumes that a large atomic orbital basis set is condensed to a much smaller molecular orbital basis. The result is the second-quantized, no-pair Hamilto- nian in molecular orbital basis that is identical in structure to the J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp second-quantized Hamiltonian encountered in nonrelativistic methods [see Eq. (2)]. The main difference is the fact that the defin- ing matrix elements in this Hamiltonian are in general complex due to the inclusion of spin–orbit coupling in the orbital generation step. As a consequence, integrals will not exhibit the usual eightfold per- mutation symmetry familiar from nonrelativistic integrals. This is even the case when higher point group symmetry is used to ren- der the integrals real, as they may be a product of two imaginary transition densities. Only for spin-free calculations, is it possible to choose phase factors for the spinors in such a way that eightfold permutational symmetry is recovered. For ease of interfacing with nonrelativistic correlation implementations, such phase factors are inserted in the final stage of the transformation when running in the spin-free mode. The primary interface files that are generated contain the (effective) one-body operator plus additional symmetry and dimensionality information needed to set up the Hamiltonian. The numerous two-body matrix elements are stored in separate files that can be distributed over multiple locations in a compute cluster environment with delocalized disk storage. The program is comple- mented by a utility program that can convert the MO integrals to formats used by other major quantum chemistry programs, such as the FCIDUMP format,54thus facilitating the interfacing55of DIRAC to other electron correlation implementations, such as MRCC,56 or even to quantum computers. With respect to the latter, a four- component relativistic quantum algorithm was reported in Ref. 57. More recently, DIRAC has been interfaced to the electronic struc- ture package OpenFermion through a Python interface,58thus allow- ing for the calculation of energies and energy derivatives on a quan- tum computer59using either the full Dirac–Coulomb Hamiltonian or the Lévy–Leblond Hamiltonian. The implementation has been revised several times over the years to account for changes in computer hardware architectures. The current default algorithm for the most demanding transforma- tion of the two-electron integrals uses an MPI type of parallelization in which half-transformed integrals are generated from recomputed AO integrals. If the total disk space is an issue, it is also possible to employ a scheme in which only a subset of half-transformed inte- grals is stored at a given time. The generation of one-body integrals is less demanding and carried out by calling the Fock matrix build routine from the SCF part of the program, with a modified density matrix that includes only contributions from the orbitals that are to be frozen in the correlation treatment. In the generation of these integrals, it is possible to account for the Gaunt correction to the Coulomb interaction, thereby making a mean-field treatment of this contribution possible.35Explicit transformation of additional inte- grals over operators needed for the evaluation of molecular prop- erties, or the inclusion of a finite strength perturbation in only the electron correlation calculation, is also possible and handled by a separate submodule. The lowest-level correlation method is second-order Møller– Plesset perturbation (MP2) theory, and an early integral-direct, closed-shell implementation was realized by Laerdahl60in 1997. This implementation focuses on efficient, parallel calculation of the MP2 energy for closed-shell systems. A more general implemen- tation that also allows for calculation of the relaxed MP2 density matrix was realized later by van Stralen61as part of the coupled clus- ter implementation discussed below. Both implementations use the conventional MP2 approach; a more efficient Cholesky-decomposeddensity matrix implementation was developed by Helmich-Paris.62 In this approach, the quaternion formalism, which has been devel- oped in earlier works,63was used to reduce the number of opera- tions. A production implementation along these lines is planned for the 2020 release. b. Configuration interaction. The first implementation (DIRRCI module) of restricted active space configuration inter- action was taken from the MOLFDIR program6,64and is briefly described in Secs. 3.4 and 4.10 of Ref. 6, with more details on the calculation of CI coupling coefficients given in Chap. 6.5 of Ref. 65. This module is mostly kept for reference purposes as a more pow- erful implementation of configuration interaction in DIRAC was introduced later by Fleig and co-workers.66A unique feature that makes the DIRRCI module still of some interest is the handling of any Abelian point group symmetry, and not just D 2hand subgroups. The implementation is capable of handling every possible Abelian group as long as the respective multiplication table is provided. This feature allows for treatment of linear symmetry (a feature lacking in the MOLFDIR program) by merely changing the dimensions of the arrays that hold the symmetry information. While the DIRRCI code is no longer actively developed, some later extensions beyond the MOLFDIR capabilities have been implemented, such as the (approx- imate) evaluation of correlated first-order properties using the unre- laxed CI density matrix by Nayak.67–69This implementation allows for the calculation of expectation values of one-electron property operators over the CI wave functions. The more recent KR-CI module is a string-based Hamiltonian- direct configuration interaction (CI) program that uses Dirac Kramers pairs from either a closed- or an open-shell calculation in a relativistic two- and four-component formalism exploiting double point-group symmetry. KR-CI is parallelized using MPI in a scal- able way where the CI vectors are distributed over the nodes, thus enabling the use of the aggregate memory on common computing clusters.70,71There are two choices for the CI kernel: LUCIAREL and GASCIP. The LUCIAREL kernel66,72is a relativistic generalization of the earlier LUCIA code by Olsen.73It is capable of doing efficient CI computations at arbitrary excitation levels, FCI, SDCI, RASCI, and MRCI, all of which are subsets of the Generalized Active Space (GAS) CI. The GAS concept74is a central and very flexible fea- ture (described in greater detail in Ref. 75) of the program that can be applied effectively to describe various physical effects in atomic matter. Apart from routine applications to valence electron correla- tion,76,77it has been used in other modern applications to efficiently describe core–valence electron correlation78and also core–core cor- relation.79These uses can be combined with excited-state calcu- lations, even when greater numbers of excited states with varying occupation types are required.80In the latter case, typical CI expan- sion lengths are on the order of up to 108terms, whereas parallel single-state calculations have been carried out with up to 1010expan- sion terms.81If desired, KR-CI computes one-particle densities from the optimized CI wave functions from which one-electron expec- tation values of any property defined in DIRAC as well as natural orbital occupations can be calculated. The complete open-shell implementation ( GOSCIP ) of MOLFDIR is used to obtain state energies after an average-of- configuration Hartree–Fock calculation. For other uses, a more J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp general and efficient GASCIP (Generalized Active Space CI Pro- gram) module was originally written by Thyssen and Jensen for KR-MCSCF and later parallelized and optimized by Jensen. This very general CI implementation is primarily used for KR-MCSCF and ESR calculations,82but it is also available for KR-CI calcula- tions. Another separate implementation is the spin-free version of LUCIAREL, LUCITA , which fully exploits spin and boson symme- try. LUCITA will consequently be faster for spin-free CI calculations than KR-CI using the LUCIAREL kernel. LUCITA has also been parallelized with MPI,83in the same way as KR-CI. c. Multiconfigurational SCF. The original KR-MCSCF imple- mentation was written by Thyssen, Fleig, and Jensen84and fol- lows closely the theory of Ref. 85. Within a given symmetry sec- tor, it allows for a state-specific optimization by taking advantage of a Newton-step-based genuine second-order optimization algo- rithm.85The KR-MCSCF module was later parallelized70,71using MPI by Knecht, Jensen, and Fleig by means of a parallelization of the individual CI-based tasks encountered in an MCSCF optimization: (i) generation of the start vector, (ii) sigma-vector calculation, and (iii) evaluation of one- and two-particle reduced density matrices (RDMs). Moreover, its extension to an efficient treatment of lin- ear symmetry (see Sec. III C)—both in the KR-CI part and in the restriction of orbital–rotational parameters—was a central element that allowed for a comprehensive study86of the electronic structure as well as of the chemical bond in the ground- and low-lying excited states of U 2based on a simultaneous, variational account of both (static) electron correlation and spin–orbit coupling. d. Coupled cluster. TheRELCCSD coupled cluster module87 is also derived from the MOLFDIR implementation, but in con- trast to the DIRRCI module, it is still under active development. The implementation uses the same philosophy as DIRRCI in demanding only a point group multiplication table to handle Abelian symmetry. Furthermore, in some cases, point group symmetry beyond Abelian symmetry can be used to render the defining integrals of the sec- ond quantization Hamiltonian real, using the scheme first outlined in Ref. 88, but adjusted to work with the quaternion algebra used elsewhere in DIRAC. The implemented algorithms work in the same way for real and complex algebra, with all time-consuming opera- tions performed by BLAS89calls that are made via a set of wrapper routines that point to the (double precision) real or complex ver- sion, depending on the value of a global module parameter. In this way, the need for maintenance of a separate code for real or com- plex arithmetic is strongly reduced. Due to the dependence on BLAS operations, shared memory parallelization is easily achieved by link- ing in a multithreaded BLAS library. Parallelization via MPI can be achieved in addition, as described in Ref. 90. For the description of electronic ground states that can be qualitatively well described by a single determinant, the standard CCSD(T) model is usually the optimal choice in terms of perfor- mance,91with the code taking the trial CCSD amplitudes from an MP2 calculation. As the RELCCSD implementation does not assume time-reversal symmetry, it is possible to treat open-shell cases as well. This is straightforward at the CCSD level of theory as the specific open-shell SCF approach used to generate the orbitals is then relatively unimportant and the implementation does not assume a diagonal reference Fock matrix. For the implemented perturbativetriple corrections,92,93a larger dependence on the starting orbitals is observed, although performance is usually satisfactory for sim- ple open-shell cases with only one unpaired electron. For more complicated cases, it is often better to use the Fock space coupled cluster (FSCC) model, in which multireference cases can also be handled. The relativistic use of the FSCC method94has been pioneered by Eliav and Kaldor,95and in the DIRAC implementation,96one is able to investigate electronic states that can be accessed by sin- gle or double electron attachment or detachment, as well as singly excited states, from a starting closed-shell reference determinant— that is, states with up to two unpaired electrons. Most calculations done with this method nowadays use their intermediate Hamilto- nian97,98(IH) schemes that remove problems with intruder states to a large extent.99–102The IH approach has been recently improved by the very efficient Padé extrapolation method.103As IH schemes often use large active spaces, and the original Fock space implementation96 was designed with small numbers of occupied orbitals in mind, these calculations can become rather memory-intensive. We have also implemented the equation-of-motion coupled cluster approach for the treatment of electron attachment (EOM- EA), ionization energy (EOM-IP), and excitation energy (EOM- EE).104EOM-IP and EOM-EE can also be used to obtain core ionization and excitation energies via the core–valence separation approach.105This complements the Fock space functionality for treating electronically excited states, especially for species that can be represented by a closed-shell ground-state configuration.106 e. Range-separated density functional theory. This method allows for grafting of wave function-based correlation methods onto density functional theory (DFT) without double counting of elec- tron correlation.107We have explored this approach combining short-range DFT with long-range MP2 theory108and CC model.109 f. Density matrix renormalization group (DMRG). If requested, KR-CI provides a one- and two-electron integral file in the FCIDUMP format,54which allows for, for the active space specified in the KR-CI input, relativistic density matrix renormalization group (DMRG) calculations110,111with the relativistic branch of the DMRG program QCM AQUIS .112,113If desired, the DMRG program computes the one-particle reduced density matrix for the optimized wave func- tion in the MO basis and writes it to a text file that can be fed back into DIRAC. This feature makes it possible to calculate (static) first-order one-electron properties in the same way as described below for SCF calculations. Moreover, this functionality also opens up further possibilities to analyze the resulting wave function and to calculate properties in real-space, as it was shown in Ref. 111 for a Dy(III) complex. For the latter functionality, the visualiza- tion module (see Sec. II E) was extended with an interface to wave functions optimized within the KR-CI/KRMCSCF framework of Dirac. C. Molecular properties A common framework for the calculation of molecular proper- ties is response theory. A less economical, but computationally easier approach is to use a finite-field approach. Implementations for both strategies are available in DIRAC. J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The property module of DIRAC has been written in a very general manner, allowing user-defined operators. More precisely, a four-component one-electron operator has the general form ˆP=∑ kfkMkˆΩk, (6) where fkis a scalar factor, Mkis a 4×4 matrix ( I4,αx,Σy,...), and ˆΩkis a scalar operator. The user can choose between 21 dif- ferent operators forms, Eq. (6) (for example, c(α⋅ˆΩ)and fγ5ˆΩ; for the full list, see the manual on the DIRAC website114). For the definition of specific scalar operators ˆΩ, DIRAC benefits from the extensive menu of one-electron operators in the HERMIT module.2 For convenience, a number of properties are predefined, as shown in Table I. 1. SCF calculations At the closed-shell SCF level, both for Hartree–Fock and for Kohn–Sham, DIRAC allows for the calculation of molecular prop- erties corresponding to expectation values as well as linear5,116and quadratic131,132response functions. In addition, first- and second- order residues of the quadratic response function have been pro- grammed, allowing for the calculation of two-photon absorption cross sections118and first-order properties of electronically excited states,133respectively. Linear response functions have been extended to complex response through the introduction of a common damping term thatremoves divergences at resonances.134This allows for not only prob- ing of second-order properties in the vicinity of resonances but also simulation of absorption spectra within a selected window of fre- quencies. In addition, complex response allows for the calculation of properties at formally imaginary frequencies, such as C 6dispersion coefficients.135 Excitation energies are available through time-dependent HF and DFT.136Restrictions may be imposed on active occupied (and virtual) orbitals, hence allowing for restricted excitation win- dow (REW) calculations47,137of x-ray absorption spectra. Another method available for core-excitation processes in molecules is the static-exchange approximation (STEX).138Transition moments may be calculated with user-specified property operators. From DIRAC20 onward, three schemes139to go beyond the electric dipole approximation in the calculation of oscillator strengths will be avail- able in DIRAC. The first is based on the full semi-classical light– matter interaction operator and the two others on a truncated inter- action within the Coulomb gauge (velocity representation) and mul- tipolar gauge (length representation). The truncated schemes can be calculated in arbitrary order in the wave vector. All schemes allow for rotational averaging. NMR shieldings as well as magnetizabilities may be calculated using London orbitals and simple magnetic balance.18For KS calcu- lations, non-collinear spin magnetization has been implemented132 and all required derivatives of exchange-correlation functionals are provided. TABLE I . Predefined molecular properties in DIRAC. EV is expectation value; LR is linear response; QR is quadratic response. Keyword Electric properties EV LR QR References .DIPOLE Electric dipole moment x .QUADRU Traceless electric quadrupole moment x .EFG Electric field gradients at nuclear positions x 115 .NQCC Nuclear quadrupole coupling constants x 115 .POLARI Electronic dipole polarizability tensor x 5 and 116 .FIRST Electronic dipole first-order hyperpolarizability tensor x 117 .TWO-PH Two-photon absorption cross sections x 118 Magnetic properties .NMR Nuclear magnetic shieldings and indirect spin–spin couplings x 15 and 119 .SHIELD Nuclear magnetic shieldings x 15 and 119 .SPIN-S Indirect spin–spin couplings x 15 .MAGNET (Static) magnetizability tensor x 120 and 121 .ROTG Rotational g-tensor (DIRAC20) x 122 Mixed electric and magnetic properties .OPTROT Optical rotation x 123 .VERDET Verdet constants x 124 Other predefined properties .MOLGRD Molecular gradient 125 .PVC Parity-violating energy (nuclear spin-independent part) x 126 and 127 .PVCNMR Parity-violating contribution to the NMR shielding tensor x 128 .RHONUC Electronic density at the nuclear positions (contact density) x 129 .EFFDEN Effective electronic density associated with nuclei (Mössbauer) x 129 .SPIN-R Nuclear spin-rotation constants x x 130 J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 2. Correlation modules a. Electronic ground state properties. The implementations for obtaining density matrices for molecular properties are still under development. For the single reference CCSD model, the current available functionality is to obtain the unrelaxed one-particle density matrix.140For the MP2 model, for which orbital relaxation effects are more important, the relaxed density matrix can be obtained.61After back-transforming to the AO basis, molecular properties can be obtained in the same way as for SCF calculations. Alternatively, one may also obtain matrix elements of property operators in the MO basis and compute expectation values of CI wave functions and/or include property operators as a finite-strength perturbation in the CI or CC wave function determination. This allows for determination of properties that break Kramers symmetry and has, for example, been used for assessing the effect of an electric dipole moment of the electron ( eEDM) in molecular systems.141 b. Excited state properties. For properties that depend also on the excited state density matrix, such as transition probabili- ties, only limited functionality is available in RELCCSD. Transi- tion intensities based on an approximate CI expression and the dipole approximation for the transition moments have been imple- mented for the Fock space coupled cluster model. Under develop- ment, and planned to be available in the 2020 DIRAC release, is an extension to the non-diagonal form of the finite field approach in the Fock space CC. This approach allows for accurate calcula- tions of the dipole moments of electron transitions in heavy atomic and molecular systems.142For KR-CI, the range of properties is larger,70and basically the same as for the electronic ground state. These include molecule-frame static electric dipole moments,70,143 E1 transition matrix elements,70,144and magnetic hyperfine inter- action constants145in electronic ground and excited states. More- over, parity-reversal (P) and time-reversal (T) violating properties are implemented as expectation values over atomic or molecular KR-CI ground- and excited-state wave functions, in particular the electron electric dipole moment interaction,146the P,T-odd scalar– pseudoscalar nucleon–electron interaction,143and the nuclear mag- netic quadrupole moment electron magnetic-field interaction.78 c. Electron propagator. The Algebraic Diagrammatic Con- struction (ADC) is an efficient, size-consistent post-Hartree–Fock method, which can be used to obtain molecular properties.147–150 With DIRAC, the calculation of single151and double152ionization as well as electronic excitation153,154spectra using the RELADC and POLPRP modules is possible. Decay widths of electronic decay pro- cesses can be obtained by the FanoADC-Stieltjes method.155The ionization spectra can be obtained at the level of ADC(2), ADC(2x), and ADC(3) plus constant diagrams, while the electronic excitation spectra are available at ADC(2) and ADC(2x) levels of accuracy. Technically, the ADC implementation and the RELCCSD code share much of their infrastructure. d. Quasi-degenerate perturbation theory using configuration interaction. A module is under development for description of properties of quasi-degenerate states as encountered in open-shell molecules. The focus has so far been on calculation of ESR/EPR g- tensors,82hyperfine couplings, and zero-field splitting. It is based on the flexible GASCIP configuration interaction module.D. Environments A great deal of information can be extracted from gas-phase electronic structure calculations, even for systems that are studied experimentally in solution or other condensed phases. Nevertheless, the environment can strongly modulate the properties of a system, such as formation/reaction energies and response properties (e.g., electronic or vibrational spectra). It can therefore be important to take into account the influence of the environment on the systems of interest. A straightforward way to include the environment is by performing calculations on large molecular systems or aggre- gates, but this quickly becomes unwieldy already for HF and DFT calculations, and for correlated electronic structure calculations (Sec. II C 2), this strategy is largely unfeasible. The alternative to such calculations is the use of embedding approaches,156in which the environment is treated in a more approximate fashion with respect to the subsystem of interest, see Fig. 1. Apart from the simplest possible embedding scheme, using fixed point charges, DIRAC offers three different classes of increasingly sophisticated embedding approaches: the implicit sol- vation polarizable continuum model (PCM), the atomistic polar- izable embedding (PE) model, and the frozen density embedding (FDE) model; the latter is also referred to as subsystem DFT. In the first two, the environment is treated classically, whereas for the latter, all fragments are treated quantum mechanically, though generally at different levels of theory. FIG. 1 . Pictorial depiction of the transition from the quantum mechanical model to one of the multiscale models available in DIRAC for the aqueous solvation ofpara-nitroaniline. Leftmost panel: a fully quantum mechanical cluster model. Upper central panel: a frozen density embedding (FDE) (explicit) model. Middle central panel: a quantum/classical discrete (explicit) model. Lower central panel: aquantum/classical continuum (implicit) model. J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 1. Polarizable continuum model (PCM) The PCM is a quantum/classical polarizable model for the approximate inclusion of solvent effects into quantum mechanical calculations.157It is a focused, implicit solvation model: the environ- ment (usually a solvent) is replaced by a dielectric with permittivity ε, and the mutual interaction between the quantum mechanical and classical regions is described by the electrostatic polarization. The model cannot describe specific, weak interactions between subsys- tems, such as hydrogen bonding, but can give the first qualitative estimate of solvation effects on many molecular properties. The quantum mechanical region is delimited by a cavity , a region of space usually constructed as a set of interlocking spheres and hosting the QM fragment, see the lower central panel in Fig. 1. Inside the cavity, the permittivity is that of vacuum ( ε= 1), while outside, the permittivity assumes the value appropriate for the envi- ronment being modeled. For example, in the case of water, the experimental value ε= 78.39 would be used. The electrostatic polar- ization is represented as an apparent surface charge (ASC), which is the solution to the integral formulation of the Poisson equation. The coupling with the QM code at the SCF level of theory is achieved by augmenting the usual Fock operator with an ASC-dependent envi- ronment polarization operator. This results in minimal, localized changes to the SCF cycle. The implementation in DIRAC is based on the PCMSOLVER library,158,159which provides a well-defined interface to a stand- alone computational backend. The PCM method is available for mean-field (HF and DFT) wave functions. Additionally, static elec- tric linear response properties can also be computed including the effect of the solvent via the PCM. 2. Polarizable embedding (PE) The PE model is a fragment-based quantum–classical embed- ding approach for including environment effects in calculations of spectroscopic properties of large and complex molecular sys- tems.160–162The effects from the classical environment on the quan- tum subsystem are included effectively through an embedding potential that is parameterized based on ab initio calculations. The molecular environment is thus subdivided into small, computation- ally manageable fragments from which multi-center multipoles and multi-center dipole–dipole polarizabilities are computed. The mul- tipoles and polarizabilities model the permanent and induced charge distributions of the fragments in the environment, respectively. For solvent environments, a fragment typically consists of an individual solvent molecule, while for large molecules, such as proteins, a frag- mentation approach based on overlapping fragments is used. The resulting embedding potential is highly accurate163and introduces an explicitly polarizable environment that thus allows the environ- ment to respond to external perturbations of the chromophore.164 The embedding-potential parameters can be conveniently produced using external tools such as the PyFraME package,165which auto- mates the workflow leading from an initial structure to the final embedding potential. The current implementation of the PE model in DIRAC can be used in combination with mean-field electronic-structure meth- ods (i.e., HF and DFT) including electric linear response and tran- sition properties where local-field effects, termed effective exter- nal field (EEF)166,167effects in the PE context, may be included.168The model is implemented in the Polarizable Embedding library (PElib)169that has been interfaced to DIRAC.168The library itself is based on an AO density-matrix-driven formulation, which facil- itates a loose-coupling modular implementation in host programs. The effects from the environment are included by adding an effec- tive one-electron embedding operator to the Fock operator of the embedded quantum subsystem. The potential from the permanent charge distributions is modeled by the multipoles, which is a static contribution that is computed once and added to the one-electron operator at the beginning of a calculation. The induced, or polarized, charge distributions are modeled by induced dipoles resulting from the electric fields exerted on the polarizabilities. This introduces a dependence on the electronic density, through the electronic electric field, and the induced dipoles are therefore updated in each itera- tion of an SCF cycle or response calculation, similar to the procedure used in the PCM.170 3. Frozen density embedding (FDE) The FDE approach is based on a reformulation of DFT whereby one can express the energy of a system in terms of subsystem energies and an interaction term (see Ref. 156 and references therein), which contains electrostatic, exchange-correlation, and kinetic energy contributions, with the latter two correcting for the non-additivity between these quantities calculated for the whole system and for the individual subsystems. As in other embedding approaches, we are generally interested in one subsystem, while all others constitute the environment. The electron density for the system of interest is determined by making the functional for the total energy stationary with respect to variations of the said den- sity, with a constraint provided by the density of the environ- ment. The interaction term thus yields a local embedding potential, vemb(r), representing the interactions between the system and its environment. The FDE implementation in DIRAC is capable of calculating vemb(r) during the SCF procedure (HF and DFT) using previously obtained densities and electrostatic potentials for the environment on a suitable DFT integration grid, as well as exporting these quan- tities. One can also import a precalculated embedding potential, obtained with DIRAC or other codes,171and include it in the molec- ular Hamiltonian as a one-body operator.172This allows for setting up iterative procedures to optimize the densities of both the system of interest and the environment via freeze–thaw cycles.173 At the end of the SCF step, the imported or calculated vemb(r) becomes part of the optimized Fock matrix and is therefore directly included in all correlated treatments mentioned above172,174as well as for TD-HF and TD-DFT. For the latter two, contributions aris- ing from the second-order derivatives of interaction energy are also available for linear response properties of electric175and mag- netic perturbations,176though the couplings in the electronic Hes- sian between excitations on different subsystems are not yet imple- mented. The interaction term for the non-additive kinetic energy contributions is calculated with one of the available approximate kinetic energy functionals that can be selected via input. E. Analysis and visualization DIRAC features Mulliken population analysis.177However, this analysis should be used with caution due to its well-known basis-set J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp dependence. An additional complication in the present case is that the analysis distributes density according to scalar basis functions, which is incompatible with two- or four-component atomic orbitals. We have therefore introduced projection analysis , similar in spirit to Mulliken analysis, but using precalculated atomic orbitals.178,179 The reference atomic orbitals may furthermore be polarized within the molecule using the intrinsic atomic orbital algorithm.180,181The projection analysis furthermore allows for the decomposition of expectation values at the SCF level into inter- and intra-atomic con- tributions, which, for instance, has elucidated the mechanisms of parity-violation in chiral molecules.127It is also possible to localize molecular orbitals, which is favorable for bonding analysis.179 The visualization module in DIRAC makes it possible to export densities and their derivatives, as well as other quantities (such as property densities obtained from response calculations), to third-party visualization software commonly used by the theoretical chemistry community such as Molden,182,183as well as by less known analysis tools such as the Topology Toolkit (TTK),184with which we can perform a wide range of topological analyses, including atoms- in-molecules (AIM)185with densities obtained with Hartree–Fock, DFT, and CCSD wave functions. DIRAC can export such data in the Gaussian cube file format, or over a custom grid. DIRAC has been extensively used for the visualization of prop- erty densities, in particular magnetically induced currents.186,187 More recently, shielding densities have been investigated in order to gain insight into the performance of FDE for such NMR proper- ties.173,176 As an illustration of the visualization module, we start from the observation of Kaupp et al.188that the spin–orbit contribution to FIG. 2 . Visualizing the analogy between the spin–orbit contribution to shieldings and the indirect spin–spin coupling: isosurface plot of the spin–spin coupling den- sityK(Hβ,I) in iodoethane (left panel; Fermi-contact + spin-dipole contributions), compared with the spin–orbit coupling contribution to the shielding density σ(Hβ) (right panel). The dihedral angle H–C–C–I is 180○.the shielding σ(Hβ) ofβ-hydrogen of iodoethane follows closely the Karplus curve of the indirect spin–spin coupling constant K(Hβ,I) as a function of the H–C–C–I dihedral angle. The DIRAC program makes it possible to isolate spin-free and spin–orbit contributions to magnetic properties;11this has allowed us to show that this connec- tion is manifest at the level of the corresponding property densities (Fig. 2). F. Programming details and installation The source code consists mostly of Fortran 77 and Fortran 90 codes, but some modules are written in C (exchange-correlation functional derivatives using symbolic differentiation, pre-Fortran- 90 memory management) and C++ (exchange-correlation func- tional derivatives using automatic differentiation, polarizable con- tinuum model). Python is used for the powerful code launcher pam, which has replaced the previous launcher written in Bash. The code base is under version control using Git and hosted on a GitLab repository server. The main development line as well as release branches is write-protected, and all changes to these are automatically tested and undergo code review. For integration tests, we use the Runtest library189and we run the test set both nightly and before each merge to the main code development branch. Since 2011, the code is configured using CMake,190which was introduced to make the installation more portable and to make it easier to build and maintain a code base with different program- ming languages and an increasing number of externally maintained modules and libraries. The code is designed to run on a Unix-like operating system, but thanks to the platform universality of the employed Python , Git and CMake tools, we have also been able to adapt the DIRAC code for the MS Windows operating system using the MinGW-GNU compiler suite. G. Code documentation The code documentation (in HTML or PDF format) is gener- ated from sources in the reStructuredText format using Sphinx191 and served via the DIRAC program website.114We track the doc- umentation sources in the same Git repository as the source code. In this way, we are able to provide documentation pages for each separate program version, which improves the reproducibility of the code and also allows us to document unreleased functionality for future code versions. In addition to a keyword reference manual, we share a broad spectrum of tutorials and annotated examples that provide an excellent starting point for users exploring a new code functionality or entering a new field. H. Distribution and user support The program is distributed in the source code form under a cus- tom open-use license. Traditionally, we have distributed the code to the community upon request, but starting with the DIRAC18 release,192we have switched to distributing the source code and collecting download metrics via the Zenodo service.193We plan to transition to an open source license (GNU Lesser General Public License) in the near future to encourage contributions and simplify the derivative work based on the DIRAC package. User support is provided on a best-effort basis using Google Groups, with presently J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 360 subscribers. Deliberate efforts in community building are also reflected in the social media presence.194 III. IMPLEMENTATION DETAILS A. Basis functions In the nonrelativistic domain, basis functions are modeled on atomic orbitals, but with adaptions facilitating integral evaluation. This has led to the dominant, but not exclusive use of Cartesian or spherical Gaussian-type orbitals (GTOs). Four-component atomic orbitals may be expressed as ψ(r)=[ψL ψS]=1 r[Pκ(r)ξκ,mj(θ,ϕ) iQκ(r)ξ−κ,mj(θ,ϕ)], (7) where Pκand Qκare real scalar radial functions and ξκ,mjare two- component complex angular functions. The first four-component relativistic molecular calculations in the finite basis approximation met with failure because the coupling of the large ψLand the small ψScomponents through the Dirac equation was ignored. Since the exact coupling is formally energy-dependent, the use of the nonrel- ativistic limit was made instead, leading to the kinetic balance pre- scription.195–197However, it is not possible to take this limit for the positive- and negative-energy solutions of the Dirac equation at the same time. Since the focus in chemistry is definitely on the positive- energy solutions, the relativistic energy scale is aligned with the non- relativistic one through the substitution E→E−mec2, whereupon the limit is taken as lim c→∞cψS=lim c→∞1 2me[1 +E−V 2mec2](σ⋅p)ψL=1 2me(σ⋅p)ψL. (8) In practice, one may choose between one- and two-component basis functions for four-component relativistic molecular calcula- tions. The latter choice allows for the straightforward realization of restricted kinetic balance,197hence a 1:1 ratio of large and small com- ponent basis functions, but requires on the other hand a dedicated integration module. In DIRAC, we opted3for Cartesian GTOs, Gα ijk(r)=Nxiyjzkexp[−αr2],i+j+k=ℓ, (9) since this gave immediate access to integrals of the HERMIT inte- gral module,2where the extensive menu of one-electron integrals boosted functionality in terms of molecular properties. DIRAC provides a library of Gaussian basis sets. The main basis sets available are those of Dyall and co-workers,198–210which cover all elements from H to Og at the double-zeta, triple-zeta, and quadruple-zeta levels of accuracy. They include functions for elec- tron correlation for valence, outer core, and inner core, as well as diffuse functions, in the style of the Dunning correlation-consistent basis sets.211In addition, many standard non-relativistic basis sets are available for use with lighter elements. All Dyall basis sets are used uncontracted; however, by default, the non-relativistic basis sets are kept contracted for non-relativistic calculations and oth- erwise for H–Kr ( Z≤36), for heavier elements, the basis sets are uncontracted to get a better description of the restricted kinetic balance in relativistic calculations. For this reason, the number of basis functions is generally considerably larger than that in non- relativistic calculations; however, this in fact gives better screeningin direct Fock matrix calculations and is therefore generally not a problem in the SCF. For correlated calculations based on HF, the usual procedure is to delete molecular orbitals with high orbital energies. B. SCF module for HF and KS calculations The SCF module has some unique features that will be described in the following. In the matrix form, the HF/KS equations read Fc=Scε, (10) where Fand Sare the Fock/KS and overlap matrices, respectively, andcrefers to expansion coefficients. Before diagonalization, the equations are transformed to an orthonormal basis, ˜F˜c=˜cε,˜F=V†FV,c=V˜c,V†SV=I. (11) As a simple example, we may take the Dirac equation in a finite basis, [VLLcΠLS cΠSLVSS−2mec2SSS][cL cS]=[SLL0 0SSS][cL cS]E. (12) After orthonormalization, it reads [˜VLLc˜ΠLS c˜ΠSL˜VSS−2mec2˜ISS][˜cL ˜cS]=[˜ILL0 0˜ISS][˜cL ˜cS]E. (13) DIRAC employs canonical orthonormalization212that allows for the elimination of linear dependencies. However, the orthonormaliza- tion step is overloaded as follows: 1. Elimination and freezing of orbitals: DIRAC allows for the elimination and freezing of orbitals. Such orbitals are provided by the user in the form of one or more coefficient files. This part of the code uses the machinery of the projection analy- sis discussed in Sec. II E. The selected orbitals can therefore be expressed either in the full molecular basis or in the basis set of some chosen (atomic) fragment. They are eliminated by transforming them to the orthonormal basis and projecting them out of the transformation matrix V. They may instead be frozen by putting them back in the appropriate position when back-transforming coefficients to the starting AO basis. An example of the use of elimination of orbitals is a study of the effect of the lanthanide contraction on the spectroscopic con- stants of the CsAu molecule.213Inspired by an atomic study by Bagus and co-workers,214the precalculated 4 f-orbitals of the gold atom were imported into a molecular calculation and eliminated. At the same time, the gold nuclear charge was reduced by 14 units, thus generating a pseudo-gold atom unaf- fected by the lanthanide contraction. An example of the freez- ing of orbitals is the study of the effect of the freezing of oxygen 2s-orbitals on the electronic and molecular structure of the water molecule.179 2. Cartesian-to-spherical transformation: As already mentioned, at the integral level, DIRAC employs Cartesian Gaussian- type orbitals (GTOs), Eq. (9), since this allowed the use of a non-relativistic integral code. Cartesian GTOs are particu- larly suited for integral evaluation due to the separability in J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Cartesian coordinates, but in non-relativistic codes, the inte- grals are typically subsequently transformed to the smaller set of integrals over spherical GTOs, Gα ℓm(r)=Nrℓexp[−αr2]Yℓm(θ,ϕ). (14) However, in relativistic calculations, the situation becomes more complicated due to the kinetic balance prescription. In the atomic case, the nonrelativistic limit of the coupling between the large and small radial functions reads lim c→∞cQκ=1 2me(∂r+κ r)Pκ, (15) which in the present case implies that if we start from a stan- dard form of the radial function for large components, we end up with a non-standard form for the small component radial function, that is, Pκ(r)=Nrℓ−1exp[−αr2] ⇒Qκ(r)=N{(κ+ℓ−1)rℓ−2−2αrℓ}exp[−αr2]. Rather than implementing the transformation to the non- standard radial part of the small component spherical GTOs at the integral level, we have embedded it in the transformation to the orthonormal basis. 3. Restricted kinetic balance: The use of scalar basis functions only allows for unrestricted kinetic balance, where the small com- ponent basis functions are generated as derivatives of the large component ones, but not in the fixed two-component linear combination of Eq. (8). This leads to the curious situation that the small component basis is represented by more functions than the large component one, e.g., a single large component s-function generates three small component p-functions. In DIRAC, we do, however, recover RKB in the orthonormal- ization step. In the first version, RKB was obtained by noting that the extra small component basis functions mean that there will be solutions of the Dirac equation with zero large compo- nents. In the free-particle case, these solutions will have energy −2mec2. RKB was therefore realized by diagonalizing the free- particle Dirac equation in the orthonormal basis, thus iden- tifying and eliminating (as described above) these unphysical solutions. It was later realized that RKB could be achieved in a sim- pler manner by embedding the transformation to the modified Dirac equation12,13 Q=[˜ILL0 01 2mec˜ΠSL] ⇒⎡⎢⎢⎢⎢⎣˜VLL 1 2me˜TLL 1 2me˜TLL 1 4m2 ec2˜WLL−1 2me˜TLL⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎢⎣˜cL′ ˜cS′⎤⎥⎥⎥⎥⎦ =[˜ILL0 01 4m2 ec2˜TLL][˜cL′ ˜cS′], (16) where ˜TLL μν=1 2me∑ γ˜ΠLS μγ˜ΠSL γν=⟨˜χμ∣p2 2me∣˜χν⟩, (17)˜WLL μν=∑ γδ˜ΠLS μγ˜VSS γδ˜ΠSL δν=⟨˜χμ∣(σ⋅p)V(σ⋅p)∣˜χν⟩, (18) where the latter equalities follow from kinetic balance.196The metric on the right-hand side of Eq. (16) indicates a non- orthonormal basis. A second canonical orthonormalization transformation ˜Vis therefore introduced, so that the total transformation, done in a single step, reads VQ˜V. 4. Elimination of spin–orbit interaction : As shown by Dyall,12 transformation to the modified Dirac equation allows for a sep- aration of spin-free and spin-dependent terms. In the quater- nion symmetry scheme of DIRAC, we obtain such a separa- tion by simply deleting the quaternion imaginary parts of Fock matrices in the orthonormal basis.13 5. X2C transformation : The transformation to the eXact 2- Component (X2C) relativistic Hamiltonian is carried out start- ing from the modified Dirac equation in the orthonormal basis. Working with a unit metric greatly simplifies the transforma- tion.215 6. Supersymmetry : At the integral level, basis functions are adapted to symmetries of D2hand subgroups. However, for linear systems, we obtain a blocking of Fock matrices in the orthonormal basis on the mjquantum number216by diago- nalizing the matrix of the ˆjzoperator in the orthonormal basis and performing the substitution V→VU m, where Umare the eigenvectors ordered on mj. This provides significant compu- tational savings, in particular at the correlated level. Recently, we have implemented atomic supersymmetry, such that the Fock matrix gets blocked on ( κ,mj) quantum numbers (to appear in DIRAC20).217 C. Symmetry considerations The DIRAC code can handle symmetries corresponding to D2hand subgroups (denoted binary groups) as well as linear and atomic (from DIRAC20) supersymmetries. At the SCF level, DIRAC employs a unique quaternion symmetry scheme that combines time reversal and spatial symmetry.4A particularity of this scheme is that symmetry reductions due to spatial symmetry are translated into a reduction of algebra, from quaternion down to complex and possi- bly real algebra. This leads to a classification of the binary groups as follows: ●Quaternion groups: C1,Ci ●Complex groups: C2,Cs,C2h ●Real groups: D2,C2v,D2h. At the SCF level, DIRAC works with the irreducible co- representations obtained by combining the above spatial sym- metry groups with time reversal symmetry.4A source of con- fusion for DIRAC users has been that occupations are given for each irreducible co-representation at the SCF level. However, one can show that starting from the binary groups, there are at most two irreducible co-representations, distinguished by par- ity. This means in practice that a single occupation number is expected for systems without inversion symmetry, whereas occu- pations for gerade and ungerade symmetries are given separately otherwise. J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp At the correlated levels, the highest proper Abelian subgroup of the point group under consideration is used and the only improper symmetry operation utilized is inversion. For the point groups implemented, this leads to the following group chains: ●D2,C2v→C2 ●D2h→C2h ●C∞v→C64 ●D∞h→C32h. The linear groups D∞hand C∞vare special as the number of finite Abelian subgroups that can be used to characterize orbitals is infinite. In practice, we map these groups to a 64-dimensional subgroup, which is more than sufficient to benefit from symmetry blocking in the handling of matrices and integrals and to identify the symmetry character of orbitals and wave functions. The group chain approach218in which each orbital transforms according to the irreps of the Abelian subgroup as well as a higher, non-Abelian group has an advantage that the defining elements of the second quantized Hamiltonian of Eq. (2) are real for the real groups, even though an Abelian complex group is used at the correlated level. The transition between the quaternion algebra used at the SCF level and the complex or real algebra used in the correlation modules is made in the AO-to-MO transformation that generates transformed inte- grals in the quaternion format, after which they are expressed and stored in a complex (or real) form.219 The use of real instead of complex algebra gives a four- fold speed-up for floating point multiplications. In RELCCSD, one generic algorithm is used for all implemented point groups, with the toggling between complex or real multiplications hidden inside a wrapper for matrix multiplications. The LUCIAREL kernel has dis- tinct implementations for real-valued and complex-valued Abelian double point groups.72For linear molecules143and atoms,80axial symmetry is useful and implemented.86For linear groups, an iso- morphic mapping between total angular momentum projection (along the distinguished axis) and group irreducible representation is possible for all practically occurring angular momenta using the 64-dimensional subgroups defined above. IV. CONCLUSIONS DIRAC is one of the earliest codes for four-component rela- tivistic molecular calculations and the very first to feature eXact 2- Component (X2C) relativistic calculations.26A strength of the code is the extensive collection of, in part, unique functionalities. This stems in part from the fact that the code has been written with gener- ality in mind. There is a wide range of Hamiltonians, and most pro- gram modules are available for all of them. The SCF module allows for Kramers-restricted HF and KS calculations using an innovative symmetry scheme based on quaternion algebra. In some situations, though, for instance, in DFT calculations of magnetic properties, an option for unrestricted calculations would be desirable in order to capture spin polarization. A number of molecular properties, such as electric field gradi- ents,115parity-violation in chiral molecules,126nuclear spin-rotation constants,220,221and rotational g-tensors,122were first studied in a four-component relativistic framework with DIRAC. The freedom of users to define their own properties combined with the availabilityof properties up to the third order means that there are many new properties waiting to be explored. Such properties may be further analyzed through the powerful visualization module. Another strength of DIRAC is the large selection of wave function-based correlation methods, including MRCI, CCSD(T), FSCCSD, EOM-CCSD, ADC, and MCSCF. As already mentioned, the latter allowed for a detailed study of the emblematic U 2molecule, demonstrating that the spin–orbit interaction reduces the bond order from five222to four.86The MRCI and CC methods imple- mented in DIRAC that account for more dynamic correlation have, combined with the experiment, provided reference values for prop- erties such as nuclear quadrupole moments,223,224hyperfine struc- ture constants,225and Mössbauer isomer shifts.226DIRAC also provides theoretical input for spectroscopic tests of fundamental physics, within both the standard model of elementary particles227 and tests of Beyond Standard Model (BSM) theories that give rise to electric dipole moments of fermions.141,143,228 In recent years, DIRAC has been extended to include several models for large environments: PCM, PE, and FDE, which opens new perspectives. For instance, recently, EOM-CC was combined with FDE to calculate ionization energies of halide ions in droplets modeled by 50 water molecules.174Another interesting development for large-scale applications is that DIRAC in 2015 was one of the 13 scientific software suites chosen for adaption to the SUMMIT supercomputer229at the Oak Ridge Leadership Computing Facility (OLCF). As of November 2019, SUMMIT was the world’s fastest supercomputer, and DIRAC production runs are currently being carried out on this machine. ACKNOWLEDGMENTS T.S. would like to thank his former advisors Knut Fægri, Jr. and the late Odd Gropen for putting him on an exciting track. L.V. acknowledges support of the Dutch Research Council (NWO) for this research via various programs. He also likes to thank his former advisors Patrick Aerts and Wim Nieuwpoort for introducing him to the wonderful world of relativistic quantum chemistry. A.S.P.G. acknowledges support from the CNRS Institute of Physics (INP), PIA ANR Project No. CaPPA (ANR-11-LABX-0005- 01), I-SITE ULNE Project No. OVERSEE (ANR-16-IDEX-0004), the French Ministry of Higher Education and Research, region Hauts de France council, and the European Regional Development Fund (ERDF) project CPER CLIMIBIO. M.I. acknowledges the support of the Slovak Research and Development Agency and the Scientific Grant Agency, APVV- 19-0164 and VEGA 1/0562/20, respectively. This research used resources of the High Performance Computing Center of the Matej Bel University in Banska Bystrica using the HPC infrastructure acquired in Project Nos. ITMS 26230120002 and 26210120002 (Slo- vak Infrastructure for High Performance Computing) supported by the Research and Development Operational Programme funded by the ERDF. R.D.R. acknowledges partial support by the Research Council of Norway through its Centres of Excellence scheme, Project No. 262695, and through its Mobility Grant scheme, Project No. 261873. M.O. acknowledges support of the Polish National Science Centre (Grant No. 2016/23/D/ST4/03217). J. Chem. Phys. 152, 204104 (2020); doi: 10.1063/5.0004844 152, 204104-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp I.A.A. acknowledges support from CONICET by Grant No. PIP 112-20130100361 and FONCYT by Grant No. PICT 2016-2936. J.M.H.O. acknowledges financial support from the Research Council of Norway through its Centres of Excellence scheme (Project No. 262695). E.D.H. acknowledges financial support from the European Commission (MetEmbed, Project No. 745967) and the Villum Foundation (Young Investigator Program, Grant No. 29412) A.S. acknowledges financial support from the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant No. 17J02767, and JSPS Overseas Challenge Program for Young Researchers, Grant No. 201880193. S.K. would like to thank Markus Reiher (ETH Zürich) for his continuous support throughout his time at ETH Zürich. M.P. gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft (DFG). T.F. acknowledges funding by the Deutsche Forschungsge- meinschaft (DFG) and the Agence Nationale de la Recherche (ANR) through various programs. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1P. A. M. Dirac, Proc. R. Soc. London, Ser. A 117, 610 (1928). 2T. Helgaker and P. R. 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Reduction of switching current distribution in spin transfer magnetic random access memories M. Iwayama, T. Kai, M. Nakayama, H. Aikawa, Y. Asao et al. Citation: J. Appl. Phys. 103, 07A720 (2008); doi: 10.1063/1.2838140 View online: http://dx.doi.org/10.1063/1.2838140 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v103/i7 Published by the American Institute of Physics. Related Articles Continuous-film vs. device-level ferromagnetic resonance in magnetic tunnel junction thin films J. Appl. Phys. 113, 083903 (2013) Reducing the writing field of L10-FePt by graded order parameter J. Appl. Phys. 113, 073912 (2013) Enhanced inverse spin-Hall effect in ultrathin ferromagnetic/normal metal bilayers Appl. Phys. Lett. 102, 072401 (2013) Heat-induced damping modification in yttrium iron garnet/platinum hetero-structures Appl. Phys. Lett. 102, 062417 (2013) Effect of antiferromagnetic thickness on thermal stability of static and dynamic magnetization of NiFe/FeMn multilayers J. Appl. Phys. 113, 063913 (2013) 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 28 Feb 2013 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsReduction of switching current distribution in spin transfer magnetic random access memories M. Iwayama,1,a/H20850T . Kai,2M. Nakayama,2H. Aikawa,2Y . Asao,1T . Kajiyama,1S. Ikegawa,2 H. Yoda,2and A. Nitayama1 1Center for Semiconductor Research and Development, Semiconductor Company, Toshiba Corporation, Yokohama, Japan 2Corporate Research and Development Center, Toshiba Corporation Kawasaki, Japan /H20849Presented on 7 November 2007; received 12 September 2007; accepted 22 November 2007; published online 12 March 2008 /H20850 In this paper, the switching current distribution by spin transfer torque is investigated for CoFeB /MgO /CoFeB magnetic tunnel junctions /H20849MTJs /H20850. The distribution of the spin transfer switching current for a MTJ with junction size of 85 /H11003110 nm2is 16% when the duration of applied pulse current is 5 ms. In the case of magnetization reversal with magnetic field induced by currentwith 5 ms pulse duration, the distribution of the switching field is 8.3%. According to ourmicromagnetic simulation, it is found that the spin transfer current switching seems to exhibit anonuniform magnetization reversal process, whereas the magnetization switching by the magneticfield exhibits a uniform magnetization reversal process. This leads to the broader distribution relatedto the repeatability. © 2008 American Institute of Physics ./H20851DOI: 10.1063/1.2838140 /H20852 I. INTRODUCTION Since the observation of spin transfer switching in mag- netic tunnel junctions /H20849MTJs /H20850,1spin transfer switching has stimulated considerable interest for magnetic random accessmemories /H20849MRAMs /H20850because of its high scalability. As a key to realizing high-density MRAMs, it is neces- sary to reduce not only average spin transfer switching cur-rent but also switching current distribution, which includes acell-to-cell variation within a cell array and switching currentfluctuation. In this paper, we investigate the fluctuation ofswitching current from both experimental results and micro-magnetic simulations. II. EXPERIMENT AND RESULTS The magnetic tunnel junctions in this study consist of Ta /H208495/H20850/PtMn /H2084915/H20850/CoFe /H208493/H20850/Ru /H208490.8 /H20850/CoFeB /H208493/H20850/MgO /H208491/H20850/CoFeB /H208492/H20850/Ta /H208495/H20850/H20849in nanometers /H20850layers. After deposition, the MTJs are annealed at 300 °C for 2 h under a magnetic field of1.5 T. A resistance area product is 40 /H9024/ /H9262m2and the mag- netoresistance ratio is 130%. The saturated magnetization ofthe free layer is 890 emu /cc. The deep submicron junctions are patterned using conventional lithography and ion beametching. Above the MTJ, there is the bit line utilized for thegeneration of the magnetic field parallel to the easy axis ofthe MTJ. The magnetization reversal of the MTJ is repeat-edly carried out by the bit line current that generates thefield. In the case of switching by spin transfer torque, thecurrent through the MTJ is applied. Figures 1/H20849a/H20850and1/H20849b/H20850 show the resistance versus bit line current and applied volt-age. The measurement is performed repeatedly, 100 times, ata 5 ms pulse duration for the MTJ with the junction size of85/H11003110 nm 2. The average coercive field is 45 Oe and the average spin transfer switching current density is around1M A /cm 2. The distribution of the bit line current corresponds to the distribution of the coercive field /H20849/H9268Hc/H20850./H9268Hc/Hcis 8.3% and the distribution of the switching current /H20849/H9268Ic/Ic/H20850is 16%. In a Stoner-Wohlfarth magnet, the thermal stability factor under the applied current /H20849I/H20850and the external field /H20849H/H20850parallel to the easy axis is expressed by2 /H9004/H11032=/H9004/H208491−H/Hc0/H208502/H208491−l/lc0/H20850. /H208491/H20850 Here, /H9004is the thermal stability factor /H20849KuV/kBT/H20850.Kuis the magnetic anisotropy energy, Vis the volume of the free layer, kBis the Boltzmann constant, Tis the temperature, and Hc0andIc0are the intrinsic switching field and current, re- spectively. The anomalous bias dependence of spin transfer torque on an MTJ is predicted in the theoretical model.3On the other hand, a linear dependence of spin transfer torque isreported from the relationship between the switching currentand the pulse duration in the MTJ of CoFeB /MgO /CoFeB. 4 The film stack of our MTJ is the same as that of the MTJ investigated for the validity of the linear dependence. Thus,we assume a linear dependence of energy barrier on spintransfer current. The thermal stability factor is determinedfrom the distribution of the switching field /H20849H c/H20850or the switching current /H20849Ic/H20850using the Eq. /H208491/H20850.5The thermal stabil- ity factor obtained from the distribution of the switching field is 34, and, in the case of the current, the thermal stabilityfactor is 24. The large /H9268Ic/Iccorresponds to the small ther- mal stability factor. In order to clarify the large distributionof the switching current, it is essential to study the micro-magnetic magnetization reversal process. III. MICROMAGNETIC SIMULATIONS To study the magnetization reversal process, we carry out the micromagnetic simulations using the Landau-a/H20850Electronic mail: masayoshi.iwayama@toshiba.co.jp.JOURNAL OF APPLIED PHYSICS 103, 07A720 /H208492008 /H20850 0021-8979/2008/103 /H208497/H20850/07A720/3/$23.00 © 2008 American Institute of Physics 103 , 07A720-1 Downloaded 28 Feb 2013 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsLifshitz-Gilbert equation with an additional spin torque term.6All parameters used in our simulations are chosen as close as possible to the parameters of above-experimentaldata /H20849the saturated magnetization M sis 890 emu /cc, thick- ness of magnetic free layer of a MTJ is 2 nm, uniaxial an-isotropy energy Kuis 2/H1100310 4erg /cc, exchange constant Ais 1/H1100310−6erg /cm, the Gilbert damping factor /H9251is 0.005, the spin polarization factor /H9257is 0.63. /H20850The unit cell size is 5 /H110035/H110032n m3. We adopt the Langevin random field HLto take account of the finite temperature effect at 300 K. We assumeno dipolar field and no current-induced Oersted field. /H9253is the gyromagnetic ratio. Jis current density. dM dt=−/H9253M/H11003/H20849M/H11003HL/H20850+/H9251 MsM/H11003dM dt+/H9253/H9257/H6036 eMS2tM /H11003J/H11003M. /H208492/H20850 In order to investigate the difference between /H9268Icand/H9268Hc, we observe the magnetization reversal process of the fieldand the current switching. Figure 2shows the magnetization reversal process induced by spin transfer current at a 10 nspulse duration for the magnetic free layer with a sample sizeof 85/H11003110 nm 2. Spin transfer current switching exhibits a nonuniform magnetization reversal process. The edge do-main of the sample exhibits the precession state independentof the center domain. This is caused by the in-plane inhomo-geneous demagnetization field. On the other hand, magneticfield switching exhibits a uniform magnetization reversalprocess. This suggests that the magnetization switching in-duced by the field exhibits coherent switching, whereas themagnetization switching induced by spin transfer current ex-hibits multidomain configuration.Since the transition from multidomain configuration to single-domain configuration 7–11depends on the sample size of the free layer,8Landau-Lifshits-Gilbert simulations are carried out for different MTJs of different sizes. Figure 3 shows the switching probability at each applied field andcurrent density at a 10 ns pulse duration for a rectangle-shaped free layer with a sample size of 30 /H1100360 nm 2and 15/H1100330 nm2. In the thermal activation obeying the classical Arrhenius formula, since the probability distribution of mag-netization switching depends on /H9004, switching probability is expressed by the following equations: P field=1−e x p /H20853−/H9270p//H92700exp /H20851−/H9004/H208491−H/Hc0/H208502/H20852/H20854, /H208493/H20850 Pcurrent =1−e x p /H20853−/H9270p//H92700exp /H20851−/H9004/H208491−J/Jc0/H20850/H20852/H20854, /H208494/H20850 where /H92700is the attempt time /H20849/H92700of 1 ns is used in this paper /H20850, /H9270pis the pulse duration of the applied field or current. The solid lines indicate fitted curves and the dashed lines are theintrinsic coercive field and the intrinsic current density esti-mated from the fitting. In the case of the sample size of 30/H1100360 nm 2, the fitted curve of Pcurrent is gentle, whereas that of Pfieldis steep as shown in Figs. 3/H20849a/H20850and3/H20849b/H20850. The thermal stability factors estimated from PfieldandPcurrent are 77.5 and 10.4, respectively. In the case of the sample size of 15/H1100330 nm 2, the thermal stability factors estimated from Pfield andPcurrent are 26.3 and 24.3, respectively. Both fitted curves are steep as shown in Figs. 3/H20849c/H20850and3/H20849d/H20850. The magnetization induced by the field exhibits coherent switching even in thecase of the sample size of 30 /H1100360 nm 2, since both groups of Pfieldare well fitted by the Eq. /H208493/H20850. The thermal stability factor estimated from Pcurrent is expected to agree with the thermal stability factor estimated from Pfieldfor the same sample, assuming coherent switching /H20849i.e., Stoner-Wohlfarth magnet /H20850. The magnetization switching by spin transfer cur- rent seems to exhibit coherent switching in the case of thesample size of 15 /H1100330 nm 2owing to almost the same ther- mal stability factor. The thermal stability factor estimatedfrom P fieldis decreased owing to the reduction of the volume of the magnetic layer. However, the thermal stability factorestimated from P current is increased. This corresponds to the reduction of the distribution due to the transition from mul-tidomain to single-domain behavior in the magnetization re-versal process since the sample size is reduced. From thefittings of Figs. 3/H20849b/H20850and3/H20849d/H20850, the obtained switching current density is around 2 MA /cm 2. In this case, the Oersted field from the current is estimated to be less than a few Oersted. FIG. 1. /H20849a/H20850Resistance vs bit line current /H20849proportional to external field /H20850/H20849b/H20850Resistance vs applied voltage. Each measurement is repeated 100 times at a pulse du-ration of 5 ms. FIG. 2. The magnetization reversal process induced by the spin transfercurrent.07A720-2 Iwayama et al. J. Appl. Phys. 103 , 07A720 /H208492008 /H20850 Downloaded 28 Feb 2013 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsThis contribution to the magnetization reversal process by spin transfer torque seems to be negligible because Hcof the calculated sample is much larger than the Oersted field.Therefore, our assumption is quite reasonable in this work. Itis claimed that the critical size of the transition from inco-herent to coherent switching is less than 100 nm from theviewpoint of the vortex formation. 8The difference between our results and their report is probably due to the smalldamping factor. It is pointed out that the total noise power,which arises from the chaotic magnetization precession, in-creases with decreasing the damping factor in currentperpendicular-to-plane read head. 9However, to achieve low switching current density without decreasing the thermal sta-bility factor, it is essential to reduce the damping factor. Torealize a high-density MRAM, it is important to decrease notonly the damping factor but also the MTJ size for reducingswitching current density and distribution of the switchingcurrent. IV. CONCLUSION On the basis of the experimental results and the micro- magnetic simulation, the fluctuation of the spin transfer cur-rent is studied. The spin transfer current switching exhibits anonuniform magnetization reversal process, whereas themagnetization switching by magnetic field exhibits a uniformmagnetization reversal process. The fluctuation of the spin transfer current is reduced by shrinking the MTJ to the sizein which coherent switching is expected. For a gigabit-density spin transfer MRAM, the distribution of the switch-ing current must be reduced and an MTJ size of a few tennanometers is a necessity. A portion of this work was sup-ported by NEDO. 1J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 2R. H. Koch, J. A. Katine, and J. Z. Sun, Phys. Rev. Lett. 92, 088302 /H208492004 /H20850. 3I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev, and W. H. Butler, Phys. Rev. Lett. 97, 237205 /H208492006 /H20850. 4M. Hosomi, H. Yamagishi, T. Yamamoto, K. Bessho, Y. Higo, K. Yamane, H. Yamada, M. Shoji, H. Hachino, C. Fukumoto, H. Nagao, and H. Kano,Tech. Dig. - Int. Electron Devices Meet. 2005, 459. 5Y. Higo, K. Yamane, K. Ohba, H. Narisawa, K. Bessho, M. Hosomi, and H. Kano, Appl. Phys. Lett. 87, 082502 /H208492005 /H20850. 6Z. Li and S. Zhang, Phys. Rev. B 68, 024404 /H208492003 /H20850. 7J. Miltat, G. Albuquerque, A. Thiaville, and C. Vouille, J. Appl. Phys. 89, 6982 /H208492001 /H20850. 8K. J. Lee and B. Dieny, Appl. Phys. Lett. 88, 132506 /H208492006 /H20850. 9J. G. Zhu and X. Zhu, IEEE Trans. Magn. 40, 182 /H208492004 /H20850. 10K. J. Lee, A. Deac, O. Redon, J. P. Nozieres, and B. Dieny, Nat. Mater. 3, 877 /H208492004 /H20850. 11Y. Acremann, J. P. Strachan, V. Chembrolu, S. D. Andrews, T. Tyliszczak, J. A. Katine, M. J. Carey, B. M. Clements, H. C. Siegmann, and J. Stohr,Phys. Rev. Lett. 96, 217202 /H208492006 /H20850. FIG. 3. The switching probability at each applied field /H20849left hand /H20850and current /H20849right hand /H20850for the sample with size of 30 /H1100360 nm2/H20849top /H20850and 15 /H1100330 nm2/H20849bottom /H20850. The applied pulse duration is 10 ns. The solid lines aretheoretical fitted curves and the dashed lines are intrin-sic switching field and current estimated from thefitting.07A720-3 Iwayama et al. J. Appl. Phys. 103 , 07A720 /H208492008 /H20850 Downloaded 28 Feb 2013 to 152.3.102.242. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

1.3610948.pdf

Inductive determination of the optimum tunnel barrier thickness in magnetic tunneling junction stacks for spin torque memory applications S. Serrano-Guisan, W. Skowronski, J. Wrona, N. Liebing, M. Czapkiewicz, T. Stobiecki, G. Reiss, and H. W. Schumacher Citation: Journal of Applied Physics 110, 023906 (2011); doi: 10.1063/1.3610948 View online: http://dx.doi.org/10.1063/1.3610948 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/110/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Influence of dynamical dipolar coupling on spin-torque-induced excitations in a magnetic tunnel junction nanopillar J. Appl. Phys. 111, 07C906 (2012); 10.1063/1.3672849 Thermal stability and spin-transfer switchings in MgO-based magnetic tunnel junctions with ferromagnetically and antiferromagnetically coupled synthetic free layers Appl. Phys. Lett. 95, 242504 (2009); 10.1063/1.3275753 Electric breakdown in ultrathin MgO tunnel barrier junctions for spin-transfer torque switching Appl. Phys. Lett. 95, 232119 (2009); 10.1063/1.3272268 Auto-oscillation and narrow spectral lines in spin-torque oscillators based on MgO magnetic tunnel junctions J. Appl. Phys. 106, 103921 (2009); 10.1063/1.3260233 Direct measurement of current-induced fieldlike torque in magnetic tunnel junctions J. Appl. Phys. 105, 113924 (2009); 10.1063/1.3143033 [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: 128.143.199.160 On: Mon, 15 Dec 2014 08:40:12Inductive determination of the optimum tunnel barrier thickness in magnetic tunneling junction stacks for spin torque memory applications S. Serrano-Guisan,1,a)W. Skowronski,2J. Wrona,2N. Liebing,1M. Czapkiewicz,2 T. Stobiecki,2G. Reiss,3and H. W. Schumacher1 1Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany 2AGH University of Science and Technology, Department of Electronics, Al. Mickiewicza 30, 30-059 Krakow, Poland 3Bielefeld University, Department of Physics, P.O. Box 100131, 33501 Bielefeld, Germany (Received 22 March 2011; accepted 12 June 2011; published online 28 July 2011) We use pulsed inductive microwave magnetometry to study the precessional magnetization dynamics of the free layer in CoFeB/MgO/CoFeB based magnetic tunneling junction stacks withvarying MgO barrier thickness. From the field dependence of the precession frequency we are able to derive the uniaxial anisotropy energy of the free layer and the exchange coupling between the free and the pinned layer. Furthermore the field dependence of the effective damping parameter isderived. Below a certain threshold barrier thickness we observe an increased effective damping for antiparallel orientation of free and pinned layer which would inhibit reversible low current density spin torque magnetization reversal. Such inductive measurements, in combination with waferprobe station based magneto transport experiments, allow a fast determination of the optimum tunnel barrier thickness range for spin torque memory applications in a lithography free process. VC2011 American Institute of Physics . [doi: 10.1063/1.3610948 ] INTRODUCTION Spin transfer torque (ST)1,2allows the realization of high density magnetic random access memories (MRAM) based on magnetic tunnel junction (MTJ) cells. In such ST-MRAM devices the cells are programmed by a highdensity current pulse that initiates spin torque precession of the free layer magnetization 3and eventually induces magnet- ization reversal.4In the MTJ stack, the thickness of the tun- neling barrier ( tMgO) plays an important role as it defines two key parameters for MRAM device applications: the resist- ance area (RA) product and the tunneling magneto resistance(TMR) ratio. However, both parameters are usually optimum at different thickness ranges of t MgO, and therefore a compro- mise must be found. In addition also the strength of the cou-pling J FLbetween the free and the reference layer depends ontMgO5–7thereby influencing the reversal properties of the ST-MRAM cells. Furthermore, this coupling could influencethe effective damping parameter aof the free layer. ais also important for ST applications as the critical current density j Cfor ST magnetization reversal is expected to be directly proportional to a.1,8Usually, jCcan only be determined in a time consuming process comprising clean room fabrication of ST nanopillars and subsequent magneto transport experi-ments on individual devices. In contrast, acan be determined by fast inductive characterization of the unpatterned MTJ stacks. Here, we study the time resolved precessional magnet- ization dynamic of the free layer in CoFeB/MgO based MTJ stacks by pulse inductive microwave magnetometry (PIMM).From a PIMM measurement at a static magnetic field, wederive the free layer precession frequency fand the effective damping parameter a. From the field dependence of fwe are able to derive the uniaxial anisotropy energy of the free layer K FLand the coupling JFLbetween the free and the reference layer. These inductively derived stack parameters are com- pared to magneto optical Kerr effect (MOKE) measurements of the same stacks. Furthermore, the ST-MRAM key param-eters TMR ratio and RA product are determined by using current-in-plane tunneling (CIPT) technique. 9The depend- ence of the derived parameters on tMgO is discussed with respect to ST-MRAM applications. Below a certain thresh- old barrier thickness, we observe an effective damping for antiparallel configurations of the MTJ stack which is largerthan for parallel configurations ( a AP>aP). This asymmetry ofa, which increases with decreasing tMgO, would also induce an asymmetric jCthereby inhibiting reversible low current density ST magnetization reversal. Therefore, in combination with wafer probe based determination of RA and TMR, such inductive measurements allow a fast deter-mination of the optimum tunnel barrier thickness range for spin torque memory applications. EXPERIMENTAL We study MTJ stacks with a wedge shaped tunneling barrier as sketched in Fig. 1(a). The stacks are deposited in a Singulus TIMARIS cluster tool on Si wafers with a layer sequence: Ta(5)/CuN(50)/Ta(3) /CuN(50)/Ta(3)/PtMn(16)/ CoFe(2)/Ru(0.9)/Co 40Fe40B20(2.3)/MgO( tMgO)/Co 40Fe40B20 (2.3)/Ta(10)/CuN(30)/Ru(7) from bottom to top (details about deposition technique were published elsewhere10). Numbers in parentheses refer to the layer thickness in nm. The MgO thickness is varied in the range of tMgO¼0.62… 0.96 nm without changing the other stack parameters usinga)Author to whom correspondence should be addressed. Electronic mail: santiago.serrano-guisan@ptb.de. 0021-8979/2011/110(2)/023906/7/$30.00 VC2011 American Institute of Physics 110, 023906-1JOURNAL OF APPLIED PHYSICS 110, 023906 (2011) [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: 128.143.199.160 On: Mon, 15 Dec 2014 08:40:12linear dynamic wedge technology. After sputter deposition, the samples are annealed at 350/C14C for 2 h with a 0.5 T mag- netic field to define the orientation of the exchange bias. Forinductive characterization pieces of 2 /C24 mm of lateral dimension and of 5 /C25 mm for magneto optical characteri- zation were cut from the wafer. Over the size of the charac-terized wafer pieces the variation of t MgOcan be neglected, as the wedge slope was 0.003 nm/mm, and hence each piece is expected to well represent a given barrier thickness. The measured dependence of RA and TMR on tMgO is presented in Fig. 1(b). For thick MgO barriers ( tMgO/C210.75 nm) the TMR ratio is very high (TMR >150%) and almost thickness independent, indicating a good quality of the tun- nel barrier and lack of pinholes, while for thinner barriers (tMgO/C200.71 nm) it drops significantly, pointing out to possi- ble barrier imperfections. On the other hand, RA increases exponentially with tMgOover a broad thickness range, in con- cordance with previous reports.11,12 PIMM measurements were performed at room tempera- ture for all MgO thicknesses. Details of our PIMM measure- ment technique are reported elsewhere.13Single PIMM measurements at a given static field deliver the precession frequency fand the effective damping parameter aof the free layer magnetization by fitting the inductive signal to adamped sinusoid. 14Figure 2shows typical time resolved PIMM data for tMgO¼0.76 nm at three different static field values along the easy axis direction. The time resolved pre-cession traces [open dots in Figs. 2(a)–(c) ] can indeed be well fitted by an exponentially damped sinusoid [lines in Fig. 2(a)–(c) ], 15showing that the observed magnetization dynam- ics is always in the linear regime. From the best fit to the exponentially damped sinusoid, the values of fandaaredetermined as described in Ref. 14. It is clear from experi- mental data that fvaries with the applied static field. This field dependence of the free layer precession frequency is plotted in Fig. 2(d)(open dots). To derive the important material parameters from Fig. 2(d), we model the precession of the free layer within the MTJ stack in a macro spin model of a coupled trilayer system consisting of a free layer, a reference layer and apinned layer [see Fig. 1(a)]. We assume a 2.3 nm thick free layer with a saturation magnetization M s, uniaxial anisotropy energy KFL, and coupling JFLbetween free layer and the ref- erence layer. Zero net magnetic moment of reference and pinned layer is assumed due to the antiferromagnetic exchange coupling between both layers. At these conditions, the total magnetic free energy Eof the system can be written as E¼/C0l0/C1MsH/C1cos//C0u ðÞ /C0 KFLcos2//C0JFL tFLcos/;(1) where tFLis the free layer thickness, and uand/are the azi- muthal coordinate of the free layer magnetization and the in- plane orientation of the applied magnetic field with respect to the easy axis. Thus, following Smit and Beljers scheme,16 the dispersion relation derived from Eq. (1)is f¼cl0 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Hexcos/þHkcos 2/þHcos//C0u ðÞ ½/C138 /C1 Hexcos/þHkcos2/þHcos//C0u ðÞ þ Ms ½/C138p ; (2) FIG. 1. (Color online) (a) Schematic picture of the MgO based MTJ stack. JFLrefers to the exchange coupling between the free layer (FL) and the ref- erence layer (RL), JAFrefers to the antiferromagnetic interlayer exchange coupling between the reference layer (RL) and the pinned layer (PL) whereas Jexrefers to the exchange bias coupling between the antiferromag- netic pinning layer (AF) and the pinned layer (PL) (b) TMR and RA product as a function of MgO barrier thickness measured by CIPT technique (Ref. 9). Line shows the exponential thickness dependence of RA. FIG. 2. (Color online) (a)–(c) PIMM data (open dots) for tMgO¼0.76 nm at different easy axis fields. Lines show fits by an exponentially damped sinusoid. (d) Static field dependence of the precession frequency derived from PIMM (open dots). Line shows the dispersion relation of a Stoner-Wolfarth single-do- main model with HK¼1.8 mT and JFL¼20.1 lJ/m2. Inset: MOKE loop (black dots) and single domain model approximation (line).023906-2 Serrano-Guisan et al. J. Appl. Phys. 110, 023906 (2011) [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: 128.143.199.160 On: Mon, 15 Dec 2014 08:40:12where Hex¼JFL tFLl0MsðÞ,Hk¼2KFL l0Msanduis obtained by mini- mizing Eq. (1). Therefore, fitting the model to the field dependence of f [line in Fig. 2(d)] allows to derive the magnetic parameters: Ms,KFL;andJFLof our MTJ stacks. Additionally, a magnetic characterization of samples was carried out along both theeasy and hard magnetization axis using MOKE magnetome- ter. An example of a minor easy axis MOKE loop of the same sample is shown in the inset in Fig. 2(d) (black dots). In order to fit such MOKE loops a complete multilayer coupled system 17is considered, even though for a small cou- pling JFL(thicker barriers), minor MOKE loops can be also fitted by considering Eq. (1). RESULTS AND DISCUSSION Msand K FL Figure 3shows the MgO thickness dependence of JFL andKFL[Fig. 3(a)] derived from PIMM and MOKE whereas a[Fig. 3(b)] derived from PIMM. A constant magnetization saturation of l0Ms/C241:35 T is obtained for all samples by both techniques (not plotted). Regarding KFL, a mismatch is observed between KFLobtained from PIMM (open squares) and MOKE (full squares). While KFLderived from PIMM is almost independent of thickness, the MOKE data show anincrease of K FLwith decreasing tMgO. This deviation up to a factor of 2 is not well understood, but could be related to the different lateral dimensions of samples used for PIMM andMOKE measurements. 18Furthermore, an over simplification of the MTJ system by our model could result in an additionaluncertainty of the derived values of KFLwhich is presently not accounted for. It is important to note, however, that this mismatch between both KFLvalues is only relevant for tunnel barrier thickness tMgO<0.80 nm. At this region [see Fig. 3(a)] the coupling contribution to the dispersion relation [third term ofEq.(1)] is at least 5 times larger than the anisotropy contri- bution [second term of Eq. (1)], suggesting that magnetiza- tion dynamics will be mostly characterized by saturationmagnetization M sandJFLcoupling. Therefore, this discrep- ancy between both KFLvalues should not question our magnetization dynamics analysis. JFLcoupling In contrast, a large ferromagnetic coupling JFLis observed for all samples, with a very good concordance between PIMM(open dots) and MOKE (full dots) measurements. For t MgO/C210.71 nm an exponential decrease of JFLas a function of the tunnel barrier thickness is observed. Such a coupling can-not be just understood as a Ne ´el “orange-peel” coupling 19,20 arising from the correlated roughness between the free layer and the reference layer. Indeed, assuming a sinusoidal rough-ness profile, the “orange-peel” effect predicts a monotonic exponential decrease of J FL:20 JFL¼p2 ffiffiffi 2ph2 k/C18/C19 /C1l0M2 FL/C1exp/C02pffiffiffi 2p /C1ts=k/C16/C17 ;(3) where handkare the amplitude and the wavelength of the roughness and tsis the barrier thickness. By fitting JFLto this equation [see red line in Fig. 3(a)]w ed e r i v e h¼1:060:7n m andk¼1:160:1 nm. These values are far too high to be compatible with the large TMR ratios observed for tMgO>0.71 nm in our high quality MTJ stacks (cp. Fig. 1). Furthermore, it has been reported that for this kind of MTJ stacks, the wave-length is 10 nm <k<30 nm. 21 This inconsistency suggests, therefore, that this large FM coupling may be understood as a signature of an interlayerexchange coupling (IEC). The IEC is an interfacial exchange interaction that appears when t wo ferromagnetic layers are separated by a thin spacer as a consequence of spin-dependentreflections at the ferromagnetic/spacer interface. 5,6Unlike for metallic spacers, where IEC oscillates with the spacer thick- ness22a monotonic nonoscillatory exponential decrease of IEC as a function of the insulating barrier thickness is expected for magnetic tunnel junctions5–7(in good concordance with our data). Previous MOKE measurements in epitaxial Fe/MgO/Fethin films have however shown an antiferromagnetic (AF) interlayer exchange coupling for the same MgO barrier thick- ness range studied here (0.5 nm <t MgO<0.9 nm).23,24Faure- Vincent et al.23were able to ascribe the magnitude and the thickness dependence of this AF coupling in the context of the spin-current Slonczewski’s model.6By doing similarly and considering EF¼2.25 eV, D¼2 . 4e V ,a n dm f¼0.75 m efor CoFeB ferromagnets,25we obtain an antiferromagnetic IEC for our samples, in clear contradiction with our data. This con-troversy could be ascribed to the limited validity of the spin- current Slonczewski’s model, which is only valid for thick FIG. 3. (Color online) MgO thickness dependence of (a) the exchange coupling JFLbetween the free layer and the reference layer (circles) as well as the uniaxial anisotropy energy KFLof the free layer (squares) and (b) the effective damping parameter. Open (full) symbols in (a) are referred to values ofJFLorKFLderived from PIMM (MOKE) measurements. Line shows the Ne´el coupling behavior derived by fitting JFLdata to Eq. (3). In (b) open trian- gles refer to the damping parameter derived at parallel configurations ( aP), open squares are referred to the damping parameter at antiparallel configura- tions ( aAP), and full squares are referred to the averaged damping parameter between both configurations. At MgO thickness tMgO>0.76 nm both configu- rations have the same effective damping parameter ( aAP¼aP:a).023906-3 Serrano-Guisan et al. J. Appl. Phys. 110, 023906 (2011) [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: 128.143.199.160 On: Mon, 15 Dec 2014 08:40:12tunnel barriers (2 /C1k/C1t>1), as pointed out by Katayama et al.24 By a full integration of the IEC over the whole Brillouin zone, they show that the strength of the IEC should be ferromagneticin epitaxial Fe/MgO/Fe thin films, and not antiferromagnetic as observed by Faure-Vincent et al. Moreover, they showed that their data can only be analyzed under Slonczewski’sapproximation for t MgO>1.2 nm, far away from the thickness range where the AF coupling was observed. These discrepan- cies therefore show that spin-dependent tunneling transporttheory in MTJs 5,6must be improved by, e.g., considering the presence of impurities and defects in the tunnel barrier. Indeed, it has been observed that TMR in MTJs is extremely sensitive to barrier impurities.26Moreover, Zhura- levet al.27showed that impurities and/or defects in the tun- nel barrier can modify the electronic density of states (DOS)and, with a proper concentration, change the strength of the IEC. 28Katayama et al. used this assumption in order to as- cribe their AF coupling due to the presence of O vacanciesin their MgO barriers. The presence of such impurities in the tunnel barrier can be produced by material diffusion from the electrode to the MgO layer during the fabrication pro-cess. Recent studies showed that a substantial concentration of BO xin the MgO layer can be obtained after annealing when MgO is deposited by radio frequency sputtering onCoFeB thin films. 29The presence of such impurities into the tunnel barrier results in an amorphous Mg-B-O layer [with thickness ranging from 1.1 to 2.1 nm (Ref. 30)] modifying the electronic properties of the MTJ and the energy band of the tunnel barrier.31This distortion induces a ferromagnetic IEC for tMgO¼1.1 nm (Ref. 31) without substantially modi- fying the large TMR ratios ( /C24150% – 200%) of such MTJ stacks.30All these results are consistent with our data and seem to support Zhuralev’s argument27in order to explain the origin of the ferromagnetic IEC in our samples. Recently, Yang et al.32studied the influence of the relaxation and the oxidation conditions of epitaxial Fe/MgO/Fe stacks on theIEC. It was found that sufficiently oxidation concentration in the Fe/MgO interface can also induce a ferromagnetic inter- layer exchange coupling. However, a further systematicstudy of this impurity-assisted IEC in the sputtered CoFeB/ MgO/CoFeB MTJs under a precise control of the parameter conditions during the growth process of the samples shouldbe carried out in order to assert this explanation. Effective damping a The dependence of the effective damping parameter a ontMgOis shown in Fig. 3(b). The behavior of the field de- pendence of asignificantly depends on tMgO as will be explained in more detail in the discussion of Fig. 4. Three different typical behaviors are found, which are marked bythe three regimes A, B, C in Fig. 3(b). For t MgO>0.76 nm (region A), no significant dependence of aontMgO is observed. Here, a¼0.01660.003 which is comparable to our previous values obtained by PIMM measurements in sin- gle CoFeB layers of similar thickness.33This implies that for tMgO>0.76 nm the influence of neighboring layers of the MTJ stack on the free layer magnetization dynamics is negli- gible, and therefore, the observed effective damping parame-teraseems to be dominated by the intrinsic properties of the CoFeB layer. The typical dynamic properties in this thickness range will be discussed with respect to Figs. 4(a)–(d). The figure shows, for tMgO¼0.88 nm, the static field dependence of the precession frequency f[Fig. 4(a)], the effective damping a [Fig. 4(b)] and the calculated free layer (FL), reference layer (RL) and pinned layer (PL) magnetization orientation forstatic fields along the easy axis [Fig. 4(c)] and along the hard axis [Fig. 4(d)]. Here, a coupled multilayer model is used in order to derive the orientation of all ferromagnetic layers ofthe MTJ stack 17by fitting this model to the measurements. Thus, RL refers to the upper CoFeB layer of the synthetic antiferromagnetic layer while PL refers to the bottom CoFelayer of the synthetic antiferromagnetic which is exchange coupled to the PtMn antiferromagnetic layer [see Fig. 1(a)]. We observe that at this tunnel barrier thickness the magnet-ization reversal of the MTJ stack is similar to a completely free layer system, where the PL and the RL stay along the easy axis while the FL is reversed. The main consequence ofthis small coupling J FLis a shift of the magnetization loop and the resonance frequency fdue to the extra effective easy axis bias field induced by the interlayer exchange couplingbetween free and reference layers through the MgO. There- fore, for t MgO>0.76 nm the magnetization dynamics of the MTJ can be interpreted just in terms of the free layer mag-netization. aHðÞis symmetric in the applied field, showing the same value for parallel and antiparallel orientations of PL and FL, whereas at low magnetic fields an enhancementofais observed (which is typically attributed to inhomoge- neous precession in the low field range 14). Note again that the value of ais comparable to the value of aderived for a single CoFeB layer. This result is of significant importance for ST-MRAM applications as the critical current density jC FIG. 4. (Color online) (a), (e) Dispersion relation and minor MOKE loops; (b), (f) effective damping dependence on easy axis magnetic fields; and (c–d), (g–h) simulated magnetic field dependence of magnetization orienta- tion of each ferromagnetic layer for (c), (f) easy axis (e.a.) and (d), (h) hard axis (h.a.) magnetic fields with tMgO¼0.88 nm [(a), (b), (c), and (d)] and tMgO¼0.71 nm [(e), (f), (g) and (h)]. (a), (e) Open dots (line) show the measured (simulated) resonance frequency. Inset: Full dots (line) shows the measured (simulated) minor MOKE loops. (c–d), (g–h) FL, RL, and PL refer to the free layer, reference layer, and pinned layer respectively.023906-4 Serrano-Guisan et al. J. Appl. Phys. 110, 023906 (2011) [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: 128.143.199.160 On: Mon, 15 Dec 2014 08:40:12for ST reversal is directly proportional to the effective damp- ingaof the free layer.1,8Hence, any additional dissipation of the ST precession by coupling of the free layer magnetiza-tion to neighboring layers would also increase j Cthereby hampering ST-MRAM applications. However, for thinner tunnel barriers (0.68 nm <tMgO <0.76 nm, region B in Fig. 3)aHðÞbecomes asymmetric and a different effective damping parameter is observed for parallel (P) and antiparallel (AP) configurations, with aAP>aPas will be shown in the following. Figures 4(e)–(h)shows, for tMgO¼0.71 nm, the static field dependence of the p recession frequency f[Fig. 4(e)], the effective damping a[Fig. 4(f)] and the FL, RL, and PL magnetization orientation for static fields along the easy axis [Fig. 4(g)] and along the hard axis [Fig. 4(h)]. We can observe from Fig. 4(g) that except for the reversal process itself, the angular magnetization configurations of the FL, RL, and PL are similar to those with tMgO¼0.88 nm. How- ever, in spite of this similarity, a much larger damping pa- rameter is observed when the fr ee layer magnetization is in the AP state ( aAP/C240.02860.004) than for P configura- tions ( aP/C240.01560.003). This implies that our macrospin model used to derive fandais not sufficient to provide a complete description of the magnetization dynamics of oursystem at such barrier thicknesses (region B). Indeed, although the frequency spectra can be well described by this approximation, a further analysis must be done in orderto understand this asymmetry on a. An intuitive explana- tion for this could be done in terms of the “orange-peel” effect (see Fig. 5). As mentioned above the Ne ´el dipolar coupling results from film roughness favoring the parallel alignment of both FL and RL magnetizations. This rough- ness leads to a large fluctuating coupling field at the inter-face of both ferromagnets which, in turn, will induce a local distribution of the magnetization. For thin MgO bar- riers, this coupling will be strong enough to freeze themagnetic moments of the free layer close to the interface parallel to the reference layer (see Fig. 5). Thus, due to the ferromagnetic nature of the coupling, for a P configurationof the stack, these local moments will be almost parallel to the FL magnetization [Fig. 5(a)], and therefore, the effect of the roughness on magnetization dynamics at firstapproximation should not be important, thereby being tun-nel barrier thickness independent, as observed. However, for AP configurations a strong inhomogeneous magneto- static field is developed in the FL [dashed area in Fig.5(b)]. At such conditions, almos t all magnetic moments of the FL are reversed, being some of them still parallel to the RL. As the way the magnetization relaxes toward equilib-rium is very sensitive to the details of the microscopic interactions, 34this large magnetic inhomogeneity will lead to an inhomogeneous dephasing of the precession andh e n c et oa ni n c r e a s eo f a.A st h eN e ´el coupling increases when t MgO decreases (see Eq. 5) this effective damping asymmetry will be larger for t hinner tunnel barriers, as observed in our experiments. With decreasing tMgO the difference of the damping in the P and AP states increases as seen in Fig. 3. Finally, for tMgO/C200.68 nm (region C in Fig. 3) magnetization preces- sion is only observed at P configurations while at AP config- urations no precessional oscillations are found in the PIMMdata (not shown). For this low barrier thickness, J FLis too strong and of the same order of magnitude as both the anti- ferromagnetic exchange coupling between the pinned layerand the reference layer ( J AF/C24/C0221lJ/m2) as well as the exchange bias coupling between the pinned layer and the pinning layer ( Jex/C24188lJ/m2).12This implies that all ferro- magnetic layers of the MTJ stack are mostly coupled leading in a macrospin model to a scissored state of the SyAF pinned layer when the free layer magnetization is being reversed.Consequently, at these conditions the whole system is involved in precessional magnetization dynamics and PIMM data can no longer be analyzed by Eq. (2). Furthermore, the hysteretic behavior of the FL magnetization reversal is not observed at such conditions (not shown). This behavior imposes therefore, a minimum tunnel barrier thickness ( t MgO >0.68 nm) that must be considered in order to ensure the existence of a bistable state suitable for MRAM applications. Note that magneto transport measurements have been previously performed on patterned MTJ nanopillars fabri- cated from the same MTJ stacks in order to study the thick- ness dependence of the threshold current density for STmagnetization reversal j c.12Such measurements reveal that jc shows similar barrier thickness dependence as the effective damping parameter derived from PIMM measurements. Fur-thermore, they also showed that for thin tunnel barriers (t MgO<0.76 nm) the evolution of the threshold current den- sity for AP-P transitions ( jcAP) is higher than for P-AP transi- tions ( jcP). This difference in jcis too large to be ascribed to a different polarization factor gon Slonczweski’s expres- sion of jc(Ref. 35) due to the different orientation of the free layer magnetization with respect to the reference layer. However, this effect could indeed be a consequence of the different damping parameters for P and AP configurationsas observed in our measurements. These results imply that, owing to the intimate relationship between j Cand the effec- tive damping parameter a,inductive measurements are an excellent tool to investigate the ST-MRAM key parameter jCwithout time consuming lithographic processes for pat- terning MTJ nanopillars and hence derive the optimumtunnel barrier thickness range for efficient ST-MRAM devices. FIG. 5. Schematic picture of the cross-sectional profile of the MTJ stack. The Ne´el coupling induces a FM coupling between the reference layer and the magnetic moments located at the “valleys” of the rough free layer. For thine n o u g hM g Ob a r r i e r st h eN e ´el coupling is so strong that those magnetic moments (small black arrows) tend to be aligned along the reference layer magnetization (white arrow). For pararallel configurations (a) the whole free layer is parallel aligned to the reference layer, whereas for antiparallel configu- rations (b) just a fraction of the free layer is reversed and a region with a large inhomogeneous distribution of the magnetization is developed (dashed area).023906-5 Serrano-Guisan et al. J. Appl. Phys. 110, 023906 (2011) [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: 128.143.199.160 On: Mon, 15 Dec 2014 08:40:12In order to develop optimum ST-MRAM devices with magnetic memory cells below 100 nm, MTJ stacks must show the following features: (i) RA product <5X/C1lm2 (for impedance matching with peripheral circuits), (ii) high enough TMR ratios (in order to generate large output sig- nals and highly enough signal-to-noise ratios36), and (iii) low ST magnetization reversal currents jc,and hence low a. In our MTJ stack under investigation such conditions are fulfilled for a tunnel barrier thickness range between 0.76nm/C20t MgO/C200.85 nm (see Fig. 1(b) and Fig. 3), in agree- ment with magneto transport experiments in patterned nano- pillars fabricated from the same MTJ stacks.12This optimum tunnel barrier thickness range can hence be deter- mined by combination of probe station based magneto transport (i,ii) and inductive measurements (iii) of unpat-terned MTJ stacks and thus in a fast and lithography free characterization process. CONCLUSIONS In summary, we have characterized sputtered CoFeB/ MgO/CoFeB magnetic tunnel junction (MTJ) stacks with different MgO thicknesses (0.61 nm /C20tMgO/C200.96 nm) by Magneto optical Kerr effect and by pulsed induced micro- wave magnetometry and wafer prober current-in-plane tun- neling. From these measurements the precession frequencyspectra f, the free layer anisotropy field K FL, the exchange coupling between the free layer and reference layer JFLand the effective damping parameter aas well as their tunnel bar- rier thickness dependence were derived. Furthermore the electrical parameters RA and TMR were determined. A large ferromagnetic exponential decrease of JFLwith decreasing barrier thickness has been observed which might arise from an impurity-assisted interlayer exchange coupling. By taking into account the thickness dependence of the TMR ratio, ofthe RA product and a, the optimum tunnel barrier thickness range for low current ST-MRAM devices is determined. For our MTJ stack under investigation this optimum thicknessrange is between 0.76 nm /C20t MgO/C200.85 nm. This fast and lithography-free determination of the optimum barrier thick- ness range is in excellent concordance with studies of thecritical current density j cdependence on tMgO derived by conventional magneto transport experiments on individual patterned nanopillars fabricated from the same MTJ stacks.This proves the potential of inductive characterization as a fast and efficient characterization tool for optimization and testing of ST materials. ACKNOWLEDGMENTS We would like to thank J. Langer and B. Ocker from Sin- gulus AG for sample growth and helpful discussions. S.S.G., N.L, and H.W.S. acknowledge funding from European com- munity’s Seventh Framework Programme, EEA-NET Plus,under IMERA-Plus Project-Grant No. 217257 and from EMRP JRP IND-08 which is jointly funded by the EMRP par- ticipating countries within EURAMET and the E.U. 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B 71, 224406 (2005). 17At such conditions the free energy of the system can be written as E¼/C0l/C1MFLH/C1cos/FL/C0u ðÞ /C0 KFLcos2/FL/C0JFL=tFLcos/FL/C0l/C1MRLH /C1cos/RL/C0u ðÞ /C0 KRLcos2/RL/C0JAF=tRLcos/RL/C0l/C1MPLH/C1cos/PL/C0u ðÞ /C0KPLcos2/PL/C0Jex=tPLcos/PL/C0KAFcos2/AFwhere MFL,MRL, and MPL refer to the saturation magnetization of the FL, RL, and PL, KFL,KRL, and KPLrefer to the anisotropy constant of the FL, RL, and PL, JFLis the inter- layer exchange coupling energy between the FL and the RL, JAFis the antiferromagnetic exchange coupling energy between the RL and the PL, Jexis the exchange bias energy between the PL and the antiferromagnetic pinning layer and KAFis the anisotropy constant of the antiferromagnetic pinning layer. 18A. Anguelouch, B. D. Schrag, G. Xiao, Y. Lu, P. L. Trouilloud, R. A.Wanner, W. J. Gallagher, and S. S. P. Parkin, Appl. Phys. Lett. 76, 622 (2000); B. D. Schrag, A. Anguelouch, G. Xiao, P. Trouilloud, Y. Lu, W. J. Gallagher, and S. S. P. 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Jansen and J. S. Moodera, Phys. Rev. B 61, 9047 (2000). 27M. Y. Zhuralev, E. Y. Tsymbal, and A. V. Vedyayev, Phys. Rev. Lett. 94, 026806 (2005).023906-6 Serrano-Guisan et al. J. Appl. Phys. 110, 023906 (2011) [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: 128.143.199.160 On: Mon, 15 Dec 2014 08:40:1228Thus, for impurity energies of 0.2 eV above the EF, a thickness depend- ence of the IEC similar to our data can be obtained for the same barrier thickness range 0.5 nm <tMgO<0.9 nm (see Fig. 3(b) Ref. 27). 29J. Y. Bae, W. C. Lim, H. J. Kim. T. D. Lee, K. W. Kim, and T. W. Kim; J. Appl. Phys. 99, 08T316 (2006); J. C. Read, P. G. Mather, and R. A. Buhrman, Appl. Phys. Lett. 90, 132503 (2007); J. J. Cha, J. C. Read, R. A. Buhrman, and D. A. Muller, Appl. Phys. Lett. 91, 062516 (2007); C. Y. You, T. Ohkubo, Y. K. Takahashi, and K. Hono, J. Appl. Phys. 104, 033517 (2008); S. Pinitsoontorn, A. Cerezo, A. K. Petford-Long, D. Mauri, L. Folks, and M. J. Carey, Appl. Phys. Lett. 93, 071901 (2008). 30J. J. Cha, J. C. Read, W. F. Egelhoff Jr., P. Y. Huang, H. W. Tseng, Y. Li, R. A. Buhrman, and D. A. Muller, Appl. Phys. Lett. 95, 032506 (2009). 31J. C. Read, J. J. Cha, W. F. Egelhoff Jr., H. W. Tseng, P. Y. Huang, Y. Li, D. A. Muller, and R. A. Buhrman, Appl. Phys. Lett. 94, 112504 (2009). 32H. X. Yang, M. Chshiev, A. Kalitsov, A. Shuhl, and W. H. Butler, Appl. Phys. Lett. 96, 262509 (2010).33S. Serrano-Guisan and H. W. Schumacher, “Enhanced effective Gilbert damping at the anisotropy field in CoFeB thin films,” AD-09, 52nd MMM conference 2007, Tampa (Florida). 34M. Czapkiewicz, T. Stobiecki, and S. Van Dijken, Phys. Rev. B 77, 024416 (2008). 35From Ref. 8 the threshold current density jccan be expressed as: jc¼/C0 e=/C22hðÞ /C1 d/C1l0M2 s=g/C1a/C11þ2Hk=Ms ðÞ where Msis the saturation magnetization of the free layer, d refers to the free layer thickness, Hk is the anisotropy field and gis the polarization factor, which is angular dependent, and can be expressed as: g¼P=2/C11þP2cosh ðÞ being h the offset angle between the free layer magnetization and the reference layer and P the spin polarization of the tunnel current. By taking into account these expressions, Skowronski et al ., (Ref. 12) cannot explain the large difference of j cbetween parallel and antiparallel configurations. 36N. Smith and P. Arnett, Appl. Phys. Lett. 78, 1448 (2001).023906-7 Serrano-Guisan et al. J. Appl. Phys. 110, 023906 (2011) [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: 128.143.199.160 On: Mon, 15 Dec 2014 08:40:12

1.5098866.pdf

Appl. Phys. Lett. 115, 092407 (2019); https://doi.org/10.1063/1.5098866 115, 092407 © 2019 Author(s).Observation of the magnetization metastable state in a perpendicularly magnetized nanopillar with asymmetric potential landscape Cite as: Appl. Phys. Lett. 115, 092407 (2019); https://doi.org/10.1063/1.5098866 Submitted: 04 April 2019 . Accepted: 11 August 2019 . Published Online: 27 August 2019 Shuichi Iwakiri , Satoshi Sugimoto , Yasuhiro Niimi , Kensuke Kobayashi , Yukiko K. Takahashi , and Shinya Kasai ARTICLES YOU MAY BE INTERESTED IN Direct observation of magneto-Peltier effect in current-in-plane giant magnetoresistive spin valve Applied Physics Letters 115, 092406 (2019); https://doi.org/10.1063/1.5120569 Anomalous spin Hall angle of a metallic ferromagnet determined by a multiterminal spin injection/detection device Applied Physics Letters 115, 092404 (2019); https://doi.org/10.1063/1.5101032 Enhancement of spin–orbit torque via interfacial hydrogen and oxygen ion manipulation Applied Physics Letters 115, 092402 (2019); https://doi.org/10.1063/1.5110206Observation of the magnetization metastable state in a perpendicularly magnetized nanopillar with asymmetric potential landscape Cite as: Appl. Phys. Lett. 115, 092407 (2019); doi: 10.1063/1.5098866 Submitted: 4 April 2019 .Accepted: 11 August 2019 . Published Online: 27 August 2019 Shuichi Iwakiri,1,2Satoshi Sugimoto,3 Yasuhiro Niimi,1,2 Kensuke Kobayashi,1 Yukiko K. Takahashi,3 and Shinya Kasai3,4,a) AFFILIATIONS 1Department of Physics, Graduate School of Science, Osaka University, Osaka 560-0043, Japan 2Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan 3Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science (NIMS), 1-2-1 Sengen,Tsukuba 305-0047, Japan 4JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan a)KASAI.Shinya@nims.go.jp ABSTRACT The effect of spin torque on magnetization is investigated in an MgO/CoFeB/W perpendicularly magnetized nanopillar with an asymmetric bistable potential landscape. While conventional pulse-current-induced magnetization switching has been implemented, a metastableswitched state is observed under direct current (DC). The mean dwell time of the metastable state depends not only on the DC amplitudebut also on the current direction. These results suggest that the damping spin Hall torque suppresses the magnetization fluctuation andrealizes the metastable state. Published under license by AIP Publishing. https://doi.org/10.1063/1.5098866 Electronic manipulation of the magnetization is one of the central themes in spintronics for realizing nonvolatile magnetic memories andspin logics. 1,2In particular, the concept that spin current exerts “spin torque” on magnetization,3,4has led to the realization of spin current- induced magnetization switching.5–7Recently, several effects due to spin–orbit interaction, e.g., the spin Hall effect8–15or the Rashba- Edelstein effect,16–19are proposed as spin torque sources besides the conventional spin transfer torque.20–23 An externally excited magnetization state such as the spin torque induced switched state is not necessarily stable mainly due to thermal activation, or thermal fluctuation of magnetization.24–26 The problem of the stability exists commonly in nanoscale magne- tization manipulation regardless of the excitation mechanism ofthe magnetization. The magnetization dynamics in the thermallyactivated regime, which is the main topic of this letter, can be char- acterized by the potential landscape of the magnetization state (typically a double-minimum landscape whose local minima corre-spond to the easy axis orientation). For example, if the potential isapproximately symmetric, the thermal fluctuation activates bothstates leading to the random switching of the magnetization across the potential barrier with a short time scale. 24–26 On the other hand, when the asymmetry of the potential becomes larger, the state with lower energy (initial state) becomesmore stable than the other state (switched state). Thus, even if theswitched state is temporarily realized by the spin torque, it would ther-mally “switch back” to the initial state when the potential barrierbecomes lower. In this case, the switched state can be either unstableor metastable depending on the relative magnitude of the thermal fluc-tuation and the potential barrier. Here, as the spin torque decreasesthe energy of the magnetization, it may be possible to modulate thethermal stability of the switched state in an asymmetric potential. In this letter, we report the magnetization switching in MgO/ CoFeB/W perpendicularly magnetized nanopillars whose potentiallandscapes have an asymmetric double minimum. In addition to theconventional pulse-induced magnetization switching through the spinHall-induced spin torque (SHT), we found a metastable switched statethat temporarily exists only under the direct current (DC) andswitches back to the initial state. In order to characterize the role of the Appl. Phys. Lett. 115, 092407 (2019); doi: 10.1063/1.5098866 115, 092407-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplspin torque, we measured the mean dwell time (MDT) of the metasta- ble state, or the average elapsed time before switching back, which canbe prolonged by spin injection up to a few tens of second. These results suggest that the damping SHT can suppress magnetization fluctuation at the metastable state, enabling the control of magnetization on awide time scale. Figure 1(a) shows a schematic of our sample and the coordinate system. A multilayered stack consisting of Ta (3 nm)/MgO (1.4 nm)/ Co 20Fe60B20(0.98 nm)/W (3 nm) was deposited on a thermally oxi- dized Si substrate by magnetron sputtering. The film stack was then annealed at 300/C14C for 30 min to improve the interface-induced per- pendicular magnetic anisotropy. The saturation magnetization Msand the anisotropy field HKare evaluated with a vibrating sample magne- tometer: Ms¼1200 emu/cm3and HK¼5 kOe. The top Ta/MgO/ CoFeB layer was patterned into a cylindrical element (400 nm in diam-eter) to suppress the in-plane demagnetization field, and the bottomW layer was etched into the Hall bar with 8 lm in width for spin injec- tion and Hall measurement. The Hall voltage V Halong the y-axis was measured with a charge current Icalong the x-axis under the magnetic field H¼ðHx;Hy;HzÞon magnetization ~m¼ðmx;my;mzÞ.I nt h e present experiment, we used two different probing systems to apply the magnetic field; one is for the in-plane ( Hin/C0x apporHin/C0y app) measure- ments and the other is for the out-of-plane ( Hout app) ones. Note that a small but finite z-component of the stray magnetic field was inelucta- bly applied only for the in-plane configuration because the sample position was slightly off-center in the electromagnet. This induces a positive and negative effective magnetic field along the z-direction forHin/C0x appandHin/C0y app, respectively. We define the effective tilted angle with respect to the x-yplane as h[seeFig. 1(a) ]. The charge current Icis converted to the spin current Iswith spin polarization ~r¼ð0;/C0s;0Þ through the spin Hall effect, which yields SHT on the magnetization. All measurements were performed at room-temperature. Figure 1(b) illustrates the potential landscape in the z-component of the magnetization under a hard-axis field Hxwith a small easy-axis perturbation Hz, which makes the potential landscape asymmetric. We denote the lower (higher) energy state as the initial (switched) state. The potential barrier is lowered with increasing Hx.N o t et h a t the potential barrier differs depending on the state due to the asymme- try induced by a finite Hz. The magnetization energy can be modu- lated by the SHT; the antidamping torque assists the thermalfluctuation, having the energy scale of /C24k BTand being important near the magnetization switching condition.27,28On the other hand, we focus on the effect of the damping torque as a way to dissipate the magnetization energy. We first show the Hout appdependence of the Hall resistance RH ¼VH=IcinFig. 1(c) . The measurement current Ic¼100lA( c u r - rent density 3 :1/C2109A=m2) is set to be sufficiently small so that neither SHT nor Oersted field, estimated as the order of 10 Oe,29 disturb the magnetization direction. The rectangular-sharp hyster-esis due to the anomalous Hall effect clearly indicates the perpen- dicular magnetic anisotropy. The switching field, H zc, is about 0.4 kOe. The Hin/C0x app andHin/C0y appdependence of RHis also plotted in Figs. 1(d) and1(e). If the magnetic field coincided with the hard axis, no magnetization switching would have been triggered and the magneti- zation would have gradually tilted toward the hard axis ( x-axis) with increasing Hin/C0x app orHin/C0y app. However, there are jumps at Hin/C0x app ffi61:7k O e ð/C176HxcÞ, indicating that the alignment is not perfect, i.e., a finite out-of-plane field component is also applied under Hin/C0x app. In addition, the switching direction for the ysweep is opposite to that for the xsweep. This can be explained the fact that Hin/C0x appandHin/C0y app include a positive and negative out of plane component, respectively. T h ee f f e c t i v et i l t i n ga n g l e hcan be evaluated by taking the ratio of two switching fields tan h¼Hz=Hx/C24Hzc=Hxc;which results in h/C2410 deg. Hereafter, RHis normalized by zero field resistance R0 for simplicity. The state with RH=R0¼/C01(þ1) corresponds to the “initial (switched) state” in Fig. 1(b) . To investigate the magnetization response to the current, we measured the injection current dependence of the Hall-resistanceR H=R0using two sequences for pulse current and DC, as shown in Fig. 2(a) . In the pulse current sequence, a pulse current Ipulseis first injected for 100 ms, and then a small measurement current Imeasure ¼0.1 mA is used to detect RH. The first pulse current is swept in the range of jIpulsej<3 mA [see the right panel of Fig. 2(a) ]. In the DC sequence, on the other hand, RHis continuously measured with sweeping DC ( jIDCj<3 mA) without any temporal modulations. In this case, the injected current is used not only as the measurement cur- rent for RHbut also as the source of the spin current into the CoFeB layer. Figure 2(b) shows the Ipulsedependence of RHwith an in-plane magnetic field of Hin/C0x app¼0.9, 1.4, and 1.9 kOe. At Hin/C0x app¼1.4 kOe, the magnetization switching is observed with increasing Ipulse up to 2.4 mA (current density 7.4 /C21010A/m2), and then switching back FIG. 1. (a) Schematic of the nanopillar sample. (b) The potential landscape of the magnetization. The in-plane field Hin/C0x app;Hin/C0y app or the out-of-plane field Hout appis applied. The spin Hall effect in W converts the charge current ICinto the spin current Is:The effect of SHT in each configuration is shown. (c) The out-of-plane field ( Hout app,h¼90 deg) dependence of the Hall resistance measured at IC¼100lA. (d) The in-plane field in the x-direction ( Hin/C0x app,h¼/C0 10 deg) and the y-direction ( Hin/C0y app,o rh¼10 deg) dependence of Hall resistance measured at Ix¼100lA.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 115, 092407 (2019); doi: 10.1063/1.5098866 115, 092407-2 Published under license by AIP Publishingoccurs at Ipulse¼/C01.0 mA. In contrast, no switching can be found at Hin/C0x app¼1:9k O e >Hxc(/C241.7 kOe), indicating that the switched state is not stable at high fields in this condition. The switching characteristics are summarized in Fig. 2(c) .O n l yt h e positive Ipulseregion is shown for simplicity. Ipulseis swept from 0 mA to 3m Au n d e r jHin/C0x appj/C202:4 kOe. The magnetization is initialized by set- ting Hin/C0x app¼2:4 kOe before every sequence to guarantee the initial state. Magnetization switching is visualized as a steep color changein the regions of I pulse/H114072:0m A a n d 1 :0k O e/H11351Hin/C0x app/H113511:7k O e . Regarding the positive field region, we observe two thresholds of Hin/C0x app, Hxufor the upper threshold and Hxlfor the lower one. Hxucorresponds toHxcinFig. 1(d) . We define the magnetization state in this region as the “bistable” state. The switching behavior is different for the DC sequence, as depicted in Fig. 2(d) . While the magnetization switching is also observed at Hin/C0x app¼1.4 kOe in the previous pulse current sequence, such a switched state remains over the upper threshold Hin/C0x app>Hxu as can be seen for Hin/C0x app¼1.9 kOe and IDC¼1.2 mA. Moreover, the switching back occurs at a small DC ( IDC¼0.5 mA), indicating that the switched state is the “metastable state” sustained only under a largeDC. The switching characteristic in Fig. 2(e) shows that the metastablestate exists under a wide range of conditions for H in/C0x app>Hxu.T h i si s the central experimental finding of this letter. The mechanism of the metastable state may be phenomenologi- cally understood as follows. First, the spin current gives the antidamp-ing/damping SHT to the initial/switched state, increasing/dissipatingthe energy of the magnetization. Below the lower threshold ( H in/C0x app <Hxl), the energy of the magnetization is smaller than the potential barrier, and the initial state remains stable even under thermal fluctua-tion. In the intermediate field ( H xl<Hin/C0x app<Hxu), the energy injec- tion with the thermal fluctuation overcomes the potential barrier andthe magnetization switches. However, above the upper threshold(H in/C0x app>Hxu), the potential barrier for the switched state becomes comparable to the thermal fluctuation. Therefore, it switches back tothe initial state in the absence of the current. In contrast, as the damping SHT dissipates energy, the magneti- zation fluctuation at the switched state is suppressed, lowering theprobability of the magnetization to overcome the barrier. Therefore, asshown in Fig. 2(e) , the metastable state can be found only in the DC sequence. To discuss the switching back process from the metastable state, we measure the mean dwell time (MDT), i.e., the average elapsed timebefore its switching back. Figure 3(a) shows the measurement sequence, and Fig. 3(b) shows a typical time evolution of R H=R0.T h e pulse current is fixed at Ipulse¼3 mA to initialize the magnetization to the switched state. The elapsed time tstarts after the end of the pulse and is measured until its switching back under various DC Imeasure val- ues. Figure 3(c) shows the Imeasure dependence of MDT for Hin/C0x app ¼1.62–1.66 kOe. The MDT increases up to a few tens of seconds under a positive Imeasure which gives the damping SHT to the magnetization. FIG. 2. (a) Left: Time sequence of the single trial for pulse hysteresis. Current is first applied Ic¼Ipulse for 100 ms, then immediately decreased to Ic¼Imeasure . Right: Current sequence of the DC and pulse hysteresis sequence. In the DCsequence, I cis swept continuously while it is applied only 100 ms in the pulse hys- teresis sequence. (b), (d) Ipulse (IDC) dependence of RH. The black arrow is the sweep direction. RHis normalized by the Hall resistance at zero field. (c) and (e) Color plot of Ipulse (IDC) and Hin/C0x app dependence of RH. The annotation indicates the magnetization state: “BS” is bistable and “MS” means metastable. FIG. 3. (a) Time sequence of the MDT measurement. Pulse current is applied Ipulse¼3 mA for 100 ms, then immediately decreased to Ic¼Imeasure . (b)The typi- cal time development at Imeasure ¼100lA and Hx¼1:63 kOe. (c) Imeasure depen- dence of the mean dwell time (MDT) under different Hx. The solid lines are the theoretical expectation with Eq. (1). (d) The in-plane field Hin/C0x app dependency of the activation energy Ea.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 115, 092407 (2019); doi: 10.1063/1.5098866 115, 092407-3 Published under license by AIP PublishingThe magnetization behavior in the vicinity of the critical con- dition can be analyzed by using the LLG equation with damping/ antidamping spin torque terms. When the current is sufficiently smaller than the threshold of the switching, the MDT is expected to follow:27 MDT /eEa kBT1þIc Ith/C0/C1b ; (1) where Ea¼D0/C2fðHxÞ¼MsVH K 2/C2fðHxÞwith fH xðÞ ad e c r e a s i n g function such as/C0 1/C0Hin/C0x app HK/C1c,a n d Ith¼2aeMsV /C22hheffHK.A s s u m i n gt h a tt h e magnetization is in the single-domain, we set b¼2 which corre- sponds to the macrospin approximation. Here, eis the elementary charge, Vis the sample volume, /C22his the Planck constant, kBis the Boltzmann constant, Tis the temperature (293 K), and heffis the effec- tive spin Hall angle which is normalized to the area ratio between the spin injecting the W electrode (8 lm/C2400 nm) and the cross section of the pillar ( p/C24002nm2)a s s u m i n gt h a tt h eb a r ea m p l i t u d eo ft h e spin Hall angle is 0.3.22 T h es o l i dc u r v ei n Fig. 3(c) is Eq. (1)whose parameters are obtained from the fitting by setting EaandIthfree parameters.27,30–33 The prefactor which corresponds to the attempt frequency is fixed to 5/C210/C07.26,30As shown in Fig. 3(d) , the field dependence of Eaesti- mated by the fitting is reasonable considering that the potential barrier is decreased by the in-plane field. In addition, Ithis estimated to be approximately 2.4 mA, which is consistent with the result in Fig. 2(c) . By using the above values, we estimated the Gilbert damping constant a/C240:06. The above discussions indicate that the damping/anti- damping SHT mechanism can explain the present experimentalresults, e.g., the MDT asymmetry to the current sign and the H in/C0x app dependence of Ea, at least qualitatively. The MDT proposed here characterizes the transient behavior of the magnetization under thermal fluctuation comparable to the poten- tial energy barrier. Note that the model adopted here describes the magnetization behavior near the critical point, but not the magnetiza-tion state or configuration itself which should be more directly investi- gated by LLG simulations. It might be another method to estimate the effect of spin torque, while it is usually evaluated by the perturbative technique such as second-harmonic measurements 34or spin-torque induced ferromagnetic resonance.15,22 Finally, we mention two factors which might affect the metastable state. First, the field-like SHT will also be applied to the magnetization under a finite current.35,36However, in our experimental conditions, it works only along the y-axis and always destabilizes the metastable state, which cannot explain the stabilization. Second, the Joule heating by the current injection to the W electrode could enhance the thermal fluctuation.33If the Joule heating were not negligible, the thermal fluc- tuation would become greater regardless of the injected current direc-tion and shorten the MDT. However, according to Fig. 3(c) ,t h eM D T becomes longer only with the positive current. In addition, as the pulse length used in our experiment is 100 ms, which is sufficiently longer than the time for magnetization switching, the effect of Joule heating is not the dominant factor. In summary, we have investigated the effect of spin injection on the magnetization of a perpendicularly magnetized CoFeB nanopillar with an asymmetric potential landscape. 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1.331629.pdf

Dynamic properties of charged walls in ion implanted garnets M. H. Kryder and B. E. Argyle Citation: Journal of Applied Physics 53, 1664 (1982); doi: 10.1063/1.331629 View online: http://dx.doi.org/10.1063/1.331629 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/53/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Abstract: Dynamic properties of charged walls in ion implanted garnets J. Appl. Phys. 52, 2370 (1981); 10.1063/1.328935 Crystalline and magnetic properties of an ionimplanted layer in bubble garnet films J. Appl. Phys. 49, 5816 (1978); 10.1063/1.324597 Magnetic bubbles, walls, and fine structure in ionimplanted garnets J. Appl. Phys. 48, 2069 (1977); 10.1063/1.323919 Wall States in IonImplanted Garnet Films AIP Conf. Proc. 34, 138 (1976); 10.1063/1.2946039 Annealing of Magnetic Properties of Ion Implanted Garnet Epitaxial Films AIP Conf. Proc. 10, 334 (1973); 10.1063/1.2946912 [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: 128.123.44.23 On: Sun, 21 Dec 2014 03:27:06Dynamic properties of charged walls in ion implanted garnets M. H. Kryder a) and B. E. Argyle IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598 (Received 30 September 1981; accepted for publication 24 November 1981) The dynamic behavior of the charged wall in the ion implanted layer of contiguous disk bubble devices is determined for the first time without the disturbing influence of a bubble domain being coupled to the charged wall. The charged wall was stabilized next to a nonimplanted region by an applied in-plane field Hx and was oscillated alongy by an rffield hy sin illt. Oscillations in Faraday magneto-optic contrast signals proportional to the wall oscillations were detected with a high-speed photomultiplier (PMT) when a focused laser beam was obliquely incident on the region of the charged wall. The drive frequency ill was provided by an rf tracking generator which was phase locked to a spectrum analyzer set to detect the fundamental amplitude response of the PMT signals. Our measurements were made on the doped yttrium iron garnet (YIG) layers Gd, Tm, Ga:YIG and Eu, Tm, Ga: YIG typically used for driving 5-and 1-}.lm bubbles, respectively. The charged wall amplitude rolloff with frequency interpreted in terms of an overdamped linear oscillator reveals a relaxation frequency near 10 MHz when Hx = 30 Oe, for example, in contrast to -0.5 MHz when the bubble is attached. The linear mobility of the charged wall }.lew = 6600 cm/sec Oe determined in Gd, Tm, Ga: YIG from the relaxation frequency contrasts with an effective mobility Pelf = 250 cm/sec Oe when the bubble is attached. These results confirm our earlier assumption [B. E. Argyle et al., IEEE Trans. MAG- 14, 593 (1978)] that the damping which affects the upper limiting frequency of operation in this contiguous disc material comes mainly from the bubble domain and almost none (i.e., :S 5%) from the charged wall. In addition, no velocity saturation effects are observed in the isolated charged wall at least up to 12.8 MHz. The maximum velocity 28 000 cm/sec inferred from the maximum harmonic amplitude at this frequency is at least 25 times the saturation velocity of an isolated bubble, e.g., 1100 cm/sec, in the nonimplanted film. A simple theoretical model taking into account restoring torques on the wall magnetization due to in-plane anisotropy and wall demagnetization predicts velocity saturation will occur at yet several times larger velocity than the maximum response we can observe in the present experiment. PACS numbers: 75.70.Kw, 75.70. -i, 76.90. + d, 85.70.Ge INTRODUCTION High mobility in velocity of charged walls in the ion implanted layer of garnet thin films is critical to obtaining good performance at high frequencies in contiguous disk de vices. Yet there has been no experimental determination of fundamental dynamic parameters or behavior for an isolated charged wall. In earlier related work 1 the dynamic response of a charged wall and bubble coupled together magnetostati cally was investigated by utilizing the dominant Faraday magneto-optic contrast of the bubble. These experiments in dicated indirectly that the charged wall mobility was high in comparison to that when the bubble is attached, but no de termination of charged wall mobility was made. Subsequent measurements2 of the dynamic properties of bubble domains in ion-implanted contiguous disk devices supported the con clusion that the damping of the bubble domain motion was large compared to the damping of the charged wall motion. Again, no measurements were made on the charged wall alone. In this paper, measurements of the mobility of charged walls with no bubble domains coupled to them are obtained even though the magneto-optical contrast from the charged wall is more than an order of magnitude weaker than that from the bubble. These measurements clearly show that the charged walls have more than one order of magni tude higher mobility than bubble domains. Further, combin ing the ferromagnetic resonanace (FMR) values of gyromag netic ratio and Gilbert damping parameter with the charged wall mobility provides a first though indirect determination of the width of the charged wall. "Present address: Department of Electrical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213. EXPERIMENTAL SAMPLES Two samples were examined in this work. Sample A, which had been previously used 1 in measurements of the mobility of a charged wall with a bubble domain coupled to it, was ofGdo9 Tm1.l Y Gao. 7 Fe4.3 012 composition, was 4.5 pm thick, and had 41TM = 195 G for nominally 5-pm bub bles. From the Curie temperature Tc = 195 ·C we estimate the exchange stiffness parameter A = 2.69 X 10 -7 erg/ cm using the procedure in Ref. 3. From FMR measurements, the values r = 1.27 X 107 Oe-1 sec-I, a = 0.044, Kl = -2100 ergs/cm3, and Ku = 8800 ergs/cm3 were ob tained for the gyromagnetic ratio, the Gilbert damping pa rameter, the crystal anisotropy and uniaxial anisotopy coef ficients, respectively. A bubble mobility of 1150 cm/sec Oe and a saturation velocity of 1100 cm/sec may be inferred from previous measurements on a similar unimplanted film 1664 J. Appl. Phys. 53(3), March 1982 0021-8979/82/031664-07$02.40 © 1982 American Institute of Physics 1664 [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: 128.123.44.23 On: Sun, 21 Dec 2014 03:27:06(number 4 in Ref. 4). This sample A was masked and im planted with lOO-keV protons at 3 X 1016/cm2 and annealed at 300·C in N2 for 1/2 h. The (unimplanted, implanted) re gions exhibited bubble collapse at (87 Oe, 103 Oe) after an nealing. The implanted layer has thickness hi = 0.6,um (Ref. 5, Fig. 22) and has an in-plane anistropy field ratio Q = 1.5 after implantation. Sample B was a double layer film of a type frequently used6 for contiguous disk devices employing l-,um diam bub bles. The unimplanted storage layer was of EUo.7 Tmo.s YI.8Gao.63 Fe4.37 012 composition, was 1.04,um thick, and had 41TM = 600G. The implanted driving layer was ofGd Y Tm Gao.4 Fe4.6 012 composition, was 0.39,um thick, and had 41T M:::::: 600 G. It was masked and implanted with 150-keVHe+ at4X1015/cm2. EXPERIMENTAL APPARATUS AND SIGNAL ANALYSIS To measure dynamic properties of charged walls with out bubble domains coupled to them required a means to detect the charged walls alone. Although the Bitter tech nique has previously been used7 for static observation of charged walls, it is not suitable for dynamic measurements because it takes a few seconds for the magnetic particles to decorate the charged wall after it has moved. Here, we chose to use magneto-optic techniques which include useful fea tures of two previous setups: the magneto-optic sensitivity to motion of a domain wall separating domains having in-plane magnetization as in the technique of Fernandex and Kryder,8 and the capability to image and control a photo multiplier (PMT) aperture, to suppress PMT noise, and to plot swept frequency response as in the setup of Argyle et al. 9 for measuring Bloch wall oscillations. Figure I illustrates the apparatus, Fig. 2 the aperture imaging relative to the charged wall and un implanted hole, and Fig. 3 an aid for analysis of signal sensitivity and its harmonic content relative to the magnetization distribution in the charged wall region. The square shape of the unim planted hole was chosen because it was found to provide a more nearly linear reponse (yo-hy) than a circular hole, for R.F. " Prism Polarizer z Sample t ~~~~-----'~~F=~~ x~y Dichroic_ Analyzer Half -Silvered ./ Mirror FIG. I. Block diagram of the experimental apparatus used to measure the dynamic properties of charged walls. 1665 J. Appl. Phys., Vol. 53, No.3, March 1982 Negative Charged Wall \ M/ ION 1M PLANTED hy Sin cut ION IMPLANTED FIG. 2. A diagram of the experiment performed with H = 20,um. The PMT received its signal through the measuring aperture with L = 14 ,urn. The relative orientation of easy stripout directions, aperture, and square were maintained for all measurements. The sign of H, determines the sign and stability of the charged wall. example. In general, the Faraday rotation of the plane of polar ization oflight is proportional to M.k, the component ofM along the light propagation vector k. Thus, for an obliquely incident laser as shown in Fig. 1 both the in-plane compo nents (Mx' My) and the perpendicular component (Mz) can cause rotation of the laser polarization. One in-plane compo nent, e.g., Mx' may be removed from the signal by choosing the direction oflaser tilt, e.g., in the yz plane as in Figs. 1 and 2. Rotation due to changes LiM caused by motion of the domain wall produces changes in light intensity when a dich roic sheet polarizer is used as an analyzer. The light beam is also split by a half-silvered mirror (Fig. 1) so that a portion going to a silicon-intensified-target T.V. camera provides for visual display on a video monitor, while the remaining portion is detected with a high-speed (2.5-ns rise time) photomultiplier (PMT). 10) Y~ •• , ~ .:5 ·T· .. · ~M -L/2 --...,----- ---, \ , , , ~ -L~~ __ ~ ____ ~' __ -L_ ;--L/2 Ib) , \ L/2 '---- Ie) FIG. 3. Illustration of drive field cycle (a) and (b) approximate component magnetization distributions of charged wall at the edge of the unimplanted hole for phases tN (N = 0,1,2,3). The signal response that would be detected using small aperture (..:1x,..:1y<H) at y = ° for each component is shown in (c). (See text). M. H. Kryder and B. E. Argyle 1665 [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: 128.123.44.23 On: Sun, 21 Dec 2014 03:27:06Additional optics (not shown in Fig. 1) place an aper ture in front of the PMT at a conjugate image plane with respect to the sample film plane, so that images of both the sample and the aperture are superimposed in the viewing optics. Precise optical alignment and mechanical control of this aperture in juxtaposition with the sample image limits the area oflight rays entering the PMT. Figure 2 illustrates a charged wall at its position of equi librium, centered with respect to the edge of an unimplanted square hole and oriented along a [211] crystallographic axis of a (111) garnet film plane. The addition of a dc field H x along this axis helps to maintain this wall equilibrium orien tation, eliminate stray walls, and establish field conditions similar to those imposed in operation of a charged wall prop agation device. The drive field hy sin wt applied orthogonal to Hx causes the charged wall to oscillate about the equilibrium positiony = O. A tracking generator (TR), a power amplifier not shown, and leveling amplifier (also not shown) provide constant rf current to the high-frequency in-plane field coils. The TR frequency was controlled and swept while being phase locked to a spectrum analyzer (SA) set to detect only the fundamental component of PMT signal. The output of the SA was recorded on an x-y recorder. Figure 3 illustrates the means by which signals due to domain wall position oscillations induced by the rf drive [Fig. 3(a)) may be detected. The solid lines in Fig. 3(b) indi cate the profiles of magnetization components Mx (y), My (y), and Mz (y) in the cross section of a charged wall at the edge of the square at t = O. The same profiles occur dynamically at the crossing times t = t2, t4, ... etc., of the drive cycle. The dashed and dotted curves illustrate displaced M distribu tions at the times of peak wall displacements t = t. and t3, respectively. Figure 3(c) illustrates the corresponding changes LlM versus time, capable of producing magneto-op tic signals at the PMT when a small aperture (Llx, Lly<.,H) is located at the equilibrium position y = O. The general conclusion evident from Fig. 3 is that sig nals resulting from convolution of the symmetric distribu tions Mx (y) and Mz (y) with an aperture shape that is symmet ric abouty = 0 can contain only even harmonics of the drive frequency, whereas the odd-symmetry component My(y) produces only odd harmonics including the fundamental. Thus, signals detected with the SA at the drive frequency using a symmetric aperture as in Fig. 2 can arise only from the My (y) of the charged wall. This signal can also be strengthened by extending the length of the aperture, i.e., Lly---+L as in Fig. 2. We also are careful to maintain symmetry with respect to y = 0 for both the aperture and the laser gaussian intensity distribution. Signals from the other components Mz(y) and Mx(y) having even distributions Mz (y) = Mz( -y) and Mx (y) = Mx ( -y) can also be obtained at the drive frequency by using a half aperture lying totally on one side of the origin (y < 0 or y> 0). These signals are, however, weak due to smaller am plitudes LlMz and LlMx relative to the background domain components and due to the small tilt angle of our laser. How ever, using the half aperture with the bubble attached to the charged wall provides strong signals because of the bubbles 1666 J. Appl. Phys., Vol. 53, No.3, March 1982 even Mz distribution opposing the Mz of the background domains and because of the larger net magneto-optic rota tion -M.k in the (thicker) bubble film. Thus, we can investi gate the bubble motion as well as the charged wall motion when they are coupled together by changing aperture sym metry with respect to y = O. An assumption inherent to this signal analysis is that magnetization symmetries of the bub ble and charged wall when isolated are maintained as well as when they are coupled together and move dynamically. The laser wavelength was chosen to give optimum sig nal-to-noise ratio with the sample being examined. As Mac Donald and BecklO showed, both the Faraday rotation and absorption increase with decreasing wavelength. Hence, in thin samples, shorter wavelength light (blue) provided better signal-to-noise ratio because the Faraday magneto-optic ro tation increases with decreasing wavelength. However, in thicker samples, it was necessary to use somewhat longer wavelengths (blue-green) to reduce absorption of the light and possible heating in the magnetic film. EXPERIMENTAL DATA In sample A, a bias field Hz = 120 Oe within the range of bubble stability was applied after collapsing all bubbles. The charged wall was stabilized at the center position (y = 0) of the unimplanted square hole, by applying the dc field Hx. An oscillating y-directed field caused the charged wall to oscillate along the edge of the square. PMT signals obtained using a symmetric aperture (Fig. 2) were processed by the SA. When plotted versus rf drive field amplitude hy, the 8 ~ f : 1 MHz ~ 6 ... " .. a. I E ::i.. 4 0 >- RF Drive Field hy (Oe-peakl FIG. 4. Charged wall oscillation amplitudey" in Sample A as a function of drive field h, for three values of stabilizing field, taken at a frequency of 1.16 MHz. M. H. Kryder and B. E. Argyle 1666 [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: 128.123.44.23 On: Sun, 21 Dec 2014 03:27:061.5 f = 12.8 MHz :; 1.0 c .. 0- I 5 E - :t 4 0 >-0.5 3 2 5 6 RF Drive Field hy(Oe-peak) FIG. 5. Charged wall oscillation amplitude Yo in Sample A as a function of drive field hy for two values of stabilizing field, taken at a frequency of 12.8 MHz. Scale change for Yo was due to change in full scale sensitivity of the spectrum analyzer. results at two frequencies, 1.16 MHz in Fig. 4 and 12.8 MHz in Fig. 5, confirm the linear response behavior above a small threshold found earlier' at lower frequencies when a bubble is attached and streak photographs were taken for measur ing the bubble displacements. Calibration for actual displacement amplitudes Yo of these oscillations (ordinates in Figs. 4-6) was performed by first reducing the aperture to a known length and noting the rf amplitude where the signal saturated. The inverse slope for the full aperture data at low frequency in Fig. 4 is there fore the restoring force coefficient. It depends on the strength of the field Hx as reported in Table I and plotted versus Hx in Fig. 11. 5 4 .¥ ! 3 I E ::l o >-2 o Hx 100e hy = 3.5 Oe-peak 5 10 15 20 25 F req u en cy (M Hz) FIG. 6. Charged wall oscillation amplitude in Sample A as a function of frequency for four values of stabilizing field H, with h, = 3.5 Oe peak. 1667 J. Appl. Phys., Vol. 53, No.3, March 1982 ~ c: :::> CI> hy = 3.5 Oe-peak > -~ CI> a: CI> "0 ::::J -Q. E <f 0 c: 0- .if) 0 5 10 15 20 25 Frequency (M Hz) FIG. 7. Charged wall oscillation amplitude in sample B as a function of frequency for three values of stabilizing field H, with hv = 3.5 Oe peak. Typical recorded outputs from the spectrum analyzer versus frequency are shown in Figs. 6 and 7 for samples A and B, respectively. The oscillation amplitude is observed to decrease with increasing frequency and increasing stabiliz ing field Hx' This general behavior is expected if one assumes that the charged wall behaves like an over damped harmonic oscillator responding to the driver field hy sin wt causing the wall to oscillate, while the field Hx' the magnetocrystalline anisotropy, and the anisotropic magnetrostriction can pro vide the balance of torques to stabilize the wall at y = O. As the frequency of hy is increased, the wall velocity and hence the viscous damping force increases, causing the charged wall oscillations to become smaller. The crystalline orientation of the sample film relative to the stabilizing field Hx has a strong influence on the oscilla tion amplitUde. ' The data in Figs. 4-7 were taken with the field Hx applied parallel to a hard direction of magnetization which, as Lin et al.7 have shown, is the crystallographically preferred, stable orientation for the charged wall. Thus, in addition to the restoring force due to the applied field there is a restoring force due to crystalline anisotropy acting on the charged wall. In Figs. 8 and 9 the field Hx is applied parallel to an easy direction of magnetization which is an unstable position for the charged wall. In Fig. 8 it is observed that with small stabilizing fields Hx, the amplitUde of the oscillations fall off more rapidly with increasing frequency than in the case (Fig. 6) with the applied field in a preferred direction for the charged wall. However, with a large (100 Oe) stabilizing field, the amplitude falls off much more slowly with increas ing frequency. In Fig. 9 it is observed that the dependence of s.ignal amplitUde on drive field amplitude hy is no longer hnear unless very large stabilizing fields Hx are applied. Although our primary emphasis has been to character ize the dynamic response of a charged wall with no bubble domain attached, data were also taken on the motion of a M. H. Kryder and 8. E. Argyle 1667 [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: 128.123.44.23 On: Sun, 21 Dec 2014 03:27:06II> ..... IV .~ ..... o IV a:: Co E « o c: 0> Vi o H = x /' -10 Oe hy = 3.5 Oe-peak /-500e ..... -IOOOe 5 10 15 20 25 Frequency (MHz) FIG. 8. Charged wall oscillation amplitude in Sample A as a function of frequency with four values of H, applied parallel to the easy direction of crystalline anisotropy (unstable direction for the charged wall) and h,. = 3.5 Oe peak. charged wall and bubble coupled together in Fig. 10. Curve I was taken using the half aperture having odd symmetry about y = 0, whereas curve II was obtained with the full aperture having even symmetry. (The SA sensitivity for curve II is three times greater than for curve I). Curve I shows that with the bubble attached, the rolloffin amplitude response at low frequencies is much more rapid than with no bubble attached. This rolloff near I MHz is consistent with earlier observations 1 on the coupled bubble and charged wall -o .. a:: .. "0 ::J Q. E « o c "" VI o f z 1 MHz • t t t I t H • -10 Oe t t x 2 3 4 RF Drive Field hy (Oe-peak) 5 FIG. 9. Charged wall oscillation amplitude in Sample A as a function of drive field amplitude hy with three values of Hx applied to the easy direction of crystalline anisotropy (unstable direction for the charged wall). 1668 J. Appl. Phys., Vol. 53, No.3, March 1982 2 c: :::> >- ~ ~ .0 ~ <l C1l '0 ~ -a. E <l c c: C' (f) I o hy = 2.5 Oe-peak Hx = 50 Oe dc 10 15 20 25 Frequency (MHz) FIG. 10. Oscillation amplitudes in Sample A as a function of frequency when a bubble is coupled to the charged wall and h,. = 2.5 Oe. H, = 50 Oe. Curve II was obtained with a symmetric aperture as in Fig. 2. while curve I was obtained with an assymetric half aperture (see text). using streak photographs of the bubble oscillations. Howev er, the resonance peaks near 8-10 MHz were not previously observed nor is bubble motion visible in our present viewing optics. These observations suggest a resonance of the charged wall oscillating within the potential well of the bub ble being detected via the even symmetry components M x or Mz • DISCUSSION The data in Figs. 4 and 6 for the motion of the naked charged wall about a preferred crystallographic direction can be analyzed approximately in terms of an overdamped linear harmonic oscillator. The driving force for the oscilla tor is provided by the drive field hy sin wt acting on the mon opole line charge of the charged wall. The restoring force or spring constant is supplied by the stabilizing restoring tor ques of the in-plane field Hx' the magnetocrystalline aniso tropy, and anisotropic magnetrostriction. Balancing the re spective forces of viscous drag, linear restoring force, and applied drive field one obtains dy+ h iw, -We Y = /-Lew ye , dt (I) where We is the relaxation frequency in radians per second and/-Lew = y/hy the effective mobility coefficient. This equa tion has the solution [ W2 ] -1/2 . Y = Yo I + w~ e'l"" _. '" I, (2) where (3) M. H. Kryder and B. E. Argyle 1668 [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: 128.123.44.23 On: Sun, 21 Dec 2014 03:27:06TABLE I. Restoring force coefficient hy/Yo, relaxation frequency We' and mobility /lew of the charged wall stabilized in film A with applied in-plane field H. parallel to the edge of an unimplanted square hole and along the hard direction of crystalline anisotropy (stable direction for the charged wall). j H. h,/yo CtJe/21T /lew (Oe) (Oe//lml (MHz) (cm/sec Oel 10 0.68 ±0.01 8±1 7400 ± 1000 30 1.0 ± 0.04 10.5 ± 1.5 6600 ± 1200 SO 1.5 ± 0.1 15.3 ± 4 6400± 1200 100 2.7 ± 0.7 19.3 ± 7 5OOO± 3000 and (4) is the low-frequency amplitude of oscillation. Equation (2) predicts that the oscillation amplitude must decrease as frequency is increased, falling to 1/v'2 times its low-frequency value at the critical relaxation fre quency We when the phase lag is 45°. As may be seen in Fig. 6, the oscillation amplitude indeed falls with increasing fre quency. A critical relaxation frequency was obtained for each spectrum after correcting the curve shape for a slight rolloffin PMT sensitivity. Both We and the restoring force coefficient (hJyo) are found to increase with Hx as listed in Table I and plotted in Fig. 11. In this case Hx is applied parallel to a crystallographic stable orientation for the charged wall. Thus, the restoring force and the critical fre quency are nonzero even for zero applied field. The ratio /-Lew/we may be determined from the inverse 25 4 20 - -N I ~ cu 9 3 15 ~ '" ~ u 2 10 3 5 50 Hx (Oe) FIG. II. Restoring force coefficient hylYo and relaxation frequency We for sample A as a function of stabilizing field Hx' 1669 J. Appl. Phys., Vol. 53, No.3, March 1982 slope hJyo of the curves in Fig. 4 as shown by Eq. (4). Then, substituting in the values of We determined from Fig. 6 one finds (Table I) that the effective mobility of the charged wall depends only weakly on the applied field Hx attaining a val ue of 7400 cm/sec Oe with an in-plane field of 10 Oe and a slightly lower value of 6600 cm/sec Oe with an in-plane field of 30 Oe. The average of all four values in Table I weighted according to the estimated accuracies is 6600 cm/sec Oe. This charged-wall mobility may be compared to the effective mobility of ;::: 250 cm/sec Oe when the charged wall is coup led to a bubble in the same material determined in an earlier publication. j Clearly the damping of the charged wall mo tion is negligible compared to the damping when the bubble domain is attached. The naked charged wall shows no saturation velocity effects at least up to 12.8 MHz as shown by the linear depen dence of oscillation amplitude on drive field amplitude in Fig. 5. The maximum velocity of a harmonic oscillation hav ing a peak displacementYmax at frequency w/2rr is given by wYmax' Hence, the largest peak velocity for the charged wall reflected in Fig. 5 is 2rr (12.8 X 106) (3.5 X 10-4) ~28, 000 cm/sec. Hence, the charged wall not only exhibits high mo bility but also a saturation velocity at least 25 times that for an isolated bubble (1100 cm/sec) in a nonimplanted material of this composition. That the data in Figs. 8 and 9 deviate radically from the harmonic oscillator model is not surprising since the stabi lizing field Hx is applied.in an unstable direction for the charged wall. In this case, magnetocrystalline anisotropy acts to pull the charged wall from the Y = 0 position to the corners of the disk. Hence, until the field Hx is larger than the effective crystalline anisotropy field, there is no linear restoring force. It is interesting to note that with a field of -100 Oe applied, the linear harmonic oscillator model seems to provide a reasonable fit. This is consistent with the observation that the effective field from crystalline aniso tropy as determined from extrapolation of the data in Fig. 11 back to zero restoring force coefficient is about 30 Oe. The observation in Fig. 10 that, when a bubble domain is coupled to the charged wall, any appreciable bubble do main motion stops by 2 MHz, whereas charged wall motion alone is well behaved to beyond 12.8 MHz in Figs. 6 and 7, provides direct evidence that the dynamic properties of the charged wall do not limit the motion of the bubble domains. Hence, the dynamic performance of contiguous disk devices is not expected to have inherent limitations different from other field accessed bubble domain devices. Another parameter that may be inferred from dynamic measurements of wall motion is the wall width. For a Bloch wall configuration the wall-width t5BW is related to dynamic precision parameters obtained in FMR by (5) where a is the Gilbert damping parameter, r is the gyromag netic ratio, and /-L is the wall mobility. A charged wall, how ever, can have complex magnetization structures with Neel wall features at the center and Bloch wall features near the film surfaces.11 Applied to a charged wall, Eq. (5) would then require slight theoretical modifications which have yet to be M. H. Kryder and B. E. Argyle 1669 [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: 128.123.44.23 On: Sun, 21 Dec 2014 03:27:06performed. Nevertheless, Eq. (5) sets a lower bound for the width of the charged wall. 12 We find Dew ;>1T(0.044) (6600 cm/sec Oe)+ 1.27 X 107 Oe-1 sec-I = O.72flm, where the values a = 0.044 and r = 1.27 X 107 Oe-1 sec-I for the un implanted material are assumed. CONCLUSION Direct magneto-optic measurements of the dynamic be havior of charged walls with no bubble domain attached show that the charged wall has an effective mobility near 6600 cm/sec Oe. This is more than one order of magnitude higher than the effective mobility (250 cm/sec Oe) of the coupled charged wall and bubble observed earlier I in the same material. Furthermore, no saturation velocity effects were observed in the charged wall motion even at a frequen cy of 12.8 MHz where a peak velocity as large as 28 000 cm/sec is detected. A maximum frequency of bubble oscilla tion when coupled to the charge wall is ~ 2 MHz. Hence, it may be concluded that the dynamic characteristics of bubble domain devices using charged walls will not be limited by the charged walls, but rather by the dynamic characteristics of the bubble domains or the driving circuitry. Measurements also showed that magnetocrystalline anisotropy and/or anisotropic magnetostriction provide a strong restoring force on the charged walls. From the mea surements made here, one would estimate that the effective crystalline anisotropy field is about 30 Oe in the A sample examined. A lower bound for the width of the charged wall ob tained dynamically, i.e., from its mobility, is Dew ~ 0.7 flm. This is appreciably larger than the width of the Bloch wall (DBW ~0.2 flm) in the un implanted region of our 5-flm bub ble material A. This large width partly explains the large, linear, unsaturated velocity response observed in the charged wall up to 28 000 cm/sec compared with the much lower saturation velocity for a Bloch wall, e.g., 1100 cm/sec in a non implanted film of this composition. Thick walls can support only small stray fields near the film surfaces, thus making the threshold velocity for a Bloch-line nucleation mechanism of saturation 1\ considerably higher. Moreover, since domains adjacent to the charged wall are magnetized nearly parallel to the film plane, the type of stray fields, due to surface poles, which cause Bloch-line nucleation in bubble materials, are largely absent. In the absence of a Bloch-line mechanism, another higher velocity saturation falls out quite naturally from the well-known "torque equation of motion" \4 for a one-dimen sional domain wall structure satisfying the Landau-Lifshitz equations. The torque equation states that the velocity of a wall must be sustained by a torque on the wall magnetiza tion. The effective field which sustains this torque in the case of a charged wall (6) 1670 J. Appl. Phys., Vol. 53, No.3, March 1982 arises from the in-plane anisotropy field Q 41TM and the wall demagnetizing field N 41TM. For a wall 0.7 flm wide and 0.6 flm tall in sample A, N is approximately 0.5, and as stated above Q = 1.5 in this implanted layer. Strictly speaking, Q relaxes downward in a region next to the unimplanted area over a distance comparable to the depth of the implanted layer. However, we neglect this effect considering our aper ture monitors charged wall motions over a distance of 3 flm from the edge of the square (Fig. 2). Thus, the effective field may be as large as (1.5 + 0.5) 195 = 390 Oe. The torque equation, if we neglect torques due to the in plane applied field Hp = 10-100 Oe and to damping (a.:ith has the simple form V = y.:i -Heff sin 2t/J, (7) 2 when t/J is the angle of inclination of the wall magnetization. Thus, velocity saturation when t/J ...... 45c can read as high as 1/2 (1.27 X 107) (0.7x 10-4)390 = 177 000 cm/sec. This is several times larger than we can induce harmonically with the implanted structure of Fig. 2. ACKNOWLEDGMENTS The authors would like to acknowledge the technical assistance ofR. E. Mundie and helpful discussions with J. C. Slonczewski and A. Hubert. M. H. Kryder would like to thank IBM Corporation for providing support and the op portunity to perform this work and also to acknowledge sup port by the National Science Foundation under Grant No. ECS-7912677 . 'B. E. Argyle, M. H. Kryder, R. E. Mundie, and J. e. Slonczewski, IEEE Trans. Magn. MAG-14, 59311978). 'I. L. Sanders and M. H. Kryder, J. App\. Phys. 50, 225211979). 'J. e. Slonczewski, A. P. Malozemoff, and E. A. Giess, App!. Phys. Lett. 24, 396 (1974). "A. P. Ma1ozemoff, J. e. Slonczew,ki, and J. e. DeLuca, AlP Conf. Proc. No. 29, Magnetism and Magnetic Materials, 58 (1975). 'Yo S. Lin, G. S. A1masi, and G. E. Keefe, IEEE Trans. Magn. MAG-I3, 1744 (1977). "Y. S. Lin, G. S. Almasi, G. E. Keefe, and E. W. Pugh, IEEE Trans. Magn. MAG-IS 164211979). 'Y. S. Lin, D. B. Dove, S. Schwerzl, and e. e. Shir, IEEE Trans. Magn. MAG·I4, 494 (1978). 'J. J. Fernandez de Castro and M. H. Kryder, Magnetism and Magnetic Materials Conference Paper BB-2, Dallas, November 1980, to be pub lished in IEEE Transactions of Magnetics. "B. E. Argyle, J. e. Slonczewski, W. Jantz, and J. H. Spreen, Magnetism and Magnetic Materials Conference Proceedings, J. App!. Phys. 52, 2353 (1981). 'OR. E. MacDonald and J. W. Beck, J. App!. Phys. 40,1429 (1969). lie. e. Shir and Y. S. Lin, J. Appl. Phys, 50, 227011979). "J. e. Slonczewski (private communication) "B. E. Argyle, J. e. Slonczewski, and F. Mayadas, AlP Conf. Proc. 5, Magnetism and Magnetic Materials, 175 (1971 J. '"A. P. Malozemoffand 1. e. Slonczewski, Magnetic Domain Walls in Bub ble Malerials (Academic, New York, 1979), Eq. 10.10, p 125. M. H. Kryder and B. E. Argyle 1670 [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: 128.123.44.23 On: Sun, 21 Dec 2014 03:27:06

1.4896027.pdf

Dynamic control of spin wave spectra using spin-polarized currents Qi Wang, Huaiwu Zhang, Xiaoli Tang, Hans Fangohr, Feiming Bai, and Zhiyong Zhong Citation: Applied Physics Letters 105, 112405 (2014); doi: 10.1063/1.4896027 View online: http://dx.doi.org/10.1063/1.4896027 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnonic band structure, complete bandgap, and collective spin wave excitation in nanoscale two-dimensional magnonic crystals J. Appl. Phys. 115, 043917 (2014); 10.1063/1.4862911 Magnonic crystal wave guide with large spin-wave propagation velocity in CoFeB Appl. Phys. Lett. 102, 222412 (2013); 10.1063/1.4809757 Micromagnetic study of spin wave propagation in bicomponent magnonic crystal waveguides Appl. Phys. Lett. 98, 153107 (2011); 10.1063/1.3579531 Optimal control of magnetization dynamics in ferromagnetic heterostructures by spin-polarized currents J. Appl. Phys. 108, 103717 (2010); 10.1063/1.3514070 Nonlinear dynamics of spin-injected magnons in magnetic nanostructures J. Appl. Phys. 91, 8046 (2002); 10.1063/1.1450819 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: 130.216.129.208 On: Fri, 05 Dec 2014 08:55:47Dynamic control of spin wave spectra using spin-polarized currents Qi Wang,1Huaiwu Zhang,1,a)Xiaoli Tang,1Hans Fangohr,2Feiming Bai,1 and Zhiyong Zhong1,b) 1State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China 2Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, United Kingdom (Received 21 July 2014; accepted 4 September 2014; published online 17 September 2014) We describe a method of controlling the spin wave spectra dynamically in a uniform nanostripe waveguide through spin-polarized currents. A stable periodic magnetization structure is observedwhen the current flows vertically through the center of nanostripe waveguide. After being excited, the spin wave is transmitted at the sides of the waveguide. Numerical simulations of spin-wave transmission and dispersion curves reveal a single, pronounced band gap. Moreover, the periodicmagnetization structure can be turned on and off by the spin-polarized current. The switching pro- cess from full rejection to full transmission takes place within less than 3 ns. Thus, this type mag- nonic waveguide can be utilized for low-dissipation spin wave based filters. VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4896027 ] A Magnonic Waveguide (MW) consists of periodic mag- netic structures. The periodic structure affects the spin wave dispersion curve by creating forbidden bands at the Brillouinzone boundaries due to Bragg reflection. In recent years, the characteristics of spin wave propagation in MWs, which were fabricated by different magnetic materials 1,2and different widths,3,4were investigated in detail. However, the spectra of spin waves in these MWs cannot be changed dynamically af- ter its fabrication. Dynamic artificial magnonic crystals arecurrently the focus of much interest because the spin wave spectra in these magnonic cyrstals can be modulated dynami- cally. It was reported that the spin wave spectra could bedynamically modulated by local Oersted fields produced by electrical currents on the surface of low-loss Yttrium iron gar- net (YIG). 5,6Nevertheless, MWs based on metal magnetic thin films are more suitable to integrate with CMOS circuits. Recently, Volkov et al. have studied the action of the strong perpendicular spin-polarized current on ferromagnetic sys-tems for two-dimensional films 7–9and a narrow one- dimensional wire.10In the both case, the stable periodic mag- netization structures induced by spin-polarized current werefound in the both case. In this letter, we present a way to modulate spin wave spectra in a uniform waveguide by spin-polarized current.The waveguide considered in this letter is presented in Fig. 1. The length, width, and thickness of the stripe are L¼2000 nm, w ¼160 nm, and h ¼10 nm, respectively. The width of the pinned layer is x p¼30 nm. A thin nonmagnetic spacer is placed between the pinned layer and the stripe. The damping constant is a¼0.01. In order to suppress spin wave reflections, the damping parameter is increased to 0.5 at the ends of the waveguide (x <20 nm and x >1980 nm). The spin-polarized current flows along the perpendicular directionto the center of the nanostripe waveguide. In order to excite spin wave within a wide frequency range, a sinc field pulse,ranging from 0 to 60 GHz, was applied to the y-axis. A static bias magnetic field H 0¼300 Oe was applied along the x-axis to avoid forming domain wall.11Our micromagnetic simula- tion is based on the Landau-Lifshitz-Gilbert equation with the Slonczewski-Berger spin torque term.12–15 Figure 2shows the spatial distributions of magnetization of MW under different conditions. In the absence of a spin- polarized current, the relaxed spatial distribution of magnet- ization is uniform in the waveguide as shown in Fig. 2(a). However, when the spin-polarized current is applied on the nanostripe waveguide, the spin transfer torque produced by the spin-polarized current creates a magnetization structurein the center of the waveguide that is periodic in the x- direction [see Fig. 2(b)]. This periodic structure consists of two chains of vortex-antivortex configurations, and is similarto the quasicrystal state shown in Fig. 2(f) in Ref. 9.W e show the z-component of the magnetization, M z, in the sys- tem along a particular line scanned along the x-axis and y-axis in Figs. 2(c) and2(d), respectively. While the magnet- ization is uniform along both the long and short axis without the spin-polarized current (Fig. 2(a)), the magnetization dis- tribution is non-uniform with the spin-polarized current applied (Figs. 2(b)–2(d) ): the spin transfer torque creates a FIG. 1. Schematic view of the three-layer waveguide. The spin polarized cur- rent flows along z-direction, i.e., perpendicular to the stripe. An alternating ex- citation field applied along the y-axis generates spin wave along the x-axisand a static bias magnetic field H 0¼300 Oe is applied along the x-axis. The length, width, and thickness of stripe are L ¼1200 nm, w ¼160 nm, and h¼10 nm, respectively. The width of pinned layer is xp¼30 nm. A thin non- magnetic spacer is placed between the pinned layer and stripe.a)hwzhang@uestc.edu.cn b)zzy@uestc.edu.cn 0003-6951/2014/105(11)/112405/4/$30.00 VC2014 AIP Publishing LLC 105, 112405-1APPLIED PHYSICS LETTERS 105, 112405 (2014) 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: 130.216.129.208 On: Fri, 05 Dec 2014 08:55:47periodic magnetization structure in the middle of the nano- stripe waveguide. The magnetization is almost parallel to the z-axis in the middle of nanostripe waveguide in a region cen-tered around the location where the spin-torque term is applied. The spin waves are not allowed to propagate in this region, because the spin-polarized current contributes aneffective damping, which is usually greater than the natural one. 7,9Hence, we focus on the edge of the nanostripewaveguide where spin waves are allowed to propagate. The effective width of this channel is about 60 nm, see Fig. 2(d). Nevertheless, the z component of magnetization along the length direction is oscillating in this channel caused byexchange and dipolar interaction. These oscillations of the magnetization component M zas a function of length (x-axis) are plotted in Fig. 2(c). From the figure, one can obtain the periodicity D ¼60 nm when spin-polarized current was applied. The spin wave spectra, characters of transmission, and dispersion curves are shown in Fig. 3. The spectra are obtained through the fast Fourier transform (FFT) of the tem-poral M z/Msalong the red line shown in Fig. 2(a). The fre- quency spectra clearly reveal allowed and forbidden bands: low values represented by blue/green represent forbiddenbands and high values associated with red and orange are the allowed bands. Figs. 3(a)–3(c) show that for zero spin- polarized current, spin wave transmission is suppressedbelow 7.5 GHz, 16which originates from the nanostripe width confinement.17,18There is no band gap above 7.5 GHz in the spectra. Figs. 3(d)–3(f) show that the spin-polarized current changes the spin-wave spectra drastically: A new gap emerges above 7.5 GHz as can be seen in Fig. 3(d). The transmission characteristic obtained by integrating the spin-wave intensity from 1400 nm to 1600 nm is displayed in Fig. 2(e), also showing the new spin-wave band gap that appears in the presence of the spin-polarized current. By calculatingthe dispersion curves numerically and analytically 17–19for spin-wave [analytical result shown as black line in Figs. 3(c) and3(f)], we find that the center frequency of the band gap is closely correlated with the first order Bragg reflection wavenumber: the center frequency of 13.2 GHz is identified to belong to the first width mode and is formed at the Braggwavenumber k D1¼p/D¼0.052 nm/C01, where D is the perio- dicity of magnetization. Other higher frequency band gaps are not found in the spectrum, because the spin-polarizedcurrent induced magnonic crystal is designed in such a way that variations of the magnetization along the length direc- tion are practically sinusoidal as shown in Fig. 2(c). Therefore, the spectrum of this MW contains only one band gap, and this phenomenon has been observed in experiment and theory. 5,20,21 Fig.3demonstrates the key result of this letter: a spin- polarized current can induce band gaps in a uniform nano- stripe waveguide, opening the path to electrically switchableband gaps. Further studies show that the center frequency and depth of the band gap is almost stable with increasing the spin-polarized current density from 22 /C210 12to 24/C21012A/m2(data not shown). The results above show that it is possible to switch the periodic structure on and off and realize dynamic control ofspin wave spectra through spin-polarized currents. We now focus on the switching time required for this process. A sig- nal with a frequency f¼f gap¼13.2 GHz was applied. The continuous spin wave signal was picked up at P point as shown in Figs. 2(a) and 2(b). A spin-polarized current (22/C21012A/m2) was applied during the first 5 ns, and then it was set to zero for the next 10 ns as shown in the inset of the Fig. 4(a). The z-component of the magnetization at point P of the transmitted spin wave signal is shown in Fig. 4(a) FIG. 3. Character of MW without spin-polarized current (top row, plots (a) to (c)) and with spin-polarized current J ¼22/C21012A/m2(bottom row, plots (d) to (f)). Figures 2(a)and2(d)show frequency spectra obtained from FFTs of M z/Msalong the red line shown in Fig. 2(a). Figures 2(b) and2(e)show spin-wave transmission characteristics obtained by integrating the spin-wave intensity of the MW (from 1400 nm to 1600 nm). The inset frequencies marked the center frequency of the band gap. Figures 2(c)and2(f)show the dispersion relation of the first modes obtained by micromagnetic simulationand analytical calculation (the black line). The black dotted lines denote the positions of the switchable band gap and the corresponding wave vectors k x. FIG. 2. (a) and (b) The remnant magnetization (M z/Ms) of stripe for (a) J¼0 (b) J ¼22/C21012A/m2, respectively. The z component of magnetiza- tion M zis extracted along (c) x-axis (red line in Figs. 2(a)and2(b)) and (d) y-axis, respectively.112405-2 Wang et al. Appl. Phys. Lett. 105, 112405 (2014) 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: 130.216.129.208 On: Fri, 05 Dec 2014 08:55:47(black line). There is no spin wave signal to be detected dur- ing the first 5 ns: the transmission of the spin wave is effec- tively suppressed by the spin-polarized current. After about 8 ns, by contrast, we observe the periodic spin wave signal.The Fourier spectrum, based on data from 8 to 15 ns, shows that the main frequency of the spin wave is equal to the fre- quency of the excitation signal. We also observe the secondand third harmonic, which is one of the most the universal phenomena appearing in nonlinear systems. 22–25The z-component of magnetization shows excitation and subse-quent decay between 5 ns and 8 ns. In order to elucidate the physical origin of this, we also measure the magnetization component M zat P point without the excitation signal as shown in Fig. 4(red line), which reveals a similar patterned. Therefore, we conclude that the attenuation is due to spin- polarized current switching and subsequent relaxation of themagnetization pattern rather than due to spin wave. It takes about 4 ns for the spin-wave propagation to be fullyestablished after the current is switched off. Fig. 4(b) shows results of the corresponding process of switching the spin- polarized current on after 5 ns: the transmission of the spin wave is effectively suppressed by the spin-polarized current,and the transition of full transmission to full rejection of spin-waves also takes approximately 3 ns. In order to study the effect of the spin-polarized current on the spin waves with frequencies far away from the center frequency of the band gap f gap¼13.2 GHz, we applied a sig- nal frequency f¼20GHz. As seen from Fig. 5, the spin wave is not suppressed by the spin-polarized current and com- pletely passes through the waveguide. The spin-polarized current can effectively suppress the spin waves whose fre-quencies are nearby the center frequency of the band gap. However, the spin-polarized current has little impact on the spin wave with other frequencies. In this letter, we have presented a dynamic magnonic crystal design in a uniform waveguide that can be switched electrically using a spin-polarized current. The spin-polarized current induces periodic magnetization structure in the uniform waveguide which leads to a pronounced spin wave band gap. This periodic magnetizaton structure isdynamically controllable, i.e., it is possible to switch the spin waves transmission at band gap frequencies on and off, using well established technology of electric currents. This type ofmagnonic crystal can be used for low-power filters and other magnonic devices. This paper was supported by the National Nature Science Foundation of China under Grant Nos. 61271037, 51171038, 51132003, and 61271038, the SRFDP under No 20120185110029, and Key Technology R&D Program ofSichuan No. 2013GZ0025. 1S. Neusser and D. Grundler, Adv. Mater. 21, 2927(2009). 2Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Jain, and A. O. Adeyeye, Appl. Phys. Lett. 94, 083112 (2009). 3K.-S. Lee, D.-S. Han, and S.-K. Kim, Phys. Rev. Lett. 102, 127202 (2009). 4S.-K. Kim, K.-S. Lee, and D.-S. Han, Appl. Phys. Lett. 95, 082507 (2009). FIG. 4. Time-dependent spin wave amplitude at P point with excitation signal (black line) and without excitation signal (red line). The insets show th e spin- polarized current as a function of time and the Fourier spectrum of out-of-plane magnetization component M z. Plot (a) shows the spin-polarized current being switched off after 5 ns, and (b) shows the opposite process of switching the current on after 5 ns. The results demonstrate that the switchable band gap c an be activated and de-activated within 3 ns. FIG. 5. Time-dependent spin wave amplitude at P point with excitation sig- nal (f¼20 GHz). The insets show the spin-polarized current as a function of time and the Fourier spectrum of out-of-plane magnetization component M z. The data show that spin-waves with frequencies away from the band-gap arenot affected by the changes of the magnetization texture originating from the spin-polarised current.112405-3 Wang et al. Appl. Phys. Lett. 105, 112405 (2014) 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: 130.216.129.208 On: Fri, 05 Dec 2014 08:55:475A. V. Chumak, T. Neumann, A. A. Serga, B. Hillebrands, and M. P. Kostylev, J. Phys. D: Appl. Phys. 42, 205005 (2009). 6A. D. Karenowska, V. S. Tiberkevich, A. N. Slavin, A. V. Chumak, A. A. Serga, and B. Hillebrands, Phy. Rev. Lett. 108, 015505 (2012). 7O. M. Volkov, V. P. Kravchuk, D. D. Sheka, and Y. Gaididei, Phys. Rev. B 84, 052404 (2011). 8Y. Gaididei, O. M. Volkov, V. P. Kravchuk, and D. D. Sheka, Phys. Rev. B 86, 144401 (2012). 9O. M. Volkov, V. P. Kravchuk, D. D. Sheka, F. G. Mertens, and Y. Gaididei, Appl. Phys. Lett. 103, 222401 (2013). 10V. P. Kravchuk, O. M. Volkov, D. D. Sheka, and Y. Gaididei, Phys. Rev. B 87, 224402 (2013). 11In general, the magnetization favors alignment parallel to the long axis of the structure due to the shape anisotropy. However, in our study, the spin- polarized current excites significant magnetization dynamics. In particular, when switching the spin-polarized current off, the magnetization may relax into undesired non-uniform configurations with domain separated by a domain wall. 12J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 13J. C. Slonczewski, J. Magn. Magn. Mater. 247, 324 (2002). 14J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70, 172405 (2004). 15We use the OOMMF code, version 1.2a5 for material parameters of Permalloy: saturation magnetization Ms ¼8.6/C2105A/m, exchange con- stant A ¼1.3/C21011J/m. The cell size is 5 /C25/C210 nm. The currentparameters are the following: polarization degree P ¼0.4 and resistance mismatch between the spacer and the ferromagnet stripe K¼2. 16The widths of the intrinsic potential barriers are different because of the width of the spin wave channels change when the current apply. 17T. W. O‘Keeffe and R. W. Patterson, J. Appl. Phys. 49, 4886 (1978). 18S. Choi, K.-S. Lee, K. Yu. Guslienko, and S.-K. Kim, Phys. Rev. Lett. 98, 087205 (2007). 19The parameters used for analytical calculations were as follows: theexchange constant A ¼13/C210 /C012, the saturation magnetization Ms¼8.6/C2105A/m, the effective width of channel w eff¼60 nm, and zero internal field. 20F. Ciubotaru, A. V. Chumak, N. Yu. Grigoryeva, A. A. Serga, and B.Hillebrands, J. Phys. D: Appl. Phys. 45, 255002 (2012). 21A. V. Chumak, V. S. Tiberkevich, A. D. Karenowska, A. A. Serga, J. F. Gregg, A. N. Slavin, and B. Hillebrands, Nat. Commun. 1, 141 (2010). 22V. E. Demidov, M. P. Kostylev, K. Rott, P. Krzysteczko, G. Reiss, and S. O. Demokritov, Phys. Rev. B 83, 054408 (2011). 23T. Sebastian, T. Br €acher, P. Pirro, A. A. Serga, B. Hillebrands, T. Kubota, H. Nagauma, M. Oogane, and Y. Ando, Phys. Rev. Lett. 110, 067201 (2013). 24J. Marsh, V. Zagorodnii, Z. Celinski, and R. E. Camley, Appl. Phys. Lett. 100, 102404 (2012). 25V. E. Demidov, H. Ulrichs, S. Urazhdin, S. O. demokritov, V. Bessonov, R. Gieniusz, and A. Maziewski, Appl. Phys. Lett. 99, 012505 (2011).112405-4 Wang et al. Appl. Phys. Lett. 105, 112405 (2014) 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: 130.216.129.208 On: Fri, 05 Dec 2014 08:55:47

1.3540411.pdf

Microscopic theory of spin torque induced by spin dynamics in magnetic tunnel junctions Daisuke Miura and Akimasa Sakuma Citation: Journal of Applied Physics 109, 07C909 (2011); doi: 10.1063/1.3540411 View online: http://dx.doi.org/10.1063/1.3540411 View Table of Contents: http://aip.scitation.org/toc/jap/109/7 Published by the American Institute of PhysicsMicroscopic theory of spin torque induced by spin dynamics in magnetic tunnel junctions Daisuke Miuraa)and Akimasa Sakuma Department of Applied Physics, Tohoku University, Sendai 980, Japan (Presented 16 November 2010; received 28 September 2010; accepted 3 November 2010; published online 24 March 2011) We studied the charge and spin currents in magnetic tunnel junctions in the presence of spin dynamics on the basis of a tight-binding scheme; the spin dynamics is assumed to be present only in one of the two ferromagnetic electrodes. The charge current is pumped by the dynamical spins having the form _m1/C1m2, where maða¼1;2Þdenotes the direction of magnetization in the electrodes and m1represents the dynamic spin. In addition, three types of spin currents are induced by the dynamical spins. One of these spin currents has the form _m1/C2m2, whose coefficient is proportional to the product of the spin polarizations of both the electrodes. This term can possiblyprevent magnetization switching, which is an effect that differs from both the Gilbert damping and spin transfer torque effects. Even in the absence of spin dynamics, the spin current exists in the form m 1/C2m2. We have confirmed that the coefficient of this static term is equal to the effective exchange interaction between the two ferromagnetic electrodes. VC2011 American Institute of Physics . [doi: 10.1063/1.3540411 ] I. INTRODUCTION The discovery of giant magnetoresistance effects1–4has generated considerable interest not only in the fields of mag- netics and magnetism but also in the field of electronics.These effects have also opened the door to a new field called “spintronics.” This field involves the development of novel devices in which the charge and spin degree of freedom ofelectrons can be manipulated individually. Recently, many theoretical and experimental analyses have been carried out to study the spin transfer torque, 5–7spin pumping,8,9and spin Hall effects.10–12These investigations have focused on the interplay between spin currents and magnetic structures, spin dynamics, and spin-orbit interaction both from thescientific and technological viewpoints. Mizukami et al . 8 performed an important experiment to clarify the interplay between the spin current and spin dynamics and demon-strated that dynamical spins can be a source of spin currents, resulting in spin damping. This experimental result resulted in considerable interest being generated in the concept ofspin pumping. The result was further developed theoretically by Tserkovnyak et al.; 9they showed that the spin current has the form m/C2_min the ferromagnetic film adjacent to non- magnetic metal layers. The form m/C2_mis the same as the Gilbert damping term in the Landau-Lifshitz-Gilbert equa- tion and can possibly enhance the damping coefficient.Recently, Ohe et al. 13have shown that the charge current is also pumped by spin dynamics in the presence of the Rashba spin-orbit interaction; this result has been extended to thecase of general spin-orbit interaction by Takeuchi et al . 14Recently, Saitoh et al .15measured the inverse spin Hall effects and experimentally confirmed the generation of spin currents by spin dynamics. In the present study, we havestudied both the charge and spin currents in magnetic tunnel junctions (MTJs) in the presence of spin dynamics. The main aim of our work is to investigate the torque acting on the dy-namical spins in the MTJs by calculating the spin current generated by spin dynamics. In this work, we have found an additional term for torque, other than the spin damping term,due to the spin current. The strategy used to calculate the currents is the nonequilibrium Green function technique that adopts the adiabatic approximation in terms of the spin dy-namics. In this calculation, we perturbatively treat the tun- neling transmittance, whereas Ohe et al . 13perturbatively treated the interaction between the localized spins and con-duction electrons. II. SYSTEM AND FORMALISM The system considered here is depicted schematically in Fig. 1. Two ferromagnetic electrodes are connected through the tunnel junction. For brevity, a one-dimensional chain is considered. The inner structure of the tunnel barrier isneglected and is represented by the transmittance coefficient T ij, where iandjdenote the sites located at either side of the junction. The magnetization in each electrode is assumed tobe spatially uniform, and the direction of magnetization is represented by m LandmRfor the left (L) and right (R) elec- trodes, respectively. We assume here that mLis time depend- ent due to, for example, an applied magnetic field. For simplicity, we treat mRas a static vector. This assumption is based on the fact that in real systems the magnetization ofeither electrode is fixed by a strong anisotropy field or an exchange bias field.a)Author to whom correspondence should be addressed. Electronic mail: dmiura@solid.apph.tohoku.ac.jp. 0021-8979/2011/109(7)/07C909/3/$30.00 VC2011 American Institute of Physics 109, 07C909-1JOURNAL OF APPLIED PHYSICS 109, 07C909 (2011)The current matrix (a hat ‘ ^’ represents a 2 /C22 matrix in the spin space) flowing from site jto site ican be represented as ^IijðtÞ:¼Tij^ULðtÞ^G< jiðt;tÞ^URþH:c:; where ^ULðtÞand ^URare 2 /C22 matrices and are defined as ^ULðtÞmLðtÞ/C1^r^ULðtÞ:¼^rzand ^URmR/C1^r^UR:¼^rz, where ^ris the Pauli matrix, ^G< jiðt;t0Þis a lesser Green function with spin components ^G< jr;ir0ðt;t0Þ:¼i /C22hc† ir0ðt0ÞdjrðtÞ/C10/C11 , and c† irðdjrÞis an operator that creates (annihilates) the rspin electron at site iðjÞ; we assume the local spin axis to be parallel to mRðmLÞ. For the calculations, the transmittance coefficient Tijis treated as the perturbation within the framework of the Keldysh-Green function. If one takes the first-order term in Tijfor ^G< jiðt;tÞ,^IijðtÞcan be expressed as ^IijðtÞ’/C0 j Tijj2ð dt0^ULðt0Þ^gr jðt;t0Þ^ULðt0Þ^UR^g< iðt0;tÞh þ^g< jðt;t0Þ^ULðt0Þ^UR^ga iðt0;tÞi ^URþH:c:; where ^gr aðt;t0Þ,^ga aðt;t0Þ, and ^g< aðt;t0Þare the unperturbed retarded, advanced, and lesser Green functions, respectively, with Tij¼0. By adopting the adiabatic approximation, we ignored the spin dynamics that occur after the electrons hop over the barrier. For convenience, we express these Green functions in the following form: ^ULðt0Þ^gjðt;t0Þ^ULðt0Þ¼ /C22gjðt;t0Þ^1þDgjðt;t0ÞmLðt0Þ/C1^r;(1) ^UR^giðt0;tÞ^UR¼/C22giðt0;tÞ^1þDgiðt0;tÞmR/C1^r; (2) where /C22ga:¼ðga"þga#Þ=2 and Dga:¼ðga"/C0ga#Þ=2. garðr¼";#Þis the Green function of the rspin electron. Using Eqs. (1)and(2)in the Fourier-transformed (fre- quency) space and taking limits (slowly varying limit andlow-temperature limit), we obtain the following relations after performing some algebraic manipulations: ^I ijðtÞ¼IC ijðtÞ^1 2þIS ijðtÞ/C1^r 2: (3) IC ijtðÞ¼/C0 4Tij/C12/C12/C12/C122F1_mLtðÞ/C1mR; (4)IS ijtðÞ ¼ /C0 4Tij/C12/C12/C12/C122F2_mLtðÞþ F3mR/C2mLtðÞ þ F4mR/C2_mLtðÞ ½/C138 ; (5) where F1:¼ðEF dEDqjðEÞD_viðEÞ/C0D_vjðEÞDqiðEÞ/C2/C3 ; (6) F2:¼ðEF dEDqjðEÞ_/C22viðEÞ/C0D_vjðEÞ/C22qiðEÞ/C2/C3 ; (7) F3:¼1 /C22hðEF dEDqjðEÞDviðEÞþDvjðEÞDqiðEÞ/C2/C3 ;(8) F4:¼pDqjðEFÞDqiðEFÞ; (9) where EFis the Fermi energy, qjðiÞrðEÞrepresents the local density of states at the surface of the left (right) elec- trode with Tij¼0. We define /C22qaðEÞ:¼½qa"ðEÞþqa#ðEÞ/C138=2, DqaðEÞ:¼½qa"ðEÞ/C0qa#ðEÞ/C138=2,/C22vaðEÞ:¼< /C22gr aðEÞ, and Dva ðEÞ:¼<Dgr aðEÞ. In Eqs. (6)–(9), the dots indicate the deriv- ative with respect to energy. If one multiples the electron charge to IC ijðtÞand /C22h=2t oIS ijðtÞ, the resultant quantities cor- respond to the charge and spin currents, respectively. III. DISCUSSION AND SUMMARY Equation (4)represents the charge pumping attributable to spin dynamics. The form mR/C1_mLðtÞindicates that its con- tribution is small when the magnetization configuration in both electrodes are nearly collinear and its contributionbecomes large when m Lis perpendicular to mR. In addition, as can be inferred from Eq. (6), if both electrodes have the same electronic structure, the coefficient vanishes and chargepumping does not occur. As seen in Eq. (5), three types of spin currents are induced by spin dynamics. The last term is dominant because the coefficient F 4is much larger than F2 and F3;_/C22vðEÞandDvðEÞin the two latter coefficients are smaller than DqðEFÞ. The coefficient F4for the term mR/C2_mLis proportional to DqjðEFÞDqiðEFÞ,w h i c hi m p l i e s that the spin current is governed by the spin polarizations at the surfaces of both electrodes and can be generated only around the interface between the two electrodes. In this sense,the present result is consistent with the theoretical results of Tserkovnyak et al . 9and Ohe et al .13However, the form mR/C2_mLimplies that the spin current is not generated in the case in which the ferromagnetic film ( mL) is adjacent to a non- magnetic metal ( mR¼0); this case was considered by Tser- kovnyak et al . In such a case, it is necessary to take into account the penetration of exchange splitting into the nonmag- netic metal through the hybridization between two layers. Tak- ing this aspect into consideration, Tserkovnyak et al. provided a phenomenological explanation to confirm that the enhance- ment in the Gilbert damping of the magnetic multilayered sys- tem was due to a spin pumping effect; this enhancement waspreviously experimentally observed by Mizukami et al. 8 In the case of the MTJ analyzed in this study, the term mR/C2_mLdoes not necessarily play a role in spin damping. For example, if mLexhibits a precessional motion in the direction ofmR, the torque attributable to mR/C2_mLforces mLtoward FIG. 1. (Color online) Schematic of one-dimensional magnetic tunnel junc- tion. Tijrepresents the transmittance coefficient between sites iandjlocated at either side of the interface. mLandmRdenote the magnetization direc- tions for the left (L) and right (R) electrodes, respectively.07C909-2 D. Miura and A. Sakuma J. Appl. Phys. 109, 07C909 (2011)mRwhen the sign of the product of the spin polarizations of both electrodes, DqjðEFÞDqiðEFÞ, is negative. This results in the generation of an effect that is similar to Gilbert damping.When Dq jðEFÞDqiðEFÞis positive, mLis forced toward /C0mR. In the case in which the spin transfer torque acts on mLbecause of a polarized current, the term mR/C2_mLpushes mLin the direction perpendicular to mRand then prevents the switching ofmLtoward the mRor/C0mRdirection. Thus, the term mR/C2_mLcan be regarded as a term representing the torque, in addition to the spin damping term, due to spin pumping. Another nonequilibrium component of the spin current, the first term of Eq. (5), corresponds to the diffusive spin current described by Takeuchi et al.14and Ohe et al.;13these research groups performed first-order perturbative calcula- tions for spin currents with respect to the interaction betweenthe localized spins and conduction electrons. Consider the second term in Eq. (5),F 3mR/C2mL; this term does not depend on spin dynamics. To gain an insightinto the physical meaning of this term, we recast the coeffi- cient F 3combined with jTijj2into the following form: 4jTijj2ðEF dEDvjðEÞDqiðEÞþDqjðEÞDviðEÞ/C2/C3 ¼1 pðEF dE=Gr" jiðEÞDiðEÞGr# ijðEÞDjðEÞhi ; (10) where Grr jiðEÞ:¼grr jðEÞTjigrr iðEÞand Da:¼gr" aðEÞ/C01 /C0gr# aðEÞ/C01;ða¼i;jÞ. It should be noted that the quantity Da describes the exchange splitting at the asite. We find that Eq.(10) is equivalent to the effective exchange constant 4 Jij in the itinerant electron system, which was first presented by Liechtenstein et al.16in the Korringa-Kohn-Rostoker scheme and later reported by us.17 In summary, we have studied the charge and spin cur- rents in magnetic tunnel junctions (MTJs) in the presenceof spin dynamics on the basis of a tight-binding scheme. The charge current is pumped by the dynamical spins having the form mL/C1mR. The spin current is also induced by the dynamical spins with the form mL/C2mR, whose coefficient is the product of the spin polarizations of both electrodes. The term mL/C2mRcan possibly prevent mag- netization switching, which is an effect that differs from both the Gilbert damping and spin pumping effects. The spin current can exist even in an equilibrium state with aform that is proportional to m L/C2mR. We have success- fully shown that the coefficient of this term is equal to the effective exchange interaction between the two ferromag-netic electrodes. 1M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Eitenne, G. Greuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988). 2G. Binasch, P. Gru ¨nberg, F. Saurenbach, and W. Zinn, Phys. Rev. Lett. 39, 4828 (1989). 3T. Miyazaki and N. Tezuka, J. Magn. Magn. Mater. 139, L231 (1995). 4J. S. Moodera, L. R. Kider, T. M. Wong, and R. Meservey, Phys. Rev. Lett. 74, 3273 (1995). 5J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989). 6E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867 (1999). 7J. Z. Sun, J. Magn. Magn. Mater. 202, 157 (1999). 8S. Mizukami, Y. Ando, and T. Miyazaki, J. Appl. Phys .40, 580 (2001). 9Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 10J. E. Hirsh, Phys. Rev. Lett. 30, 1834 (1999). 11S. Takahashi, H. Imamura, and S. Maekawa, in Concepts in Spin Electron- ics, edited by S. Maekawa (Oxford University Press, New York, 2006), pp. 343–367. 12T. Ishida, T. Kimura, and Y. Otani, Phys. Rev. B 74, 014424 (2006). 13J. Ohe, A. Takeuchi, and G. Tatara, Phys. Rev. Lett. 99, 266603 (2007). 14A. Takeuchi and G. Tatara, J. Phys. Soc. Jpn. 77, 074701 (2008). 15E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett. 88, 182509 (2006). 16A. I. Liechtenstein, M. I. Katsunelson, V. P. Antropov, and V. A. Gubanov,J. Magn. Magn. Mater . 67, 65 (1987). 17A. Sakuma, IEEE Trans. Mag. 35, 3349 (1999).07C909-3 D. Miura and A. Sakuma J. Appl. Phys. 109, 07C909 (2011)

1.4934566.pdf

Effect of temperature variations and thermal noise on the static and dynamic behavior of straintronics devices Mahmood Barangi and Pinaki Mazumder Citation: Journal of Applied Physics 118, 173902 (2015); doi: 10.1063/1.4934566 View online: http://dx.doi.org/10.1063/1.4934566 View Table of Contents: http://aip.scitation.org/toc/jap/118/17 Published by the American Institute of PhysicsEffect of temperature variations and thermal noise on the static and dynamic behavior of straintronics devices Mahmood Barangia)and Pinaki Mazumder Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109, USA (Received 25 May 2015; accepted 13 October 2015; published online 4 November 2015) A theoretical model quantifying the effect of temperature variations on the magnetic properties and static and dynamic behavior of the straintronics magnetic tunneling junction is presented. Four common magnetostrictive materials (Nickel, Cobalt, Terfenol-D, and Galfenol) are analyzed todetermine their temperature sensitivity and to provide a comprehensive database for different appli- cations. The variations of magnetic anisotropies are studied in detail for temperature levels up to the Curie temperature. The energy barrier of the free layer and the critical voltage required for flip-ping the magnetization vector are inspected as important metrics that dominate the energy require- ments and noise immunity when the device is incorporated into large systems. To study the dynamic thermal noise, the effect of the Langevin thermal field on the free layer’s magnetizationvector is incorporated into the Landau-Lifshitz-Gilbert equation. The switching energy, flipping delay, write, and hold error probabilities are studied, which are important metrics for nonvolatile memories, an important application of the straintronics magnetic tunneling junctions. VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4934566 ] I. INTRODUCTION The discovery of tunnel magnetoresistance (TMR) in magnetic tunneling junction (MTJ)1was followed by a pleth- ora of theoretical and practical studies in research labs to de-velop a new generation of nonvolatile magnetic memories, called magnetic random access memories (MRAM). 2–4 However, in the early stages, industry did not warmly wel- come MRAM, as the proposed methods for writing into the magnetic cell were energy hungry and area inefficient, fail- ing to compete with charge-based memories at the time.Field induced magnetization switching (FIMS), 2as the first proposed method for writing data into MTJ, relied on the magnetic field generated by the current flow in a neighboringwire. High requirements of static current in order to generate a strong magnetic field and the possibility of half-select errors were the main shortcomings of this method. The dis-covery of spin transfer torque (STT) for MTJ switching, 5 which relies on spin-polarized currents, revitalized MRAMresearch and development. 6–10STT is a much more energy efficient method than FIMS and is scalable with the comple- mentary metal-oxide-semiconductor (CMOS) integrated cir- cuits (IC).11,12 Both FIMS and STT, however, employ a flow of static current to achieve magnetization vector switching in the free layer of the MTJ. The use of static charge flow essentiallynullifies the inherent energy advantage of the magnetic logic (E min/C0charge¼N/C2Emin/C0magnetic ,Nbeing the number of charge carriers, and Emin/C0charge and Emin/C0magnetic being the minimum energy required to switch the state of a charge- based logic and magnetic logic, respectively13). In order to maximize the energy efficiency, the amount of chargeemployed for MTJ switching should be minimized. To this end, the recently-proposed straintronics principle, a combi- nation of piezoelectricity and magnetostriction, is an alterna-tive approach that overcomes the aforementionedobstacle. 14–18The amount of charge consumed for switching the MTJ’s state in straintronics is well below STT and FIMS.15,16 Temperature variations can severely impact both static and dynamic responses of straintronics devices. The former is affected due to the strong dependency of the saturationmagnetization, shape anisotropy, magnetocrystalline anisot-ropy, and magnetostriction coefficient on temperature. 19–22 While these parameters assume a fairly fixed value at low temperatures, when approaching the Curie temperature, TC, they fall dramatically, bringing the free layer close to a para- magnetic state. The energy barrier (EB) and, as a result, the critical flipping voltage of the free layer in the straintronicsdevice, are strong functions of temperature. It is specifically worthwhile to investigate the variations of the above parame- ters at temperature ranges between 200 K and 400 K, as thisis the operating range of a wide variety of integratedcircuits. 23 The dependency of the device’s dynamic response is realized by incorporating the Langevin thermal noise field,representing the thermal noise, into the Landau-Lifshitz- Gilbert (LLG) equation. The random noise field has three important impacts on the dynamic behavior: (i) it assistswith the magnetization vector’s flipping—without it, themagnetization will stagnate at relaxation state and will not respond to the applied stress; 16(ii) a larger thermal noise leads to larger fluctuations of the magnetization vector,resulting in a faster response and reducing the write error probability (WEP); and (iii) fluctuations can also lead to hold error probabilities (HEP), also known as retention a)Email: barangi@umich.edu 0021-8979/2015/118(17)/173902/11/$30.00 VC2015 AIP Publishing LLC 118, 173902-1JOURNAL OF APPLIED PHYSICS 118, 173902 (2015) errors, which are hazardous to straintronics-based MRAM design. Due to its crucial importance, the effect of thermal noise on the dynamic behavior of the magnetization in a nanomag-net has been the subject of study in the literature. 24–27A gen- eral study of the dynamics in a single domain magnet underLangevin thermal noise has been published previously, 24 providing a comprehensive statistical analysis on the mag-netization dynamics with and without the effect of externalmagnetic field. Analysis of the dynamics in strain-inducedmultiferroics has also been the subject of study recently. 25–27 These works mainly focus on the effect of dynamic thermal noise on the switching behavior of a single magnet understress and investigate the switching reliability under differentstress removal conditions. While the study of the thermalnoise is of significant importance, a comprehensive modelthat investigates the effect of temperature fluctuations andthermal noise on both static and dynamic behaviors of thestraintronics device has yet to exist. In this paper, we perform an in-depth analysis on the temperature dependency of the static and dynamic metrics ofthe straintronics MTJ. In search for the proper material forstraintronics-based integrated circuits, we investigate fourcommon magnetostrictive materials. The effect of theLangevin thermal field on the initial magnetization angle andthe delay metrics of the straintronics device, and the result-ing WEP and HEP are studied in detail. The flipping energyand the energy-delay trade-off for the straintronics-basedsystem design are analyzed. The rest of the paper is organ-ized as follows: Section IIintroduces the device architecture and the principle of operation; Section IIIintroduces the de- pendency of the magnetic anisotropies on temperature;Section IVanalyzes the energy barrier and the critical volt- age and their variations with temperature; Section Vintrodu- ces the Langevin field and its effect on the magnetizationvector’s dynamic response; Section VIpresents WEP, HEP, and energy-delay trade-off as important metrics for memorydesign; and Section VIIconcludes the paper. II. THE STRAINTRONICS DEVICE Strain-assisted switching and details of read and write operations of the device are studied in detail in previousworks. 15,16,28In this section, a brief introduction of the devi- ce’s architecture and the switching principle is provided. Thearchitecture of the straintronics MTJ is given in Fig. 1(a). The device is made by placing a piezoelectric layer (PZT) ontop of the magnetostrictive free layer of an MTJ. The deviceis shaped as a rectangle with the major and minor axes lyingalong the z-axis and y-axis, respectively, as demonstrated in Fig.1(a). The PZT is modeled as a parallel plate capacitance, and the MTJ is a variable resistor. The resistance of the MTJis a function of the relative orientation of the free layer’smagnetization vector compared to the magnetization of thepinned layer. The resistance assumes its minimum and maxi-mum in parallel and antiparallel orientations, respectively. The state of the MTJ can be read via sensing the resistance level by sending a small current through the free layer intothe MTJ. 29The PZT interface occupies most of the free layerto assure efficient transfer of the strain.29In this paper, a complete transfer of strain is assumed between the PZT and the free layer. A shift in the critical switching voltage of the straintronics device may result in the case of partial strain transfer. Nevertheless, such a shift will not affect the thermal analysis procedure used in this paper. The thickness of the free layer is chosen to be 10 nm and the major and minor axes are 120 nm and 80 nm, respectively. In the absence of an external stress, the intrinsic mag- netic energy of the free layer creates an EB between parallel (h¼0) and antiparallel ( h¼p) orientations as demonstrated in Fig. 1(a). Therefore, the magnetization vector prefers to stay along the major axis. An applied voltage across the straintronics device creates a strain in the PZT, which will be transferred to the free layer of the MTJ. The stress in the magnetostrictive free layer will reduce the magnetic energy barrier. A higher stress level can eliminate the energy barrier, forcing the magnetization vector to rotate and settle along the minor axis ( h¼p=2) as demonstrated in Fig. 1(b). Switching the state of the magnetization vector from the major axis to the minor axis, denoted as the write event, is, therefore, possible by applying a voltage across the device. This is the principle of straintronics magnetization switching. The use of voltage instead of current brings major energysavings to the table. This energy efficiency, however, comes at the expense of more complicated write algorithms and iterative methods in memory applications. 28The latter is because a write operation in MTJ-based memories consists of flipping of the magnetization from parallel to antiparallel orientation or vice versa, which is warranted in STT-based switching, but requires iterative methods in straintronics memories.28Nevertheless, the straintronics method still demonstrates remarkable advantages over STT switching when it comes to energy-delay products29(a metric to evalu- ate the tradeoff between energy and delay), making it a promising candidate for future memory applications. The static metrics of the discussed device, including the intrinsic and stress anisotropy energies, the energy barrier, and the critical voltage required for switching, are strongfunctions of temperature, which will be comprehensively discussed Sections IIIandIV. III. DEPENDENCY OF STATIC BEHAVIOR ON TEMPERATURE The magnetic energy of the straintronics device has three major components:15(i) shape anisotropy ( Esh), which is the tendency of the magnetization vector to settle along a certain direction due to the shape of the free layer; (ii) uniax- ial anisotropy ( Eu), also called magnetocrystalline anisot- ropy, which is primarily due to the spin-orbit interactions, and magnetizes the free layer in a certain direction; and (iii) stress anisotropy ( Er), which is due to the applied stress across the magnetostrictive free layer. Hence, the total mag- netic energy, Emag, of the free layer can be expressed as15 Emag¼EshþEuþEr; (1) where173902-2 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)Esh¼l0 2M2 sNshV; (2) Eu¼KuVsin2h; (3) Er¼3 2ksrVsin2hr: (4) In the above equations, l0is the permeability of vac- uum, Msis the saturation magnetization, Nshis the demag- netization factor, Kuis the uniaxial anisotropy coefficient, ks is the magnetostrictive expansion at saturation, ris the applied stress, Vis the free layer’s volume, and hris the angle of the magnetization vector with the minor axis. Itshould be noted that combining shape and uniaxial anisot-ropy energies gives intrinsic magnetic energy to the freelayer, while the stress anisotropy is the external magneto-strictive force that switches the state of the free layer usingthe straintronics principle. In order to study the static metrics of the straintronics device, the variations in the magnetic parameters and energylevels should be examined. 30Modeling the effect of temper- ature on the saturation magnetization of the free layer is ana-lyzed in the supplementary material. 31Next, we will inspect the temperature dependency of different terms in the totalmagnetic energy of the straintronics device. In this work, thedownfall of exchange interactions at temperatures close to Curie temperature is not accounted for. The latter can com- promise the single domain assumption of the nanomagnet attemperatures around T Cand should be handled with care whenever necessary. A. Shape anisotropy Shape anisotropy, as formulated in (2), is one of the major decision makers of the free layer’s energy barrier. From (2), the variations in M2 swith temperature can bepredicted using the Brillouin function.31However, the varia- tions of NshandVdue to thermal expansion should also be further investigated. The demagnetization factor, Nsh¼Nzzcos2hþNyysin2h sin2uþNxxsin2hcos2u, for the device in Fig. 1, assumes its maximum along the x-axis and its minimum along the y-axis. The parameters Nxx;Nyy;andNzzare obtained using the fol- lowing equations:15 Nzz¼p 4t a1/C01 4a/C0b a/C18/C19 /C03 16a/C0b a/C18/C192 ! ; (5) Nyy¼p 4t a1þ5 4a/C0b a/C18/C19 þ21 16a/C0b a/C18/C192 ! ; (6) Nxx¼1/C0ðNyyþNzzÞ; (7) where a,b, and tare the free layer’s major axis, minor axis, and thickness, respectively. Variations in temperature, T, will lead to compression or expansion. However, the relative ratios of t=aandða/C0bÞ=a, which are decision makers in (5) and(6), will stay constant, assuming a linear thermal expan- sion (DL=L¼aLDT,aLbeing the material’s expansion coef- ficient, LandDLbeing the length and change in length, respectively, and DTbeing the temperature variations). Lastly, due to the small value of aL, the variations in volume due to thermal expansion are negligible compared tothe changes in M sðTÞ. For example, Nickel exhibits merely 0:4% increase in its volume for every 100/C14increase in temperature. As a result of the above discussion, the shape anisotro- py’s dependency on temperature can be summarized as EshTðÞ Esh0¼MsTðÞ Ms0/C18/C192 ; (8) FIG. 1. (a) View of the straintronics device and the demonstration of the equivalent electrical model, GPandGAPare the conductances of the MTJ in parallel and antiparallel orientations; the resistance of the MTJ, RMTJis simply obtained as RMTJ¼1=GMTJ, (b) dynamic flipping of the magnetization vector under stress; the magnetization rotates and settles along the minor axis when the stress is retained between 5 ns and 15 ns.173902-3 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)where Esh0is the value of shape anisotropy at near-zero temperatures. B. Magnetocrystalline anisotropy According to Callen and Callen’s theory,32the depend- ence of the uniaxial anisotropy constant on temperature orig-inates from the changes in M sðTÞ, and can be expressed as32 KuTðÞ Ku0¼MsTðÞ Ms0/C18/C19m ; (9) where Ku0is the uniaxial anisotropy’s constant near absolute zero temperature. For cubic and uniaxial crystals,m¼3 and m¼10, respectively. 33Therefore, Nickel and Cobalt will have the powers of 3 and 10 in the above equa-tion, respectively. Although Callen and Callen’s theory predicts the tem- perature dependency of the magnetocrystalline anisotropyfairly well for pure element crystals, it is shown that it fails to predict the temperature dependency of K ufor alloys.33 Hence, the variations in the uniaxial coefficient for Galfenol and Terfenol-D should be investigated separately. Given the crystal structure of Galfenol ( Fe1/C0xGax;0:13 /C20x/C200:24), using the power m¼2:1 provides a fairly accu- rate estimation.34–37Terfenol-D ( ðTb;DyÞFe2), however, isconsidered as a rare-earth 3d-transition-metal alloy. For these alloys, the magnetic anisotropy transits through threedifferent phases: 38,39 (i) When the temperature of the alloy is below the spin reorientation temperature, TSR, the magnetic anisot- ropy follows the famous power law in (9), in which m¼lðlþn/C02Þ=ðn/C01Þ. For lowest order anisot- ropy l¼2, and assuming a planar model in which n¼2, we will have m¼4. The value of TSRfor Terfenol-D is /C24/C010/C14C,39,40which means that, up to this temperature, the power law is enforced. (ii) For the values above spin-reorientation temperature, the behavior is mostly dominated by the rare-earth elements and is given by38 KuTðÞ Ku0¼J2 SR nnþ2ðÞ k2T2; (10) where kis the Boltzmann constant and JSRis an alloy- dependent constant and can be obtained by assuminga continuous transition of K uðTÞat the spin reorienta- tion temperature. (iii) When the temperature approaches the Curie tempera- ture, (10) fails to predict the behavior. The behavior, at this point, can be expressed as38 KuTðÞ Ku0¼1/C0T TC: (11) By combining the three regions above, the uniaxial ani- sotropy of Terfenol-D can be predicted. Our simulations onthe magnetic anisotropy of Terfenol-D closely follow reportsin the literature. 40,41 Fig. 2contains the simulation data on the normalized variations of shape and uniaxial anisotropies, as the tempera- ture increases for four materials. The values are also re-plotted for a 200 K–400 K IC temperature range in Fig. 3, and the percentages of anisotropy reduction for the fourmaterials along with their magnetic properties 16,42–47used in our simulation model are listed in Table I. Dramatic reduc- tions of both shape anisotropy and uniaxial anisotropy revealthe critical influence of temperature on the device’s energy barrier, an important metric for non-volatile memory design. C. Magnetostriction expansion at saturation The magnetostriction expansion at saturation, ks, plays a major role in determining the critical stress required for FIG. 2. The dependency of shape and uniaxial anisotropies on temperature up to the Curie levels for different materials; as the Curie temperature is reached, the materials lose their intrinsic magnetic energies and approach a paramagnetic state. FIG. 3. Further demonstration of the(a) shape and (b) uniaxial anisotropies’ variations within 200 K and 400 K.173902-4 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)flipping the magnetization state of the straintronics device. The dependency of this parameter on temperature is expressed using the reduced hyperbolic Bessel function48,49 ksTðÞ ks0¼^I5 2uðÞ; (12) where coth ðuÞ/C01=u¼MsðTÞ=Ms0. The simulation results are plotted in Fig. 4for the four magnetostrictive materials. The simulation results are in fair accordance with thereported behavior in the literature. 21,48–51In fact, it is dem- onstrated that the hyperbolic Bessel function in molecular- field approximation holds accurately at all temperatures upto Curie temperature, 32while at low temperatures, the mag- netostriction coefficient follows the same power laws asmagnetic anisotropy. The percentages of variations in E rdue toksvariations, when the temperature rises from 200 K to 400 K, are tabulated in Table Ifor the sake of comparing dif- ferent materials. From the obtained metrics in Table I, it is understood that Cobalt and Galfenol show the least amount of variation in the temperature range of interest, while Nickel andTerfenol-D show dramatic variations in their magnetic pa-rameters. This is mainly due to the high Curie temperature ofCobalt and Galfenol, which might make them the preferredcandidates to be integrated into electronic circuits. Terfenol-D, although demonstrating fast response and low switchingvoltage, 15,16is not an ideal candidate for temperature- sensitive straintronics-based integrated circuits, as its mag- netic properties vary dramatically with temperature varia- tions, a phenomenon that frequently occurs in circuit chips. IV. ENERGY BARRIER AND CRITICAL FLIPPING VOLTAGE The energy barrier of the device, arising from its intrinsic shape and uniaxial anisotropies,15is a measure of the device’s immunity against the thermal noise and magnetic interfer- ences. Assuming the rotation of the magnetization vector within the y-zplane, which is enforced by shape anisotropy, the energy barrier is defined as EB¼Emagðh¼p=2Þ /C0Emagðh¼0ðorpÞÞ. From the discussions in Section III,i ti s naturally expected that the barrier will reduce as the tempera-ture increases due to the fall in the magnetic anisotropies. Thisis demonstrated in Fig. 5(a), where the energy barrier is plot- ted for Nickel as a function of temperature in the absence of stress. A contour map of the energy barrier’s graph is re-plotted in Fig. 5(b)to further demonstrate the energy behavior as a function of temperature. From the two graphs, the follow-ing conclusions can be drawn: (i) the intrinsic magneticenergy assumes its minimum in the parallel and antiparallelorientation and its maximum when the magnetization is ori- ented along the minor axis; (ii) the energy barrier reduces and eventually vanishes as the temperature approaches the Curielevel, where the material reaches a paramagnetic state; and(iii) the absolute value of the energy at any orientation of themagnetization vector (for example, at h¼0) also reduces as temperature increases. For example, from Fig. 5(a),a th¼0, the magnetic energy at near-zero temperature is eliminated asthe temperature approaches T C. It is particularly worthwhile to investigate the effect of stress and temperature on the device’s thermal stability,D¼EB=kT, which is an important data retention metric in non-volatile memory design. Usually, a thermal stability fac-tor larger than 40 is required for storage class memories. 52 The thermal stability of the straintronics device, with Galfenolas the magnetostrictive material, is demonstrated as a functionof temperature for different stress values in Fig. 6.I ti s observed that as the temperature merges with T C, a sharp reduction in the thermal stability is observed. Furthermore,increasing stress reduces the thermal stability linearly, whichis expected from (1)and(4). In general, it is observed that Galfenol keeps its thermal stability well above 40, within a 200 K to 400 K temperature range, even at stress values closerto its critical stress ( r C/C25180 MPa for Galfenol in our simulations). Lastly, the effect of temperature on the minimum volt- age required for the magnetization flipping, also called the critical voltage, VC, should be analyzed. By equating anisot- ropy energies, it is concluded that VC¼l0 2M2 sNyy/C0Nzz ðÞ þKu/C18/C19 tPZT 3 2ksYd31; (13)TABLE I. Materials’ properties and the percentage of reduction in shape, uniaxial, and stress energies of different magnetostrictive materials when thetemperature is raised from 200 K to 400 K. Nickel Cobalt Terfenol-D Galfenol M s(kA/m) 510 1400 912 1340 Ku(kJ/m3) 12 16 1.6 5 jksj(ppm) 20 20 600 200 TC(K) 627 1400 652 972 Esh(%) 21.7 0.4 18.8 5.8 Eu(%) 31.6 1.8 59.7 6.1 Er(%) 30.8 0.6 26.6 8.2 FIG. 4. The dependency of the magnetostriction coefficient on temperature as predicted by the hyperbolic Bessel function.173902-5 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)where tPZTis the thickness of the PZT, Yis the Young Modulus of the free layer, and d31is PZT’s dielectric coeffi- cient. The dependency of VCon temperature is simulated in Fig.7for different magnetostrictive materials. By observing the graphs closely, the critical voltage goes through two dif-ferent slope phases as the temperature increases. First, at lowtemperatures, V Cslightly reduces as temperature increases. Then, an increase in the value of the critical voltage is observed at higher temperatures. This behavior can be ana- lyzed by taking the derivative of (13) with respect to temperature dVC dT¼AdM s dTþBdKu dT/C0Cdks dT 3 2ksYd31/C18/C192; (14a) A¼3 2ksYd31l0Nyy/C0Nzz ðÞ MstPZT; (14b) B¼3 2ksYd31KutPZT; (14c) C¼3 2Yd31l0 2M2 sNyy/C0Nzz ðÞ þKu/C18/C19 tPZT: (14d)The saturation magnetization starts degrading at lower temperatures compared to the magnetostriction coefficient.As a result, when T/C28T C, we have dVC=dT<0, and a slight reduction of the critical voltage is observed. This behavior ismore noticeable for Cobalt on the graphs, mainly due to its high M sand very low ks. As the temperature rises, ksstarts decreasing according to (12), while MsandKucontinue to fall as predicted by the saturation magnetization’s behavior31 and(12)–(14), respectively. When the slope of dks=dTis large enough to fulfill A/C2dMs=dTþB/C2dKu=dT/C0C /C2dks=dT>0, the critical voltage will begin to rise. From the inset of Fig. 7, it is also concluded that Galfenol and Cobalt keep their critical voltage at a fairlyconstant level, while Terfenol-D and Nickel show almost 7%and 4% increases in V Cwithin a 200 K to 400 K temperature range, respectively. This can come in handy when consider- ing a straintronics-based system design for temperature-sensitive applications. FIG. 5. (a) and (b) The dependency of the energy barrier of Nickel on temper- ature; as the temperature rises, both the energy barrier and the absolute values of energy are reduced. FIG. 6. The dependency of thermal stability of Galfenol on temperature and applied stress; the graph shows two fast regions: (i) at low temperatures, where the parameter kTrises, and (ii) at temperatures close to TC, where the energy barrier approaches zero. FIG. 7. The dependency of the critical flipping voltage on temperatures upto the Curie levels for four magnetostrictive materials; the variations within 200 K–400 K are demonstrated in the inset of the figure, showing that the four materials maintain an almost-constant critical voltage within the range of interest; the results are normalized to V C0, the critical flipping voltage near absolute zero temperature.173902-6 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)V. DYNAMIC THERMAL NOISE FIELD The dynamic response of the magnetization vector in a straintronics device is predicted using the LLG equation given by53 dM dt¼/C0c0 1þa2 ðÞ~M/C2~H ðÞ /C0c0 Ms/C2aþ1 a/C18/C19 ~M/C2~M/C2~H ðÞ ; (15) where Mis the magnetization vector, c0is the gyromagnetic ratio, ais the Gilbert damping factor, and H¼Hr^rþHh^h þHu^uis the net magnetic field due to shape and uniaxial anisotropies and the applied stress.15The effect of thermal noise is modeled by following the same procedure developed by Brown54and Grinstein.55The thermal flux density can be incorporated in (15)by including the Langevin thermal noise field, HN, in the total magnetic field; i.e., Htot¼HþHN, where HNis a Gaussian random noise field variable31with a strength of D¼2kTa=l0c0MsV, and a correlation of hHiðx;tÞHjðx0;t0Þi ¼ Ddij/C2dðx/C0x0Þdðt/C0t0Þ: (16) Therefore, the thermal noise field to be incorporated in (15)can be expressed as HN;i¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2akT l0c0MsVs XitðÞ i¼x;y;zðÞ ; (17) where XiðtÞ’s are uncorrelated zero-mean unit-variance Gaussian random variables in the direction of Cartesianaxes. The relative ratio of the thermal noise field to the net magnetic field of the device (i.e., H N/C0rms=H) can be simu- lated to observe the strength of the thermal noise. It is expected that as we increase the stress level, the net magnetic field forcing the magnetization vector to stay along the easyaxis ( Hh) becomes weaker.31It is also shown31that as we increase the stress, the value of Hu, which forces the magnet- ization to stay in plane (within the y-zplane of Fig. 1(a)), increases slightly. Therefore, an increase in stress increases HN/C0rms=Hhas demonstrated in Fig. 8, allowing the magnet- ization to fluctuate more easily around the easy axis. As thestress approaches its critical value, the thermal noisebecomes significantly stronger owing to the fact that 31 limr!rCHh¼0. It is also observed from Fig. 8that as we increase the stress, HN/C0rms=Huslightly reduces. This means that while the magnetization vector’s fluctuations around themajor axis ( z-axis) increase at higher stress levels, its tend- ency to stay within the y-z plane increases slightly, leading to more in-plane fluctuations. The flipping delay of the straintronics device (also called the alignment delay in some works) is a strong function ofthe initial magnetization angle, h i, which is mainly due to the thermally stimulated agitations. It is shown that the initialmagnetization angle has a zero-mean Gaussian distributionwith the strength of 56 hi/C0rms¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kT l0VM sHs : (18) Due to the dependency of the flipping delay on the ini- tial magnetization angle, Gaussian fluctuations of hilead to variations in the flipping delay, td. This is demonstrated in Fig. 9, where our thermally-incorporated model based on (15)–(17) is simulated at room temperature. The dynamic waveforms of the magnetization flipping for N¼200 sam- ples and the resulting histogram for the flipping delays aredemonstrated. The results indicate an average delay of 197ps with a standard deviation of 52 ps. The delay histogram isslightly skewed due to the lower limit on the flipping delay. Fig.10illustrates dependency of h i/C0rmson temperature. As the temperature increases and approaches the Curie level, it is expected that the fluctuations increase since H!0a s temperature approaches TC. By plotting the value of hi/C0rms between 200 K and 400 K in Fig. 10, it is observed that Nickel and Terfenol-D demonstrate more fluctuations mainlyowing to their lower T Cvalues. The higher fluctuations will FIG. 8. The effect of stress on the relative strength of the thermal noise; as the stress increases, HN=Hhrises, leading to more fluctuations around the z-axis, while HN=Hudecreases slightly (inset), increasing the magnetization vector’s tendency to stay within the y-zplane. FIG. 9. Due to the random nature of the initial angle, the flipping delay varies with a skewed Gaussian distribution as demonstrated in the inset of the figure; at room temperature, the mean value of the delay is observed to be 197ps with merely 52ps of standard deviation; the left inset is the voltage pulse, applied at t¼1 ns, and the right inset shows the histogram of the delay values on 200 plotted dynamic waveforms.173902-7 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)assist with the easier flipping of the magnetization vector. Another parameter that can dramatically alter the value of hi/C0rmsis the applied stress, as demonstrated in Fig. 11(a) .A s the stress levels reach their critical value for the four simu-lated materials, the initial angle approaches the value of p=2, owing to the stress-reduced energy barrier. From the basics of the straintronics principle, it is expected that whenr>r C, the magnetization settles along the minor axis, where h¼p=2 and the magnetization vector will now fluctu- ate around this axis. The dynamic waveforms and histogramsof the magnetization’s fluctuations around the major axis along with their histograms at different stress levels below critical stress are also plotted in Fig. 11(b) . The dependency of the flipping delay on h i/C0rmsis simu- lated and demonstrated in Fig. 12for temperature ranges between 200 K and 400 K. As we increase the temperature,the value of h i/C0rmsincreases, leading to easier magnetizationflipping and, therefore, a lower delay. The analytical data on the graph are the expected results from (18) and the simu- lated data are obtained from our Verilog-A model based on the thermally incorporated LLG dynamics in (15)–(17). The accuracy of the developed model can also be confirmed bycomparing the analytical and simulated results. The flipping delay of different materials, besides depending on the initial angle, is a strong function of theapplied voltage (and therefore stress) across the straintronics device. In our previous work, 15we simulated different mate- rials’ flipping delay as a function of the applied voltagewhile assuming the same thermal noise for all the materials. Here, we analyzed the voltage dependency while including the materially-dependent thermal noise. The four materialsare simulated at room temperature and the results are recorded in Fig. 13, where it is observed that Terfenol-D has a very fast response owing to its high h i/C0rms(as expected from Fig. 10) and ks, while Cobalt shows a slow response due to its low hi/C0rmsandks. Nickel, although demonstrating a higher initial angle in Fig. 10, fails to compete with Galfenol and Terfenol-D due to its low ks. This confirms the FIG. 10. The dependency of the initial magnetization angle on temperature; a higher temperature leads to more fluctuations due to the higher thermal noise. FIG. 11. (a) The dependency of the initial magnetization angle on the applied stress; as the stress approaches the critical values, the initial angle a pproaches p=2, as predicted by the stress anisotropy (b) dynamic waveforms and histograms of the initial angle of Galfenol for different stress levels, showing mu ch larger fluctuations at high stress values. FIG. 12. Simulations results on Galfenol, showing the dependency of the initial angle and flipping delay on temperature along with the analytical datafrom (18); as temperature rises, the initial angle increases and the delay decreases slightly.173902-8 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)suitability of Galfenol for integrated circuit applications due to its low critical flipping voltage, low flipping delay, andlow variations of static features across temperatures between 200 K and 400 K as discussed earlier in Section III. VI. TEMPERATURE DEPENDENCY OF DYNAMIC METRICS In Section VIIof this paper, some of the important met- rics related to non-volatile memory design, an importantapplication of straintronics devices, will be discussed. Theeffect of thermal noise and temperature variations on WEPand the speed-WEP trade-off will be analyzed. A writemethod that improves the energy and performance of thestraintronics-based memories will be proposed. The effect ofstress on the flipping delay and the HEP of the device will beanalyzed in order to investigate the reliability and advantagesof the proposed method. A. Write error probability One of the important obstacles in memory design is the probability of write error during the write operation, abbrevi-ated as WEP. Consider any memory with a certain write pul-sewidth, demonstrated in the inset of Fig. 14(a) . The duration of the pulsewidth indicates the write speed of thememory. If a higher speed is desired, the pulsewidth can bereduced. However, since the speed of writing in any memory cell is limited, there is a lower bound, beyond which the pul-sewidth cannot be reduced. This lower bound is usually selected according to the memory’s write error tolerance. For example, consider the straintronics device of Fig. 1(a). The application of a pulse with an amplitude higher than V C will force the magnetization vector to settle along the minor axis ( h¼p=2) as demonstrated in Fig. 1(b). Due to the ran- dom nature of the Langevin thermal noise, the flipping delaycan take a range of values as demonstrated in Fig. 14(a) . Write error is associated with cases, where the delay ishigher than the write pulsewidth, in which the magnetizationvector will fail to flip. Due to the Gaussian distribution of the flipping delay, demonstrated in Fig. 9, the WEP is expected to reduce signif- icantly as we increase the write pulsewidth, which is demon-strated in Fig. 14(b) . On the other hand, a longer pulsewidth is associated with a slower memory. Therefore, there is atrade-off between speed and WEP. As can be seen in thegraphs, a reduced write speed from 0.2 ns to 0.4 ns leads tomore than 1000X lower WEP at room temperature. In mem- ory applications, the pulsewidth does not need to be increased further than the system’s WEP requirements. The effect of temperature on WEP can also be observed in Fig. 14(b) , where we simulated Galfenol for different pulsewidths at different temperatures. A lower WEP athigher temperatures is mainly due to the increased h i/C0rms from 200 K to 400 K, as expected from (18). B. A proposed write method, the energy-performance trade-off, and hold error probability When it comes to memory design, energy and perform- ance are two of the most important metrics. A considerableamount of research has been going on to reduce the writeenergy while retaining the speed of the MTJ-basedmemories. 57–60 The switching energy, associated with the flipping of the straintronics device, can be formulated as16 E¼CPZTDV2þEd; (19) where CPZTis the capacitance of the piezoelectric layer, DV is the voltage swing across the device, and Edis the dissi- pated energy due to the Gilbert damping.61For the devices with high energy barriers, the critical voltage is high enough FIG. 13. Flipping delay for different magnetostrictive materials as a function of applied voltage’s amplitude, showing the significant effect of high stresson flipping time of the nanomagnet. FIG. 14. Dynamic waveforms for Galfenol, demonstrating the possibility of write error due to late flipping; the inset of the figure shows the voltage pulse, applied at t¼1 ns, and (b) WEP as a function of pulsewidth and tem- perature; it is evident that as the pulse-width is increased, the WEP decreases dramatically; increasing temperature will also reduce the WEP slightly for a given pulsewidth due to the depend- ency of the initial angle of temperature in(18).173902-9 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)to assure that the capacitive switching will consume the ma- jority of the total switching energy. The switching energy can be significantly reduced if the voltage swing across the device is reduced, as demonstrated in the inset of Fig. 15. Increasing the value of Vlowto the levels closer to VChas two main advantages: (i) as DV¼Vhigh/C0Vlowreduces, the capacitive switching energy will drop as demonstrated forGalfenol in Fig. 15, where we fixed V highslightly above VC and started sweeping Vlowfrom 0 to VC. When Vlow/C25VC, the capacitive switching will consume negligible energy; (ii) the flipping delay will reduce as Vlowincreases as demon- strated in Fig. 15. The latter is expected since a higher Vlow will create some stress across the device, reducing the energy barrier and increasing hi/C0rmsaccording to (18). Therefore, a higher Vlowleads to a higher hi/C0rms, which is associated with a faster flipping. This is further demonstrated in Fig. 16, where the delay histograms are plotted. The mean of the dis- tributions moves towards smaller delays when the value of Vlowis raised. Note that in the simulations of Figs. 15and 16,Vhighis set to be slightly higher than VC. Should the value of Vhighbe increased, the delay will reduced signifi- cantly, as already discussed in Section V.In order to analyze the reliability of the proposed method, we simulated the HEP of our straintronics device asan important data retention related property for non-volatilememories. It is expected that as we increase V low, the HEP will reduce due to the increased thermal noise fluctuations.This phenomenon is demonstrated in Fig. 17, where we increased V lowto values close to VCand plotted the resulting HEP of the device in two cases. First, we only assumed thepresence of the Langevin thermal noise, and then weincluded 1% fluctuations of the applied V low, which can fre- quently happened due to clock feedthrough in the ICs.62In the first case, the HEP is negligible as long as Vlowis kept below 0 :97VC. In the second case, the HEP is noticeably higher compared to the first case, but reduces to negligiblevalues as V lowgoes below 0 :95VC. In the above simulations, the possibility of dimension changes due to process variations is not considered. Assumingthat the effect of process variations on the device’s dimensionsis included, the value of V lowwill decrease accordingly. In any event, from the above discussions, it can be concludedthat reducing the voltage swing while retaining the value of V lowreliably below VCwill increase the energy efficiency and performance of the system while providing enough noise mar-gin to keep the HEP well below the system’s error tolerance.Nevertheless, it should be noted that the value of HEP is astrong function of the device’s energy barrier. Should theenergy barrier be decreased, the value of HEP in Fig. 17will increase. VII. CONCLUSION A comprehensive study was performed on the effect of temperature on the magnetic properties of the straintronicsdevice. The effect of temperature variations up to the Curietemperature on the energy barrier and the critical voltage ofthe device were analyzed. Four different magnetostrictivematerials were simulated in order to provide a comprehen- sive platform for different applications. The effect of thermal noise was examined by incorporating the Langevin randomfield into the LLG equation and investigating the effect oftemperature and stress on the initial magnetization angle. FIG. 15. By increasing the value of Vlowcloser to the critical voltage of Galfenol, the capacitive switching energy and flipping delay decrease. FIG. 16. Histograms of the flipping delays, demonstrating the reduction inthe flipping delay due to higher V low. FIG. 17. HEP as a function of Vlowin the presence of thermal noise only, and in the presence of both thermal noise and 1% voltage node fluctuations.173902-10 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)Lastly, the effect of temperature and thermal noise on some of the important metrics for the nonvolatile memory design was studied and an energy efficient write method was intro-duced that can reliably reduce the capacitive switching energy and the flipping delay of the straintronics devices. ACKNOWLEDGMENTS This work was done partially under NSF NEB Grant No. ECCS-1124714 (PT106594-SC103006) and partially under AFOSR Grant No. FA9550-12-1-0402. 1M. Julliere, Phys. Lett. A 54, 225 (1975). 2D. Tang, P. Wang, V. Speriosu, S. Le, and K. Kung, IEEE Trans. Magn. 31, 3206 (1995). 3T. Kenneth, D. Denny, and W. P. Kang, U.S. patent 5,343,422 (1994). 4Y. Zheng and J.-G. Zhu, IEEE Trans. Magn. 32, 4237 (1996). 5J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 6R. Scheuerlein, in IEEE International Solid-State Circuits Conference (San Fransisco, Feb 9, 2000). 7Z. Li and S. Zhang, Phys. Rev. B 68, 024404 (2003). 8Y. Huai, F. Albert, P. Nguyen, M. Pakala, and T. 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Rev. 130, 1677 (1963). 55G. Grinstein and R. Koch, Phys. Rev. Lett. 90, 207201 (2003). 56S. Manipatruni, D. Nikonov, and I. Young, IEEE Trans. Circuits Syst. I 59, 2801 (2012). 57R. Heindl, W. Rippard, S. Russek, and A. Kos, Phys. Rev. B 83, 054430 (2011). 58Z. Sun, X. Bi, H. Li, W. Wong, and X. Zhu, IEEE Trans. VLSI Syst. 22, 1281 (2014). 59W. Wang, M. Li, S. Hageman, and C. Chien, Nature Mater. 11, 64 (2011). 60W. Wen, Y. Zhang, Y. Chen, Y. Wang, and Y. Xie, IEEE Trans. Comput.- Aided Des. Integr. Circuits Syst. 33, 1644 (2014). 61B. Behin-Aein et al. ,IEEE Trans. Nanotechnol. 8, 505 (2009). 62J. Rabaey, A. Chandrakasan, and B. Nikolic /C19,Digital Integrated Circuits (Pearson Education, Upper Saddle River, NJ, 2003).173902-11 M. Barangi and P . Mazumder J. Appl. Phys. 118, 173902 (2015)

1.370414.pdf

Observation of ferromagnetic resonance in the time domain R. J. Hicken and J. Wu Citation: Journal of Applied Physics 85, 4580 (1999); doi: 10.1063/1.370414 View online: http://dx.doi.org/10.1063/1.370414 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/85/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Ultrafast laser excitation of spin waves and the permanent modification of the exchange bias interaction in Ir Mn ∕ Co J. Appl. Phys. 103, 07C104 (2008); 10.1063/1.2830013 Ferromagnetic resonance study of polycrystalline cobalt ultrathin films J. Appl. Phys. 99, 08N503 (2006); 10.1063/1.2151832 Multiple ferromagnetic resonance in mesoscopic permalloy rings J. Appl. Phys. 97, 10A712 (2005); 10.1063/1.1851932 Use of microscale coplanar striplines with indium tin oxide windows in optical ferromagnetic resonance measurements J. Appl. Phys. 97, 10R304 (2005); 10.1063/1.1849071 Kerr detected time average of magnetization precession in ferromagnetic resonance Appl. Phys. Lett. 86, 112506 (2005); 10.1063/1.1883326 [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: 131.181.251.130 On: Sat, 22 Nov 2014 12:07:01Instrumentation and Measurement I Steve Arnold, Chairman Observation of ferromagnetic resonance in the time domain R. J. Hicken and J. Wu School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom Optical pump–probe spectroscopy has been used to observe damped ferromagnetic resonance ~FMR !oscillations in thin film Fe samples. The FMR was pumped by magnetic field pulses generated by an optically triggered photoconductive switch, and probed by means of time resolvedmeasurements of the magneto-optical Kerr rotation. The photoconductive switch structure consistedof a parallel wire transmission line, of 125 mm track width and separation, defined on a semi-insulating GaAs substrate. The biased transmission line was optically gated at one end so thata current pulse propagated along the transmission line to where the sample had been overlaid. Themagneticfieldassociatedwiththecurrentpulseisspatiallynonuniform.Byfocusingtheprobebeamonthesampleatdifferentpointsabovethetransmissionlinetheeffectoftheorientationofthepumpfieldhasbeenstudied.ThegyroscopicmotionofthemagnetizationhasbeenmodeledbysolvingtheLandau–Lifshitz–Gilbert equation and the magneto-optical response of the sample has beencalculated by taking account of both the longitudinal and polar Kerr effects. The calculated andmeasured magneto-optical Kerr rotations are found to be in reasonable agreement. © 1999 American Institute of Physics. @S0021-8979 ~99!78608-1 # I. INTRODUCTION It has recently been shown that optical pump–probe spectroscopy can be used to observe dynamical magneticprocesses in thin film ferromagnetic samples on picosecondtime scales. 1–3The sample is pumped by a magnetic field pulse generated by a photoconductive switch and the re-sponse of the magnetization is probed by means of a timeresolved measurement of the magneto-optical Kerr rotation.In this article we explore how the orientation of the pulsedmagnetic field affects the subsequent motion of the magne-tization and hence the observed magneto-optical response ofthe sample. II. EXPERIMENT The optical pump–probe apparatus used to observe the FMR is shown in schematic form in Fig. 1. A mode-lockedTi:Al 2O3laser was used to generate 100 fs pulses at a wave- length of 800 nm, a repetition rate of 82 MHz, and an aver-age power of about 800 mW. Each pulse was divided intopump and probe parts by a 90/10 beam splitter and thep-polarized probe beam was intensity stabilized 4to 0.1%. The pump and probe beams were focused with lenses offocal length 15 cm and 8 cm, respectively, giving a probespot diameter of approximately 50 mm. The time delay be- tween the pump and probe pulses at the sample was variedby reflecting the pump beam from a corner cube reflectormounted on a stepper motor-driven translation stage. Thesample was placed on top of a photoconductive switch struc-ture such as that shown in Fig. 2. This was then mounted ona small translation stage so that the position of the probe spoton the sample could be easily adjusted. An electromagnetwas used to apply a static magnetic field either parallel or perpendicular to the plane of incidence of the probe beam atthe sample. The detector consisted of a Glan–Thompson po-larizing beam splitter and two photodiode detectors thatformed an optical bridge. The magneto-optical Kerr effect~MOKE !causes the reflected probe beam to be elliptically polarized. The difference of the two photodiode outputs isproportional to the rotation of the major axis of the ellipse.Light from a 633 nm HeNe laser was added to the mainbeam to aid alignment of the apparatus. FIG. 1. Schematic representation of the optical pump–probe apparatus.JOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 8 15 APRIL 1999 4580 0021-8979/99/85(8)/4580/3/$15.00 © 1999 American Institute of Physics [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: 131.181.251.130 On: Sat, 22 Nov 2014 12:07:01The design of the photoconductive switch structure is shown in Fig. 1 and is similar to that used in Refs. 1 and 2.A Au transmission line of 125 mm track width and separa- tion, and thickness of order 0.1 mm, was lithographically defined on an intrinsic semi-insulating GaAs substrate withlateral dimensions of approximately 2 31 cm. The switch was triggered by focusing the pump beam on the open end ofthe transmission line. The rise time of the current is expectedto be of the order of 1 ps ~Ref. 2 !while the decay time is determined by the recombination time of the GaAs substrate.A bias voltage of amplitude 20 V and frequency 2 kHz wasapplied to the transmission line. Figure 3 ~a!shows the results of current autocorrelation measurements. 1The average cur- rent in the sense line was measured after being gated with theprobe beam. In addition an oscilloscope 5was connected across a surface mount resistor in the main line in order tomonitor the shape of the current pulses and confirm that thefocused pump spot was in the optimum position. The risetime of the trace is limited by the bandwidth of the oscillo-scope. Fits to the current autocorrelation measurements andthe oscilloscope trace indicate that after the first few picosec-onds the time dependent current is well described by theexpression I5I 0exp(2t/t), where I0andtare approxi- mately equal to 20 mA and 2 ns, respectively. The largevalue of tmeans that the autocorrelation curve is rather in- sensitive to the rise time of the current and so its value can-not be reliably determined. All current autocorrelation andtime resolved Kerr rotation measurements were made in aphase sensitive manner using the transmission line bias volt-age as a reference. The sample chosen for the present study was a 500 Å film of Fe grown by electron beam evaporation, at a back-ground pressure of 10 26mbar, onto a glass substrate and capped with a 300 Å layer of Au. The sample was placedface down on the transmission line structure and probedthrough the substrate. A piece of insulating tape of thickness60 mm was placed between the sample and the switch to prevent the metallic sample from short circuiting the trans-mission line. III. THEORY The motion of the sample magnetization may be de- scribed by the Landau–Lifshitz–Gilbert equation of motion]M ]t52uguM3Heff1a MSM3]M ]tD, ~1! whereMis the magnetization vector, gis the magneto me- chanical ratio, and ais the damping constant. The total ef- fective magnetic field Heffincludes the static applied mag- netic field, the pulsed magnetic field h, and the thin film demagnetizing field. The equation of motion may be solvednumerically after being cast into a suitable form. 6 The magneto-optical Kerr rotation of the reflected probe beam is dependent upon the components of the magnetiza-tion in the plane of incidence through the longitudinal andpolar Kerr effects. Zak et al.have calculated expressions for the complex reflection coefficients r ppandrspat the inter- face between a magnetic medium and a nonmagnetic me-dium for the case that the magnetization is canted relative tothe interface. 7By combining the solution of Eq. ~1!with Eqs. ~19!and~22!of Ref. 7 we are able to calculate the time dependent Kerr rotation. This simple treatment may be ap-plied to the present study because the thickness of the Fe filmis large compared to the optical skin depth, and because mul-tiple reflections in the glass substrate may be ignored due tothe small value of the reflection coefficient at the air–glassinterface. IV. RESULTS Measurements were performed with a static magnetic field of 1.04 kOe applied along the xaxis in Fig. 2, i.e., parallel to the length of the transmission line and perpendicu-lar to the plane of incidence. The probe beam was incidenton the glass substrate at an angle of approximately 20°. Thefocused spot was scanned across the transmission line andthe time dependent Kerr rotation recorded. It was necessaryto subtract a background signal from the rotation data. Webelieve that the background is due to motion of the sample.This occurs because the static magnetic field exerts a forceon the alternating current in the transmission line. The lifetime of an individual current pulse is long com- pared to the time taken for the pulse to travel the length ofthe transmission line. The magnetic field associated with thecurrent can therefore be calculated by assuming a uniformcurrent density in the Au tracks and using the value of thecurrent described in Sec. II. The pulsed magnetic field vectorlies perpendicular to the length of the transmission line. At aheight of 60 mm above the center of one of the Au tracks we FIG. 2. The photoconductive switch structure is shown in schematic form. FIG. 3. ~a!A typical current autocorrelation signal. ~b!An oscilloscope trace indicating the time dependence of the current in the main line.4581 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 R. J. Hicken and J. Wu [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: 131.181.251.130 On: Sat, 22 Nov 2014 12:07:01obtain peak field values of uhyu50.48Oe and uhzu 50.12Oe, while at the same height above the center of the transmission line structure we obtain values of 0 and 0.53Oe. The time dependent Kerr rotations measured for thesetwo probe spot positions correspond to traces ~a!and~c!in Fig. 4. Traces ~b!and~d!were calculated by assuming: the values of the pulsed magnetic field above; bulk values of2.09 and 1710 emu/cm 3for thegfactor and the magnetiza- tion of the Fe film; a value of 0.1 for the damping constant a; a value of 1.55 for the refractive index of glass; values of3.0313.69iand 0.0575 20.00768ifor the refractive index 8 and the magneto-optic constant9of Fe; and an infinitesimally short rise time for the pulsed magnetic field. While the rela-tive phase of the two experimental curves is expected to becorrect, the absolute phase could not be accurately deter-mined. Therefore the experimental data has been shifted soas to align the first maxima in curves ~a!and~b!in Fig. 4. We see that the amplitude and frequency of the calculatedcurves are in reasonable agreement with the experimentaldata. The frequency of oscillation in the experimental data isapproximately 13 GHz while in the calculation a frequencyof 14.2 GHz is obtained as expected from the expression v 5gAH(H14pM). Better agreement between calculation and experiment might be obtained by varying the values ofthe material parameters and by allowing the pump field tohave a finite rise time as in Ref. 3. The calculated yandzcomponents of the vector u5M/Mhave been plotted in Fig. 5. The maximum devia- tion of the magnetization from the equilibrium position isseen to be an order of magnitude larger in Fig. 5 ~a!where the direction of the pulsed magnetic field vector lies close to theplane of the sample. The Kerr rotation depends upon u y through the longitudinal Kerr effect and on uzthrough the polar Kerr effect. Although the polar Kerr effect is an orderof magnitude larger than the longitudinal Kerr effect in Fe,u yis an order of magnitude larger than uzin Fig. 5 ~a!and so uyanduzmake contributions of similar magnitude to the total Kerr rotation in Fig. 4 ~a!. However in Fig. 5 ~b!uyand uzare of similar magnitude and so uzdominates the total rotation signal.V. SUMMARY In conclusion, we have observed FMR oscillations from a thin Fe film in the time domain. A photoconductive switchwas used to apply the pump magnetic field both parallel andperpendicular to the plane of the sample. A larger deviationof the magnetization was achieved with the in-plane pumpand we have found that it is essential to take account of boththe longitudinal and polar Kerr effects when calculating themagneto-optical response of the sample. The calculated Kerrrotation is in reasonable agreement with the experimentalcurves given that the values of a large number of materialparameters have been assumed. ACKNOWLEDGMENTS The authors gratefully acknowledge the support of the EPSRC and the LSF laser loan pool, the fabrication ofsamples by D. J. Jarvis, and discussions with Dr. C. D. H.Williams and Dr. T. W. Preist. 1M. R. Freeman, R. R. Ruf, and R. J. Gambino, IEEE Trans. Magn. 27, 4840 ~1991!. 2M. R. Freeman, M. J. Brady, and J. Smyth, Appl. Phys. Lett. 60, 2555 ~1992!. 3W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 ~1997!. 4J. A. C. Bland, M. J. Padgett, R. J. Butcher, and N. Bett, J. Phys. E 22, 308 ~1989!. 5Tektronix DSA 601A with an 11A72 plug-in. 6L. He and W. D. Doyle, J. Appl. Phys. 79, 6489 ~1996!. 7J. Zak, E. R. Moog, C. Liu, and S. D. Bader, J. Appl. Phys. 68, 4203 ~1990!. 8Handbook of Chemistry and Physics, 75th ed. ~Chemical Rubber, Boca Raton, 1994 !. 9G. S. Krinchik and V. A. Artem’ev, Sov. Phys. JETP 26, 1080 ~1968!. FIG. 4. The measured and calculated time dependent Kerr rotations are plotted for the case that the probe spot is: ~a!,~b!halfway between the two Au tracks; ~c!,~d!above the center of a Au track. FIG. 5. The trajectories of the unit vector uare plotted. Panels ~a!and~b! correspond to the traces ~b!and~d!, respectively, in Fig. 4.4582 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 R. J. Hicken and J. Wu [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: 131.181.251.130 On: Sat, 22 Nov 2014 12:07:01

1.4967170.pdf

UV excitations of halons Ljiljana Stojanović, , Abdulrahman O. Alyoubi , Saadullah G. Aziz , Rifaat H. Hilal , and Mario Barbatti, Citation: J. Chem. Phys. 145, 184306 (2016); doi: 10.1063/1.4967170 View online: http://dx.doi.org/10.1063/1.4967170 View Table of Contents: http://aip.scitation.org/toc/jcp/145/18 Published by the American Institute of Physics THE JOURNAL OF CHEMICAL PHYSICS 145, 184306 (2016) UV excitations of halons Ljiljana Stojanović,1,a)Abdulrahman O. Alyoubi,2Saadullah G. Aziz,2Rifaat H. Hilal,2,3 and Mario Barbatti1,b) 1Aix Marseille Univ, CNRS, ICR, Marseille, France 2Chemistry Department, Faculty of Science, King Abdulaziz University, Jeddah B.O. 208203, Saudi Arabia 3Chemistry Department, Faculty of Science, Cairo University, Giza, Egypt (Received 19 August 2016; accepted 19 October 2016; published online 11 November 2016) In the present study, we examined the UV excitations of a newly introduced molecular set, Halons-9, composed of nine gaseous halon molecules. The performance of the density functional-based multi- reference configuration interaction method (DFT /MRCI) and time-dependent density functional theory with CAM-B3LYP functional (TD-CAM-B3LYP) in the computation of singlet and triplet excited states of this set was evaluated against coupled-cluster with singles and doubles (CCSD). Excited states up to the corresponding ionization limits, including both localized and delocalized exci- tations, have been benchmarked. TD-CAM-B3LYP significantly underestimates excitation energies of the higher mixed valence-Rydberg and Rydberg states, with computed mean absolute deviations from the equation of motion (EOM)-CCSD results 1.06 and 0.76 eV , respectively. DFT /MRCI gives a significantly better description of higher excited states, albeit still poor, compared to the TD-CAM-B3LYP. The mean absolute deviations of mixed valence-Rydberg and Rydberg states from the reference EOM-CCSD values are 0.66 and 0.47 eV , respectively. The performance of DFT/MRCI for description of strongly correlated states with valence-Rydberg mixing is still not satisfactory enough. On the other hand, oscillator strengths of most of singlet states obtained with both methods are close to the EOM-CCSD values. The largest deviations, occurring in the case of several high-lying multiconfigurational states, are of an order of magnitude. Published by AIP Publishing. [http: //dx.doi.org /10.1063 /1.4967170] I. INTRODUCTION One of the main challenges in quantum chemistry is the reliable prediction of excited-state properties. Numerous theoretical methods for computation of excited states have been introduced and assessed, among them are several popular single-reference methods (e.g., the linear-response time-dependent density functional theory (TD-DFT)1,2and coupled-cluster-based methods), as well as multi-reference methods (e.g., the complete active space second-order perturbation theory (CASPT2)3and the multi-reference configuration interaction (MRCI)4method). However, select- ing an appropriate electronic structure method for excited state computation in a particular case, in terms of accuracy and computational cost, is not a straightforward problem. A number of computational studies have been devoted to performance evaluation of di fferent quantum chemistry methods for prediction of various molecular properties on different molecular sets.5–11Since the focus of this study is on excited states, we here mention few well-established molecular sets, which stand out among numerous sets used in method assessments. The molecular set by Thiel and his co-workers,5an assembly of 28 small representative organic molecules, is one of the most widely used molecular sets. It has been employed the for evaluation of excited states of a)Electronic mail: stojanovicmljiljana@gmail.com b)Electronic mail: mario.barbatti@univ-amu.frseveral multi- and single-reference methods.5–8Furthermore, Grimme and his co-workers introduced several molecular sets.9–12The first of them9comprises 14 middle-sized to large molecules, including organic and inorganic molecules, where the studied excited states included the lowest-lying localized and Rydberg states. The molecular set composed of 14 small- to medium-sized molecules, introduced by Leang et al. , was used for the benchmarking of TD-DFT with various density functionals.13Winter et al. introduced a molecular set of 66 organic molecules, including porphine derivatives, polycyclic aromatic hydrocarbons, heterocyclic organic compounds, and aromatic alcohols, and assessed several single-reference methods for the excited-state computations.14 By inspecting these and other prominent molecular sets, we noticed that a large field of halo-organic compounds remained not properly covered by any of them. This motivated us to introduce a new molecular set, Halons-9, composed of nine halon molecules: bromomethane (CH 3Br), dichloroflu- oromethane (CHCl 2F), bromodifluoromethane (CHBrF 2), bromochlorodifluoro-methane (CBrClF 2), dichlorodifluoro- methane (CCl 2F2), bromotrifluoromethane (CBrF 3), tetraflu- oromethane (CF 4), 1,2-dichlorotetrafluoroethane (C 2Cl2F4), and hexafluoroethane (C 2F6). Among all halon molecules, we have chosen to investigate the excited states of only those which are gaseous under standard conditions of temperature and pressure, primarily because of their potential importance in atmospheric chemistry. The structures of the studied molecules alongside their names according to the special 0021-9606/2016/145(18)/184306/13/$30.00 145, 184306-1 Published by AIP Publishing. 184306-2 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) FIG. 1. The molecular structures of the nine studied halon molecules (Halons-9 molecular set). halons’ naming convention are shown in Figure 1. To our knowledge, none of the existing molecular sets included any of these molecules. Apart from CH 3Br, whose low-lying excited states are examined in several studies,15–17excited states of the other molecules in the set have not yet been studied computationally. Nevertheless, there are numerous experimental studies of the UV spectra of halons (for CH 3Br Ref. 18 and the references cited herein; for CBrF 3Refs. 19–21 and the references cited herein; for CF 4Refs. 22 and 23; for CHCl 2F Ref. 24; for CHBrF 2Ref. 21; for CBrClF 2Refs. 25, 21, and 26; for CCl 2F2Refs. 24 and 27; for C 2F6Ref. 28; for C2Cl2F4Ref. 29). This omission may impact the evaluation of computa- tional methods and properties by introducing a bias against halogen compounds. We particularly felt that in recent simulations of halo-compounds of atmospheric interest,30–34 where we could not count on performance evaluation for any of the target molecules. Our aim in this study is to start to close this knowledge gap, first, by proposing a clear set of molecules to be tested, and second, by evaluating them using few popular methods. As a first study, we limited ourselves to the investigation of spin-free vertical excitation energies for the whole series. Important topics such as adiabatic excitation energies and spin-orbit coupling will be left for further studies. We probed the performance of two DFT-based quantum chemical methods against equation of motion coupled-cluster with singles and doubles (EOM- CCSD)35–38in the computations of excited states of these systems.We focus on the performance of density functional theory-based multi-reference configuration interaction method (DFT /MRCI)39and TD-DFT in combination with CAM- B3LYP functional40,41for the description of excited states of halons. The 240 excited states included in our study comprise a small number of valence excited states in low-energy regions and numerous highly correlated Rydberg and mixed valence-Rydberg states in high-energy regions converging to the ionization limits. TD-DFT stands out as one of the most popular computational methods due to its good compromise between accuracy and computational cost. Several di fferent flavors of density functionals have been developed, and their performance has been benchmarked against high-level ab initio or experimental data so far, probing their performance for a range of excited states, from localized to delocalized ones.6,42–46One of the main drawbacks of TD-DFT is the description of systems with a pronounced multi-reference character of ground state. Moreover, within the usual linear- response and adiabatic approximations, TD-DFT is incapable of reproducing doubly excited states.47 DFT/MRCI overcomes a large number of shortcomings of TD-DFT because of its multi-reference character. It is better suited to treat highly correlated systems and to capture the excited states of double-excited character. The DFT/MRCI method has been tested for a number of organic molecules6,13,39,48,49and transition metal complexes,50applied for simulation of UV /vis spectra (e.g., Refs. 48, 51, and 52), and recently assessed for the computation of spin-orbit 184306-3 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) coupling in organic molecules.53It has been shown that it is able to reproduce excitation energies with errors within few tenths of an eV . However, most of the available benchmark studies of excited states with DFT /MRCI are usually focused on valence excited states of n π* orππ* character, and do not include higher states characterized by σbond excitations, valence-Rydberg mixing, and Rydberg states. EOM-CCSD computations of excited states are taken as the reference data for the comparison with DFT /MRCI and TD-DFT results. Although it is one of the most reliable methods available for routine excited state calculations, we should mention that, just like TD-DFT, it is not able to describe multi-reference and double excitations. Therefore, concerning these properties, we should take DFT /MRCI as the reference. II. METHODS AND COMPUTATIONAL DETAILS The ground state molecular structures of the nine molecules in the Halons-9 molecular set have been optimized using Møller-Plesset perturbation theory up to the second order (MP2)54–57with def2-TZVPP basis set58without symmetry constraints. First ionization potentials (IPs) of the studied molecules are computed at their optimized structures using coupled cluster with singles and doubles (CCSD) method59–62 with the triple-zeta def2-TZVPP basis set, as the di fference between the ground state energy of the Nand the N−1 electron systems. Cartesian coordinates for the optimized structures are given in the supplementary material (Tables I-IX) and vertical ionization potentials (IPs) are given in Table I. Vertical excitation energies of singlet and triplet excited states up to around the ionization limits are computed at the optimized structures at three di fferent levels of theory— EOM-CCSD,35–38DFT/MRCI,39and linear response TD-DFT. Since excited states with pure Rydberg or mixed valence-Rydberg character are expected to occur among the computed excited states, TD-DFT computations are performed employing range-separated (rs) hybrid Coulomb attenuated method CAM-B3LYP functional.40,41As it is a well-known63generalized gradient approximation (GGA) and hybrid exchange-correlation (XC) functionals usually describe accurately local excitations, with an accuracy of few tenths of an eV . However, they significantly underestimate excitation TABLE I. Vertical ionization potentials (IP in eV) computed on CCSD /def2- TZVPP level at the MP2 /def2-TZVPP optimized geometries and negative val- ues of HOMO energies ( −EHOMO in eV) obtained with CAM-B3LYP /def2- TZVPP of the Halons-9 molecular set. Molecules I P(eV) −EHOMO (eV) CH3Br 10.54 9.29 CHCl 2F 11.86 10.54 CHBrF 2 11.44 10.24 CBrClF 2 11.48 10.29 CCl 2F2 12.21 10.87 CBrF 3 11.68 8.70 CF4 16.60 14.24 C2Cl2F4 12.60 11.24 C2F6 14.67 12.99energies of non-local states, such as Rydberg or charge- transfer excitations,64with the typical errors of order of an eV . The origin of this problem is already very well-known—it lies in an incorrect description of long-range exchange functionals with GGA and hybrid functionals.65–67This problem can be overcome using range-separated functionals, which describe short- and long-range exchange functionals by di fferent terms. One of the most widely used rs functionals, CAM-B3LYP, comprises 0.19 Hartree-Fock (HF)-like and 0.81 Becke 1988 (B88) GGA exchange at short range; the long-range region is represented by 0.65 HF-like plus 0.35 B88 exchange; and the transition between short- and long-range regions is modulated by an error function with parameter 0.33 a−1 0.40Assessed for the computation of excitation energies of Rydberg and charge transfer states, CAM-B3LYP generally shows good performance.68,69 To analyze the delocalization of the excited states obtained on the TD-CAM-B3LYP level, we employ the Λdiagnostic, introduced by Peach et al. ,70as a method to quantify the degree of delocalization of states within the TD-DFT method. TheΛvalues are computed as sums of spatial overlaps between all occupied and virtual orbital pairs involved in TD-DFT excitations, weighted by their contributions to the specific excited state. These contributions are obtained by solving Casida’s equations.70TheΛvalue is a dimensionless number which takes a value between 0 and 1; small Λvalues signify excitations with strong delocalized characters, whereas large values signify local excitations. All presented TD-DFT computations were performed with GAMESS-US code.71,72 We tested the influence of size of the basis set on excited state energies of CF 4by comparing their values obtained with TD-CAM-B3LYP with the def2-TZVPP and quadruple- zeta def2-QZVPP58bases (see Table X in the supplementary material). This brief test indicates that basis set e ffects may be significant in the Halons-9 set, even for basis sets as large as def2-TZVPP. The def2-QZVPP basis set decreases the vertical excitation energies of CF 4molecule. The root mean square deviation of vertical excitation energies obtained using def2-TZVPP from those ones obtained with def2-QZVPP set is 0.35 eV . Since we are primarily interested to compare the performances of di fferent electronic structure methods for the same basis set, we decided to use smaller def2-TZVPP set in all computations. The basic idea of the combined density functional theory /multi-reference configuration interaction (DFT /MRCI) method is to perform DFT computations of a closed- shell reference ground state and to use the obtained one- particle Kohn-Sham (KS) orbitals as a basis to subsequently build configuration state functions (CSFs) with short MRCI expansions.39,73In the CI matrix, the Hamiltonian matrix elements computed between the CSFs with the same space and same spin parts (diagonal elements) are the most important because a CI matrix is usually diagonal-dominant. They account for most of the dynamical electronic correlation included via DFT. On the other hand, the o ff-diagonal Hamiltonian matrix elements are computed between CSFs with same space but di fferent spin parts or between CSFs with one-electron or two-electron di fferences in space parts. In DFT /MRCI o ff-diagonal matrix elements are scaled using 184306-4 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) exponential damping function, which depends on the energy difference between diagonal elements of two CSFs. This scaling is introduced to avoid large extent of double counting of electron correlation which is already included via diagonal terms. It also ensures e fficient selection of energetically close CSFs, whose interaction gives rise to non-dynamic correlation effects which are not taken into account within DFT. A few semi-empirical screening parameters are employed in the DFT/MRCI Hamiltonian, including parameters used for fitting the exponential damping function for scaling procedure and parameters used for the screening of two-electron integrals in diagonal matrix elements. In our DFT /MRCI computations, the reference configura- tions were initially generated by single and double excitations of ten active electrons within the active space composed of ten orbitals. We discard the configurations with energies above 1 hartree. The number of created configurations spans from 946 in the case of CF 4molecule to 10 139 in the case of C 2Cl2F4molecule. We employ the original sets of five empirical parameters for singlet and triplet excited states for the DFT /MRCI Hamiltonian,39,74which are available in combination with the BH-LYP functional.75 The initial DFT calculations are performed with TURBOMOLE program,76and the subsequent MRCI computations are done with the MRCI code developed by Grimme and Waletzke39and recently updated by Lyskov and co-workers.73The two-electron integrals in DFT /MRCI computations were approximated using the resolution-of-the- identity approximation with auxiliary basis set76from the TURBOMOLE library. The spin-restricted EOM-CCSD computations and the MP2 geometry optimizations are performed with Gaussian 09 program.78 III. RESULTS AND DISCUSSION A. Vertical excitations Computed vertical excitation energies, state assignments, and their Λvalues are compiled in Table II for the singlet and in Table III for the triplet excited states. Oscillator strengths for the singlet transitions are given as well. For the sake of brevity, we only show the singlet excited states with oscillator strengths larger than 0.001 and only the first 5 triplet excited states for each molecule. Complete tables including all computed excited states are given in the supplementary material (Tables XI and XII). The assignment of the excited states computed with the applied methods is done by analyzing the nature of orbitals involved in the main excitations contributing to each particular state. The states computed with di fferent methods were correlated according to their determined types and weights of main configurations. In the case of several high-lying multiconfigurational states, correlation of the excited states computed with EOM-CCSD and the DFT-based methods according to these criteria is non-trivial due to a pronounced configuration mixing within EOM-CCSD, which is not particularly well reproduced with DFT /MRCI and TD-CAM-B3LYP methods. In these cases, the comparison ofthe oscillator strength values and the symmetry representation of the states were also helpful for the assignment. Weights of the main configurations obtained with EOM-CCSD are given in the tables in the cases where it was possible to determine them. For numerous states, however, arising from the excitations between mixed orbitals (represented as linear combinations of di fferent orbital types, for instance n σ* and Rydberg orbitals), the configuration weights are not computed. According to the DFT /MRCI results, all computed states are single-reference and singly excited, with negligible contributions of doubly excited configurations (less than 5 %). This implies that all states obtained with DFT /MRCI could in principle be also computed with EOM-CCSD and TD- DFT methods, which are unsuited to capture states with predominantly multi-reference and doubly excited character. Based on the assignment, the benchmark set of 240 excited states was split into 64 localized valence states, 91 Rydberg states, and 85 states of mixed valence- Rydberg character, and into 110 singlet and 130 triplet states. The correlation graphs between the TD-CAM- B3LYP and DFT /MRCI vertical excitation energies with the corresponding EOM-CCSD values are shown in Figure 2, separately for singlet and triplet excited states. Some of the features of the computed states are noticeable from first sight. Both DFT /MRCI and TD-CAM-B3LYP underestimate excitation energies of all computed states. DFT /MRCI excitation energies lie in between EOM-CCSD and TD-CAM- B3LYP values, significantly closer to the EOM-CCSD values. The standard deviation (SD) of DFT /MRCI vertical excitation energies from the corresponding EOM-CCSD values is 0.49 for singlet and 0.51 for triplet states, whereas for TD-CAM- B3LYP these values are 0.72 and 0.79, respectively. Both methods describe slightly better singlet than triplet states. For most of the molecules, several pure valence excited singlet and triplet states occur in the low energy region. They are usually of n σ* type, but there are also several states of σσ* or mixed n σ∗ −σσ* type. These states emerge upon excitations from the highest HOMO orbitals of non-bonding orσtype to the lowest-lying LUMO and LUMO +1 orbitals. In these cases, the lowest-lying LUMO orbitals are antibonding σ*(C–Br) or σ*(C–Cl). Exceptions to this behavior are CF 4 and C 2F6molecules, where the LUMO orbitals are Rydberg ones. This induces the lowest-lying excited states of CF 4 and C 2F6to be Rydberg. Since the energies of the σ*(C–F) orbitals are higher than the energies of the Rydberg orbitals, states of pure n σ* andσσ* character do not occur, whereas states with mixed Rydberg and n σ* configurations occur only in the upper parts of their spectra, contrary to the other molecules in the set (Tables XI and XII of the supplementary material). In all cases, the medium part of the excited state spectrum is characterized by numerous multiconfigurational excited states featured by considerable valence-Rydberg mixing. The highest excited states are mostly purely Rydberg states. In several cases, few delocalized states of n σ* character appear in the upper part of the spectrum. These states arise from excitations from the compact non-bonding orbitals of fluorine to the σ*(C–Br) or σ*(C–Cl) orbitals, which have a small spatial overlap. 184306-5 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) TABLE II. Computed vertical excitation energies (eV), oscillator strengths, assignment of the states, and Λvalues of the singlet excited states for the Halons-9 molecular set. Only states with oscillator strength above 0.001 are shown. EOM-CCSD DFT /MRCI TD-CAM-B3LYP State Ev f Configurations Weights Ev f E v f Λ CH3Br S0 0.00 . . . gs 1.00 0.00 . . . 0.00 . . . . . . S1 6.64 0.0020 nσ∗+nRydb 0.62, 0.22 6.37 0.0002 6.39 0.0024 0.532 S2 6.64 0.0020 nσ*+nRydb 0.62, 0.22 6.37 0.0002 6.39 0.0024 0.527 S3 8.91 0.0087 nσ∗+nRydb 0.20, 0.69 8.50 0.0077 8.50 0.0063 0.375 S4 8.91 0.0087 nσ∗+nRydb 0.20, 0.69 8.50 0.0077 8.50 0.0063 0.382 S6 10.40 0.0031 nRydb 0.88 9.79 0.0014 9.87 0.0023 0.278 S7 10.40 0.0031 nRydb 0.88 9.80 0.0015 9.87 0.0023 0.348 S8 10.41 0.0020 nRydb+nσ∗ 0.77, 0.10 9.82 0.1241 9.89 0.0198 0.367 S9 10.59 0.8266 σσ∗+nRydb 0.63 10.15 0.9426 10.32 0.7346 0.676 CBrF 3 S0 0.00 . . . gs 1.00 0.00 . . . 0.00 . . . . . . S1 6.84 0.0000 nσ∗ 0.90 6.68 0.0014 6.54 0.0001 0.578 S2 6.84 0.0000 nσ∗ 0.90 6.68 0.0014 6.54 0.0001 0.580 S3 10.33 0.2644 σσ∗+nRydb 0.71, 0.16 10.03 0.2605 9.79 0.2075 0.686 S4 11.34 0.1550 nRydb 0.84 10.88 0.1401 10.78 0.0889 0.412 S5 11.34 0.1550 nRydb 0.84 10.88 0.1401 10.78 0.0889 0.406 S7 11.48 0.0040 nRydb 0.77 10.92 0.0489 10.83 0.0350 0.358 S8 11.48 0.0040 nRydb 0.77 10.92 0.0489 10.83 0.0350 0.358 CF4 S0 0.00 . . . gs 1.00 0.00 . . . 0.00 . . . . . . S1 14.23 0.0000 ns 0.75 14.41 0.0000 13.51 0.0000 0.405 S2 14.23 0.0000 ns 0.75 14.41 0.0000 13.51 0.0000 0.368 S3 14.23 0.0000 ns 0.75 14.41 0.0000 13.51 0.0000 0.392 S4 15.57 0.0000 nRydb+nσ∗ 0.68, 0.13 15.17 0.0003 14.78 0.0000 0.455 S5 15.57 0.0000 nRydb+nσ∗ 0.68, 0.13 15.17 0.0003 14.78 0.0000 0.407 S6 15.65 0.2455 nRydb 0.68 15.24 0.2060 14.83 0.2103 0.461 S7 15.65 0.2455 nRydb 0.68 15.24 0.2060 14.83 0.2103 0.476 S8 15.65 0.2455 nRydb 0.68 15.24 0.2060 14.83 0.2103 0.473 CHCl 2F S0 0.00 . . . gs 1.00 0.00 . . . 0.00 . . . . . . S1 7.58 0.0159 nσ∗ 0.82 7.19 0.0149 7.27 0.0147 0.596 S3 7.93 0.0038 nσ∗+nRydb 7.53 0.0028 7.61 0.0005 0.563 S4 8.35 0.0007 nσ∗+nRydb 7.95 0.0013 8.06 0.0005 0.574 S5 9.52 0.0017 nRydb+nσ∗ 8.96 0.0011 8.83 0.0002 0.565 S6 9.81 0.0569 nRydb+nσ∗ 9.29 0.0758 9.12 0.0389 0.584 S7 10.05 0.0156 nσ∗ 9.44 0.0084 9.24 0.0085 0.562 S8 10.23 0.0056 nσ∗ 9.69 0.0044 9.60 0.0025 0.376 S9 10.54 0.0104 nσ∗ 9.96 0.0542 9.78 0.0128 0.515 S10 10.58 0.0129 nσ∗ 10.04 0.0243 9.94 0.0095 0.469 S11 10.64 0.0348 nσ∗ 10.12 0.0028 9.97 0.0095 0.489 S12 10.72 0.0011 nσ∗ 10.19 0.0064 10.04 0.0006 0.381 S13 10.86 0.0026 nσ∗ 10.39 0.0042 10.26 0.0029 0.483 S14 10.98 0.1553 nσ∗ 10.47 0.2121 10.32 0.0091 0.425 S15 11.17 0.0125 nσ∗+σσ∗ 10.64 0.0157 10.36 0.1059 0.554 S16 11.41 0.0381 nσ∗+σσ∗ 10.85 0.0434 10.70 0.0271 0.412 S17 11.62 0.7122 nσ∗+σσ∗ 11.01 0.9102 10.98 0.4998 0.489 CHBrF 2 S0 0.00 . . . gs 1.00 0.00 . . . 0.00 . . . . . . S1 6.62 0.0012 nσ* 0.81 6.39 0.0022 6.34 0.0014 0.565 S3 9.27 0.2670 σσ* 0.71 8.97 0.2732 8.72 0.2130 0.662 S4 10.22 0.0317 nσ* 0.80 9.53 0.0340 9.60 0.0273 0.318 S5 10.27 0.0146 nσ* 0.81 9.58 0.0148 9.62 0.0157 0.238 S6 10.33 0.0114 nσ* 0.82 9.67 0.0013 9.70 0.0037 0.327 S7 10.63 0.1441 nσ* 0.64 10.00 0.1853 10.00 0.0905 0.465 S8 11.24 0.0107 nσ∗+σσ∗+nRydb 10.56 0.0109 10.83 0.0793 0.453 S9 11.32 0.0170 nσ∗+σσ∗+nRydb 11.09 0.0018 10.61 0.0110 0.301 S10 11.33 0.0424 nσ∗+σσ* 10.77 0.1722 10.53 0.0025 0.460 184306-6 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) TABLE II. (Continued. ) EOM-CCSD DFT /MRCI TD-CAM-B3LYP State Ev f Configurations Weights Ev f E v f Λ CBrClF 2 S0 0.00 . . . gs 1.00 0.00 . . . 0.00 . . . . . . S1 6.32 0.0095 nσ* 0.86 6.02 0.0088 6.02 0.0089 0.602 S4 7.86 0.0247 nσ* 0.81 7.52 0.0206 7.48 0.0196 0.580 S5 9.16 0.0846 nσ* 0.77 8.35 0.0172 8.35 0.0157 0.524 S6 9.50 0.0012 nσ* 0.84 8.57 0.0000 8.59 0.0005 0.404 S7 9.50 0.1985 nσ∗+σσ* 9.05 0.3428 8.84 0.1751 0.639 S9 10.13 0.0554 nσ* 0.72 9.47 0.0840 9.37 0.0778 0.580 S11 10.98 0.1863 nσ∗+nRydb+σσ* 10.34 0.3504 10.31 0.1084 0.586 S12 11.16 0.1566 nσ∗+nRydb+σσ* 10.62 0.0226 10.37 0.0893 0.563 S13 11.22 0.0915 nσ∗+nRydb+σσ* 10.76 0.0782 10.49 0.0008 0.246 S14 11.36 0.0905 ns 0.82 10.79 0.0815 10.48 0.0589 0.457 CCl 2F2 S0 0.00 . . . gs 1.00 0.00 . . . 0.00 . . . . . . S1 7.49 0.0171 nσ* 0.88 7.13 0.0146 7.10 0.0150 0.611 S6 10.30 0.0011 nσ* 0.84 9.62 0.0000 9.27 0.0005 0.568 S8 10.31 0.2091 (n+σ)σ∗+nRydb 0.79, 0.08 9.97 0.2481 9.62 0.1128 0.624 S9 10.89 0.0442 nσ∗+nRydb 0.68, 0.16 10.23 0.1152 10.04 0.0675 0.583 S11 11.68 0.0150 (n+σ)σ+nRydb 0.41, 0.40 11.21 0.0089 10.89 0.0159 0.682 S12 11.82 0.1841 nRydb+nσ* 0.69, 0.10 11.22 0.3770 11.00 0.1927 0.537 S13 11.93 0.1942 nRydb 0.81 11.40 0.0110 11.05 0.0026 0.260 S15 12.21 0.0040 nRydb 0.74 11.63 0.0000 11.27 0.0052 0.616 C2F6 S0 0.00 . . . gs 1.00 0.00 . . . 0.00 . . . S1 12.66 0.0427 (n+σ)σ* 0.84 12.49 0.0663 11.89 0.0354 0.631 S2 12.66 0.0427 (n+σ)σ* 0.84 12.49 0.0661 11.89 0.0354 0.643 S4 14.11 0.0278 nσ* 0.76 13.77 0.0182 13.11 0.0249 0.495 S5 14.11 0.0278 nσ* 0.76 13.78 0.0220 13.11 0.0249 0.477 S8 14.31 0.2143 nσ∗+ns 0.58, 0.14 13.92 0.0000 13.36 0.0000 0.410 C2Cl2F4 S0 0.00 . . . gs 1.00 0.00 . . . 0.00 . . . . . . S1 8.29 0.0009 nσ* 7.81 0.0003 7.88 0.0006 0.582 S3 8.38 0.0020 nσ* 7.91 0.0033 8.01 0.0001 0.600 S5 10.34 0.0863 nσ∗+σσ* 9.9 0.0715 9.39 0.0456 0.609 S6 10.38 0.1039 nσ∗+σσ* 9.84 0.0909 9.45 0.0502 0.604 S7 10.75 0.0034 nRydb 10.05 0.0024 9.65 0.0025 0.588 S8 10.87 0.0052 nRydb 10.14 0.0044 9.68 0.0004 0.596 S9 11.09 0.0462 nRydb 10.46 0.0676 9.98 0.0364 0.606 S10 11.17 0.0447 nRydb 10.52 0.0577 10.18 0.0344 0.622 S11 11.37 0.0020 nRydb 10.77 0.0040 10.41 0.0045 0.427 S13 11.62 0.0016 nRydb+nσ* 10.92 0.0144 10.73 0.0030 0.498 S14 11.66 0.0183 nRydb 10.97 0.0115 10.82 0.0007 0.459 S15 11.89 0.0003 nRydb 11.12 0.0004 10.89 0.0037 0.552 S16 11.95 0.0076 nRydb 11.25 0.0196 10.93 0.0061 0.551 S17 12.06 0.0052 nRydb+nσ* 11.28 0.0342 11.12 0.0017 0.431 S19 12.19 0.0053 nRydb 11.72 0.0225 11.38 0.0046 0.537 S20 12.50 0.1489 nRydb 12.02 0.1056 11.44 0.0001 0.342 This pattern occurring in the excited state spectrum could be explained by the energy ordering of molecular orbitals, common to all molecules, σ(C–F)< σ (C–Cl)< σ (C–Br) <n(F)<n(Cl)<n(Br)< σ*(C–Br)< σ*(C–Cl)<Rydberg < σ (C–F). The compositions of the lowest Rydberg orbitals vary, but in most of the cases they are represented as linear combinations of several atomic orbitals rather than as a pure one. Based on this ordering of orbitals, it could be predicted that in the molecules containing C–Cl and C–Br bonds, thelowest-lying excited states would be of n σ* character, while in molecules containing only C–F and C–C bonds, the lowest lying excited states would be Rydberg states. B. TD-DFT results A summary of statistical evaluation including mean errors (MEs), mean absolute errors (MAEs), standard deviations (SDs), maximum and minimum errors for TD-CAM-B3LYP, 184306-7 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) TABLE III. Computed vertical excitation energies (eV), assignment of the states, and Λvalues of the triplet excited states for the Halons-9 molecular set. EOM-CCSD DFT /MRCITD-CAM- B3LYP State Ev Configurations Weights Ev Ev Λ CH3Br T1 6.01 nσ* 0.87 5.70 5.62 0.529 T2 6.01 nσ* 0.87 5.70 5.62 0.529 T3 8.05 σσ* 0.80 7.91 7.76 0.702 T4 8.73 nσ* 0.89 8.27 8.27 0.377 T5 8.73 nσ* 0.89 8.27 8.27 0.384 CBrF 3 T1 6.08 nσ* 0.90 5.87 5.65 0.575 T2 6.08 nσ* 0.90 5.87 5.65 0.578 T3 7.91 σσ* 0.87 7.85 7.52 0.725 T4 10.76 np 0.76 10.29 10.20 0.539 T5 10.90 ns 0.80 10.46 10.28 0.407 CF4 T1 13.87 ns+nσ* 0.60, 0.11 13.60 12.95 0.415 T2 13.87 ns+nσ* 0.60, 0.11 13.60 12.95 0.384 T3 13.87 ns+nσ* 0.60, 0.11 13.60 12.95 0.391 T4 14.57 nRydb+nσ* 0.54, 0.31 14.23 13.74 0.445 T5 14.57 nRydb+nσ* 0.54, 0.31 14.23 13.74 0.470 CHCl 2F T1 6.79 nσ* 6.38 6.30 0.591 T2 7.14 nσ* 6.71 6.63 0.612 T3 7.17 nσ* 6.74 6.68 0.564 T4 7.46 nσ* 7.04 6.99 0.571 T5 8.71 nσ*+σσ* 8.39 8.14 0.571 CHBrF 2 T1 5.95 nσ* 0.82 5.66 5.53 0.564 T2 5.96 nσ* 0.81 5.69 5.55 0.543 T3 7.16 σσ* 0.86 7.06 6.72 0.693 T4 9.91 nσ* 0.76 9.25 9.33 0.348 T5 9.92 nσ* 0.78 9.28 9.34 0.450 CBrClF 2 T1 5.61 nσ* 0.97 5.27 5.19 0.601 T2 5.77 nσ* 0.86 5.45 5.35 0.549 T3 6.89 nσ* 0.83 6.44 6.44 0.591 T4 7.06 nσ* 0.89 6.57 6.56 0.465 T5 7.63 nσ*+σσ* 7.41 7.18 0.618 CCl 2F2 T1 6.66 nσ* 0.90 6.29 6.16 0.610 T2 7.00 nσ* 0.90 6.61 6.49 0.567 T3 7.28 nσ* 0.90 6.88 6.79 0.567 T4 7.42 nσ* 0.87 7.00 6.93 0.597 T5 8.70 nσ*+σσ* 8.52 8.21 0.597 C2F6 T1 11.70 (n+σ)σ* 0.89 11.42 10.82 0.633 T2 11.70 (n+σ)σ* 0.89 11.42 10.82 0.645 T3 13.29 ns 0.65 12.91 12.35 0.471 T4 13.46 nσ*+ns 0.49, 0.11 13.08 12.61 0.491 T5 13.45 nσ* 0.67 13.11 12.45 0.483 C2Cl2F4 T1 7.41 nσ* 6.96 6.91 0.587 T2 7.45 nσ* 7.00 6.95 0.598 T3 7.48 nσ* 7.01 6.97 0.600 T4 7.49 nσ* 7.03 6.99 0.586 T5 9.00 nσ*+σσ* 8.72 8.37 0.652 184306-8 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) FIG. 2. Correlation diagrams of vertical excitation energies (in eV) for the Halons-9 molecular set obtained with DFT /MRCI and TD-CAM-B3LYP methods, with vertical excitation energies obtained with EOM-CCSD method. and DFT /MRCI results with respect to EOM-CCSD results is given in Table IV and Figure 3, separately for valence, mixed valence-Rydberg, Rydberg, and all studies states. Comparing TD-CAM-B3LYP vertical excitation energies with the corresponding EOM-CCSD values, it could be noticed that among all the excited states, valence states are described the best on TD-CAM-B3LYP level. Their MAE from the EOM-CCSD results is 0.58 eV . Analyzing the valence state compositions, we find that the lowest-lying valence states, arising upon excitations from the highest occupied non- TABLE IV . Mean error (ME), mean absolute error (MAE), standard devia- tion (SD), and maximum and minimum deviations (in eV) of the DFT /MRCI and TD-CAM-B3LYP excitation energies from EOM-CCSD results for the Halons-9 molecular set. TD-CAM- B3LYPDFT- MRCI Valence (64 states) ME −0.58 −0.44 MAE 0.58 0.44 SD 0.63 0.48 Max. error 1.10 1.02 Min. error 0.27 0.06 Mixed (85 states) ME −1.06 −0.66 MAE 1.06 0.66 SD 0.80 0.51 Max. error 1.33 0.80 Min. error 0.28 0.18 Rydberg (91 states) ME −0.76 −0.47 MAE 0.76 0.47 SD 0.85 0.54 Max. error 1.20 0.80 Min. error 0.51 0.19 Total (240 states) ME −0.82 −0.53 MAE 0.82 0.53 SD 0.78 0.51 Max. error 1.33 1.02 Min. error 0.27 0.06bonding orbitals to the LUMO orbital, are described fairly well. The MAE of their vertical excitation energies from EOM- CCSD values is 0.38 eV for singlet (18 states) and 0.48 eV for triplet (20) excited states. These states are mostly n σ* states represented mainly by a single configuration (Tables I and II in the main text and also Tables XI and XII in the supplementary material). The description of the higher excited states deteriorates in the upper region of the spectrum. In several cases, the ordering of the states obtained with TD-CAM-B3LYP di ffers from that obtained with EOM-CCSD. The excitation energies of higher valence, mixed valence-Rydberg, and Rydberg states are severely underestimated. The high-energy n σ* valence excited states occurring in CCl 2F2, CBrClF 2, and C2Cl2F4are placed significantly lower compared to the EOM- CCSD results (their maximum deviation is 1.1 eV). The most dominant configurations contributing to these states involve transitions from the highest HOMO orbitals to LUMO +1 FIG. 3. The mean absolute errors of the vertical excitation energies from the corresponding EOM-CCSD values (in eV) of valence, mixed valence- Rydberg, Rydberg, and all studied states of the Halons-9 molecular set obtained with TD-CAM-B3LYP and DFT /MRCI. 184306-9 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) orbitals, which is the σ*(C–Cl) orbital (in the cases of CCl 2F2 and C 2Cl2F4this is the higher σ*(C–Cl) orbital). The excitation energies of the pure Rydberg states are also systematically underestimated compared to the EOM- CCSD results, with MAE of 0.76 eV and maximum error of 1.20 eV . We employ the Λdiagnostic to analyze the correlation between the degree of delocalization of the states with dominant Rydberg character and the errors of their excitation energies. The obtained Λvalues of the studied Rydberg states are in the range between 0.2 and 0.65 (Figure 1 in the supplementary material). Analyzing the deviations of TD-CAM-B3LYP excitation energies of Rydberg states against the corresponding Λvalues for Rydberg excited states, it could be noticed that in the cases of three molecules (C 2F6, CBrF 3, and C 2Cl2F4) there is no correlation between excitation energy errors and Λvalues. In the majority of cases, however, the deviations of the excitation energies are larger in the cases of more delocalized states, which are characterized by smaller Λvalues. The excitation energy errors for most of the molecules slowly reduce upon increasing of Λvalues (Figure 1 and Table XI in the supplementary material), but they still remain large (close to 1 eV) even for Rydberg states with relatively large Λvalues. A tendency of improvement of the excitation energies for the states with larger Λvalues could indicate that the CAM-B3LYP functional does not describe correctly the long-range behavior of the exchange functional and that additional asymptotic correction of the functional is necessary. This behavior of CAM-B3LYP functional has already been observed before in several di fferent benchmark studies (e.g., Ref. 70). However, in the case of halons, the MAE of the Rydberg states’ excitation energies (0.76 eV) is close to the one of the valence states (0.56 eV), and it is smaller than the corresponding value of the mixed valence-Rydberg states (1.06 eV). This indicates that the eventual incorrect description of the long-range behavior of the exchange functional with TD-CAM-B3LYP is not the most important source of the errors of excitation energies of halons. The excitation energies of the states with mixed valence- Rydberg character are also severely underestimated on the TD-CAM-B3LYP level. The MAE of the excitation energies compared to EOM-CCSD results is 1.06 eV , whereas the maximum deviation is 1.33 eV . Such large excitation energy errors are due to underestimation of the excitation energies of Rydberg orbitals but also seem to be correlated to the pronounced multiconfigurational character of these states. TheΛvalues of the states with mixed valence-Rydberg character are spread in similar region as pure Rydberg states (Figure 1 of the supplementary material), from 0.2 to 0.7, but their mean Λvalue (0.73) is larger than the mean Λvalue for pure Rydberg states (0.65). Also, their mean deviation from the EOM-CCSD results (1.06 eV) is larger than one for Rydberg states (0.76 eV). The low-lying valence states have Λvalues concentrated in the region from 0.55 to 0.65 (Figure 1 of the supplementary material). One of the interesting features occurring in the case of CCl 2F2and CBrF 3molecules is the several n σ* excited states arising from excitation from non-bonding orbital of fluorine to antibonding σ*(C–Cl) or σ*(C–Br) orbitals, which couldbe classified as delocalized. These states are featured by very smallΛvalues, since the overlaps between localized non- bonding fluorine orbitals and antibonding σ* orbitals are small in these molecules. According to EOM-CCSD results, these states occur in the highest part of the spectrum, above Rydberg states. TD-CAM-B3LYP underestimates significantly the excitation energies of these states. TD-CAM-B3LYP performs the best in the case of the excited states of CH 3Br molecule (MAE is 0.65 eV), whereas it gives the largest errors for the excited states of CF 4(1.03 eV) and C 2F6(0.93 eV), which are mainly Rydberg states. Considering all states, TD-CAM-B3LYP systematically underestimates the excitation energies in the Halons-9 set, with a MAE of 0.82 eV (Table IV). This error is astonishingly large given the usual 0.3 eV MAE usually observed in benchmarks based on organic molecular sets,13which are biased towards low-lying nπ* andππ* states. This deterioration of the TD- DFT energies for high excitations was recognized long ago by Casida et al. ,64who traced it back to a wrong asymptotic behavior of the exchange-correlation (XC) potential. Since then, diverse schemes have been proposed to fix the asymptotic behavior,64,79,80although none of them has yet been adopted as standard in TD-DFT calculations. One of the features of this problem with the XC potential is the strong underestimation of−EHOMO (the negative of the HOMO energy), lying much lower than the true IP, with a consequent collapse of the higher excited states in the region between −EHOMO and the IP.47It goes beyond the scope of this work to attempt to correct this feature here, as our focus is at the performance of standard methods. However, given the strong deviation between −EHOMO and the IP at CAM-B3LYP level (see Table I), we note that an adequate treatment of the XC potential may help to improve the TD-DFT description of the excited state of halons. Finally, the obtained oscillator strengths agree well with the EOM-CCSD values. The largest discrepancies are found in the cases of several multiconfigurational states, which are poorly described with TD-CAM-B3LYP. C. DFT/MRCI results The DFT /MRCI excitation energies for all studied states are also underestimated compared to the EOM-CCSD values, but they provide much better description of the excited states; the DFT /MRCI excitation energies lie much closer to the EOM-CCSD values. The ordering of the excited states according to DFT /MRCI coincides with the one predicted by EOM-CCSD. Similarly as in the case of TD-CAM-B3LYP, the description of the lowest-lying valence states, which arise upon excitations from the several highest occupied orbitals to the LUMO orbital, is fairly good. However, the description of the higher excited states deteriorates. The deviations of the excitation energies from the corresponding EOM- CCSD values are larger for the higher valence, Rydberg, and mixed valence-Rydberg states. The largest deviations occur in the case of the higher valence states, where the maximum deviation from the EOM-CCSD results is 1.02 eV (Table IV, Figure 3). These states mainly arise due to 184306-10 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) excitations to LUMO +1 orbitals. The MAE of all valence states (including those formed by excitations to LUMO and to LUMO +1 orbitals) is 0.44 eV , reflecting an overall satisfying performance of DFT /MRCI for these states. The medium and upper parts of the excited state spectra of halons are mainly featured by states with strong valence- Rydberg mixing. As a CI-based method, it is expected that DFT/MRCI should be flexible enough to treat properly these states. In principle, the DFT /MRCI description of these states is more reliable compared to TD-CAM-B3LYP, but the obtained deviations from EOM-CCSD are still significant; the MAE of the states with a mixed valence-Rydberg character is 0.66 eV and their maximum error is 0.80 eV . The description of Rydberg states, on the other hand, is significantly improved compared to TD-CAM-B3LYP results; the MAE of the Rydberg states is 0.47 eV and the maximum error is 0.80 eV . Insufficiently good performances of DFT /MRCI in the cases of multiconfigurational highly correlated excited states of several systems have already been noted before.50,73 DFT/MRCI describes slightly better singlet than triplet excited states. The mean average error of the triplet states (0.53 eV) is a little bit larger compared to the error of the singlet states (0.47 eV). The trend of inferior description of triplet states, common to all DFT-based methods, has already been noticed in the case of some other benchmarking sets.6,74 DFT/MRCI describes very well excited states of CF 4, C2F6, CF3Br, and CH 3Br molecules, whereas the largest deviations are found in the cases of CCl 2F2, C 2Cl2F4, and CBrClF 2molecules. The bad performance in these cases is due to occurrence of the highly correlated multiconfigurational valence and valence-Rydberg states, for which the performance of DFT /MRCI is not good enough. Globally, considering all states the MAE is 0. 53 eV .Most of the computed DFT /MRCI oscillator strengths are in the range of the corresponding EOM-CCSD values. Only in a number of high-lying Rydberg and valence-Rydberg states with strong configuration mixing, which are not described very well with DFT /MRCI, there are sometimes discrepancies between DFT /MRCI and EOM-CCSD oscillator strengths. D. Comparison with experimental data The computed vertical excitation energies and the positions of band peaks of available experimental UV photoabsorption spectra are compiled in Table V. However, comparison of the computed excitation energies with the peak positions is problematic due to influences of spin-orbit effects which were not taken into account in computations of excitation energies. It is expected that spin-orbit coupling would be particularly intensive in the case of molecules which contain bromine, which is a relatively heavy element. Man et al.19computed the absorption spectrum of bromomethane using time-dependent approach to resonance Raman scattering. They deconvoluted the A-band in gas-phase absorption spectra into three transitions:3Q1at 45 260 cm−1 (5.61 eV),3Q0at 48 272 cm−1(5.98 eV), and1Q1at 51 372 cm−1(6.37 eV). The positions of3Q0and1Q1bands are close to the values of EOM-CCSD excitation energies of the first triplet (6.01 eV) and singlet (6.64 eV) states, respectively (Table V). In the cases of other molecules, the recorded UV spectra are not resolved into spin components. In several cases, the positions of band peaks are closer to S1and in some cases toT1excitation energies. In the cases of highly symmetric molecules (CF 4, C 2F6, CCl 2F2, CBrF 3, and CH 3Br) with doubly degenerate S1andT1states, comparison of the reference EOM-CCSD results with the experimental ones is even more TABLE V . Computed vertical excitation energies (E vin eV) compared to the corresponding positions of the band maxima in the available experimental UV photoabsorption spectra of the Halons-9 set. The computed oscillator strengths for the transitions are also given (f). EOM-CCSD DFT /MRCI TD-CAM-B3LYP Expt. Molecule State E v f E v f E v f E v CH3BrT1 6.01 . . . 5.70 . . . 5.62 . . . 5.9819 S1 6.64 0.0020 6.37 0.0002 6.39 0.0024 6.3719 CBrF 3T1 6.08 . . . 5.878 . . . 5.65 . . .6.0520 S1 6.84 0.0000 6.68 0.0014 6.54 0.0001 CF4T1 13.87 . . . 13.60 . . . 12.95 . . .13.6, 13.922 S1 14.23 0.0000 14.41 0.0000 13.51 0.0000 CHCl 2FT1 6.79 . . . 6.38 . . . 6.30 . . .7.2921 S1 7.58 0.0159 7.19 0.0149 7.27 0.0147 CHBrF 2T1 5.95 . . . 5.66 . . . 5.53 . . .6.5221 S1 6.62 0.0012 6.39 0.0022 6.34 0.0014 CBrClF 2T1 5.61 . . . 5.27 . . . 5.19 . . .6.0221 S1 6.32 0.0095 6.02 0.0088 6.03 0.0089 CCl 2F2T1 6.66 . . . 6.29 . . . 6.19 . . .6.9724 S1 7.49 0.0171 7.13 0.0146 7.10 0.0150 C2F6T1 10.70 . . . 11.42 . . . 10.82 . . .12.128 S1 12.66 0.0427 12.49 0.0663 11.89 0.0354 C2Cl2F4T1 7.41 . . . 6.96 . . . 6.91 . . .7.2129 S1 8.29 0.0009 7.81 0.0003 7.88 0.0006 184306-11 Stojanović et al. J. Chem. Phys. 145, 184306 (2016) difficult because of vibronic interactions which influence the excitation energies of these states. For these reasons, we cannot draw any conclusion about validity of the reference EOM-CCSD method. A more comprehensive study of excited states with inclusion of spin-orbit and adiabatic e ffects is necessary in order to compare ab initio and experimental data. IV. CONCLUSIONS Given the complete lack of excited state information about halo-organic compounds in excited state benchmarks, we have introduced a new molecular set tailored for the evaluation of molecular properties of such systems. The Halons-9 set is composed of nine small gaseous halons, including F, Cl, and Br elements. Vertical excitation energies were assessed with two DFT-based methods, TD-DFT with CAM-B3LYP functional and DFT /MRCI methods, taking CCSD as the reference method. The present benchmark shows that the evaluated methods only provide semi-quantitative results, with errors in the order of several tenths of an eV . Overall, we observed a superior performance of multi-reference DFT /MRCI method in the calculation of the excited states of halons (MAE is 0.53 eV), compared to the TD-CAM-B3LYP method (MAE is 0.82 eV) (Table IV, Figure 3). The TD-CAM-B3LYP method performs the best for the lowest-lying valence states represented mainly by single configurations (MAE is 0.56 eV). In the case of higher valence states, states with pronounced valence-Rydberg mixing, and Rydberg states, TD-CAM-B3LYP vertical excitation energies are significantly underestimated compared to EOM-CCSD values. The description of the mixed valence-Rydberg states with TD-CAM-B3LYP is the most challenging; the MAE of the mixed valence-Rydberg states is the largest (1.06 eV), whereas the MAE of the pure Rydberg states (0.76 eV) is considerably larger than the one of the valence states (0.56 eV). On the other hand, the DFT /MRCI method performs reasonably well for most of the studied states. It gives much more reliable description of states with pronounced multiconfigurational character, which appear in the middle and upper parts of the spectrum, compared to TD-CAM- B3LYP. The largest deviations from the EOM-CCSD values are observed in the case of higher-lying multiconfigurational mixed valence-Rydberg states. Contrary to expectations based on previous numerous benchmarking studies, where valence excited states are usually described with errors around 0.2–0.3 eV with TD-DFT or DFT/MRCI, the valence excited states of Halons-9 molecular set are described only semi-quantitatively with both methods (the MAE of valence states’ excitation energies obtained with TD-CAM-B3LYP and DFT /MRCI is 0.58 and 0.44 eV , respectively). The reason for this is in the fact that most of the benchmark studies focus on organic molecules with low-lying nπ* orππ* excited states. By construction, the excited states in the Halons-9 molecular set ( nσ*,σσ*, n-Rydbergs, and mixed valence-Rydbergs) are much higher. In this spectral region, the e ffects of wrong asymptotic behavior of the XCpotential are much more pronounced, leading to a deterioration of the results. Oscillator strengths for most of the transitions are described satisfactorily with both methods. The exceptions are strongly correlated multiconfigurational states, where, in the cases of several states, oscillator strengths strongly deviate from the EOM-CCSD values. All molecules in the Halons-9 set are featured by electronic transitions in far- and extreme UV-region, implying that these transitions cannot be photoactivated close to the Earth’s surface, although they may play a role in the photochemistry of the high atmosphere. Our experience with the Halons-9 set tells us that there is a long way to go before reaching the precision we are used to when computing excited states of light aromatic systems. We have seen that there are di fferent challenges to be faced, including basis set dependence, spin orbit couplings, strong correlation, and proper XC potential description. In the future, we expect that significant progress in the description of the Halons-9 set may be achieved through the inclusion of XC potential corrections as in the Casida-Salahub approach.64 The use of the time-dependent multiconfigurational short- range density functional theory (TD-MC-srDFT)81also o ffers a promising alternative for dealing with these systems. We were not able to evaluate the validity of the EOM- CCSD method, which was used as the reference because the UV excitations of most molecules in the Halons-9 set should be strongly a ffected by spin-orbit couplings. For this reason, we limited our analysis to computational methods. The e ffect of spin-orbit coupling and adiabatic e ffects on excited states of Halons-9 set will be investigated in our future work. SUPPLEMENTARY MATERIAL See the supplementary material for the optimized ground state geometries of molecules, the comparison of vertical excitation energies obtained with TD-CAM-B3LYP applying two di fferent basis sets, the complete tables containing data about all computed states, and the correlation diagram between excitation energy errors and Λvalues. ACKNOWLEDGMENTS This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. 43-130-35-RG. The authors, therefore, acknowl- edge with thanks DSR support for Scientific Research. Also, the authors appreciate the kind cooperation provided by the Deanship of Scientific Research (DSR), King Abdulaziz University. M.B. and L.S. thank the support of the A*MIDEX Grant (No. ANR-11-IDEX-0001-02) and the project Equip@Meso (Grant No. ANR-10-EQPX-29-01), both funded by the French Government “Investissements d’Avenir” program. 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1.4705289.pdf

Nanoscale spin wave valve and phase shifter Y. Au, M. Dvornik, O. Dmytriiev, and V. V. Kruglyak Citation: Appl. Phys. Lett. 100, 172408 (2012); doi: 10.1063/1.4705289 View online: http://dx.doi.org/10.1063/1.4705289 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v100/i17 Published by the American Institute of Physics. Related Articles Highly (001)-oriented thin continuous L10 FePt film by introducing an FeOx cap layer Appl. Phys. Lett. 102, 062420 (2013) Remedying magnetic hysteresis and 1/f noise for magnetoresistive sensors Appl. Phys. Lett. 102, 054104 (2013) Physical properties of Al doped Ba hexagonal ferrite thin films J. Appl. Phys. 113, 043903 (2013) Electric-field effect on coercivity distributions in FePt magneto-electric devices Appl. Phys. Lett. 102, 012409 (2013) Fabrication of single-dot planar nano-devices and the application to the exchange bias characterization in nano- pillar devices Appl. Phys. Lett. 101, 222406 (2012) Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 20 Feb 2013 to 146.201.208.22. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsNanoscale spin wave valve and phase shifter Y. Au ,a)M. Dvornik, O. Dmytriiev, and V. V. Kruglyak School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom (Received 15 February 2012; accepted 4 April 2012; published online 24 April 2012) We have used micromagnetic simulations to demonstrate a method for controlling the amplitude and phase of spin waves propagating inside a magnonic waveguide. The method employs ananomagnet formed on top of a magnonic waveguide. The function of the proposed device is controlled by defining the static magnetization direction of the nanomagnet. The result is a valve or phase shifter for spin waves, acting as the carrier of information for computation or data processingwithin the emerging spin wave logic architectures of magnonics. The proposed concept offers such technically important benefits as energy efficiency, non-volatility, and miniaturization. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4705289 ] The short wavelength of spin waves propagating in nanoscale magnonic waveguides at microwave frequencies coinciding with nowadays communication standards couldpotentially provide a device miniaturization opportunity via accomplishing certain tasks in a more efficient manner when compared to semiconductor circuitry. 1,2For example, spin waves have been proposed to be utilized in GHz filters,3 logic devices,4,5amplifiers,6–8mixers,9etc. The utilization of spin waves, within the paradigm of so called“magnonics,” is also strongly relevant to exciting such phe- nomena recently observed in adjacent disciplines as spin transfer torque oscillations, 10,11current induced domain wall motion,12spin Hall effect,13spin Seebeck effects,14etc. In light of the burgeoning research in magnonics and related topics, the spin wave technologies are expected to provide acompetitive alternative route for performing tasks that are currently accomplished by semiconductor electronics. How- ever, one of the important challenges on the way to fulfill-ment of this promise of magnonics is to learn to manipulate the spin wave phase and perhaps even magnitude, with this manipulation itself being highly relevant to data encodingand logical operation. Historically, several spin wave phase shifting mecha- nisms have been proposed. These include the use of domainwalls, 15magnetic material nonuniformity,16local electrical voltage,2and flow of electrical current.5,17However, the flow of electrical current produces Joule heating and con-sumes electrical power. The feasibility of use of piezoelec- tric elements to control spin waves is yet to be demonstrated while the nucleation of domain wall inside a uniformly mag-netized waveguide is energetically costly. At the same time, we have recently demonstrated efficient conversion of a global spatially uniform microwave field into a spin wavepropagating in a magnonic waveguide by a reservoir contin- uous film 18and then by a stripe transducer element placed on top of the waveguide.19In this Letter, we demonstrate that a mechanism that is reciprocal to the latter method of spin wave emission19can be used to provide controllable spin wave absorption or phase shift. In particular, a spinwave travelling inside the waveguide resonantly excites thenanomagnet (passive in this case). The excited nanomagnet provides a back action on the incident spin wave, either absorbing its energy or shifting its phase. The action dependsupon the static magnetization of the nanomagnet. In particu- lar, the absorption or phase shifting behavior disappears as soon as the static magnetization direction is flipped. Weargue that the observed effects could allow one to build a spin wave valve or a spin wave phase shifter, with the static magnetization of the nanomagnet used as an on and offswitch for the valve or phase shifter function. Fig. 1(a) shows the ground state at zero applied mag- netic field of a nanomagnet (“resonator”) placed on the topof a magnonic waveguide, both made of Permalloy. 20The simulations are performed using object oriented micromag- netic framework ( OOMMF ) (Ref. 21) with the cell size of 5/C25/C25n m3. The waveguide is 2200 nm long, 100 nm wide, and 10 nm thick, while the resonator is 150 nm long, 50 nm wide, and 30 nm thick. Damping factor ais set to 0.005 for both the waveguide and resonator. The ground state is prepared by relaxing the system under zero applied field ( Hext¼0 Oe) from a state at which the waveguide and resonator are uniformly magnetized in positive xandydirec- tions, respectively. The gray area on the right hand side of the waveguide depicts the part of the waveguide(50/C2100/C210 nm 3inx,y,zdirections, respectively) sub- jected to a rf magnetic field oscillating at 11.5 GHz, with am- plitude of 1 Oe and polarized in ydirection. The field excites two spin waves with wavelength of about 100 nm propagat- ing in the waveguide away from the excitation source towards positive and negative x directions, as shown in Figs.1(b)–1(d). The rf frequency (11.5 GHz) is chosen so as to coincide with that of the uniform precession (fundamental mode) of the resonator, deduced from a separate pulse exci-tation simulation. The damping near the two ends of the waveguide is increased to 0.05 to prevent spin wave reflec- tion. The vertical separation between the resonator andwaveguide (in zdirection) is 5, 20, and 50 nm in Figs. 1(b)–1(d), respectively. The incident spin wave drives the resonator into resonance. In turn, the reverse action from theprecessing resonator upon the incident spin wave alters its propagation in the waveguide, in a manner that depends strongly on the resonator-waveguide separation. For a)Electronic mail: y.au@exeter.ac.uk. 0003-6951/2012/100(17)/172408/4/$30.00 VC2012 American Institute of Physics 100, 172408-1APPLIED PHYSICS LETTERS 100, 172408 (2012) Downloaded 20 Feb 2013 to 146.201.208.22. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsexample, the spin wave in Fig. 1(b) undergoes an almost 180 degree phase shift while the phase of the spin wave in Fig.1(d) is largely unaffected. Moreover, for an appropriately set value of the resonator-waveguide separation (20 nm in this particular case), the spin wave in the waveguide is predomi-nantly absorbed by the resonator, as demonstrated in Fig. 1(c). The observed phenomena could be highly useful for magnonic applications since their occurrence depends heav- ily on the static magnetization of the resonator. In particular, the magnetization points towards positive yin Figs. 1(b)–1(d). Fig. 1(e) shows the ground state of the same de- vice but with the magnetization of the resonator toggled to negative ydirection. In this case, the resonator does not cou- ple at all to the spin wave propagating in the waveguide, so that its amplitude and phase remain largely unaffected (Fig. 1(f)). Hence, by changing the static magnetization direction of the resonator, it is possible to control the amplitude and phase of the spin wave propagating under the resonator. Depending on the vertical separation between the resonatorand the waveguide, these switchable, non-volatile devices could serve as either spin wave valves or phase shifters. Since the absorption and phase shifting effects arise due to the resonant excitation of the resonator, the time required to bring the excited resonator into steady state of precession would largely define the transient time duration required bythe valve or phase shifter to react to change in resonator static magnetization, i.e., operation speed of the device. This “reaction time” in turn is determined mainly by the dampingfactor of the resonator’s material, which has to be artificially adjusted (e.g., by doping) to improve the device perform- ance. Hence, we investigate the device absorption and phaseshifting effects as a function of Gilbert damping factor aof the resonator and of the vertical separation between the reso- nator and the waveguide, while keeping damping factor aof the waveguide at 0.005 as before. Figs. 2(a) and 2(b) display the relative magnitude of spin wave transmitted in the waveguide under the resonator as measured at 200 nm away from the central axis of the res-onator to negative xdirection, for the static magnetization of the resonator pointing towards positive and negative y direc- tion, respectively. The displayed “relative magnitude” is nor- malized to that measured at the same position on the waveguide in a separate simulation (not shown) that is iden-tical to that discussed but with the resonator removed (“unperturbed case”). In Fig. 2(a), we observe a dark diago- nal belt representing absorption maxima that run towardslower values of vertical separation between the resonator and waveguide (“spacing”) as aincreases. When the static mag- netization in the resonator is flipped, the belt largely disap-pears, as shown in Fig. 2(b). The situation becomes clearer if we inspect the relative phase (again, referenced to the unper- turbed case) of the transmitted spin wave, depicted in Fig.2(c). The phase shift is close to zero on one side of the dark belt of Fig. 2(a), while it is almost 180 degrees on the other side. Again this phase shift phenomenon disappears once thestatic magnetization of the resonator is flipped (Fig. 2(d)). We interpret the observations by assuming that the excited resonator emits a spin wave with a phase that isshifted relative to that of the incident spin wave by 180 degrees. The interference between the incident and emitted waves then determines the amplitude and phase of the trans-mitted wave. Let us consider the case of a¼0.005 as an example. For small values of the resonator-waveguide FIG. 1. (a) Ground state of the micromagnetic structure considered. The entire box of view in (a) represents a 2200 /C2600 nm region. The magnonic waveguide of 100 nm width and of 10 nm thickness is separated by 5 nm spacing from the overlaid 50 nm wide, 150 nm long, and 30 nm thick resona- tor. Little arrows inside the waveguide and transducer represent local mag- netic moment direction. (b)–(d) Out of plane magnetization (m z) inside the waveguide (static background subtracted) at the same relative simulation time for a vertical spacing between the resonator and the waveguide kept at 5 nm and changed to 20 and 50 nm, respectively. These images wererecorded after the system has attained dynamic steady state. Inset of (a): Color scale for M yin (a) and mzin (b)–(d) with range of /C040 to 40 Oe and /C00.2 to 0.2 Oe, respectively. (e) Ground state of the same structure but with the resonator magnetization flipped to the opposite direction. (f) Waveguide mzfor case of opposite magnetized resonator at the same aforementioned relative simulation time. Resonator-waveguide spacing equals 5 nm both in (e) and (f). FIG. 2. (a) and (b): Phase diagram of relative magnitude of the spin wavetransmitted underneath the resonator measured at 200 nm in negative x direction away from the central axis of the resonator for static magnetization of the resonator pointing towards positive and negative y direction respec- tively. Note that ais plotted in logarithmic scale. (c) and (d): Identical to (a) and (b) but with magnitude of the spin wave replaced by oscillation phase (in radian). (e) and (f): Phase diagram of the precession magnitude of the resonator (averaged over the resonator volume) for static magnetization ofthe resonator pointing towards positive and negative y direction, respectively.172408-2 Au et al. Appl. Phys. Lett. 100, 172408 (2012) Downloaded 20 Feb 2013 to 146.201.208.22. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsspacing, the transmitted wave is dominated by the resonator emitted wave. Indeed, the emitted spin wave results from the dynamic dipolar stray field of the resonator, which is strong if the resonator is positioned near to the waveguide. Hence,the transmitted spin wave demonstrates a 180 degree phase shift. For large values of the resonator-waveguide spacing, the influence of the resonator stray field on the waveguide diminishes, and so, the incident wave dominates. Hence, the transmitted wave has a zero phase-shift. However, at spacing of 20 nm, the effects of the incident and emitted waves become equal, and so, a complete destructive interferenceoccurs, leading to the observed minimized transmission. As the damping factor of the resonator increases, the minimum moves to smaller values of the resonator-waveguide spacingbecause the rise of asuppresses the precession amplitude in the resonator. Consequently, the resonator has to be placed nearer to the waveguide in order to compensate the weaken-ing of its stray field and to achieve the same complete de- structive interference between the incident and emitted waves. This interpretation is further supported by the de-pendence of the amplitude of precession in the resonator upon aand spacing, as depicted in Figs. 2(e)and2(f). In order to account for the startling difference of the res- onator’s behavior for the opposite directions of its static magnetization (positive and negative y), we plot in Figures 3(a)to3(d) the dynamic dipolar stay field of the resonator in x-z plane across the middle of the resonator for the relative simulation time of 0, 0.125, 0.25, and 0.375 T, respectively, where period T is equal 0.087 ns. The static magnetization ofthe resonator points towards positive y, and the waveguide is removed for simplicity. If we pay attention to the regionbelow the resonator, we observe a pattern of the dipolar field that starts from the right and moves towards left from (a) to (d), echoing a wave travelling towards negative xdirection. As the static magnetization inside the waveguide pointslargely towards positive x, the magnetization in the wave- guide will only be sensitive to the zcomponent of the resona- tor’s stray field ( h z). The situation becomes even clearer if we Fourier transform hzalong xdirection, as shown in Fig. 3(e). The Fourier transform displays a striking imbalance between positive and negative kxvalues, with the majority of the Fourier amplitude residing in the negative region of kx. Hence, the resonator is capable of interacting with the spinwave travelling only in one direction (negative xin this case) but not the other (positive x). When the static magnetization of the resonator is flipped, the chirality of rotation of the res-onator’s dipolar stray field also flips. This in turn leads to a mirror flip of the Fourier transform of h zin Fig. 3(e)relative to the kx¼0 axis, thereby completely modifying the effect of the resonator upon the spin wave in the waveguide. In summary, we have proposed a method by which to achieve magnetically switchable absorption or phase shiftingof spin waves travelling in a magnonic waveguide. The mechanism, which relies solely on the state of the static mag- netization of a nanomagnet resonator, exhibits advantages oflow energy consumption and non-volatility against other mechanisms relying on continuous application of electrical current or voltage. 2,5,17The direct current induced domain wall motion12and related spintronic phenomena intensively studied in recent years can become natural methods to pro- vide the static magnetization switching of resonator requiredin our proposed concept. The concept acts on exchange dominated spin waves in backward volume mode geometry. The latter geometry is preferred for building magnonic cir-cuitry because of its ability to operate at remanence. Further- more, spin waves in this geometry are able to follow the physical bending of the magnonic waveguide, 5,22while the shortness of the spin wave wavelength offers an opportunity to miniaturize magnonic devices. The functionality of the de- vice relies only on a single nanomagnet and is therefore spa-tially very compact. We have explained the engineering prospects of the concept and have revealed the physical ori- gin of the phenomenon by analyzing the dependence of thewave vector Fourier spectrum of the stray field produced by the resonator. The discovered effects could help to build switchable bandpass filters or even serve as means to controlthe synchronization of spin transfer torque nano-oscillators. We believe that the proposed devices could provide the foun- dation for the emerging magnonic nanotechnology. The research leading to these results has received fund- ing from the EC’s Seventh Framework Programme (FP7/ 2007-2013) under GAs 233552 (DYNAMAG) and 228673 (MAGNONICS) and from the EPSRC of the UK (EP/E055087/1). 1V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D: Appl. Phys. 43, 264001 (2010). 2A. Khitun and K. L. Wang, J. Appl. Phys. 110, 034306 (2011). 3S. K. Kim, K. S. Lee, and D. S. Han, Appl. Phys. Lett. 95, 082507 (2009). 4T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L. Stamps, and M. P. Kostylev, Appl. Phys. Lett. 92, 022505 (2008). FIG. 3. (a)–(d) Dipolar stray field of the resonator at relative simulation time equal 0, 0.125, 0.25, and 0.375 T (T ¼1/11.5 ns), respectively. (e) Spatial Fourier transform along x direction of the stray field z component atdifferent vertical distance from the lower surface of the resonator (vertical axis, z).172408-3 Au et al. Appl. Phys. Lett. 100, 172408 (2012) Downloaded 20 Feb 2013 to 146.201.208.22. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions5K. S. Lee and S. K. Kim, J. Appl. Phys. 104, 053909 (2008). 6M. Bao, A. Khitun, Y. Wu, J. Y. Lee, K. L. Wang, and A. P. Jacob, Appl. Phys. Lett. 93, 072509 (2008). 7A. Khitun, D. E. Nikonov, and K. L. Wang, J. Appl. Phys. 106, 123909 (2009). 8E. Padron-Hernandez, A. Azevedo, and S. M. Rezende, Phys. Rev. Lett. 107, 197203 (2011). 9Y. Khivintsev, J. Marsh, V. Zagorodnii, I. Harward, J. Lovejoy, P. Krivo- sik, R. E. Camley, and Z. Celinski, Appl. Phys. Lett. 98, 042505 (2011). 10V. E. Demidov, S. Urazhdin, and S. O. Demokritov, Nature Mater. 9, 984 (2010). 11M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti, G. Gubbiotti,F. B. Mancoff, M. A. Yar, and J. Akerman, Nature Nanotechnol. 6, 635 (2011). 12S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 13Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010).14K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Mae- kawa, and E. Saitoh, Nature 455, 778 (2008). 15R. Hertel, W. Wulfhekel, and J. Kirschner, Phys. Rev. Lett. 93, 257202 (2004). 16S. V. Vasiliev, V. V. Kruglyak, M. L. Sokolvskii, and A. N. Kuchko, J. Appl. Phys. 101, 113919 (2007). 17V. E. Demidov, S. Urazhdin, and S. O. Demokritov, Appl. Phys. Lett 95, 262509 (2009). 18Y. Au, T. Davison, E. Ahmad, P. S. Keatley, R. J. Hicken, and V. V. Kru-glyak, Appl. Phys. Lett. 98, 122506 (2011). 19Y. Au, E. Ahmad, O. Dmytriiev, M. Dvornik, T. Davison, and V. V. Kru- glyak, “Resonant microwave-to-spin-wave transducer” (unpublished). 20Saturation magnetization Ms¼800 Oe, exchange stiffness A¼1.3/C210/C011 J/m, and zero magnetocrystalline anisotropy. 21M. Donahue and D. G. Porter, OOMMF User’s guide, Version 1.0, Inter- agency Report NISTIR 6376, NIST, Gaithersburg, MD, 1999. 22S. Bance, T. Schrefl, G. Hrkac, A. Goncharov, D. A. Allwood, and J.Dean, J. Appl. Phys. 103, 07E735 (2008).172408-4 Au et al. Appl. Phys. Lett. 100, 172408 (2012) Downloaded 20 Feb 2013 to 146.201.208.22. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

5.0033239.pdf

Appl. Phys. Lett. 118, 022404 (2021); https://doi.org/10.1063/5.0033239 118, 022404 © 2021 Author(s).Unidirectional current-driven toron motion in a cylindrical nanowire Cite as: Appl. Phys. Lett. 118, 022404 (2021); https://doi.org/10.1063/5.0033239 Submitted: 14 October 2020 . Accepted: 21 December 2020 . Published Online: 11 January 2021 Qiyang Hu , Boyao Lyu , Jin Tang , Lingyao Kong , Haifeng Du , and Weiwei Wang COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Parametron on magnetic dot: Stable and stochastic operation Applied Physics Letters 118, 022402 (2021); https://doi.org/10.1063/5.0038946 Strain-induced cooling-heating switching of anisotropic magneto-Peltier effect Applied Physics Letters 118, 022403 (2021); https://doi.org/10.1063/5.0034858 Metal organic framework/layer double hydroxide/graphene oxide nanocomposite supercapacitor electrode Applied Physics Letters 118, 023901 (2021); https://doi.org/10.1063/5.0030311Unidirectional current-driven toron motion in a cylindrical nanowire Cite as: Appl. Phys. Lett. 118, 022404 (2021); doi: 10.1063/5.0033239 Submitted: 14 October 2020 .Accepted: 21 December 2020 . Published Online: 11 January 2021 Qiyang Hu,1Boyao Lyu,2JinTang,2 Lingyao Kong,3Haifeng Du,2and Weiwei Wang1,4,a) AFFILIATIONS 1Institutes of Physical Science and Information Technology, Anhui University, Hefei 230601, China 2Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory of Chinese Academy of Sciences, and University of Science and Technology of China, Hefei 230031, China 3School of Physics and Materials Science, Anhui University, Hefei 230601, China 4Key Laboratory of Structure and Functional Regulation of Hybrid Materials of Ministry of Education, Anhui University, Hefei 230601, China a)Author to whom correspondence should be addressed: wangweiwei@ahu.edu.cn ABSTRACT A magnetic toron is a spatially localized three-dimensional spin texture composed of skyrmionic layers with two Bloch points at its two ends. The magnetic toron can, thus, be stabilized in chiral magnets using external fields. In this work, we studied the toron dynamics induced byelectric currents in a cylindrical nanowire using micromagnetic simulations. We show that the toron performs a unidirectional motion in ananowire where the current is applied along the wire direction. The current-induced toron motion can be divided into three regions: staticregion for a small current due to the pinning effect, toron moving region for a large current, and toron annihilation region for a large reversal current. Moreover, the moving direction can be tuned by the sign of Dzyaloshinskii–Moriya interaction. Such peculiar dynamics indicates that the magnetic toron is a possible candidate as an information carrier. Published under license by AIP Publishing. https://doi.org/10.1063/5.0033239 Magnetic structures, such as magnetic vortex and domain walls, play fundamental roles in future spintronics. The presence ofDzyaloshinskii–Moriya interaction 1,2(DMI), including both the interfa- cial DMI at the interface of multilayers and the bulk DMI in chiral mag- nets, enriches the formation of magnetic (topological) structures such asmagnetic skyrmions. 3–6Magnetic skyrmions are two-dimensional struc- tures with a skyrmion number Q¼61 and are promising candidates as information carriers for spintronic applications such as logic devices,7 racetrack memory,5and neuromorphic computation.8In chiral mag- nets, such as MnSi4and FeGe,9the bulk DMI also supports three- dimensional topological structures such as magnetic bobbers10,11and magnetic hopfions.12–14Two magnetic bobbers together can form a new structure—a magnetic toron.15,16The toron is not symmetric in the external field direction in the sense that two Bloch points (BPs) with dif-ferent charges are attached at the two (top and bottom) ends of thetoron. Spin-transfer torque (STT) emerges 17,18when an electron passes through the non-uniform magnetic structures and, thus, is an effective method to manipulate magnetic structures. For example, the STT candrive the domain walls, 17magnetic skyrmions,5,19,20magneticbobbers,21and magnetic hopfions14,22effectively. Therefore, it is expected that the STT can move the toron as well when the current is applied in the plane of the skyrmion since each layer of the magnetictoron is a magnetic skyrmion. However, as the toron is an asymmetric 3D structure and is characterized by two BPs, it is not clear how the toron responds to the STT when the current is applied perpendicularto the plane of the skyrmion. Spatial symmetry breaking may result in a unidirectional motion. For example, the ratchet-like transport phenomena are expected withperiodic driving forces by breaking the spatial symmetry. 23–25In par- ticular, the unidirectional motion may occur for the magnetic domain wall even without explicit asymmetry in the presence of DMI.26At the same time, the explicit symmetry breaking does not necessarily lead to a unidirectional motion when driven by STT. An example is the mag- netic skyrmions with symmetry breaking by an in-plane field.Therefore, it is natural to ask whether the magnetic toron will have a unidirectional motion. In this work, we investigate a cylindrical nanowire with bulk DMI in which the magnetic toron could be stabilized when the applied field is parallel to the nanowire. We show that the magnetic toron Appl. Phys. Lett. 118, 022404 (2021); doi: 10.1063/5.0033239 118, 022404-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplperforms a unidirectional motion along the nanowire when an electri- cal current is applied: the toron moves forward when a large positive current is applied in the wire direction and the toron will be annihi- lated if the current direction is reversed. Most importantly, this asym-metric response to currents can be tuned by DMI: if the sign of DMI isnegative, the toron will be annihilated for a positive current and will move backward for a negative current, that is, the response of the toron to currents is reversed. We consider a FeGe nanowire along the z-axis with exchange interaction, bulk DMI, Zeeman energy, and an effective anisotropy.The overall effective anisotropy contains the contributions of the demagnetization field, magnetocrystalline anisotropy, and magnetoe- lastic anisotropy. Therefore, the total free energy density of the wirereads E¼AðrmÞ 2/C0Km2 zþDm/C1ð r/C2 mÞ/C0l0MsH/C1m; (1) where mis the unit vector of the magnetization, Ais the exchange constant, Kis the effective anisotropy coefficient, and Dis the DMI constant. The external field is applied in the z-axis, i.e., H¼Hzez, which has the same direction as the nanowire. We use finite difference methods to compute the energy density, Eq.(1), and the central derivatives to evaluate the exchange energy,which makes Eq. (1)equivalent to the classical spin model with nearest-neighbor exchange interaction. A graphics processor unit (GPU)-supported micromagnetic package JuMag27was used to per- form the simulation. In the simulation, the parameters of FeGe are chosen as follows:28the exchange constant A¼8:78/C210/C012J=m, the DMI constant jDj¼1:58/C210/C03J=m2, the saturation magnetiza- tion Ms¼3:84/C2105A=m, and the easy axis anisotropy K¼2 /C2105J=m3. The radius of the wire is fixed to be R¼50 nm, and the lengths of the wires are L¼600 nm for the toron dynamics and L¼800 nm for the elongated toron simulations. The cell size is chosen to be 2 /C22/C22n m3. The formation of the toron depends on the external fields. Figure 1(a) shows magnetic torons using the isosurface with mz¼0. A suitable external field, such as Hz¼160 mT, can stabilize the magnetic toron, and a larger external field results in a shorter toron.15The length of the toron depends on the external fields and the strength of the anisotropy.15Two Bloch points with charges q¼þ1a n d q¼–1 are located at the top and bottom of the toron, respectively, where thecharge (also known as topological charge or the wrapping number 29) for the BP is defined as30–32 q¼1=ð8pÞð dSi/C15ijkm/C1@jm/C2@km; (2) FIG. 1. (a) A magnetic toron represented by the isosurface with mz¼0. The toron is stabilized by the bulk DMI and an external field Hz¼160 mT. Two ends of the toron are BPs with charges q¼þ 1a n d q¼–1, respectively. The second (third) toron is obtained for a positive (negative) DMI, and the circulating direction is determined by the sign of DMI. (b) Top side view of the magnetization in the middle of the toron ( xy-plane), and each layer of the toron is a magnetic skyrmion. (c) The magnetization distribu- tion in the xz-plane of the toron where two Bloch points are clearly depicted. (d) Plots of spatial z-component magnetization mzwith various radii. (e)–(j) The snapshots of the magnetization to describe the dynamics of the toron inside the nanowire where both the external field and the electric current are applied with the w ire direction, i.e., the zdirection. In (e)–(g), the positive DMI is used, and in (h)–(j), the negative DMI is used. (e) A negative current u¼/C0 60 m=s for the positive DMI scenario: the q¼þ 1 side moves faster than the q¼–1 side and the toron becomes smaller and eventually is eliminated by the current. (f) For the positive current u¼60 m=s, the toron becomes slightly larger and is pinned. (g) The toron moves forward along the wire when u¼120 m =s. (h) A negative current u¼/C0 60 m=s failed to move the toron when the sign of DMI is negative. (i) A current u¼/C0 60m=s eliminates the toron for a negative DMI. (j) The toron moves backward (– z-direction) for a negative current u¼/C0 120 m =s. The scale bar is 100 nm.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 022404 (2021); doi: 10.1063/5.0033239 118, 022404-2 Published under license by AIP Publishingwhere i;j;k¼x;y;zand/C15is the Levi-Civita tensor. The integration is taken over a closed surface Ssurrounding the BP. Equation (2)can also be written as q¼ð1=4pÞÐ G/C1dS, where the gyrovector Gis defined as G¼r cosH/C2rUif the unit vector of the magnetization is expressed using polar angle Hand azimuthal angle U.T oc o m p u t e the charge of the BP at the top end of a magnetic toron, we could con- struct a cylinder surrounding the BP, as shown in Fig. 1(a) ;b o t ht h e integrations over the top surface and the lateral surface are zero, andthus, the charge of the BP equals the skyrmion number of the bottom surface, which gives q¼þ1 as each layer of the toron is a magnetic skyrmion. The divergence theorem allows us to define the volume topologi- cal density qas q¼ð1=4pÞr /C1G¼ð1=4pÞ@ xGxþ@yGyþ@zGz/C2/C3; (3) where Gi¼ð1=2Þ/C15ijkm/C1@jm/C2@km. Note that the integration of Gz¼m/C1ð@xm/C2@ymÞgives the definition of skyrmion number Q. The volume topological density qis convenient for the calculation of topological charge of the BP numerically. For the toron or bob-ber shown in Figs. 1(a) and2(a),o n l yt h e G zterm has a contribu- tion, i.e., ð1=4pÞÐ V@zGzdV¼61. Moreover, similar to the guiding center of the skyrmions, the center of the BP can be computed as rc¼Ð VqrdV=Ð VqdV. For a magnetic skyrmion, the sign of DMI determines the chiral- ity of the skyrmion, i.e., clockwise or counterclockwise. Similarly, the BPs attached at two sides of the toron could be clockwise or counter- clockwise circulating, as shown in Fig. 1(a) .Figure 1(b) shows the top side view of the magnetization of the xy-plane in the middle of the toron. It can be seen that the layer is a magnetic skyrmion withQ¼–1. Meanwhile, the magnetization in the xz-plane is plotted in Fig. 1(c) . In the core of the toron, the magnetization is antiparallel to the external fields. The corresponding z-component magnetization mz is plotted in Fig. 1(d) , which shows that mzchanges abruptly close to the BPs. The dynamics of the magnetization in the presence of a spin polarized current is governed by the extended Landau–Lifshitz–Gilbert (LLG) equation with spin-transfer torque: 17,18 @m @t¼/C0cm/C2Heffþam/C2@m @t/C0u@m @zþbum/C2@m @z;(4) where cis the gyromagnetic ratio, Heff¼/C0 ð 1=l0MsÞ/C1dE=dmis the total effective field, and ais the Gilbert damping. The parameter u¼/C0jzPglB=ð2eMsÞdenotes the strength of spin-polarized current, where jzis the current density along the zaxis, gis the Land /C19e factor, lB is the Bohr magneton, Pis the spin polarization rate, eð>0Þis the electron charge, and brepresents the strength of the nonadiabatic tor- que. In this work, we have fixed a¼b¼0:1 to demonstrate the dynamics of the toron. Figures 1(e)–1(j) show the snapshots of the toron dynamics in the presence of electric currents, and the positive DMI case is showninFigs. 1(e)–1(g) . When a negative current u¼–60 m/s is applied, the toron becomes smaller and smaller and eventually disappears, asshown in Fig. 1(e) . However, the toron is not eliminated for a positive current with u¼þ60 m/s; as depicted in Fig. 1(f) ,t h et o r o nb e c o m e s slightly larger and is pinned by BPs without moving further. A largecurrent with u¼120 m/s is able to overcome the pinning effect and moves the toron, as shown in Fig. 1(g) . The animations of the toron dynamics are shown in the supplementary material (I.mp4). The skyrmion velocity is independent of the skyrmion chirality when driven by electric currents using STT. 19Surprisingly, the sign of DMI influences the toron dynamics significantly. The skyrmion torondynamics for a negative DMI is shown in Figs. 1(h)–1(j) .I nt h i ss i t u a - tion, a negative current with u¼–60 m/s does not eliminate the toron, as shown in Fig. 1(h) .I n s t e a d ,t h et o r o ni sp i n n e da st h ec a s et h a t u¼þ60 m/s with positive DMI. Similarly, the positive current u¼60 m/s eliminates the toron, which corresponds to Fig. 1(e) with a negative current. Further, a negative current u¼–120 m/s moves the toron backward along the— z-direction. To understand the response of the toron to electric currents such as the toron elimination and the DMI-induced asymmetric toronmotion as described in Fig. 1 , we present the bobber dynamics here because it looks like that a toron can be considered as a combinationof two bobbers. An upward bobber and a downward bobber are showninFigs. 2(a) and2(b), respectively, where a BP with q¼–1 (q¼þ1) is attached to the upward (downward) bobber. Their velocities drivenby STT are shown in Fig. 2(c) . Two observations are presented in order. (i) The BP with q¼–1 (q ¼þ1) attached at the end of the upward (downward) bobber only moves forward (backward). A nega-tive current u<0f a i l e dt om o v et h e q¼–1 BP due to the pinning effects. One of the pinning effects originates from the atomic lattice, 30 corresponding to the finite difference discretization in the micromag- netic simulations. The pinning effect is also influenced by the anisot- ropy and external fields.33Here, the pinning effect is asymmetric in thez-direction since the magnetic profile of the BP is asymmetric in thez-direction. (ii) The sign of DMI does not induce the asymmetry for bobber motion, which is totally different from the toron case. FIG. 2. (a) An upward bobber and (b) a downward bobber are stabilized by the same external field Hz¼160 mT. Only one BP is attached at the end of each bob- ber. (c) The velocity of the BPs attached at the end of upward and downward bob-bers as a function of the current parameter ufor both positive and negative DMI constants.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 022404 (2021); doi: 10.1063/5.0033239 118, 022404-3 Published under license by AIP PublishingTherefore, a magnetic toron is not simply a combination of two bobbers. Based on the bobber dynamics and the toron dynamics, we can make the following assumption: Both the toron elimination and toronmovement are induced by the velocity difference of two BPs, that is, aBP alone with charge q¼þ1(q¼–1) only moves backward (forward). The sign of DMI combined with the current direction may stabilize the toron when two BPs are close enough, thus resulting in atoron motion. We can check this assumption using an elongatedtoron: For a positive current u¼120 m/s, in the first stage, the bottom BP with q¼–1 should move forward, while the top BP with q¼þ1 will almost remain static, and thus, the toron becomes shorter. For the positive DMI case, once the length of the toron is shorter than a criticallength, the top BP with q¼þ1 will also move and, thus, leads to the toron motion. Meanwhile, if the sign of DMI is negative, the top BPwith q¼þ1 will still remain static and, thus, results in the elimination of the toron. The corresponding simulation results are shown in Fig. 3 (see the supplementary material (II.mp4) for the animations of the elongated toron); it is found that the dynamics of the elongated toronagrees with the aforementioned predictions.Now, a remaining question is where the unidirectionality of DMI-induced toron motion comes from. While the static profiles forboth positive and negative DMI constants are equivalent (clockwise or counterclockwise circulating), we guess it comes from the precessional direction of the LLG equation. A macrospin rotates clockwise in thepresence of an effective field when viewed from the field direction.We perform simulations based on a modified LLG equation in which macrospin rotates counterclockwise, i.e., ð1þa 2Þ@tm¼cm /C2Heff/C0acm/C2ðm/C2HeffÞ/C0ð 1þabÞu@zmþðb/C0aÞuðm/C2@zmÞ. Figure 4(a) shows the simulation results using an orange arrow for a positive DMI, which are the same for the negative DMI case based on the standard (correct) LLG equation. Therefore, the observed unidirec-tionality is related to the precessional direction of a spin in the pres- ence of external fields. Figure 4(a) summarizes the response of the toron to currents for both positive and negative DMI constants. There are three regions split by two critical currents u c1anduc2.I nt h er e g i m eo f uc1<u<uc2, the toron is pinned for both positive and negative DMI constants.The other two regions [( u<u c1¼/C060 m/s for toron elimination) and ( u>uc1¼120 m/s for toron moving)] are switched if the sign of the DMI is changed. It can be seen that the asymmetry isinduced mainly by the DMI since other parameters remain the same. Figure 4(b) plots the toron centers as a function of the time for both positive and negative DMI scenarios; the toron performs a steadymotion after initial stimulations. The toron center is defined as z c¼ÐQðzÞzdz=ÐQðzÞdz,w h e r e Q(z) is the skyrmion number for the layer at z. Note that the toron is not strictly rigid—its length oscil- lates slightly accompanying the moving, which can be seen from theanimations (I.mp4 and II.mp4) in the supplementary material . Figure 4(c) shows the toron velocity as a function of current ufor different bvalues. The toron velocity almost scales linearly with the current strength u. The velocities are fitted using straight lines, and the fitted slopes are 0.58, 1.28, and 1.44 for b¼0.05, 0.1, and 0.15, respec- tively. The b-dependent velocity for u ¼150 m/s and a¼0:1i ss h o w n inFig. 4(d) .A sbincreases, the toron velocity increases linearly, and a large damping adecreases the toron velocity, which is similar to the current-velocity relation for a skyrmion in antiferrimagnets. In our previous simulations, the demagnetization field is not explicitly included. We also performed the full micromagnetic simula- tions including the demagnetization field. The used external field is H z ¼210 mT, and the anisotropy is set to zero ( Ku¼0). The simulation result is summarized as the animation (III.mp4) in the supplementary material , which shows that the main conclusions remain valid in the presence of the demagnetization field. FIG. 3. The simulation results of an elongated toron driven by current with u¼þ 120 m/s. (a) The snapshots of the toron with a positive DMI, and the elon- gated toron becomes shorter and then moves forward. (b) When the DMI constant is negative, the toron becomes shorter and shorter and eventually disappears. FIG. 4. (a) The phase diagram of the toron response to the current for both positive and negative DMI constants. The orange arrow shows the simulation results b ased on a modified LLG equation in which the macrospin rotates counterclockwise when viewed from the field direction. (b) The toron position changes with time fo ru¼120 m/s with D>0 and u¼–120 m/s with D<0. (c) The toron velocity as a function of currents for different bvalues with a positive DMI constant. The velocity-current relations are fitted using straight lines. Both the slope and the x-intercept depend on b. (d) The toron velocity as a function of bfora¼0:1 and u¼150 m/s.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 022404 (2021); doi: 10.1063/5.0033239 118, 022404-4 Published under license by AIP PublishingThe toron structure can also be found in chiral liquid crystals,16,34 in which the torons can be moved using an oscillating electric field. Considering the asymmetric pinning effect of the BPs of the toron, an oscillating electric current may induce interesting toron dynamics. In summary, we have studied the toron dynamics driven by elec- tric currents using micromagnetic simulations. We find that the toron c a nb em o v e db yt h ee l e c t r i cc u r r e n t sw h e nt h ec u r r e n td i r e c t i o ni s parallel to the toron. The response of the toron to the current is asym-metric, and the moving direction of the toron depends on the sign ofthe DMI constant. The toron only moves forward for a positive DMIand backward for a negative DMI in the presence of currents. The uni- directional motion of the magnetic toron enables the designing of diodes based on 3D spin textures with topological characteristics. See the supplementary material for the animations of the dynam- ics of a toron (I.mp4), an elongated toron (II.mp4), and the toron inthe presence of demagnetization fields (III.mp4). The authors acknowledge the High-performance Computing Platform of Anhui University for providing computing resources. L.K. acknowledges the financial support of the National Natural Science Foundation of China (Grant No. 11974021) and the Natural Science Foundation of Anhui Province (Grant No.2008085QA48). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1I. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958). 2T. Moriya, Phys. Rev. 120, 91 (1960). 3U. K. R €oßler, A. N. Bogdanov, and C. Pfleiderer, Nature 442(5), 797 (2006). 4S. M €uhlbauer, B. Binz, F. Jonietz, C. 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Assessment of the fatigue transformation zone in bulk metallic glasses using positron annihilation spectroscopy M. Liu, R. S. Vallery, D. W. Gidley, M. E. Launey, and J. J. Kruzic Citation: Journal of Applied Physics 105, 093501 (2009); doi: 10.1063/1.3120784 View online: http://dx.doi.org/10.1063/1.3120784 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Resolving ensembled microstructural information of bulk-metallic-glass-matrix composites using synchrotron x- ray diffraction Appl. Phys. Lett. 97, 171910 (2010); 10.1063/1.3506694 Enhancing plasticity of Zr 46.75 Ti 8.25 Cu 7.5 Ni 10 Be 27.5 bulk metallic glass by precompression Appl. Phys. Lett. 95, 071906 (2009); 10.1063/1.3211112 Characterization of fatigue-induced free volume changes in a bulk metallic glass using positron annihilation spectroscopy Appl. Phys. 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Kruzic3,a/H20850 1Department of Physics, Randall Laboratory, University of Michigan, Ann Arbor, Michigan 48109, USA 2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 3Materials Science, School of Mechanical, Industrial, and Manufacturing Engineering, Oregon State University, Corvallis, Oregon 97331, USA /H20849Received 21 November 2008; accepted 23 March 2009; published online 1 May 2009 /H20850 Depth-profiled Doppler broadening spectroscopy of positron annihilation on fatigue fracture surfaces of two amorphous Zr 44Ti11Ni10Cu10Be25metallic glass specimens reveals the presence of a layer of increased free volume induced by cyclic deformation, as compared to surfaces that havebeen etched to remove any surface damage. The damage layer, or fatigue transformation zone /H20849FTZ /H20850, is generated by the propagating fatigue crack tip and the deduced size of that zone is similar to the predicted cyclic plastic zone size at a number of locations where the crack grew at differentstress intensities. The presence of the FTZ is independent of the initial amount of bulk free volume,which was varied between the two specimens by structural relaxation via annealing, and the freevolume sites generated in the zone are distinct from those typical of the bulk, as evidenced by thehigher Sparameter. Such observations support the concept that the mechanically induced free volume within the FTZ zone controls the fatigue crack growth rates rather than the initial freevolume of the bulk material. © 2009 American Institute of Physics ./H20851DOI: 10.1063/1.3120784 /H20852 I. INTRODUCTION Bulk metallic glasses /H20849BMGs /H20850are a relatively new class of engineering materials with unique and unusual propertiesthat make them potential candidates for many structuralapplications. 1Favorable properties include near theoretical strengths combined with reasonable fracture toughness, lowdamping, large elastic strain limits, and the ability to be ther-moplastically formed into precision shaped parts with com-plex geometries, 2,3all of which are generally distinct from, or superior to, corresponding crystalline metals and alloys.One property which has been perceived as a limitation forBMGs has been low fatigue resistance relative to crystallinemetallic materials; however, not all studies to date have beenin agreement with this point. For the most studied BMG,Zr 41.25Ti13.75Ni10Cu12.5Be22.5,4the reported 107cycle fatigue strengths vary by a factor of 7,5,6and fatigue thresholds vary by a factor of three.5While some of the reported scatter may be explained by different testing configurations,7this does not account for all the observed variations, for example,those within single studies. 5,8Results on crack initiation have also varied widely, with some published reports indicatingalmost immediate crack initiation at the onset of cycling, 9 while other results show a significant part of the fatigue lifecan be spent initiating a crack for a different BMG withsimilar composition. 10Clearly there is a need for more fun- damental understanding of fatigue mechanisms and the fac-tors that govern fatigue behavior. It is now widely accepted that metallic glasses are dense, more or less random arrangements of efficiently packed clus-ters of various atoms with different sizes, often exhibiting well-defined medium range order /H20849MRO /H20850on a scale of 1–2 nm. 11–14Like crystalline metals, fatigue crack growth in BMGs is known to be governed by plastic deformation at thecrack tip. 10,15,16Spaepen originally proposed that the plastic deformation of metallic glasses utilizes rearrangement of at-oms and free volume regions, where the free volume is theatomic volume in excess of the ideal densely packed but stillamorphous structure. 17Furthermore, inhomogeneous defor- mation is accommodated by the creation of additional freevolume via a mechanism of flow-induced dilatation, whichcauses shear softening. 17Experimental assessments of excess volume by positron annihilation spectroscopy18,19/H20849PAS /H20850and x-ray synchrotron radiation20provided evidence of such deformation-induced dilatation. Macroscopic deformationoccurs within shear bands that are thought to initiate by thecooperative reorganization of atomic regions that involvetens of atoms referred to as shear transformation zones/H20849STZs /H20850. 21–23The plastic deformation response is therefore expected to depend on the initial state of the STZs, i.e., theinitial local MRO and free volume. 24As such, the amount of free volume has been shown to affect the deformation andfracture behavior of BMGs, 10,25–27and recent studies demon- strated a pronounced effect on the fatigue life.8,10Surpris- ingly, one mechanical property, namely, the fatigue crackgrowth, has been found to be largely unaffected by bulk free volume differences. 10 Because deformation of metallic glasses is accommo- dated by plastic rearrangement of STZs,21it may be expected that the intense deformation near a fatigue crack tip willgenerate a local increase in free volume. It has been pro- posed that this deformation induced free volume determinesthe local flow properties, rendering fatigue crack growth be- a/H20850Author to whom correspondence should be addressed. Tel.: /H110011-541-737-7027. FAX: /H110011-541-737-2600. Electronic mail: jamie.kruzic@oregonstate.edu.JOURNAL OF APPLIED PHYSICS 105, 093501 /H208492009 /H20850 0021-8979/2009/105 /H208499/H20850/093501/6/$25.00 © 2009 American Institute of Physics 105 , 093501-1 [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: 138.26.31.3 On: Mon, 22 Dec 2014 10:37:08havior relatively insensitive to initial bulk free volume differences.10PAS provides an experimental capability to ex- amine subatomic open volumes and is therefore useful fordirectly analyzing the free volume in amorphous materials.Indeed, PAS has recently been used to demonstrate the pres-ence of a fatigue transformation zone /H20849FTZ /H20850of enhanced free volume that is generated by a propagating fatigue crack tip. 28 Those results suggested that the zone of enhanced free vol-ume is roughly equal in size to the extent of reversed plasticflow during each cycle, i.e., the cyclic plastic zone. 28How- ever, that initial study only looked at a single location oneach sample and did not systematically characterize the ex-tent of the FTZ over regions with different cyclic plasticzone sizes. Accordingly, the purpose of this manuscript is to more fully characterize the FTZ, which forms during fatigue crackpropagation in a Zr–Ti–Ni–Cu–Be based BMG. Specifically,by utilizing a focused positron beam, the Doppler broadeningspectroscopy /H20849DBS /H20850technique of PAS is used to depth- profile fatigue fracture surfaces along several millimeters ofcrack growth to characterize the FTZ and its evolution dur-ing stable fatigue crack propagation. II. EXPERIMENTAL METHODS A. Materials The two fully amorphous Zr 44Ti11Ni10Cu10Be25compact tension C /H20849T/H20850specimens were heat treated differently to achieve residual stress free samples with different amountsof free volume prior to fatigue testing. The annealing proce-dures are described in detail elsewhere 10and a brief descrip- tion is given here. One sample was isothermally relaxed at610 K for 10 /H9270to lower the free volume relative to the as-cast condition, where /H9270represents the structural relaxation time /H20849/H9270=438 s at 610 K /H20850.29After annealing for 10 /H9270, the BMG was confirmed to be fully amorphous by high resolution trans-mission electron microscopy and is assumed to be fully re-laxed into its metastable equilibrium, or lowest free volumestate, with a normalized free volume difference relative tothe original as cast state, /H9004 /H9263f//H9263m, of 0.044%.29Here/H9004/H9263fis the average free volume difference per atom and /H9263mis the atomic volume near the liquidus.30For the second sample, only a residual stress relief /H20849SR/H20850annealing treatment /H20849573 K for 120 s /H20850was applied and, unlike the 10 /H9270sample, no free volume relaxation occurred due to the low temperature andshort time, as confirmed by differential scanning calorimetry/H20849DSC /H20850experiments. 10 In order to compare the PAS results for the FTZ to a nominally undeformed condition, two additional fatiguetested samples that were heat treated as described abovewere chemically etched using an acid solution/H20849HF:HNO 3:H2O=10:45:45 by volume /H20850at 24 °C for 30 s to remove both the fatigue induced deformation on the frac-ture surfaces as well as the mechanical polishing damage onthe polished surfaces. Roughly 10 /H9262m of material was re- moved from the surfaces by etching. The etching procedurewas found to be necessary as initial results indicated, in somecases, there is a surface damage layer associated with me- chanical polishing that is smaller than, or similar in size to,that found on the fatigue fracture surfaces. 28 B. Cyclic fatigue-crack growth rate measurements Each C /H20849T/H20850sample was cycled until complete fatigue fail- ure /H20849Fig. 1/H20850. Fatigue crack growth experiments were con- ducted in general accordance with ASTM standard E647/H20849Ref. 31/H20850using a computer controlled servohydraulic test machine with a frequency /H9263of 25 Hz /H20849sine wave /H20850and a constant load ratio /H20849ratio of minimum to maximum applied load, R=Pmin /Pmax/H20850of 0.1. The C /H20849T/H20850samples were of stan- dard dimensions with thicknesses B=2.2 mm and width W =25.4 mm. Fatigue crack growth rates da /dNwere mea- sured as a function of the applied stress intensity range,/H9004K=K max−Kmin, where KmaxandKminare the maximum and minimum stress intensity experienced during the loadingcycle. After complete sample fracture, free volume was char-acterized along the cyclically deformed fracture surfaceswhere the maximum applied stress intensity K maxranged from /H110113.2 to /H110111.6 MPa /H20881m. The gradient of applied stress intensities was achieved by growing the fatigue cracks underdecreasing /H9004Kloading conditions with a normalized /H9004K-gradient, d/H9004K/da /d/H9004K, of −0.08 mm −1.31 C. Bulk positron annihilation lifetime spectroscopy To better understand the size distribution of free volume voids in the bulk of the SR and 10 /H9270samples, positron anni- hilation lifetime spectroscopy /H20849PALS /H20850was performed on each of them far away from the fracture surface. A smalldrop of radioactive 22Na in saline solution was deposited and allowed to dry at the center of a polished face. The corre-sponding second half piece was placed over this insuring thatall the beta-decay positrons from the source stop in the bulkof the sample /H20849the positrons will penetrate only 0.1–0.2 mm deep /H20850. This sample-source sandwich arrangement was placed in a typical fast-timing PALS spectrometer with time reso-lution of 270 ps. Typically 4–10 million events were col-lected in each lifetime histogram. Standard lifetime fitting FIG. 1. /H20849Color online /H20850Typical half C /H20849T/H20850specimen after complete fatigue failure. Depth-profiled DBS experiments were performed on the polishedface and the cyclically loaded fracture surface of the fatigue specimens.Some samples were examined after etching to remove any damage layersfrom fatigue or mechanical polishing.093501-2 Liu et al. J. Appl. Phys. 105 , 093501 /H208492009 /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: 138.26.31.3 On: Mon, 22 Dec 2014 10:37:08programs were used to fit the lifetime spectrum to three life- time components, which correspond to three different sizesof open volume voids /H20849to be discussed /H20850. 32 D. Depth-profiled Doppler broadening spectroscopy DBS of positron annihilation is a standard positron an- nihilation spectroscopic technique to characterize the freevolume defects and voids in materials. DBS monitors theenergy spectrum of the annihilation radiation in a high en-ergy resolution Ge gamma ray detector. The Doppler shifts inthe 511 keV annihilation radiation resulting from the mo-mentum of the annihilating positron-electron pair is mani-fested in the spectrum as a Doppler energy broadening of the511 keV peak. The broadening of the 511 keV annihilationpeak is characterized by the Sparameter: the ratio of the number of counts within a central region of the peak to thetotal number of counts in the full peak. In this experiment,this central peak region for all DBS energy spectra was set tobe between 0.88 keV above and below the 511 keV peak, aregion which gave a fitted Sparameter of nominally 0.5. Positron beams with variable positron implantation energiesEin the range 1–10 keV can depth profile the defect charac- teristics up to several hundreds of nanometers below a BMGsurface where significant deviations from bulk behavior mayoccur. The mean positron implantation depth for a normalincidence beam on a BMG with nominal density of 6 g /cm 3 isz¯/H20849nm/H20850=6.7 /H20851E/H20849keV /H20850/H208521.6. Details of depth-profiled PAS can be found elsewhere.33–35 In the present experiment, a focused positron beam with an on-target diameter of /H110212 mm was used to depth profile and compare both the deformed fracture surfaces and thepolished faces of the two specimen types, SR and 10 /H9270, before and after etching. As mentioned previously, the etchedsamples were free of any fatigue induced or mechanical pol-ishing damage, hence, they were used as baselines for ourexperiments. Beam-based DBS experiments were conductedat a series of positions where the crack grew at differentstress intensities /H20849nominally 2 mm apart, see Fig. 1/H20850along the deformed fracture edges /H208492.2 mm thick /H20850of both SR and 10 /H9270samples. The two fractured pieces of each sample were stacked side-by-side with their corresponding fracture edgesaligned to give a total width of 4.4 mm, ensuring that nopositrons missed the fracture surface. At each position alongthe fracture, a series of positron beam energies, from 1.1 to10 keV, were used in order to profile the free volume varia-tions from depths of /H110118t o /H11011270 nm. On the etched samples, one central position /H20849x=8 mm from left edge in Fig.1/H20850on the fatigue fracture surface was depth profiled by DBS for comparison along with the polished and etched sideface /H20849Fig.1/H20850. III. RESULTS The fatigue crack-growth rates are found to be insensi- tive to free volume variations /H20849Fig.2/H20850. Here the results are presented in terms of the maximum stress intensity of theloading cycle K maxrather than the traditional stress intensityrange /H9004Ksince the former corresponds with the maximum extent to the plastic deformation zone, as will be discussedbelow. The bulk PALS results for both the 10 /H9270and SR samples are identical. Consistent with Ref. 32, three lifetime compo- nents were fitted: a short 177.1 /H110060.4 ps lifetime with which 98.4% of the positrons annihilate, an intermediate lifetime of420/H1100615 ps with 1.4% intensity, and a long 2.1 /H110060.2 ns lifetime with only 0.2%. This almost complete dearth of an-nihilation from the two longer lifetimes indicates they areessentially free of deformation induced flow defects andnanovoids. 32The difference in free volume between the two samples is indistinguishable in PALS because it apparentlyjust involves differences in concentrations of the smallestvoids from which virtually all the positrons annihilate re-gardless of concentration. TheSparameters deduced from the depth-profiled DBS experiments were recorded as a function of mean positronimplantation depth. In Fig. 3, the Sparameter depth profiles at three different positions along the fatigue crack of the SRsample are shown /H20849the same procedures were used on the 10 /H9270sample and the detailed depth profiles are not shown here for clarity /H20850. For comparison, the Sparameters for the polished and etched surface and etched fatigue fracture sur-face /H20849atx=8 mm /H20850are shown. These etched surface Sparam- eters have been shifted by /H110020.011 and /H110020.007, respectively, since these data were acquired subsequent to an apparentshift in the gamma detector’s energy resolution. More impor-tantly and contrary to all the fractured surface depth profiles,the etched surfaces show no indication of an increased S parameter at low implantation depth. Indeed, there is a clearsurface effect at depths less than 50 nm that is considerablymore pronounced than the corresponding surface effect onthe fractured surfaces. This is not surprising as the samplesurface may be chemically altered by etching and the posi-tron diffusion length in the fractured sample may be reducedby defect trapping, hence reducing the depth from whichpositrons can return to annihilate near the surface. The fracture surface depth profiles were fit to a two layer model, ignoring data below 20 nm mean implantation depth FIG. 2. /H20849Color online /H20850Fatigue crack-growth rates, da /dN, plotted as a func- tion of the maximum applied stress intensity, Kmax, for two different free volume states of Zr44Ti11Ni10Cu10Be25.093501-3 Liu et al. J. Appl. Phys. 105 , 093501 /H208492009 /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: 138.26.31.3 On: Mon, 22 Dec 2014 10:37:08as being too strongly affected by an uninteresting surface state. In this simple model, a deformed region, i.e., the FTZ,with a uniformly higher Sparameter due to annihilation in larger open volume fracture-induced defects is assumed toexist on top of an unaffected bulk region with a lower S parameter /H20849as shown by PALS the bulk is quite free of deformation-induced defects /H20850. The observed Sparameter is assumed to be an average of the two Sparameters weighted by the fraction of positrons stopping in each layer /H20849FTZ or bulk /H20850as determined from the positron implantation profile. 33–35Positron diffusion after implantation is taken to be negligible. The fitted parameters are the corresponding S parameters of the FTZ and in the bulk along with the depth/thickness of the transformation zone. Ideally, the bulk Spa- rameters would be identical for each case, but minor differ-ences in positron backscattering and detector resolutioncause small variations that can be accounted for by fittingthis bulk Svalue for each profile. The results of fitting to the two-layer model are shown in Fig. 3. The fitted values of the FTZ thickness at each location along the fracture surface foreach sample is then plotted in Fig. 4versus the calculated maximum stress intensity K maxat that specific location /H20849to be discussed below /H20850. IV. DISCUSSION To explain the insensitivity of fatigue crack growth rates to the bulk free volume of the BMG, it has been proposedthat the large cyclic plastic strains that occur near a fatiguecrack tip cause a local increase in free volume, or FTZ. 10,28 Propagation of the crack requires it to grow through the zone of enhanced free volume and as such the local deformationproperties are governed by the free volume within that zonerather than the bulk free volume of the material determinedby the thermal history. This theory predicts the depth of the fatigue-induced transformation layer should correspond with the expected ex-tent of plastic deformation due to cyclic loading, i.e., the0 50 100 150 200 2500.5160.5180.5200.5220.5240.5260.5280.5300.5320.534S Parameter Mean Implantation Depth (nm)Position along the deformation zone 10mm, Kmax= 2.49 8mm, Kmax= 2.15 4mm, Kmax= 1.71 After etching 8mm Bulk surfaceSR FIG. 3. /H20849Color online /H20850Sparameter profiles and two- layer model fits of selected positions along the fatiguedeformation zone of sample SR. The higher Sparam- eters correspond to increased free volume induced byfatigue. Distances corresponding to each curve are mea-sured from the left edge of the sample in Fig. 1and each corresponds to a location where the fatigue crackwas grown at a different stress intensity K max. The stress intensity values are also given for each position /H20849in MPa /H20881m/H20850. Results for 10 /H9270are similar to those shown above. 1.2 1.6 2.0 2.4 2.8 3.2020406080100120140 Eq. (2) ExperimentalTransformation zone thickness [nm] Maximum Stress Intensity, Kmax[MPam1/2]A SR Kmax,th 1.2 1.6 2.0 2.4 2.8 3.2020406080100120140Trans formation zone thickness [nm] Maximum Stress Intensity, Kmax[MPam1/2]B 10τ Eq. (2) Experimental Kmax,th(a) (b) FIG. 4. /H20849Color online /H20850Thickness of the FTZ plotted as a function of the maximum stress intensity Kmaxfor /H20849a/H20850the stress relieved sample SR /H20849as- processed free volume /H20850,a n d /H20849b/H20850the sample 10 /H9270/H20849relaxed lower free volume state /H20850. Reasonable correlations are found between the experimental data from depth-profiled DBS experiments and the theoretical estimate definedby Eq. /H208492/H20850.093501-4 Liu et al. J. Appl. Phys. 105 , 093501 /H208492009 /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: 138.26.31.3 On: Mon, 22 Dec 2014 10:37:08FTZ size should increase with increasing stress intensity. This is clearly confirmed by the results in Fig. 4, where both samples show a trend of increasing FTZ size with increasingstress intensity. Furthermore, it is interesting to compare thePAS measured FTZ thicknesses to plastic zone size estimatesfrom linear elastic fracture mechanics. For plane strain con-ditions, the overall plastic zone extent normal to the crackplane /H20849i.e., into the fracture surface /H20850is roughly three times that in the direction ahead of the crack tip, i.e., 36 rp=1 2/H9266/H20873Kmax /H9268Y/H208742 , /H208491/H20850 where Kmaxis the maximum stress intensity of the loading cycle and /H9268Yis the yield stress of the material /H20849/H110111900 MPa /H20850.37During cyclic loading, there is a smaller cyclic plastic zone where reversed plastic flow occurs during each cycle. It is generally accepted that this cyclic plastic zone is roughly1 4the total plastic zone size.38This would predict the depth of the FTZ into each fracture surface tocorrespond roughly to r FTZ=1 8/H9266/H20873Kmax /H9268Y/H208742 . /H208492/H20850 This concept is indeed consistent with the DBS results. Fig- ure4shows Eq. /H208492/H20850/H20849with no fitted parameters /H20850plotted along with the depth profiled DBS results and the agreement isquite reasonable. The present theory catches the key featuresassociated with the FTZ formation: /H20849i/H20850fatigue cycling in- duces a FTZ ahead of the crack tip that has distinctly largerfree volume voids than the bulk and /H20849ii/H20850the thickness of the FTZ is roughly similar to the expected cyclic plastic zone.Furthermore, such results help explain the surprising insen-sitivity of fatigue crack growth rates to the initial free vol-ume of the BMG. Indeed, the local free volume appears to bedetermined by the cyclic deformation rather than the thermalhistory. There are some discrepancies between the data and pre- dictions that should be discussed, however, such as the factthat the DBS measured data do not appear to rise as fast withthe stress intensity as the prediction does for the SR samplein Fig. 4/H20849a/H20850and the zone size is somewhat under predicted for the 10 /H9270sample in Fig. 4/H20849b/H20850. Here it is important to note some limitations of the present analysis. First, to minimizethe number of fitted parameters, the two layer model used todetermine the layer thickness values assumed the FTZ layersto have a constant Sparameter, when in fact it is likely graded from a peak value near the surface to a value closer tothe bulk near the interface. Second, the traditional fracturemechanics methods for determining plastic zone sizes haveall been developed for crystalline metals. There are severalmechanistic differences in the plastic flow behavior of crys-talline versus amorphous metals. While crystalline metals de-form by dislocation motion at constant volume, usually ac-companied by strain hardening, amorphous metals deform atlow temperatures by highly localized flow in shear bandswith locally increasing volume and strain softening. Finally,the plastic zone is affected by the constraint of the surround-ing elastic material and in this case there are distinct differ-ences in the surrounding elastic materials due to the different heat treatments. This latter factor could help explain whyagreement appears to be better in the SR case than the 10 /H9270 case. Thus, while the key features of the theory are capturedin the present work, there is clearly more work to be done tofully understand the cyclic plastic zone formed in amorphousmetals. While it appears that the expected size of the FTZ is nominally described by Eq. /H208492/H20850, less is known about the in- ternal structure. As mentioned earlier, recent PALS and DBSresults suggest that the size distribution of free volume sitesmay be trimodal. 32Indeed, three characteristic positron life- times have been observed suggesting three distinct sizeranges for free volume elements. The smallest free volumesize corresponds to Bernal holes /H20849tetrahedral interstitial holes /H20850in the densely packed structure. Bernal holes are in- trinsic to the glassy structure and cannot relax out completelywith annealing. The intermediate size corresponds to largerflow defects, while the longest lifetime corresponds to thelifetime of orthopositronium in nanovoids with radii of a fewangstroms. 32The average size of the small and intermediate sized sites is thought to decrease with extensive deformation,while the largest sites remain roughly constant in size. Also,the concentrations of the larger two sites increase with largedeformations at the expense of the smallest sites. 32 In the present experiments, both bulk PALS and depth profiled DBS are not sensitive to the free volume changesfrom structural relaxation. For the latter case, the depth pro-filed data asymptotes to roughly the same value of the S parameter in both the 10 /H9270and SR cases. This suggests that every positron is finding a free volume element to trap in andfree volume changes due to structural relaxation do not sig-nificantly change the size of those free volume elements.Using atomic models of amorphous Pd–Ni–P, Sietsma andThijsse 39addressed the effect of structural relaxation on the volume distribution of the free volume elements. They foundthat the difference between as-quenched and annealed BMGpredominantly lies in the quantity of the flow defects. Morespecifically, the number of intrinsic voids, or Bernal holes,increases, whereas the number of flow defects decreases afterannealing. The interpretation is that during structural relax-ation the flow defects break up into two or more Bernalholes. Therefore, it is unlikely that a high concentration offlow defects and nanovoids would exist in the glass afterextensive structural relaxation. This is completely consistentwith our PALS data. Accordingly, structural relaxation ap-pears to affect the concentration of the smallest free volumeelements /H20849Bernal holes /H20850, rather than the size, while cyclic deformation produces distinctly larger free volume sites, asevidenced by the higher Svalues. V. CONCLUSIONS Based on depth profiled PAS results characterizing mul- tiple positions along the fatigue fracture surfaces for a Zr–Ti–Ni–Cu–Be BMG, the following conclusions can be made: /H208491/H20850The existence of a layer of locally increased free vol- ume, i.e., a fatigue transformation zone /H20849FTZ /H20850, is con- firmed at every point observed along the fatigue fracture093501-5 Liu et al. J. Appl. Phys. 105 , 093501 /H208492009 /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: 138.26.31.3 On: Mon, 22 Dec 2014 10:37:08surfaces. Indeed, when compared to the depth profile for an undeformed etched surface, the fatigue fracture sur-face shows a distinctly higher Sparameter, which de- creases asymptotically with depth to the nominal bulkvalue. /H208492/H20850The increased free volume of the FTZ almost certainly is comprised of larger flow defects and/or nanovoids.PALS results conclusively show that very few of theselarger voids exist in the undeformed bulk state of eithersample. Since DBS is found to be insensitive to differingconcentrations of the smallest, intrinsic voids /H20849Bernal holes /H20850it is concluded that larger defects are responsible for the increased Sparameter in the FTZ. Future mea- surements with high time resolution depth-profiledPALS could confirm this deduction. /H208493/H20850By fitting the depth profiled Sparameter data to a simple two layer model, it was found that the deduced size ofthe FTZ corresponds reasonably well with estimates ofthe cyclic plastic zone size from linear elastic fracturemechanics. A trend of increasing zone size with increas-ing maximum stress intensity is clearly seen, which isroughly equal to that predicted based on a simple esti-mate of the variation in the cyclic plastic zone size;however, future work is likely needed to understand allof the details involved in the cyclic plastic zone forma-tion in metallic glasses. ACKNOWLEDGMENTS M.E.L. and J.J.K. thank Dr. A. Peker and Dr. J. Schroers for supplying the material and Dr. R. Busch for many usefuldiscussions. D.W.G. gratefully acknowledges the support ofthe University of Michigan for positron research. 1C. J. Byrne and M. 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1.5120565.pdf

Appl. Phys. Rev. 7, 011304 (2020); https://doi.org/10.1063/1.5120565 7, 011304 © 2020 Author(s).Ferroic tunnel junctions and their application in neuromorphic networks Cite as: Appl. Phys. Rev. 7, 011304 (2020); https://doi.org/10.1063/1.5120565 Submitted: 20 July 2019 . Accepted: 01 October 2019 . Published Online: 06 January 2020 Rui Guo , Weinan Lin , Xiaobing Yan , T. Venkatesan , and Jingsheng Chen COLLECTIONS Note: This paper is part of the special collection on Brain Inspired Electronics. This paper was selected as an Editor’s Pick Ferroic tunnel junctions and their application in neuromorphic networks Cite as: Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 Submitted: 20 July 2019 .Accepted: 1 October 2019 . Published Online: 6 January 2020 RuiGuo,1,2,a)Weinan Lin,1,a)Xiaobing Yan,1,2,a)T.Venkatesan,1,3,4,5,6and Jingsheng Chen1,b) AFFILIATIONS 1Department of Materials Science and Engineering, National University of Singapore, Singapore 117575, Singapore 2College of Electron and Information Engineering, Hebei University, Baoding 071002, China 3NUSNNI-Nanocore, National University of Singapore, Singapore 117411, Singapore 4Department of Electrical and Computer Science Engineering, National University of Singapore, Singapore 117583, Singapore 5Department of Physics, National University of Singapore, Singapore 117542, Singapore 6Integrative Science and Engineering, National University of Singapore, Singapore 119077, Singapore Note: This paper is part of the special collection on Brain Inspired Electronics. a)Contributions: R. Guo, W. Lin, and X. Yan equally contributed to this work. b)Author to whom correspondence should be addressed: msecj@nus.edu.sg ABSTRACT Brain-inspired neuromorphic computing has been intensively studied due to its potential to address the inherent energy and throughput limitations of conventional Von-Neumann based computing architecture. Memristors are ideal building blocks for artificial synapses, whichare the fundamental components of neuromorphic computing. In recent years, the emerging ferroic (ferroelectric and ferromagnetic) tunneljunctions have been shown to be able to function as memristors, which are potential candidates to emulate artificial synapses for neuromor- phic computing. Here, we provide a review on the ferroic tunnel junctions and their applications as artificial synapses in neuromorphic net- works. We focus on the development history of ferroic tunnel junctions, their physical conduction mechanisms, and the intrinsic dynamicsof memristors. Their current applications in neuromorphic networks will also be discussed. Finally, a conclusion and future outlooks on thedevelopment of ferroic tunnel junctions will be given. Our goal is to give a broad review of ferroic tunnel junction based artificial synapses that can be applied to neuromorphic computing and to help further ongoing research in this field. Published by AIP Publishing. https://doi.org/10.1063/1.5120565 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. FERROELECTRIC TUNNEL JUNCTIONS AND THEIR APPLICATIONS AS ARTIFICIAL SYNAPSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 A. Introduction of ferroelectric tunnel junctions . . . . 3 B. Physical mechanism of ferroelectric tunnel junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 C. Transport mechanisms in ferroelectric tunnel junctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Direct tunneling . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Fowler-Nordheim tunneling . . . . . . . . . . . . . . . . 63. Thermionic injection . . . . . . . . . . . . . . . . . . . . . . 6 D. Multiferroic tunnel junction . . . . . . . . . . . . . . . . . . . 6E. Mechanisms of the ferroelectric tunnel memristor 6 1. Ferroelectric domain switching . . . . . . . . . . . . . 72. Ferroelectric field effect . . . . . . . . . . . . . . . . . . . . 7 3. Migration of oxygen vacancies . . . . . . . . . . . . . . 8 F. Performance of the ferroelectric tunnel memristors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 G. Ferroelectric tunnel memristor for neuromorphic network applications . . . . . . . . . . . . . . . . . . . . . . . . . 10 III. MAGNETIC TUNNEL JUNCTIONS AND THEIR APPLICATION AS ARTIFICIAL SYNAPSE . . . . . . . . . 14 A. Introduction of magnetic tunnel junction . . . . . . . 14 B. Mechanism of the magnetic tunnel junction . . . . . 15 C. Fundamental physics and the basic principle for the current-induced torques for manipulating the magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 D. Spintronic devices for synapse application. . . . . . . 17E. Spintronic devices for neuron application. . . . . . . . 18 IV. CONCLUSION AND FUTURE OUTLOOK . . . . . . . . . 20 Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-1 Published by AIP PublishingApplied Physics Reviews REVIEW scitation.org/journal/areI. INTRODUCTION The continuing development of computers based on traditional memories is facing great challenges due to the end of Moore’s law in sight, the bottlenecks of power efficiency, and bandwidth caused by the Von-Neumann architecture with separated processors and memory. On the contrary, brainlike neuromorphic computing systems have great potential to address the inherent energy and throughput limitations of conventional von-Neumann computing architecture and hold promise to achieve high efficiency in performing cognitive and data-intensive tasks.1–7Artificial synapses and neurons are the fundamental functional components for neuromorphic computing systems, where the synapse transmits information from one neuron to another neuron. A synapse is essentially a two-terminal device, which has striking similarities to theelectrical device known as a memristor. Similar to the synaptic plasticity of a biological synapse, the conductance of a memristor could be tuned continuously based on the history of applied voltage or current. 8 Furthermore, the memristors are found to be able to function as artifi-cial neurons. Therefore, memristors and devices with similar features are considered as promising building blocks for artificial synapses and neurons and can thus be utilized to construct future storage and hard- ware neural networks for neuromorphic computing. 9,10 To date, applications for artificial synapses have been extensively investigated in many memristive devices, with resistive random-access memory (RRAM) and phase change memory (PCM) as the most stud- ied ones. RRAM and related memristors have been broadly studied for decades using different types of materials, for example, binary metal oxides, such as HfO 2,11,12TiO 2,13–15WO x,16SiO 2,7,17,18and Ga 2O3.19 Tremendous research efforts have been made to improve the deviceperformance and to address the challenges associated with their large-scale implementations. Studies have shown that RRAM-based mem- ristors can be scaled down to nanosize, 20with subnanosecond switch- ing speed,21,22low power consumption,23good data retention properties,1and long cycling endurances.24Despite the advantages, RRAM also has its drawbacks. For filamentary type RRAM, the intrin- sic inhomogeneity and randomness of filament formation and the device-to-device and cycle-to-cycle variations in devices are the major drawbacks for their practical applications. Compared with filamentary RRAM, nonfilamentary type oxide memristors have better switching uniformity, due to their different resistive switching mechanisms.25,26 Nevertheless, studies have shown that they usually have a small OFF/ ON ratio and also have the same problem as filament RRAM in the linearity and symmetry of the conductance change.10Detailed reviews on resistive switching and their applications in neuromorphic comput- ing can be referred to in Refs. 1,10and27–38 . Different from RRAM, the resistive switching of PCM is induced by the phase transition between amorphous and crystalline phases. PCM based memristors usually have large OFF/ON ratios and fast switching speed.39–42 However, their thermal effect based switching mechanism results inthe inherent unipolar switching characteristic and high power con- sumption due to the Joule heat. 43–45Detailed reviews on PCM and their applications in neuromorphic computing can be referred to inRefs. 46–51 . In addition to RRAM and PCM, ferroelectric and magnetic tun- nel junctions have demonstrated their significant importance in the data-storage field due to the switchable electric polarization (magneti- zation) states of ferroelectric (ferromagnetic) materials, which allows binary information storage. 52–54Electron tunneling is a quantummechanical phenomenon where electrons can pass through a potential barrier that is higher than its kinetic energy. Many useful electronic devices are built based on this phenomenon.55Although the concept of ferroelectric tunnel junctions (FTJs) with an ultrathin ferroelectric layer sandwiched by two different electrodes has been proposed in 1971,56only in the 2000s, were experimental demonstrations of FTJs with different kinds of ferroelectric oxide layers reported due to the advance of thin film deposition technology, which enables the fabrica-tion of a ferroelectric oxide layer with the ferroelectricity down to a few unit cells. 57,58The polarization switching of the ferroelectric thin film in an FTJ leads to a large change of the junction resistance, whichoffers a new strategy to the electrical switching of resistance that can be utilized in noncharge-based memory and logic devices. 59,60Besides, FTJs use a nondestructive reading method, which is to read the resis- tance of the junction by applying electrical voltages that are much smaller than the coercive voltages of the ferroelectric thin film.61–64 This solves the biggest problem of the conventional ferroelectric ran- dom access memory (FeRAM),65which opens the door for faster and energy-efficient nonvolatile random access memories. In very recent years, it has been discovered that, in contrast to the common belief that ferroelectric materials can only exhibit binary switching behavior,ferroelectric-based memories have been reported to show memristive behaviors too. 66–68FTJ-based memristors were then demonstrated using different ferroelectric thin films. By applying different externalvoltage biases, multilevel conductance states could be simply obtained. As an emerging type of memristor, FTJ-based memristors possess the advantages of large OFF/ON ratios, fast switching speed, and good cycling endurance. 63,69–71Furthermore, compared with the conven- tional RRAM and PCM, the electroforming process is not needed. Inaddition, FTJ-based memristors have quite simple physical mecha- nisms and clear microscopic processes of the resistive switching, which are crucial for memristors to achieve reliable and predictable nanode- vices. 68,72All these mentioned merits of FTJ-based memristors enable them to be promising candidates as the artificial synapse for neuro-morphic computing. The FTJ-based memristors are based on the electron’s charge degree of freedom of the investigated materials to explore theirpotential application in neuromorphic computing. The spin degree of freedom of the electron can also be exploited for potential applications in the new computing architecture. Spin-dependenttunneling in magnetic tunnel junctions (MTJs) has attracted exten- sive attention recently. Different from the FTJ-based memristors, where the multistates come from the ferroelectric barrier, the func- tionalities of the MTJ rely on the employed ferromagnetic electro- des. The advantage of the MTJ-based spintronic devices is due tothe versatile material functionality available with the spin degree of freedom and a flurry of physics phenomena involved, such as current-induced spin torque, current-induced deterministic or stochastic magnetic switching, and domain wall formation and its propagation. This flexibility may allow the implementation of differ-ent processes for neuromorphic computing, i.e., synapse and neuron, in a structure with the same material stacks. The intrinsic property of the spin degree of freedom may endow spin-based neuromorphiccomputing with merits of high speed, high integration scalability, low power consumption, and excellent endurance. 73Furthermore, with the recent successful development of the spin transfer torque magnetic random-access memory (STT-MRAM), it is promising toApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-2 Published by AIP Publishingdevelop complementary metal-oxide-semiconductor (CMOS)-com- patible spin-based neuromorphic computing architecture. This review focuses on recent ferroics-based explorations for their possible application for the promising neuromorphic computing architecture, which is composed of two parts. The first part provides a detailed overview of ferroelectric tunnel junction based memristorsand their applications as artificial synapses in the neuromorphic com- puting. The second part gives a summary of the development of MTJs and spin-based devices as components for neuromorphic computing.We focus on the physical mechanisms of ferroic tunnel memristors and try to cover their state-of-the-art research and their applications as artificial synapses and neurons. In the end, we will conclude our review by summarizing the two parts and giving our opinions on the future development of the devices reviewed in this work. II. FERROELECTRIC TUNNEL JUNCTIONS AND THEIR APPLICATIONS AS ARTIFICIAL SYNAPSES A. Introduction of ferroelectric tunnel junctions An FTJ is a two-terminal device, which has a simple structure: an ultrathin ferroelectric thin film layer sandwiched between two electro- des, as shown in Fig. 1(a) . The polarization of the ultrathin ferroelec- tric thin film can be switched either toward the top electrode or toward the bottom electrode by an external electric field, leading to dif- ferent electroresistance states, as shown in Fig. 1(b) . The history of FTJs is briefly summarized in Fig. 2 . The basic idea of an FTJ (called a polar switch) was initially formulated very early in 1971 by Esaki et al. 56Although a change in the tunnel resistance upon ferroelectric polarization reversal had already been predicted at that time, this con- cept had been forgotten for about 30 years, due to the belief that ferro- electricity could not exist in films with thicknesses allowing tunnelingto happen. Only in the 2000s did the concept of an FTJ begin to attract people’s attention again, because the epitaxial deposition methods and characterization techniques had developed to be able to realize high- quality ultrathin ferroelectric films. 59,61–63,74–79Furthermore, because the electrical resistance is coupled to the ferroelectric polarization direc-tion, FTJs provide a simple nondestructive reading method with a relaxed constraint on the capacitor size, which represents important advantages over conventional ferroelectric random access memory(FeRAM). 80–82As a result, the topic of FTJs has attracted more andmore researchers’ attention since their first experimental demonstra- tion, and many patents have been filed using the concept of FTJ as thememory element. Our group has also done a lot of work on ferroelec- tric tunnel junctions, using different ferroelectric thin films, such as BaTiO 3(BTO),70,76,82–84BiFeO 3(BFO),81,85and Hf 0.5Zr0.5O2(HZO).86 With the deepening of research on FTJs, the concept of the ferro- electric tunnel memristor (FTM) was demonstrated experimentally first in year 2012,67,68which kick started research in FTJ based memo- ries. Subsequently, functional FTJs have been shown to be compatiblewith silicon substrates, the results of which predict their potential prac- tical applications in silicon-based nonvolatile memories. 83,87In very recent years, along with the research progress in brainlike neuromor-phic computing, plasticity in synaptic learning in FTJ-based artificial synapses has already been demonstrated using different ferroelectric thin films. 70,88,89Nowadays, flexible electronic devices are considered as emerging electronic technologies for information storage and other fields;90–95in the meantime, new deposition and transferring techni- ques have enabled the realization of flexible perovskite oxide ferroelec-tric thin films, 96–98which makes it likely that flexible neuromorphic architectures based on FTJs will be realized in the future. B. Physical mechanism of ferroelectric tunnel junctions Representative theoretical study on the interplay between the electron transport in an FTJ with an ultrathin ferroelectric barrier layer and the polarization state of the barrier was done by Zhuravlev et al. in 2005.99Using a model which takes into account the screening of polar- ization bound charges in metallic electrodes and direct quantum tunneling across the ferroelectric layer, the change in the tunnelingelectroresistance associated with the polarization switching can be cal- culated. The electrical resistance of an FTJ is strongly dependent on the polarization orientation of the ultrathin ferroelectric film, the phe-nomenon being called the tunneling electroresistance (TER) effect. 60 The physical mechanism responsible for the TER effect is the changeof the electrostatic potential profile u(z) induced by the reversal of the ferroelectric polarization, as illustrated in Fig. 3 . When connected with metal electrodes, the polarization surface charges are screened by the screening charges per unit area ( r s). Using the Thomas-Fermi model of screening, the screening potential withintwo metal electrodes can be given by FIG. 1. (a) Schematic of the FTJ structure. (b) Schematic of the two resistance states ROFFand RONof FTJs through polarization switching.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-3 Published by AIP PublishinguzðÞ¼rsd1 e0e/C0zjj d1/C0/C1 z/C200 /C0rsd2 e0e/C0z/C0d d2ðÞz/C21d;8 >>>< >>>:(1) r s¼Pd ebd1 e1þd2 e2/C18/C19 þd; (2) where d1andd2are the Thomas-Fermi screening length of the metal electrode 1 and 2, e1,eb,a n d e2are the static dielectric constant of elec- trode 1, barrier, and electrode 2, rsis the surface bound-charge den- sity, Pis the ferroelectric polarization, and dis the ferroelectric thickness. On the other hand, the incomplete screening of the ferroelectric surface bound charges by the electrodes generates the depolarizationelectric field Edpin the ferroelectric film. The electrostatic potential associated with this Edpdepends on the polarization direction of the ferroelectric thin film. Because the two different top and bottom elec- trodes have different screening lengths, the potential profile for the opposite polarization direction is asymmetric, as shown in Fig. 3(b) . As a result, the potential seen by transport electrons changes with the ferroelectric polarization switching, leading to the TER effect, which can be written as TER¼R"/C0R# R#; (3) where the two arrows represent the two polarization directions of the ferroelectric. Usually, the TER (OFF/ON) ratio can also be cal- culated as ROFF/RONsince the resistance changes in FTJs are usu- ally large. FIG. 2. A general summary of the history of FTJs and a simple prediction of their future applications in flexible neuromorphic computing.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-4 Published by AIP PublishingGenerally, the change of the electrostatic potential profile due to the polarization reversal could explain the TER effect in FTJs well; however, both theoretical and experimental studies have shown that other factors, such as the interface effect, strain effect, and migration of oxygen vacancies, could also contribute to the resistive switching mechanisms in FTJs.59,100,101The polarization reversal changes posi- tions of ions at the interfaces that influence the atomic orbital hybrid- izations at the interface and thus the transmission probability. The inverse piezoelectric effect existing in the ferroelectric thin film can result in a change in its thickness and therefore a change in the tunnel- ing barrier width, which does not only influence the screening in the electrodes but also the transmission probability. As for the oxygen vacancies, they will move though the ferroelectric film under an elec- tric field, which might form conducting filaments or change an interfa- cial Schottky barrier.84,102Besides, other factors, such as the electrodes and the electrode/ferroelectric interface, also affect the resistive switch- ing of FTJs.76,103–108 C. Transport mechanisms in ferroelectric tunnel junctions Upon the application of an external electric field, several electron transport mechanisms can contribute to the current across the ferro- electric tunnel barrier: direct tunneling, Fowler-Nordheim tunneling,and thermionic injection.63,109Schematics in Fig. 4 illustrate the three electron transport mechanisms in FTJs. 1. Direct tunneling Direct tunneling is a quantum mechanical phenomenon, which happens in very thin films with thicknesses within several nanometers.As discussed above, screening of the polarization surface charges willchange the electrostatic potential profile, which is the interfacial energybarrier. Moreover, different screening lengths of the top and bottom electrodes lead to different interfacial barrier heights u 1andu2,w h i c h causes an asymmetric barrier profile, as shown in Fig. 4(a) .F o rat r a p e - zoidal potential barrier using the Wentzel-Kramers-Brillouin (WKB)approximation, the current density j DTcan be written as follows: jDTVðÞ¼/C04em/C3 e;b 9p2/C22h3exp aVðÞ u2/C0eV 2/C18/C193=2 /C0u1þeV 2/C18/C193=2()"# aVðÞ½/C1382ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2/C0eV 2r /C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1þeV 2r"#2 /C2sinh3eV 4aVðÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u2/C0eeV 2r /C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u1þeV 2r() "# ; (4) FIG. 3. (a) Charge distribution, (b) the respective electrostatic potential profile, and (c) the band diagram of a M1/FE/M2 junction with different polariza tion directions. Here, it is assumed that the screening length of metal 1 is smaller than that of metal 2 ( d1<d2), which results in the asymmetry in the potential profile. Yellow and purple colors repre- sent different polarization directions. FIG. 4. Schematics of band diagrams of (a) direct tunneling, (b) Fowler-Nordheim tunneling, and (c) thermionic injection.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-5 Published by AIP Publishingwhere aðVÞ¼4dffiffiffiffiffiffiffiffi2m/C3 e;bp 3/C22hu1þeV/C0u2 ðÞ,m/C3 e;bis the tunneling electron effective mass in the barrier, e is the electron charge, /C22his the reduced Planck constant, u1,2are the left and right energy barrier heights, d is the bar- rier thickness, and V is the applied voltage. 2. Fowler-Nordheim tunneling Fowler-Nordheim tunneling (FNT) is generally the same physical phenomenon as direct tunneling but in a higher voltage regime. When the applied voltage exceeds the interfacial barrier height, part of the energy barrier profile will lie beneath the Fermi energy level of the electrode, and therefore, the effective tunneling barrier width is modi- fied. As shown in Fig. 4(b) , FNT is tunneling across a triangular- shaped potential barrier. The widely used equation for the current density in the FNT regime is given by jFNEðÞ¼e3E2 16p2/C22hubexp/C04ffiffiffiffiffiffiffiffiffi 2m/C3p ub3=2 3e/C22hE/C20/C21 ; (5) where ubis the potential barrier height and m /C3is the effective mass of the tunneling charge carrier. 3. Thermionic injection Thermionic injection (TI) describes the phenomenon that charge carriers have the possibility to overcome the potential barrier by ther- mal excitation when temperature is above zero. The barrier height is lowered by image force lowering, which is called the Schottky barrier, as shown in Fig. 4(c) . The current density under an applied electric field can be described by jSchVðÞ¼A/C3/C3T2exp/C01 kBTub/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi e3E 4pe0eifls 0 @1 A2 43 5; (6) where A/C3/C3i st h ee f f e c t i v eR i c h a r d s o nc o n s t a n t , ubis the potential bar- rier height, and eiflis the permittivity of the ferroelectric responsible for image force lowering. The three mechanisms concurrently govern the transport of carriers with the major contribution depending on the ferroelectric film thickness, applied voltage, the polarization direction, and other parameters related to the materials used in the FTJs. The magnitude and sign of TER are determined by the voltage polarity and the major transport mechanism contributing to the current. For all transport mechanisms, a higher TER ratio will be generated with a thicker ferroelectric thin film; however, the device performance might worsen due to the high resistance at larger film thickness. TER ratios can be optimized by choosing the metal electrodes as metal electrodes with significantly different screening lengths can lead to resistance change up to a few orders of magnitude.99 D. Multiferroic tunnel junction A multiferroic tunnel junction (MFTJ) can be viewed as a partic- ular type of FTJ, which utilizes two ferromagnetic metals as the elec- trodes.55In the meantime, the concept of an MFTJ also combines a magnetic tunnel junction (MTJ), which will be discussed in the next chapter. Electron tunneling from a ferromagnetic electrode through a thin barrier is spin-dependent. Figure 5(a) shows the schematic of theMFTJ structure. The key character of MFTJs is the coexistence of the TER and tunneling magnetoresistance (TMR) effects, which was pre- dicted by Velev et al.110and Zhuravlev et al.111Because of the ferro- magnetic and ferroelectric components, electron and spin tunneling can be controlled, and a four-state resistance device can be realized by polarization and magnetization switching, as illustrated in Fig. 5(b) . As shown in Fig. 2 , the first experimental demonstration of TER is by Gajek et al. in 2007 using a structure of La 0.67Sr0.33MnO 3 (LSMO)/La 0.1Bi0.9MnO 3/Au.112In that work, except the TER effect, a tunneling magnetoresistance ratio of 81% was also obtained due to the ferromagnetic LSMO electrode. Velev et al. theoretically studied the resistive switching of MFTJ using an SRO/BTO/SRO structure.110 Their density functional calculation results show that the TMR effectresults from the wave-function symmetry conservation across the epi- taxial SRO/BTO interface, and the TER effect originates from the asymmetric interface termination sequence, which causes a different polarization profile when the ferroelectric polarization is reversed. Experimental demonstration of the four resistance states of MFTJs was first reported by Garcia et al. using an MFTJ structure of Fe/BTO/ LSMO through the spin polarization control by ferroelectric polariza- tion switching. 113Four nonvolatile resistance states or the coexistence of TER and TMR have also been reported in other MFTJs.114–123 Utilizing the ferroelectric and ferromagnetic switching, MFTJs are ableto control the electron and spin tunneling, the property of which could be used to design novel functional devices. Detailed reviews especially on MFTJs can be found in Refs. 55and124–126 . E. Mechanisms of the ferroelectric tunnel memristor As introduced above, polarization switching of the ultrathin fer- roelectric layer could lead to two different resistance states, high FIG. 5. (a) Schematic of the MFTJ structure. (b) Schematic of the 4 resistance states realized by polarization and magnetization switching.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-6 Published by AIP Publishingresistance state ROFF, and low resistance state RON,r e s p e c t i v e l y . However, it is known that the polarization of ferroelectric thin films does not switch all of a sudden but through the nucleation of domains of opposed polarity and the growth of these new domains.88During this process, the resistance of the device is neither ROFForRON,b u ti n between. This brings up a new concept–ferroelectric tunnel memristor (FTM), in which multiple resistance states could be obtained during the polarization switching process of the ferroelectric ultrathin layer. FTJ therefore can be reviewed as the building block of the FTM. Stable multiple resistance levels in FTJ were first reported in BTO based tun- nel junctions by Chanthbouala et al.68Afterward, many other studies have been reported on FTMs in different ferroelectric thin films, like BFO,72,127Pb(Zr,Ti)O 3(PZT),66,128HZO,86,129of which the switching mechanisms of FTJs are summarized below. 1. Ferroelectric domain switching As introduced above, the voltage-controlled domain configura- tion in ferroelectric tunnel barriers could lead to the memristive behavior in FTJs. In FTJs, the ferroelectric polarization switching hap- pens through the nucleation and propagation of domains of opposed polarity, as illustrated in Fig. 6(a) . Therefore, when the switching volt- age is cycled, the ferroelectric domain configuration varies accordingly, during which process the resistance is between ROFFand RON,a s shown in Fig. 6(b) . Chanthbouala et al. first demonstrated that the continuous range of multilevel resistances between tunneling OFF and ON states could be obtained by controlling the domain configuration of an FTJ with a Co/Au/BTO/LSMO structure.68To reveal the micro- scopic mechanisms responsible for the memristive behaviors of their FTJs devices, they investigated the correlations between the resistance value and the ferroelectric domain configuration using piezoresponse force microscopy (PFM). Their results show that the resistance of thejunction shows a continuous variation with the relative fraction of downward domains extracted from the PFM images. Following the above study on BTO-based FTJs, in the next year, Yamada et al. reported memristive behavior in supertetragonal BFO- based FTJs with giant electroresistance up to 104.72They also demon- strate that the resistance changes in FTJs scale with the nucleation and growth of ferroelectric domains in the tetragonal phase BFO. In both papers, their results could be interpreted using a simple model where down and up ferroelectric domains conduct in parallel, as shown in Fig. 7(a) , and the intermediate resistances Rcould be expressed as 1 R¼S ROFFþ1/C0S RON; (7) where s is the fraction of the downward domains (in their works, upward polarizations correspond to resistance ON state, and down- ward polarizations correspond to resistance OFF state). The multilevel intermediate resistance states of FTJs could be explained well by this model which qualifies FTJs to be applied as memristors. Besides, Chanthbouala et al. also demonstrate that the very fine resistance tun- ing of the FTJ could not only be achieved by one pulse of appropriate amplitude but also by applying a successive series of constant- amplitude voltage pulses, as shown in Figs. 7(b) and 7(c),w h i c h matches the definition of memristive devices. The fact that the FTJ resistance could be tuned by an appropriate number of consecutive pulses of a fixed voltage enables FTJs to be integrated into brain- inspired neuromorphic computing architectures. Memristive behav- iors of FTJs controlled by the reversal of ferroelectric domains have also been reported in other FTMs.66,130–132 2. Ferroelectric field effect Although the mechanism of ferroelectric domain switching could explain the memristive behavior well in the mentioned works above, another way to realize memristive switching in FTJs was reported in the next few years, which uses a ferroelectric field effect to modulate the tunneling resistance.127,133The effect of the ferroelectric field effect on FTJs was first reported in year 2013 by Wen et al.134In their work, the authors proposed a tunneling structure by replacing one of the metal electrodes in a normal FTJ (metal/FE/metal) with a semiconduc- tor [Nb:SrTiO3 (NSTO)]. In these metal/FE/semiconductor FTJs, it was found that not only the barrier height but also the barrier width can be electrically modulated as a result of the ferroelectric field effect, which leads to a greatly enhanced TER ratio. Using the same concept, FTMs grown on NSTO substrates were reported, which use BTO and BFO as the tunneling barrier, respectively.127,133The mechanism of the ferroelectric field effect governed FTMs is illustrated in Fig. 8 , where n-type semiconductor NSTO is assumed to be the bottom elec- trode. When the ferroelectric polarization is switched upward toward the top metal electrode, except the tunneling barrier, an extra barrier exists in the semiconductor surface due to the depletion of carriers by the ferroelectric bound charges. As shown in Fig. 8 , the profile of the extra barrier in the depletion region of NSTO surface could be modu- lated continuously in parallel with the tunneling barrier height of the ferroelectric layer through the change of the effective polarization asso- ciated with the ferroelectric domain switching. That is, the ferroelectric polarization reversal could tune the tunneling width of the junction by FIG. 6. (a) Schematic of the ferroelectric domain switching process from downward to upward direction. (b) Schematic of the corresponding R-Vswitching loops.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-7 Published by AIP Publishingmodulating the width of the depletion region in NSTO, and thereby controlling the tunneling resistance. The ferroelectric field effect could not only modulate the tunnel- ing barrier width, but also induce charge redistribution at the ferroelec-tric/electrode interface which results in the modulation of the interfacebarrier height and therefore leads to the memristive switching behav-ior, as reported by Kim et al. 67In their work, the resistance in Co/ BTO/(La,Sr)MnO 3FTJs could be tuned, which was attributed to the modulation of the barrier height in a possible CoO xinterfacial layer as a result of ferroelectric field-induced charge migration and aggregation.3. Migration of oxygen vacancies For oxide thin films deposited by the pulsed laser deposition (PLD) technique, defects are usually generated during the thin filmdeposition process. In particular, oxygen vacancies are the most likelydefects because the deposition environment in PLD deposition is gen- erally in a reducing nature due to the deficient oxygen pressure. Migration of oxygen vacancies plays an important role in the resistiveswitching process of oxide resistive switching devices. 135–137 Memristive switching of FTJs led by the oxygen vacancy migration has FIG. 7. (a) Scaling of the resistance state with the fraction of downward domains. The resistance is measured followed by a PFM image of the ferroelectric doma ins. The device resistance as a function of the fraction of downward domains is in a log-log scale. The black line is calculated using the parallel conduction mo del sketched in the right side. Reproduced with permission from V. Garcia and M. Bibes, “Ferroelectric tunnel junctions for information storage and processing,” Nat. Commun .5, 4289 (2014). Copyright 2014 Springer Nature. (b) Evolution of the junction resistance as a function of the different voltage pulse sequences. (c) The applied volt age pulse sequences. (b) and (c) Reproduced with permission from Chanthbouala et al. , “A ferroelectric memristor,” Nat. Mater. 11, 860–864 (2012). Copyright 2012 Springer Nature.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-8 Published by AIP Publishingbeen reported in FTJs grown on NSTO substrates.69,70,128In their works, the authors found that the resistance of the FTJs could be mod- ulated by a large negative bias without being saturated, which makes theR-Vswitching loops asymmetric. They attribute the resistance modulation by negative voltage biases to the drifting of oxygen vacan-cies under the negative electric field. Under negative electric field, the migration of oxygen vacancies toward the ferroelectric/NSTO interface will widen the depletion layer and consequently enhance the electrore-sistance. As a result, the gradual increase in the negative voltage biasleads to the continuous increase in the OFF-state resistance, which results in the memristive behavior of the FTJs. Typical R-Vswitching loops of oxygen vacancy migration induced memristive switching inFTMs are shown in Fig. 9(b) , where a giant OFF/ON ratio up to 10 7 was obtained due to the contribution of the migration of oxygen vacancies to ROFF, offering a useful method to enlarge the resistive switching window through oxygen vacancy modulation. The evidence of the contribution of oxygen vacancies to the resis- tive switching was provided by our group by investigating the influ-ence of the pulse width on the resistive switching behavior. Figure 9(c) shows the resistive switching process as a function of the pulse width. It is known that the switching time scale of ferroelectric polarizationcan be as short as nanoseconds, but the time scale of oxygen vacancy migration is around the magnitude of microseconds. The plateau ofthe resistance during the ON-to-OFF switching process indicates the saturation of the ferroelectric polarization switching. With increasing switching pulse width, the oxygen vacancies started to migrate, and finally, the resistance of the devices saturated at the pulse width of 1 ms. The ON/OFF ratio caused by ferroelectric polarization switching is around 10 3, and the migration of oxygen vacancies enhances the ON/OFF ratio significantly to the order of 106. F. Performance of the ferroelectric tunnel memristors As a new type of memristor, FTJs exhibit many advantages com- pared with conventional RRAM and PCM for neuromorphic applica- tions. First, because the ferroelectric polarization switching speed can be as fast as ns, FTJs have a higher operation speed than the biological counterpart. Second, because of the very small size of the ferroelectric domain, very fine tuning of the resistance can be achieved in FTJs. Third, unlike conventional RRAM and PCM, the electroforming pro- cess is not needed for an FTJ-based memristor. This characteristic is important, since the forming process would result in discrete and abrupt resistive transition during the SET or/and RESET process in many cases. FTMs also have their advantages, such as the reported giant OFF/ON ratios. However, they also have some disadvantages. The main disadvantage of FTM arises because ferroelectric thin films FIG. 9. (a) Typical I-Vswitching curve of the FTM. (b) Typical R-Vloops of the FTM with increasing negative voltage. (c) Resistive switching as a function of the pulse width. (a)–(c) Reproduced with permission from Guo et al. , “Control of synaptic plasticity learning of ferroelectric tunnel memristor by nanoscale interface engineering,” ACS Appl. Mater. Interfaces 10, 12862–12869 (2018). Copyright 2018 American Chemical Society. FIG. 8. Schematics for the evolution of the barrier profile with the polarization switching from downward to upward. (a) Resistance ON state when ferroelectr ic polarization points toward NSTO. The interface of FE/NSTO is at the electron accumulation state. The dots represent electrons. (b) and (c) When resistance is switc hed to the OFF state, the FE/NSTO interface is at the electron repletion state. The width of the depleted space charge region on the NSTO surface is controlled by the domain s witching in the ferro- electric layer, which leads to the memristive behavior of the FTJs. The circled plus symbols represent ionized donors.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-9 Published by AIP Publishingare usually grown on perovskite substrates like STO, which makes it d i f fi c u l tf o rF T M st ob ec o m p a t i b l ew i t ht h ec u r r e n tS it e c h n i q u e .T h e performances of reported FTMs are summarized in Table I . G. Ferroelectric tunnel memristor for neuromorphic network applications As discussed above, the conductance of an FTM could be tuned continuously based on the history of applied voltage or current, which enables it to be an electronic equivalent of the synapse for artificial neural networks. Besides, an FTM can be scaled down to nanoscale size, with nonvolatile, fast, low-energy electrical switching properties.Because of these properties, FTMs become potential candidates for applications of artificial synapses. For synaptic plasticity learning, spike-timing-dependent plasticity (STDP) is a very important memo- rization mechanism, which determines how the weight of a synapse in a neural network evolves. The overlapping-waveform mechanism is usually utilized to measure STDP in FTMs. The important advantage of overlapping-waveform mechanism is that, it allows testing a wide variety of different STDP forms just through changing the shape of the pre- and postneuron waveforms, and the weight change is only deter- mined by the overlapping waveforms at the FTM. 138Using this mech- anism, STDP, together with other key synaptic functions like long andshort-term potentiation and depression, and paired-pulse facilitation have been demonstrated in FTMs with different ferroelectric materials. 70,86,88,129,131,132,139 Among the mentioned works above, our group has done system- atic studies on the synaptic learning of FTJs using different ferroelec-tric thin films. Using BTO as the tunneling layer, we realized the control of synaptic plasticity learning of FTM by nanoscale interface engineering. 70It was found in our study that nanoscale interface engi- neering can tune the intrinsic band alignment of the BTO/NSTO het- erostructure over a large range of 1.28 eV, which consequently results in different memristive and STDP properties of FTMs. As shown in Fig. 10 , FTMs grown on TiO 2-terminated NSTO substrates show good fitting of the two learning rules, while FTMs on SrO-terminatedNSTO substrates display abrupt changes in Dw. The different synaptic learning characteristics of FTMs grown on different terminated NSTO substrates are due to their gradual or abrupt conductance modulationproperty. The diverse STDP forms of the different types of FTMs may play different specific roles in various spike neural networks (SNNs). FTMs on TiO 2-terminated NSTO substrates could emulate biological synapses well with precise learning results, and thus can function as a decent artificial synapse in most of the present SNN. Nevertheless, in some specific cases, FTMs on SrO-terminated NSTO substrates maybe preferred in SNN when the synapse is required to learn easily but forget with much more difficulty. Therefore, we successfully realized the control of the synaptic learning properties by tuning the band alignments at the FTM interface through tailoring of the device interfaces. In addition to BTO, our group also studied the synaptic learning behaviors of HZO-based FTJs. 86Doped-HfO 2thin films with ferro- electricity have attracted intensive attention due to their high dielectricconstant, good compatibility with silicon substrates, and absence of toxic elements like Pb. The incorporation of ferroelectric HfO 2-based films in field effect transistors is already well developed and imple-mented in complementary metal-oxide-semiconductor (CMOS) tech- nology. Based on this background, our group has successfully fabricated an epitaxial single orthorhombic phase ferroelectric HZO thin film with good ferroelectric-based resistive switching behavior as well as good synaptic learning behavior. Long-term potentiation(LTP) and long-term depression (LTD) behaviors were measured as shown in Fig. 11(a) . The results show clearly that the samples exhibit bidirectional gradual conductance changes, which can be modulatedby positive or negative pulse trains. The typical asymmetric Hebbian learning rule was also measured, as shown in Fig. 11(b) , which reflects a large window of multiple resistance states available in synaptic mem-ory applications. Furthermore, the paired-pulse facilitation (PPF) ratio w a sm e a s u r e da ss h o w ni n Figs. 11(c) and11(d) . Reinforcement of the synaptic weight is observed when pulse trains are applied consecu- tively to the samples, which is in good agreement with a biologicalTABLE I. Summary of the performances of different FTMs. References Structure OFF/ON Switching speed Endurance Retention Synaptic behavior Boyn et al.89Pt/Co/BFO/CCMO >104/C240.1ls >103cycles Not specified Yes Chanthbouala et al.68Co/Au/BTO/LSMO /C24300 10 ns Not specified Not specified … Yamada et al.72Pt/Co/BFO/CCMO >104Not specified >2000 cycles >3 days … Huet al.127Pt/BFO/NSTO /C24103Not specified Not specified >1000 s … Quindeau et al.66Co/PZT/LSMO >100 /C240.1ls Not specified Not specified … Hou et al.128Pr or Ag/PZT/Pt /C24106/C245ls >100 cycles >105s… Yoong et al.86Pd/HZO/LSMO /C24160 Not specified /C24106cycles >6 h Yes Chen et al.129TaN/HZO/Pt /C245 Not specified >103cycles Not specified Yes Huang et al.130LSMO/BTO/LSMO >5 6 ns Not specified Not specified … Majumdar et al.131Au/P(VDF-TrFE)/NSTO /C2410420 ns Not specified >10 mins Yes Wang et al.132Co/BTO/LSMO >104<10 ns Not specified Not specified Yes Wen et al.133Pt/BTO/NSTO /C24105Not specified Not specified Not specified … Kim et al.67Au/Co/BTO/LSMO /C24103Not specified Not specified >100 s … Guo et al.70Pt/BTO/NSTO /C24106/C2410 ns >105cycles >1 h Yes Huet al.70Pt/Sm 0.1Bi0.9FeO 3/NSTO /C24105/C2410 ns /C24106cycles >20 h …Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-10 Published by AIP PublishingFIG. 10. (a) Schematic of the Pt/BTO/NSTO FTMs with different terminations. (b) Schematic of how an FTJ represents a synapse. (c) and (d) STDP properties with a typical antisymmetric Hebbian learning rule of Pt/BTO/NSTO FTMs grown on TiO 2-and SrO-terminated NSTO substrates, respectively. (e) and (f) STDP properties with a typical anti- symmetric anti-Hebbian learning rule of Pt/BTO/NSTO FTMs grown on TiO 2- and SrO-terminated NSTO substrates, respectively. (a)–(f) Reproduced with permission from Guo et al. , “Control of synaptic plasticity learning of ferroelectric tunnel memristor by nanoscale interface engineering,” ACS Appl. Mater. Interfaces 10, 12862–12869 (2018). Copyright 2018 American Chemical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-11 Published by AIP Publishingsynapse. This study will stimulate the study of HfO 2-based thin films and their potential applications in the artificial electronic synapses. Another representative research group on synaptic plasticity learn- i n go fF T J si st h eg r o u po fB o y n et al., whose work shows that synaptic learning could be modeled through nucleation-dominated reversal ofdomains. 88In their work, first, they emulate the spikes from pre- and postneurons using the waveforms with the shape of rectangular voltage p u l s e s( s m a l l e rt h a n Vth) followed by smooth slopes of opposite polar- ity, as shown in Fig. 12(a) .Vthrefers to the threshold voltage amplitude as shown in Fig. 12(b) , below which domain reversal cannot happen when the pulse width is fixed. When both pre- and postneuron spikes arrive at the memristor with a timing difference Dt, their superposition generates the waveforms ( Vpre/C0Vpost). As shown in Fig. 12(c) ,t h e i r results show that the combined waveform leads to an increase in con- ductance when Dt>0, and a decrease in conductance when Dt<0. Second, in order to model the shape of the STDP curve, they investigate t h ep h y s i c a lp r o c e s su n d e r l y i n gt h ec o n d u c t a n c ec h a n g ei nF T M sw h e nthe waveform is applied, and they found that the conductance changesin FTMs due to ferroelectric switching can be described by a well- established nucleation-limited model. Using this model, they realized the predictive modeling of synaptic learning, as shown in Fig. 12(d) . When applying different voltage waveforms to their memristors to emulate various types of pre- and postneuron activities, they can predict the conductance changes for these specific types of STDP, and the pre- dictions agree well with the measured conductance variations associated with different STDP waveforms. Third, they use the nucleation-limitedmodel to simulate unsupervised learning in a spiking neural network, which is built around a crossbar of 9 /C25 ferroelectric synapses, as shown in Fig. 12(e) . Their simulations demonstrate that arrays of ferro- electric artificial synapses can autonomously learn to recognize image patterns in a predictable way, as shown in Fig. 12(f) .B e s i d e s ,t h e ya l s o show that successful unsupervised learning highly depends on the exact shape of the pre- and postneuron spike waveforms. This representative work sheds light on the potential of FTMs being used as artificial synap- ses and paves the way toward implementations of predictable and reli-able artificial synapses in future brain-inspired computers. FIG. 11. Synaptic learning properties of FTJs with Hf 0.5Zr0.5O2as the tunneling barrier. (a) Long-term potentiation (LTP) and depression (LTD) behaviors upon application of increasing positive and negative pulse trains. (b) Asymmetric Hebbian learning rule with the fitted results in solid lines. (c) and (d) Paired-pulse f acilitation (PPF) ratios of posi- tive and negative pulse trains, respectively, with fitted results in solid lines. The insets of each figure show the pulse waveforms used for the measure ments. (a)–(d) Reproduced with permission from Yan et al. , “Memristor with Ag-cluster-doped TiO2 films as artificial synapse for neuroinspired computing,” Adv. Funct. Mater. 28, 1–10 (2018). Copyright 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinherm.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-12 Published by AIP PublishingExcept the unsupervised learning of ferroelectric synapses using BFO, a parallel supervised learning has also been demonstrated in a1T1R crossbar neuromorphic network with FTMs using BTO, where the working principle of the FTM is also based on Eq. (7)in Sec. II E 1 132,139as shown in Fig. 13(a) .I nt h e i rw o r k ,t h en e u r o m o r p h i c network is arranged with a crossbar structure, in which each cross-point forms a synapse consisting of a MOS transistor and an FTM, as shown in Fig. 13(c) . Based on this architecture, they managed to dem- onstrate the STDP scheme as shown in Figs. 13(b) and13(d) .B e s i d e s , a parallel supervised learning circuit shown was also demonstrated in Ref.137. Both the functionalities were validated through transient sim- ulation using a compact model of FTM and CMOS 40 nm design kit. FIG. 12. (a) Schematic of FTM with BFO is sandwiched between two electrodes. The synaptic transmission is modulated by the causality ( Dt) of neuron spikes. (b) Hysteresis loop of the ferroelectric memristor, which displays the voltage thresholds Vthþand Vth/C0. (c) Modulation of the device conductance DG as a function of the time delay Dt between pre- and postsynapse. (d) Examples of STDP learning curves with different shapes. (e) Simulated spiking neural network comprising a crossba ro f9/C25 ferroelectric memristors. The inputs are noisy images of the patterns to recognize. (f) Recognition rate as a function of the number of presented images for differen t noise levels. The col- ored images are conductance maps of the memristors in each line and show their evolution for a noise level of 0.3. (a)–(f) Reproduced with permission fr om Garcia et al. , “Learning through ferroelectric domain dynamics in solid-state synapses,” Nat. Commun. 8, 14736 (2017). Copyright 2017 Springer Nature.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-13 Published by AIP PublishingIn addition to the works mentioned above, CMOS compatible 3D vertical FTMs using HZO were demonstrated for hardware neuralnetwork applications. 129In the study, the authors show that the imple- mentation of pattern training in hardware with strong tolerance to input faults and variations, as well as the pattern classification and rec-ognition, can be realized in the 3D vertical FTMs array. To concludethis capture here, although FTMs have many advantages as being arti-ficial synapses, the study on the applications of FTMs in neuromor-phic networks is still in an early stage. However, the state-of-artresearch results of FTMs provide an avenue for more complex hard-ware neural network applications with potential for a significantimpact on the next-generation nonvolatile memory developments. III. MAGNETIC TUNNEL JUNCTIONS AND THEIR APPLICATION AS ARTIFICIAL SYNAPSE A. Introduction of magnetic tunnel junction An MTJ consists of two layers of magnetic metal separated by an ultrathin barrier layer of insulator, as shown in Figs. 14(a) and14(b) .The insulating layer is so thin that electrons can tunnel through the barrier when a voltage bias is applied to the junction. Unlike FTJs, the tunneling current in MTJs depends on the relative orientation of mag- netizations of the two ferromagnetic layers, which can be changed by an applied magnetic field, as illustrated in Figs. 14(c) and 14(d) . Extensive studies of MTJs have been done because of their potential applications in spintronic devices. The magnetic tunneling effect was originally discovered in 1975 by Julliere in the junctions of Fe/Ge-O/Co with a resistance change around 14%. 140Then, amorphous Al 2O3was used as the tunnel bar- rier.141Great effort has been made to enhance the magnetic tunnel effect through improving the properties of the ferromagnetic electro- des and the amorphous barrier layer. Since this century, magnetic tun- nel barriers of oxide MgO have been developed theoretically and experimentally. In year 2001, ab initio calculations have predicted that using MgO as the insulator, and Fe as the ferromagnets, the TMR ratio can reach several thousand percent.142,143Experimentally, demonstra- tion of a significant TMR in a MTJ (Fe/MgO/FeCo) was reported by FIG. 13. (a) The structure of the FTJ and the model of the FTM based on ferroelectric switching dynamics. (b) The signal sequences used in the work and the measure d change of synaptic weight as a function of relative timing of neuron spikes. (c) Schematic of 1T1R synapse between pre- and postneuron. (d) A typical neuromorph ic network for STDP applica- tion based on a 2 /C22 crossbar configuration. (a)–(d) Reproduced with permission from Wang et al. , “Compact modelling of ferroelectric tunnel memristor and its use for neuromorphic simulation,” Appl. Phys. Lett. 104, 053505 (2014). Copyright 2014 AIP Publishing.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-14 Published by AIP PublishingM. Bowen in the same year,144and a TEM over 200% at room temper- ature was reported by Parkin et al. by using Fe/MgO/Fe junctions in year 2004.145Many other studies had been done using MgO since then, and large TMR ratios have been obtained at room tempera- tures.146The TMR effect can be used in read head, sensor, magnetic random access memory (MRAM), etc. B. Mechanism of the magnetic tunnel junction As mentioned above, the phenomenon that the tunneling current depends on the relative magnetization orientation of the two ferro- magnetic electrodes is called the TMR effect, which is a consequence of the spin-dependent tunneling. The TMR ratio can be calculatedusing the following formula: TMR ¼RAP/C0RP RP; (8) where AP and P represent the magnetizations of the two magnetic metals antiparallel or parallel, respectively. The TMR effect was explained by Julliere with the spin polarizations of the ferromag-netic electrodes. 140The origin of TMR arises from the difference in the electronic density of state (DOS) at the Fermi lever EFbetween spin-up n"(EF) and spin-down n#(EF) electrons.147,148As electrons will preserve their spin orientations during the tunneling process,electrons can only tunnel into the subband with the same spinorientation, as illustrated in Fig. 15 . Therefore, the tunneling cur- rent is proportional to the product of the electrode DOS at the Fermi level. In the ferromagnetic electrodes, the ground-state energy bands in the vicinity of the Fermi level are shifted in energy, generating separate majority and minority bands for electrons withopposite spins. When assuming spin conservation for the tunneling electrons, there are two parallel currents with spin-up and spin- down, respectively. As a result of all these aspects, the tunnel current through the electrodes with the same magnetization orienta- tion will be higher than that of electrodes with opposite magnetization.Using this model, the TMR ratio can be given by TMR ¼ 2P1P2 1/C0P1P2; (9) where P1andP2are the polarization factors for the two electrodes, respectively. The polarization factors are defined as P¼n"EFðÞ /C0 n#EFðÞ n"EFðÞ þ n#EFðÞ; (10) where nrepresents the spin dependent DOS at the Fermi energy. This formula clearly demonstrates the presence of a magnetoresistance effect and the relevance of the magnetic character for the spin polariza-tion of the tunneling electrons. FIG. 14. (a) and (b) Schematics of the MTJ structures with antiparallel and parallel magnetizations of the two ferromagnetic metal electrodes. (c) Resistive switching loop as the function of the applied magnetic field. (d) Two TMR resistance states with the magnetization directions of the two metals parallel and antiparalle l, respectively.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-15 Published by AIP PublishingC. Fundamental physics and the basic principle for the current-induced torques for manipulating the magnetization The driving force for the MTJ-based spintronic device is that the manipulation of the magnetic moment can be achieved by the charge- induced spin current,149,150instead of the external magnetic field, w h i c hi sd e v e l o p i n gi nt h ep a s tt h r e ed e c a d e s .T h em a g n e t i cd y n a m i c under the polarized current can be well described by the Landau- Lifshitz-Gilbert (LLG) equation151,152 dm dt¼/C0cm/C2Heffþam/C2dm dtþc MsT; (11) where m¼M=Msis the magnetization unite vector, Msis the satura- tion magnetization, c¼glB=/C22his the gyromagnetic ratio, where g is the Lande g-factor and lBis the Bohr magneton, and ais the Gilbert damping constant. The first term on the right side of Eq. (11)accounts for the precession of the magnetization about the effective magnetic field, while the second term accounts for the relaxation of the magneti-zation toward the effective field direction. The third term describes the torque ( T) that is induced by the charge current. Such torques can be generally divided into two orthogonal components, i.e., one is thefieldlike (FL) torque and the other the dampinglike (DL) torque, as shown in Fig. 16(a) . The FL and DL torques act like the ones described in the first and second terms, respectively, in Eq. (11), and they can be described as s FLm/C2randsDLm/C2ðm/C2rÞ,w h e r e sFLðDLÞare the magnitudes of the FL (DL) torque, and ris the unit vector of the spin direction which depends on the source of the spin current. With all these forces present, the magnetic moment can have either a determin- istic switching by the external current, or persistent precession due tothe balance of all the torques.The first typical spintronic device is the spin transferred torque (STT) based MTJs, as shown in Fig. 16(b) , which makes use of a refer- ence magnetic layer to achieve the polarized charge current for switch- ing. In such a structure, the reference magnetic layer is with magnetic moment pinned at one direction, while a free layer is with momentswitchable by an external charge current. When the magneticmoments in the reference and free layers are not parallel, a charge cur-rent injected from the free layer will be polarized by the magnetizationof the reference layer, i.e., there will be net spin current with the polari-zation direction parallel to the fixed magnetization, and then, thepolarized current will execute a DL on the moment in the free layer toalign it to the direction of the moment in the reference layer. In gen-eral, the FL torque is smaller than DL in the conventional MTJ config-uration, with the ratio of FL/DL around 30%. 153The magnetic moment in the free layer can also be switched to the antiparallel direc-tion by injecting the charge current from the reference layer, as theelectrons with opposite spin direction will be reflected at the interface between the barrier and the reference layer, and thus execute a DL on the moment at the free layer to flip its direction. As introduced inSec.III B, the relative orientations of these two magnetic moments can be indicated by the TMR with a relative small reading current acrossthe junction. 154–156 As described above, a reference magnetic layer is necessary to polarize the passing electrons for the STT-MTJ device. The spin cur-rent can also be generated from the heavy metal (HM) via the spinHall effect (SHE), as shown in Fig. 16(c) , where the spin polarization direction ris determined by J C/C2JS.157The resulting torques by this pure spin current contain DL and FL torque, which will drive the mag- netization switching under certain conditions149,158and induce the precession of the magnetic moment, respectively. As their physical FIG. 15. Schematic of the TMR effect. Electrons can only tunnel to the subband with the same spin orientation. A change from (a) the parallel to (b) the antiparal lel magnetiza- tions of the two metals leads to an exchange of the spin subband for metal 2, and results in a corresponding change in the conductance.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-16 Published by AIP Publishingmechanism originates from the spin–orbit interaction (SOI) in the structure, the torques are generally called spin–orbit torque (SOT). S i m i l a r l y ,t h eT M Rc a nb eu s e dt oi n d i c a t et h es t a t u so ft h ej u n c t i o nabove the HM layer. Furthermore, a simple magnetic layer with per-pendicular magnetic anisotropy (PMA) can replace the junction as itdoesn’t require a second magnetic layer to generate the spin current,and thus the anomalous Hall effect (AHE) of the FM can be utilized as a reading indicator too. 159It is noted that besides the SHE, many other exotic physics can also generate the transverse spin current, such as theRashba effect at the oxide interface 160and spin-momentum locking in the topological insulator.161In the SOT-MTJ structure, the route of the writing current to manipulate the magnetization is decoupled from the reading current, different from that of the STT-MTJ structure. The last spin device is the one based on the antiferromagnetic (AFM) materials, which is regarded as a promising candidate for spin- tronic devices due to its insensitivity to the external magnetic field andits potential ultrafast switching. 162The ultrafast switching of the Neel order of the AFM are able to be induced by the current-inducedSOT. 163At the beginning, the feature was reported in the materialswith a specified crystal structure, such as CuMnAs163and Mu 2Au,164 which hold local asymmetry centers and thus can generate staggered spin accumulation in the sublattice center when a current is applied, thus leading to bulk Neel SOT. This Neel SOT is a FL torque, which will drive the Neel vector of the spin pair to the transverse direction of the current path, as shown in Fig. 16(d) . Later, the DL torque from a neighbor HM is reported to be capable of manipulating the Neel vec- tors too, however, aligning it to the current direction.165The Neel order of the AFM can be detected electrically by the anisotropic mag- netoresistance (AMR) or planar Hall effect (PHE). D. Spintronic devices for synapse application Based on the spintronic devices described above, there are a few synapse-oriented applications demonstrated so far, which essentially rely on their capability to achieve the multiresistance states. The first spintronic device proposed as synapse is the STT-MTJ. As described in Sec. III B, the resistance of the junction depends on the relative orientations of the magnetizations in the reference and free FIG. 16. (a) Spin torques on the magnetic moment that have precession around the effective magnetic field. The damping torque tends to drive the magnetic moment to the effective field direction, while the antidamping torque have the opposite effect and can be utilized to switch the moment direction. The fieldlike torq ue will drive the moment into precession around the effective field. Both antidamping and fieldlike torque can be generated by the injected polarized-current. (b) MTJ with a fixed ma gnetic layer and a free layer sandwiching an oxide barrier. The resistance across the junction depends on the relative orientations of the magnetic moments in the reference and free layer. In such a structure, an injected charge current will be polarized to achieve a net spin current when passing the reference magnetic layer, and execute a spin tra nsfer torque (STT) to the free magnetic layer, resulting in the manipulation of the magnetization in the free layer and thus the junction’s resistance. (c) Spin orbit torque (S OT) based MTJ. The spin cur- rent is generated from the nonmagnetic heavy metal via the spin Hall effect (SHE). In this structure, the writing current to manipulate the magnetic mo ment in the free layer decouples with the reading current, which crosses through the tunnel junction. (d) Antiferromagnetic (AF) device with the fieldlike toque executed o n the Neel vector, which will drive the AF staggered moments into the direction of the effective field.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-17 Published by AIP Publishinglayers, which gives two extreme values of the resistance when their magnetizations align in P and AP directions. It is natural to expect that multistates can be obtained if there are domains in P and AP states with that of the reference layer; thus, the resistance of the middle states will depend on the ratio of the P and AP components in the freelayer. Therefore, the domain wall was introduced into the magnetic free layer, whose position will decide the ratio of the P and AP compo- nents and hence the resistance of the junction, as schematized in Fig. 17(a) . 166To facilitate the formation and pinning of the domain walls, a long T-shape MTJ structure was synthesized [ Fig. 17(b) ], where the current-induced STT will execute spin torques on the domain walls and drive their movement, and the domain walls will be pinned to the randomly generated pinning centers or impurity sites after removing the injected current.166Consequently, the shape of the MTJ structure was introduced as a degree of freedom too for achieving the multistate easily.167The energy required to overcome the pinning barrier varies for different sites, and therefore, multiresistance states can be obtainedupon increasing the amplitude of the injected current, as shown in Fig. 17(c) . In these cases, both DL and FL torques contribute to the manip- ulation of the domain wall, depending on the relative energy of the H k andHcof the domain wall in the free layer.153,166 In contrast to the STT-MTJ, where the writing and reading cur- rents share the same path, the recently developed SOT technique is able to decouple the writing and reading current paths via introducing the HM underlayer into the MTJ structure, which stimulates explora- tions for new designs to emulate the synapse behavior.158,168As most of the ferromagnetic free layer in a SOT structure is generally with PMA, the anomalous Hall effect is utilized as the reading indicator for the proof-of-concept research, a typical device structure of which isshown in Fig. 17(e) . Similar to the STT-MTJ, the multistate is achieved via introducing the magnetic domains into the magnetic free layer, however, not in the same method. Fukami et al. introduced the AFM PtMn as the SOT source and exchange bias source to switch the [Co/ Ni] multilayer without applying external magnetic field [ Fig. 17(d) ], and demonstrated the memristive behavior [ Fig. 17(f) ]. 169They ascribed the mechanism to the domain nucleation during magnetiza- tion reversal by the exchange bias assisted SOT,169,170which may be due to the polycrystallinity of the AFM layer [ Fig. 17(d) ]. Later, Zhang et al. demonstrated that the domain wall propagation by the SOT under the in-plane magnetic field could also result in the memristive behavior in the Ta/CoFeb/MgO structure.171Furthermore, Cao et al. found that capping the PMA layer with an in-plane magnetic layer is capable of tuning the efficiency of the SOT to drive domain wall prop- agation too, and thereby obtaining the memristive behavior.172 AFM favors multistable domain configurations due to the lack of the internal dipolar field, compared to their FM counterparts. Figure 17(g) shows a typical multidomain structure of the AFM CuMnAs, imaged by the X-ray magnetic linear dichroism–photoelectron emis- sion microscopy (XMLD-PEEM) technique.163,173With the recent demonstration of current-induced switching of the AFM Neel vector and other additional features, as described in the section above, theAFM-based memory devices have attracted great interest. Wadley et al. first demonstrated that the CuMnAs AFM can be electrically written and read by staggered FL torques with multistable states in ambient enviroments. 163Olejnik et al. further explored the device and showed that the memory behavior can be up to thousands of states and the reproducible resistance states depend on the number andduration [ Fig. 17(i) ] which can be downscaled to 250 ps, of the current pulses.174Similar features in AFM Mn 2Au were reported by Bodnar et al. , which is also characterized with a large AMR readout.164 Although these seminal works demonstrated advanced memory behavior of the AFM device, the requirement of the staggered SOT limit the choice of the materials, i.e., the AFM with the structure char- acterized with the global centrosymmetry plus broken sublattice inver-sion symmetry. Moriyama et al. overcame this by utilizing the DL torque from the HM to switch the Neel vector in the CoGd syntheticantiferromagnetic. 175Recently, it was reported that the AFM insulator can also be manipulated by the DL torque from the spin current gen- erated in the neighboring HM, which might further reduce the energyconsumption. 165 E. Spintronic devices for neuron application Different from other counterparts, the spintronic device is fre- quently proposed as a promising candidate to implement the neuronfunction in the new neuromorphic computing architecture theoreti- cally and experimentally. 73,176–184This is because of their flexible spin dynamic process, which functions like the biologically inspired leakedand fire-spiking process. In CMOS-based architecture, it requiresmore than 20 transistors to emulate this behavior, which is area andenergy consuming, while the spintronics-based neuron promises analternative pathway with energy- and area-efficiency. 177,184 The typical spin device to emulate the neuron function is the MTJ. While a deterministic switching process between the P and AP states of the MTJ, when the spin torque (ST) is large enough, is required for the memory function, it is the stochastic switching processwith a moderate ST that lays the ground for the neuron application,which is caused by thermal fluctuations. 185There is a critical current (Ic) that generates the ST, for an MTJ with certain energy barrier ( DE) between two stable states [P and AP, Fig. 18(b) ], below which a sto- chastic switching happens. Consequently, the switching probability between the stable states depends on the magnitude, pulse length, andnumbers of the injected current [or voltage, Fig. 18(e) ]. In general, the stochasticity will be enhanced as the free layer dimension scale downs(thus reducing the DE), which also further reduces the operation cur- rent or voltage for the neuron application. However, if the energy bar- rier scales down further comparable to the thermal energy K BT( K Bis the Boltzmann constant), the MTJ will enter the superparamagneticregime, where the thermal fluctuation will cause repeated and stochas-tic switching between two states at room temperature [ Figs. 18(a) and 18(d) ], i.e., telegraphic switching. 186,187The dwell time in either stable state (P or AP) can be influenced by the direction and amplitude of the injected current.186,187The opposite extreme condition is that a stable state of the MTJ is pinned by a large external magnetic field,equivalent to a very large DE. In this scenario, a large enough injected current will induce sustained precession of the magnetization of thefree layer, resulting in an oscillating resistance (or voltage output) of the MTJ [ Figs. 18(c) and18(f)], 188which is called spin torque nano- oscillator (STNO). As discussed in Sec. III C, the sustained precession is set by the balance among the torques executed on the moment. Theamplitude of the oscillating output depends on the magnitude of theinjected current. 181 The stochastic switching of the MTJ has been well demonstrated to show the key neuron function, i.e., leaky integrate-and-fire func-tion. 176,178It is noted that the stochastic switching is also possible toApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-18 Published by AIP PublishingFIG. 17. (a) Schematic of a typical MTJ in side-view, where a domain wall is presented in the magnetic free-layer. Reproduced with permission from Lequeux et al. , “A mag- netic synapse: multilevel spin-torque memristor with perpendicular anisotropy,” Sci. Rep. 6, 31510 (2016). Copyright 2016 Springer Nature. (b) Scanning electron microscope (SEM) images of two MTJ devices to facilitate the introduction of domain wall operation. Left image: Reproduced with permission from Lequeux et al. , “A magnetic synapse: multilevel spin-torque memristor with perpendicular anisotropy,” Sci. Rep. 6, 31510 (2016). Copyright 2016 Springer Nature; Right image: Reproduced from Cai et al. , “Multilevel storage device based on domain-wall motion in a magnetic tunnel junction,”Appl. Phys. Lett. 111, 182410 (2017), Copyright 2017 AIP Publishing. (c) The multiresist- ance states in an MTJ with domain wall present. Reproduced with permission from Lequeux et al. , “A magnetic synapse: multilevel spin-torque memristor with perpendicular anisotropy,” Sci. Rep. 6, 31510 (2016). Copyright 2016 Springer Nature. (d) Schematic of the antiferromagnetic/ferromagnetic (AFM/FM, PtMn/[Co/Ni] here) bilayer (upper panel) and the potential multidomain structure in antiferromagnetic layer. Up image: Reproduced with permission from Fukami et al. ,“Magnetization switching by spin–orbit tor- que in an antiferromagnet–ferromagnet bilayer system,” Nat. Mater. 15, 535 (2016). Copyright 2016 Springer Nature; Bottom image: Reproduced from Kurenkov et al. , “Device-size dependence of field-free spin–orbit torque induced magnetization switching in antiferromagnet/ferromagnet structures,” Appl. Phy s. Lett. 110, 92410 (2017), Copyright 2017 AIP Publishing. (e) Experimental set-up used for the SOT switching with the SEM image of the device structure. Reproduced with permiss ion from Borders et al., “Analogue spin–orbit torque device for artificial-neural-network-based associative memory operation,” Appl. Phys. Express 10, 13007 (2016). Copyright 2016 Japan Society of Applied Physics. (f) Memristorlike behavior of the AFM/FM structure. Reproduced with permission from Borders et al. , “Analogue spin–orbit torque device for artificial-neural- network-based associative memory operation,” Appl. Phys. Express 10, 13007 (2016). Copyright 2016 Japan Society of Applied Physics. (g) Multidomains in CuMnAs AFM imaged by X-ray magnetic linear dichroism–photoelectron emission microscopy (XMLD-PEEM) technique. Reproduced with permission from Wadley et al. , “Electrical switching of an antiferromagnet,” Science 351, 587–590 (2016). Copyright 2016 AAAS. (h) Optical microscopy image of the AFM single-layer device. Reproduced with permission from Olejn/C19ıket al. , “Antiferromagnetic CuMnAs multi-level memory cell with microelectronic compatibility,” Nat. Commun. 8, 15434 (2017). Copyright 2017 Springer Nature. (i) The Readout signal of the AFM single-layer device as function of the write pulse length. Reproduced with permission from Olejn /C19ıket al. , “Antiferromagnetic CuMnAs multi-level memory cell with microelectronic compatibility,” Nat. Commun. 8, 15434 (2017). Copyright 2017 Springer Nature.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-19 Published by AIP Publishingimplement a synaptic function, as demonstrated by Vincent et al.189 Recently, the biofunctions have been further explored by forming an integrated system with MTJ working at the telegraphic switching or persistent oscillation regime. For the telegraphic switching, a group of nanoscale MTJs, demonstrated by Mizrahi et al.,182were fabricated to emulate neurons operated under the population coding principle, which showed the capability to generate cursive letters and realized nonlinear resilient transformations with designed hybrid magnetic- CMOS systems. The neurons in brain can also be regarded as nonlin- ear oscillators,190which stimulates the exploration to using the STNO to emulate its function. Torrejon et al. Demonstrated such a function by utilizing a single STNO to achieve spoken-digit recognition with good accuracy.181Later, the synchronizing function of the STNO was further exploited to recognize spoken vowels.180 IV. CONCLUSION AND FUTURE OUTLOOK In conclusion, the development of the ferroic tunnel junction based memristive devices has made significant progress, with competi- tive advantages comparable to other state-of-the-art memristors. The following are future outlook on the ferroic tunnel junctions and their applications based on their current development.First, although the basic principles that explain the memristive switching mechanisms of FTMs are generally understood, systems with an ultrathin ferroelectric layer still provide a large opportunity for research. For example, the domain wall in ferroelectric films has been reported to be conductive in specific cases (e.g., BFO) although it is only around nano- meters thick.191–196The conductive domain wall has been identified as a possible transport mechanism in ferroelectric thin films, which might con- tribute additionally to the tunneling mechanisms and therefore the mem- ristive switching mechanism.197The different domain wall types and quantities during the ferroelectric polarization switching mechanism might correspond to different resistance states of the FTMs. However, until now, no demonstration of a contribution of conductive domain wall in FTJs has been reported. Therefore, future studies on the role of the domain wall in the resistive switching process of FTJs become necessary.Moreover, the design of memristive devices that could take advantage of the domain wall conductivity should be very interesting and useful. Second, compared with PCM and RRAM that are well compati- ble with the CMOS process, as well as with MTJ that has been widely used as a magnetic sensor in hard disk drive and as a basic storage unit in the commercialized STT-MRAM, FTJs have a number of challengesneeded to be solved for the practical application: (1) the current depo- sition technique for most of the ferroelectric perovskite oxide films is FIG. 18. (a)–(c) The stable state of the MTJ with various energy barrier heights. For ultralow energy barrier, the thermal fluctuation is able to flip the moments between the two stable states (telegraphic regime, a), while the other two (b) and (c) require spin torque (ST) to achieve the switching and precession, respectively . (d) Time evolution of the electrical resistance of a MTJ in telegraphic regime measured at room temperature (upper panel), and the frequency of the signal depends on the inject ed current (lower panel). Reproduced with permission from Mizrahi et al. , “Neural-like computing with populations of superparamagnetic basis functions,” Nat. Commun. 9, 1533 (2018). Copyright 2018 Springer Nature. (e) The switching probability of the MTJ at stochastic regime as a function of the pulse duration, for different pulse amplitude. Reproduced with permission from Vincent et al. , “Spin-transfer torque magnetic memory as a stochastic memristive synapse for neuromorphic systems,” IEEE Trans. Biomed. Circuits Syst. 9, 166–174 (2015). Copyright 2015 IEEE. (f) The oscillating voltage from MTJ at sustained precession (upper panel), and the voltage amplitude show non linear behavior with the input current (lower panel). Reproduced with permission from Torrejon et al. , “Neuromorphic computing with nanoscale spintronic oscillators,” Nature 547, 428 (2017). Copyright 2017 Springer Nature.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 011304 (2020); doi: 10.1063/1.5120565 7, 011304-20 Published by AIP Publishingthe pulsed laser deposition, which has the limitations such as a small deposited area with uniform properties, low deposition rate, etc.; (2)the high temperature deposition process, which is not compatible withCMOS process; (3) strict requirements on the crystallographic struc-ture and lattice constant of the substrates, which limit their integrationwith CMOS in the back-end of line (BEOL). In the future work, devel-opment of a low temperature, large scale deposition technique is nec-essary. It is worth noting that the newly discovered ferroelectric filmssuch as HfZrO 2,198–204which can be fabricated by industry preferred magnetron sputtering or atomic layer deposition, are very promisingfor the FTJ based nonvolatile memory or memristors/neurons.Beyond the above challenges for practical applications, some funda-mental questions such as what is the smallest lateral size of the ferro- electric film that still has the domain nucleation and expansion dominated reversal mechanism are also needed to be addressed. Third, regarding the MTJ based devices, the flurry of intrinsic physics and their flexibility responding to the external conditions inthe spintronic devices promise a huge possibility to develop fast,energy- and area-efficient spin-based neuromorphic computing sys-tems. However, with explorations presented above, there is still a lot ofscope to improve, such as the accuracy and repeatability of the multi-resistance states, the small readout of the output signal in some spin-tronic devices, and exploration of faster and more energy-efficientcandidates. All these remaining problems may be overcome with fur-ther studies of the physics involved and utilization of new functionalmagnetic materials. The recently proposed skyrmion-based spintronicdevices 205and electric-field manipulation of the magnetic moments206 are good examples for such directions. Finally, flexible electronics have attracted more and more atten- tion in recent years due to their distinguished portability, conformalcontact property, and human friendly interfaces compared to conven-tional bulk Si technology. 91,207,208Particularly, flexible memory is regarded as an integral component and a key element for system-on-plastic applications due to its crucial role in data processing, storage,and information communications with external devices. 92Currently, although some groups have already demonstrated memories on flexi-ble substrates, the study is still in the early stage, and most of thestudies are mainly focused on organic materials. 94,95,209Since ferroelectric-based memories possess many advantages that organic-based memories do not have, it is meaningful to study flexible ferroictunnel junction artificial synapses in the future. ACKNOWLEDGMENTS This research was supported by the Singapore National Research Foundation under CRP Awards Nos. 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5.0035586.pdf

Appl. Phys. Lett. 118, 032406 (2021); https://doi.org/10.1063/5.0035586 118, 032406 © 2021 Author(s).Collective spin dynamics under dissipative spin Hall torque Cite as: Appl. Phys. Lett. 118, 032406 (2021); https://doi.org/10.1063/5.0035586 Submitted: 30 October 2020 . Accepted: 04 January 2021 . Published Online: 21 January 2021 Yaroslav Tserkovnyak , Eran Maniv , and James G. Analytis COLLECTIONS Paper published as part of the special topic on Spin-Orbit Torque (SOT): Materials, Physics, and Devices ARTICLES YOU MAY BE INTERESTED IN Current switching of interface antiferromagnet in ferromagnet/antiferromagnet heterostructure Applied Physics Letters 118, 032402 (2021); https://doi.org/10.1063/5.0039074 Gradual magnetization switching via domain nucleation driven by spin–orbit torque Applied Physics Letters 118, 032407 (2021); https://doi.org/10.1063/5.0035667 Parity-controlled spin-wave excitations in synthetic antiferromagnets Applied Physics Letters 118, 032403 (2021); https://doi.org/10.1063/5.0037427Collective spin dynamics under dissipative spin Hall torque Cite as: Appl. Phys. Lett. 118, 032406 (2021); doi: 10.1063/5.0035586 Submitted: 30 October 2020 .Accepted: 4 January 2021 . Published Online: 21 January 2021 Yaroslav Tserkovnyak,1,a) Eran Maniv,2,3 and James G. Analytis2,3 AFFILIATIONS 1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 2Department of Physics, University of California, Berkeley, California 94720, USA 3Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Note: This paper is part of the Special Topic on Spin-Orbit Torque (SOT): Materials, Physics and Devices. a)Author to whom correspondence should be addressed: yaroslav@physics.ucla.edu ABSTRACT Current-induced spin torques in layered magnetic heterostructures have many commonalities across broad classes of magnetic materials. These include not only collinear ferromagnets, ferrimagnets, and antiferromagnets but also more complex noncollinear spin systems. Wedevelop a general Lagrangian–Rayleigh approach for studying the role of dissipative torques, which can pump energy into long-wavelengthmagnetic dynamics, causing dynamic instabilities. While the Rayleigh structure of such torques is similar for different magnetic materials,their consequences depend sensitively on the nature of the order and, in particular, on whether there is a net magnetic moment. The latter endows the system with a unipolar switching capability, while magnetically compensated materials tend to evolve toward limit cycles, at large torques, with chirality dependent on the torque sign. Apart from the ferromagnetic and antiferromagnetic cases, we discuss ferrimagnets,which display an intricate competition between switching and limit cycles. As a simple case for compensated noncollinear order, we considerisotropic spin glasses and a scenario of their coexistence with a collinear magnetic order. Published under license by AIP Publishing. https://doi.org/10.1063/5.0035586 The spin Hall effect and the associated torque on magnetization dynamics encompass a vast array of nonequilibrium phenomena indiverse magnetic heterostructures with different microscopic origins. 1 Here, we attempt to establish a common thread for thinking about theconsequences of these torques on magnetic dynamics and switching in different families of magnetic materials: ferro-, ferri-, and antiferro- magnets and spin glasses. We focus on the most generic dissipative torques unconditioned by the crystalline symmetries. Through spin–orbit interactions, these are exerted by electrical currents on col- lective dynamics of magnetic degrees of freedom, steadily pumping energy into the latter. While the formal structure of the theory is distilled down to uni- versal ingredients, irrespective of the details of the magnetic order, the ensuing magnetic dynamics differ significantly for different families ofthe magnetic materials. In particular, we emphasize a general tendency for (anti)ferromagnetic correlations to align along (perpendicular) the effective spin-accumulation direction produced by the dissipative spin torque. Ferrimagnets exhibit a competition between these opposite propensities, while magnetically compensated materials, such as spin glasses, share much commonality with antiferromagnets. The situationcan become especially intriguing when there is a coexistence of com- peting magnetic orders in the same material. As a central generic model, we consider low-temperature classical dynamics of an ordered collinear ferrimagnet, 2i nt h ep r e s e n c eo f Gilbert damping3and spin Hall torque.4Following Ref. 5, we write the magnetic Lagrangian density L½n;m/C138(focusing on the dominant kinetic and energy terms) as L¼/C0 saðnÞ/C1@tnþm/C1n/C2@tn/C0m2 2vþðmþsnÞ/C1b/C0Eð nÞ:(1) The collective dynamics are parameterized by the directional order parameter nðtÞ(s.t.,jnj/C171) and a small transverse spin density mðtÞ (obeying the constraint m/C1n/C170 and realizing generators for the order-parameter rotations6)sis the (uncompensated) equilibrium lon- gitudinal spin density along the order parameter (zero in the purely antiferromagnetic limit), and v/J/C01is the transverse spin suscepti- bility (where Jis the microscopic Heisenberg exchange assumed to be the largest energy scale in the problem). The first term is the ferro-like (Wess–Zumino) kinetic term, expressed in terms of a vector potential aðnÞproduced on a unit sphere by a magnetic monopole of unit Appl. Phys. Lett. 118, 032406 (2021); doi: 10.1063/5.0035586 118, 032406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplcharge. The second term is the antiferro-like r-model kinetic term. Both of these kinetic terms stem from the Berry phases summed over the individual spins.7The remaining terms consist of Zeeman energy proportional to b/C17cB, in terms of the gyromagnetic ratio cand magnetic field B,a n de n e r g y EðnÞthat includes all other order- parameter-dependent terms, such as dipolar interactions (whens6¼0), anisotropies, and exchange-stiffness terms (in the case of order-parameter inhomogeneities). Lagrangian (1)constitutes a minimal model to describe any (strongly) collinearly ordered magnet. Pure antiferromagnetic dynam- ics (i.e., the standard nonlinear rmodel 7) is recovered by setting s!0, while the Landau–Lifshitz equation9for the ferromagnetic case would be obtained by v!0. Generic dissipation can be introduced into the model through the following Rayleigh dissipation function:5 R¼a 2ð@tnÞ2þg 2ð@tn/C0l/C2nÞ2; (2) which complements the above Lagrangian. ais the Gilbert damping constant,3which describes the viscosity of the reorientational order- parameter dynamics, and gis the spin-mixing conductance,10which describes the dissipative coupling between the spin accumulation l (here induced by the Edelstein/spin Hall effect) and magnetic dynam-ics. For a Rashba-type, i.e., C 1v, symmetry breaking normal to a film’s (yz) plane, for example, l/x/C2j, in terms of the applied current density j(seeFig. 1 for a schematic). Rayleigh function (2)boils down to the dissipative torque (i.e., the net angular momentum transfer onto the collective magneticdynamics from the nonmagnetic/incoherent degrees of freedom), s/C17/C0n/C2@R @@tn¼n/C2ðgl/C2n/C0a@tnÞ; (3) where a!aþgis, henceforth, the total effective damping, including also the spin-mixing-conductance contribution (known as spin pump-ing 10) According to the schematic shown in Fig. 1 ,w ew i l ls e t l!z, lumping the current and the effective spin Hall angle,4which deter- mine l,i n t o g(whose sign, thus, depends on the current direction). Other torques can be added to Eq. (3), if the structural symmetries are reduced further11(depending on the details of the crystal and device structure), as in Ref. 12, for example. We are, furthermore, omitting some of the other torques allowed in the present high-symmetrycase13( l i k et h efi e l d - l i k et o r q u e /n/C2land the less common torques /nzn/C2jand/nzn/C2j/C2n), which are typically less important for large-angle reorientational dynamics. Minimizing the Lagrangian (1)with respect to m,w efi n dt h e usual expression,7 m¼vn/C2ð@tn/C0n/C2bÞ; (4) for the transverse spin density. The other Euler–Lagrange–Rayleigh equation gives the equation of motion for the total spin density, ð@tþb/C2ÞðsnþmÞþn/C2@nE¼ s: (5) Setting v!0 recovers the Landau–Lifshitz–Gilbert equation with the (damping-like) spin Hall torque, while setting instead s!0 gives the standard r-model equation for the N /C19eel order (including spin-transfer torque and relaxation). The expression for the torque, Eq. (3),i st h e same in both cases.5The equation of motion (5)describes Larmor- type dynamics of the total spin density, snþm, in the presence of var- ious torques: Zeeman, anisotropy, spin-transfer, damping, etc. The above equations of motion, Eqs. (4)and(5), establish the general starting point for collinear-order spin dynamics. As a simpleillustrative example, let us now specialize it to analyze stability of thefixed points along the 6zorientations, if zis the easy axis, i.e., EðnÞ¼/C0 Kn2 z 2; (6) with K>0. The two coupled equations of motion (setting b!0, hereafter) are @tn¼1 vm/C2n; @tm¼n/C2Knzzþgz/C2nþ1 vðsmþan/C2mÞ/C20/C21 ;(7) which could be integrated up, starting from an arbitrary initial config- uration, with m?n. Linearizing these equations close to the two equi- libria, n!6z, and switching to the complex notation: n/C17nxþiny andm/C17mxþimy,w eg e t @tn¼7i vmand@tm¼7ðgþiKÞn/C0a7is vm: (8) Writing these as @t~s¼/C0 ^A~s,w h e r e ~s/C17ðn;mÞand^Ais the associated response matrix (that we can think of as a non-HermitianHamiltonian describing the small-angle dynamics), which is read outfrom the linearized equations (8)we are interested in the eigenvalue of^Awith the smallest real part k.k<0 would signal a (spin- torque-induced) instability. This smaller (real part of the) eigenvalue is given by k¼ a 2v/C0Reffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a7is 2v/C18/C192 /C0K/C0ig vs ; (9) where the square root is evaluated for its principal value (yielding the non-negative real part). As a sanity check, we verify that k/C210i fg¼0. For the ferromagnetic (F) case, v!0, we get kF¼ReK/C0ig a7is¼aK6gs a2þs2: (10)FIG. 1. The schematic of a current-biased planar structure: the magnetic film is in theyzplane. The reflection symmetry is broken along the film’s normal, x. The elec- tric current jis applied in the ydirection, and an effective nonequilibrium spin accu- mulation l(according to the Edelstein effect,8in the presence of the C1v symmetry relative to the vertical xaxis) is induced along the zaxis: l/x/C2j. This spin accumulation can be generated by a nonmagnetic heavy-metal capping layer, the substrate, or the magnetic film itself. In the case of an axial symmetry about the zaxis, the spin accumulation needs to exceed the gap xin the lowest magnon band, in order to induce a magnetic instability. (The sign of l, furthermore, needs to be consistent with the chirality of the excited mode.).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 032406 (2021); doi: 10.1063/5.0035586 118, 032406-2 Published under license by AIP PublishingThe instability sets in when kF!0, which corresponds to g¼7aK s¼7axF; (11) where xF/C17K=sis the ferromagnetic resonance frequency. For the equilibrium noriented along the 6zaxis, we, thus, need a negative (positive) torque gin order to trigger the F instability. This results in the familiar unidirectional switching of the F orientation toward the 7 orientation (which is then stable against the torque). The above thresh-old makes sense thermodynamically: in the absence of intrinsic damp-ing (so that ais determined by the spin-mixing conductance g), the spin accumulation lmust overcome the intrinsic gap in the magnon spectrum, i.e., x F, while oriented parallel to the spin angular momen- tum of individual magnons.14A finite intrinsic damping raises the instability threshold further. For the antiferromagnetic (AF) case, s!0, kAF¼a 2v/C0Reffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2v/C18/C192 /C0K/C0ig vs /C25a 2v/C0Reffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C0K/C0ig vs ¼a 2vþImffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K/C0ijgj vs /C25a 2v/C0jgj 2ffiffiffiffiffiffiKvp ; (12) where the approximations are based on the assumptions that a/C28ffiffiffiffiffiffivKp(which physically corresponds to the quality factor of the AF resonance Q/C291) and jgj/C28 K(which, in fact, follows from a/C28ffiffiffiffiffiffivKp, at the onset of the instability: kAF!0). Independent of the sign of the torque g, we reach the instability at15 jgj¼affiffiffiffi K vs ¼ax AF: (13) xAF/C17ffiffiffiffiffiffiffiffiffi K=vp is the AF resonance frequency. As in the F case, this result makes sense thermodynamically: In the absence of intrinsicdamping (so that ais determined by the spin pumping /g), the spin accumulation lmust overcome the intrinsic gap in the magnon spec- trum, i.e., x AF.T h es i g no f gdetermines which of the two magnon branches of the spectrum goes unstable. Beyond the threshold of insta-bility, the order parameter nreaches a stable precession within the xy plane, which we obtain according to Eq. (7)(with s!0) as @ tn¼g az/C2n;m¼vg az: (14) For this trajectory, the torque (3)vanishes, so that the work done on the magnetic dynamics by the spin transfer gis dissipated by Gilbert damping a. The corresponding precession frequency is x¼g=a (recall that gis proportional to the applied electric current). For a spin glass, we expect the dynamics similar to that of the above AF case.16Namely, beyond the threshold current set by aniso- tropies (which may be randomized and, thus, reduced by disorder),the SO(3)-valued state variable (which is rooted in the Edwards–Anderson order parameter 17) will precess around the zaxis at the frequency xSG¼g a; (15)where gis the appropriate spin-mixing conductance, which is closely analogous to the collinear case,5and ais the spin-glass Gilbert damping. At the same time, a nonequilibrium spin polarization m builds up, as in Eq. (14), determined by the spin-glass susceptibility v. This spin polarization, with magnitude vxSG, makes sense from the rotating-frame perspective. Since the F and AF cases showed qualitatively different behavior at the instability threshold—namely, the F order tends to undergo aunipolar switching along the easy axis, while the AF order settles at anequatorial limit cycle perpendicular to the easy axis—it is interestingto study the interplay between these different tendencies in ferrimag- nets. This, in particular, can provide us a blueprint for driven dynam- ics in complex systems with competing magnetic orders. We, thus, return to discuss the instability threshold in the general (ferrimagnetic) case, Eq. (9), with both vandsbeing nonzero. In anal- ogy to the above F and AF cases, we expect the threshold to be reached when gapproaches ax, in terms of the respective resonance frequency x(supposing that the overall damping is weak, so that the quality fac- tor of the dynamics is /C291), which is also easily verified directly from Eq. (9). The normal-mode frequencies for solutions /e ixtare obtained from Eqs. (8), after setting a;g!0, x¼rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K vþs2 4v2s 6s 2v; (16) with two magnon branches labeled by r!6.A sb e f o r e ,t h eo t h e r 6 in Eq. (16)stands for the initial 6zorientation of the order parameter. The positive (negative) frequency (corresponding here to r!6)i s associated with magnons carrying spin angular momentum 6/C22h, thus requiring a positive (negative) torque gto reach the threshold. Using Eq.(16),w er e c o v e rt h eFl i m i tw h e n s=ffiffiffiffiffiffivKp!1 and the AF limit when s=ffiffiffiffiffiffivKp!0. Let us now start from the AF limit, and consider the weakly ferri- magnetic case, s/C28ffiffiffiffiffiffivKp. The normal-mode frequencies (16) (corre- sponding in the continuum description to the gaps of the respectivemagnon branches) then become x/C25rffiffiffiffi K vs 6s 2v¼rxAF6s 2v: (17) When sis increased, the lower of these two frequencies would eventu- ally approach the ferromagnetic resonance frequency ( xF¼K=s), according to Eq. (16).E q u a t i o n (17)shows that the threshold of insta- bility depends on the orientation of nand the sign of g. For instance, when g>0, the x>0 branch is excited and the critical current corre- sponds to the r!þ (i.e., spin-up magnon) mode, which is now split bys=v, depending on the initial up or down orientation of nalong the zaxis [denoted by 6in Eq. (17)]. We, thus, conclude that when g=a<xAF/C0s=2v, both n!6zare stable fixed points; when xAF/C0s=2v<g=a<xAFþs=2v,n!zis stable, while n!/C0 zis unstable (allowing for the unidirectional switching, as in the ferromag- net); and when g=a>xAFþs=2v, both are unstable (resulting in a stable precessional state, as in the AF case, albeit somewhat canted outof the xyplane). In Fig. 2 , we show a schematic of how this dynamical s y s t e me v o l v e sa saf u n c t i o no f g>0. In summary, this Letter aims to establish basic aspects and intui- tion for thinking about spin-torque instabilities and the ensuingApplied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 032406 (2021); doi: 10.1063/5.0035586 118, 032406-3 Published under license by AIP Publishingdynamics of the ordered magnets and spin glasses, under the generic dissipative torque (3). To summarize, beyond an instability threshold, which is determined by the anisotropies, the order parameter generally has a tendency to precess according to the right-hand rule around the spin accumulation l/z.T h i si ss e e nf r o mt o r q u e (3), which imparts positive work /lfor the right-hand precession. As a result of the instability, the order parameter either switches (cf. the ferromagneticcase) or precesses steadily (cf. the antiferromagnetic and spin-glassphases), in which case the steady input of work is dissipated by Gilbertdamping. The ferrimagnetic case (with both s;v6¼0) combines both of these aspects, as sketched in Fig. 2 . Generally, when under the action of the damping torque (3),w e expect for the F spin order to tend to point along the applied spinaccumulation (depending on its sign). In this case, the magnons asso-ciated with a potential destabilization of the order, which carry spinopposite to the order parameter, are thermodynamically biased to beejected by the spin accumulation. 14The AF order, on the other hand, tends to dynamically orient normal to the spin accumulation (inde-pendent of its sign). Such order-parameter reorientations can result incharacteristic electrical magnetoresistivity or x-ray dichroism signa-tures. 15,18,19In the multidomain case, the possible reorientation of domains depends crucially on the dynamics within the domain walls,whose motion is likewise driven by a positive gain of the work by thespin torque /l(thus being sensitive to the chiral structure of the moving domain walls 19). It is interesting to consider a situation in which a torque-driven precessional state is reached in a spin glass (SG) coexisting withanother magnetic order. 20In the case of a (coarse-grained) rotational symmetry about the zaxis, furthermore, the threshold for SG dynam- ics could be low. The steady-state SG frequency is given by Eq. (15).I n the rotating frame of reference, other degrees of freedom would getpolarized by the fictitious Zeeman field of x SGalong the rotation (i.e., z) axis. This, in turn, would facilitate the switching toward the zaxis in the F case or precession toward the xyplane in the AF case. The details would depend on the exact interaction of the coexisting SG and collin-ear magnetic components (and could, in principle, be handled withour Lagrangian–Rayleigh approach 5and subject to a decomposition of the magnetic state into the SG and ferrimagnetic variables). In thissense, the SG dynamics can be viewed as enhancing the torque transfer from the electronic to the magnetic degrees of freedom.20 Finally, we wish to emphasize that, in our analysis, we have retained only the most generic Slonczewski-like21torque (3)due to the Edelstein/spin Hall effect, as a means to pump energy into magnetic precession and cause dynamic instabilities. Other types of torques, as discussed below Eq. (3), may need to be included in more specialized cases, especially if the symmetries are reduced by crystalline order. This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences under Award No. DE-SC0012190. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1A. Hoffmann, IEEE Trans. Magn. 49, 5172 (2013). 2B. A. lvanov and A. L. Sukstanskii, Sov. Phys. JETP 57, 214 (1983). 3T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 4Y. Tserkovnyak and S. A. Bender, Phys. Rev. B 90, 014428 (2014). 5Y. Tserkovnyak and H. Ochoa, Phys. Rev. B 96, 100402(R) (2017). 6A. F. Andreev and V. I. Marchenko, Sov. Phys. Usp. 23, 21 (1980). 7A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer-Verlag, New York, 1994). 8V. M. Edelstein, J. Phys.: Condens. Matter 7(1), 1 (1995). 9E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2 , Course of Theoretical Physics, 3rd ed. (Pergamon, Oxford, 1980), Vol. 9. 10Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). 11R. Zarzuela and Y. Tserkovnyak, Phys. Rev. B 95, 180402(R) (2017). 12J./C20Zelezn /C19y, H. Gao, K. V /C19yborn /C19y, J. Zemen, J. Ma /C20sek, A. Manchon, J. Wunderlich, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 113, 157201 (2014). 13K .G a r e l l o ,I .M .M i r o n ,C .O .A v c i ,F .F r e i m u t h ,Y .M o k r o u s o v ,S .B l €u g e l ,S .A u f f r e t , O. Boulle, G. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8, 587 (2013). 14S .A .B e n d e r ,R .A .D u i n e ,a n dY .T s e r k o v n y a k , Phys. Rev. Lett. 108, 246601 (2012). 15X. Z. Chen, R. Zarzuela, J. Zhang, C. Song, X. F. Zhou, G. Y. Shi, F. Li, H. A. Zhou, W. J. Jiang, F. Pan, and Y. Tserkovnyak, Phys. Rev. Lett. 120, 207204 (2018). 16H. Ochoa, R. Zarzuela, and Y. Tserkovnyak, Phys. Rev. B 98, 054424 (2018). 17S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975).FIG. 2. Evolution of the dynamical state for the order parameter nðtÞaccording to Eq. (7), in the ferrimagnetic case [cf. Eq. (17)]. Stable (unstable) fixed points and limit cycles are shown in black (red). For small positive g, two stable fixed points at nz!61 coexist with an unstable limit cycle (in the lower hemisphere) near the xyplane. For large g, these fixed points become unstable, while the limit cycle stabilizes (in the upper hemisphere), analogous to the antiferromagnetic case. At intermed iate values of g, only one fixed point is stable, as in the ferromagnetic case.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 032406 (2021); doi: 10.1063/5.0035586 118, 032406-4 Published under license by AIP Publishing18P. Wadley, B. Howells, J. /C20Zelezn /C19y, C. Andrews, V. Hills, R. P. Campion, V. Nov/C19ak, K. Olejnik, F. Maccherozzi, S. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich, F. Freimuth, Y. Mokrousov, J. Kune /C20s, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds, B. L. Gallagher, and T.Jungwirth, Science 351, 587 (2016); P. Wadley, S. Reimers, M. J. Grzybowski, C. Andrews, M. Wang, J. S. Chauhan, B. L. Gallagher, R. P. Campion, K. W. Edmonds, S. S. Dhesi, F. Maccherozzi, V. Novak, J. Wunderlich, and T.Jungwirth, Nat. Nanotech. 13, 362 (2018).19L .B a l d r a t i ,O .G o m o n a y ,A .R o s s ,M .F i l i a n i n a ,R .L e b r u n ,R .R a m o s ,C . Leveille, F. Fuhrmann, T. R. Forrest, F. Maccherozzi, S. Valencia, F. Kronast, E. Saitoh, J. Sinova, and M. Kl €aui,P h y s .R e v .L e t t . 123, 177201 (2019). 20E. Maniv, N. Nair, S. C. Haley, S. Doyle, C. John, S. Cabrini, A. Maniv, S. K. Ramakrishna, Y.-L. Tang, P. Ercius, R. Ramesh, Y. Tserkovnyak, A. P. Reyes, and J. Analytis, “Antiferromagnetic switching driven by the collective dynam- ics of a coexisting spin glass,” Sci. Adv. 7, eabd8452 (2021). 21J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 118, 032406 (2021); doi: 10.1063/5.0035586 118, 032406-5 Published under license by AIP Publishing

1.4771683.pdf

Direct imaging of phase relation in a pair of coupled vortex oscillators Andreas Vogel, André Drews, Markus Weigand, and Guido Meier Citation: AIP Advances 2, 042180 (2012); doi: 10.1063/1.4771683 View online: http://dx.doi.org/10.1063/1.4771683 View Table of Contents: http://aipadvances.aip.org/resource/1/AAIDBI/v2/i4 Published by the American Institute of Physics. Related Articles The magnetic structure of exchange coupled FePt/FePt3 thin films J. Appl. Phys. 113, 013909 (2013) Promising ferrimagnetic double perovskite oxides towards high spin polarization at high temperature AIP Advances 3, 012107 (2013) Magnetocaloric effect of an Fe-based metallic glass compared to benchmark gadolinium J. Appl. Phys. 112, 123918 (2012) Magnetostructural transformation in Mn1+xNi1−xGe and Mn1−xNi1+xGe alloys J. Appl. Phys. 112, 123911 (2012) Magnetic interactions in Ge1−xCrxTe semimagnetic semiconductors J. Appl. 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See: http://creativecommons.org/licenses/by/3.0/AIP ADV ANCES 2, 042180 (2012) Direct imaging of phase relation in a pair of coupled vortex oscillators Andreas Vogel,1,aAndr ´eD r e w s ,1,2Markus Weigand,3and Guido Meier1 1Institut f ¨ur Angewandte Physik und Zentrum f ¨ur Mikrostrukturforschung, Universit ¨at Hamburg, 20355 Hamburg, Germany 2Arbeitsbereich Technische Informatik Systeme, Universit ¨at Hamburg, 22527 Hamburg, Germany 3Max-Planck-Institut f ¨ur Intelligente Systeme, 70569 Stuttgart, Germany (Received 27 September 2012; accepted 28 November 2012; published online 5 December 2012) We study the magnetization dynamics in a stray-field coupled pair of ferromagnetic squares in the vortex state. Micromagnetic simulations give an idea of the mediating stray field during vortex gyration. The frequency-dependent phase relation between the vortices in the spatially separated squares is studied using time-resolved scanningtransmission x-ray microscopy while one element is harmonically excited via an alternating magnetic field. It is shown that the normal modes of coupled vortex- core motion can be understood as an attractive (low-frequency) and a repulsive(high-frequency) mode of the effective magnetic moments of the microstructures. Copyright 2012 Author(s). This article is distributed under a Creative Commons Attribution 3.0 Unported License .[http://dx.doi.org/10.1063/1.4771683 ] Due to potential applications in new concepts of high-density and ultrafast nonvolatile data stor- age devices, 1–5information-signal processing devices,6,7logical units,8–10and microwave emission sources,11,12the dynamic properties of ferromagnetic micro- and nanostructures have gained a broad scientific interest. Recently, the magnetization dynamics in neighboring ferromagnetic structures in the vortex magnetization state13–15coupled via their stray fields attracted a lot of attention.16–24On their specific time and length scales, the structures reveal characteristics known from other funda- mental physical systems. It has been shown that the resonance frequency of vortices is strongly influenced by the strength of the magnetostatic interaction given by the distance between the ele-ments and the relative configuration of the core polarizations, i.e., the directions of the out-of-plane magnetization components. In densely-packed arrays of permalloy disks, a relative broadening of the absorption peak in broadband ferromagnetic-resonance measurements varying with the inverse-sixthpower of the normalized center-to-center distance between the elements 25and the size of the array26 has been observed. Stray-field coupled vortex gyrations in a pair of ferromagnetic elements havebeen demonstrated via time-resolved magnetic transmission x-ray microscopy, 27–29time-resolved photoemission electron microscopy,30and electrical measurements.31,32The system behaves like damped coupled harmonic oscillators. In a pair with opposite core polarizations, a beating pattern attributed to the superposition of two characteristic normal modes has been observed after initialdeflection of one of the cores. 28,29A mode splitting has been detected electrically via excitation of one element with an ac current by Sugimoto et al.31Interestingly, the inverse-sixth power law for the dependence of the frequency shift on the inter-element distance observed in densely-packedarrays 25and predicted by the so-called rigid vortex model17could not be confirmed for a small inter- element distance in experiments and micromagnetic simulations on pairs. This has been attributed to a modification of the core trajectories and the magnetization configuration near the edge for strong magnetostatic coupling.31A distortion of the magnetization pattern also becomes clear considering the distance-dependent strength of the stray field, as shown in Fig. 1for a micron-sized square. Due aElectronic mail: andreas.vogel@physnet.uni-hamburg.de 2158-3226/2012/2(4)/042180/7 C/circlecopyrtAuthor(s) 2012 2, 042180-1 Downloaded 09 Jan 2013 to 152.14.136.96. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/042180-2 Vogel et al. AIP Advances 2, 042180 (2012) FIG. 1. Micromagnetic simulation of the stray field during the vortex gyration outside a square. (a) The amplitude of the stray field along the center line in positive xdirection at a distance 2 d/lfrom the center of the structure is shown for three core positions (solid, dashed, and dotted lines). The corresponding core deflections in negative x, positive x, and positive y direction are illustrated in the insets via the magnetization pattern using the same color code as in (b) where arrows outsidethe square indicate the direction of the stray field and arrows inside the square mark the magnetization direction in thedomains for a core deflection in negative xdirection. The xandycomponents of the stray field during one cycle for a distance 2d/l=2.2 are plotted against each other in (c). For comparison, the gray dashed line indicates the shape of a rotational field with constant amplitude. Black dots mark the stray fields corresponding to vortex core deflections along the xandyaxis. to the spatial extent of the elements, the stray field at the edge closer to the other element is different to the stray field at the edge that is further apart. The gradient in the strength of the stray field that the neighboring element is exposed to breaks the symmetry of its magnetic confining potential.Magnetic volume charges have to be considered and the rigid vortex model assuming an unchanged static structure of the vortex does not hold anymore. On the other hand, placing another element on the opposite side leads to a symmetric effective stray field seen by the element in between, whichcould explain the different distance dependences observed in pairs and in densely-packed arrays of elements. Here, we focus on the frequency-dependent phase relation between the vortices in a stray-field coupled pair of ferromagnetic squares where only one element is directly excited. Square-shaped structures are of particular interest since they provide a strong dependence of the coupling on therelative orientation of the vortex cores in neighboring elements 20,28which can be attributed to the in- plane stray field arising during vortex gyration. Micromagnetic simulations of the stray field outside a square for deflections of the vortex core in different directions are presented. Time-resolved scanningtransmission x-ray microscopy (STXM) is used to image the magnetization dynamics in a pair of squares in real space and in this way to resolve the phase difference between the vortices. It is shown that the modes of coupled vortex-core motion can be understood as an attractive (low-frequency)and a repulsive (high-frequency) mode of the effective magnetic moments. To get an idea of the stray field that is responsible for the magnetostatic coupling, micromagnetic simulations have been performed using the MicroMagnum code. 33Typical parameters for permalloy, i.e., a saturation magnetization MS=8×105Am−1, an exchange constant A=13×10−12Jm−1, a Gilbert damping parameter α=0.01, and an anisotropy constant K=0 are assumed. In the simulations, a square with an edge length of l=1μm and a thickness of 50 nm is excited via an alternating magnetic field with an amplitude of 0.3 mT. Figure 1(a) shows the amplitude of the stray field/vectorHintemerging along the center line on the xaxis at a normalized distance 2 d/lbetween 1.0 and 4.6 from the center of the structure for three characteristic core positions. Note that 2 d/l=1.0 Downloaded 09 Jan 2013 to 152.14.136.96. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/042180-3 Vogel et al. AIP Advances 2, 042180 (2012) FIG. 2. (a) Scanning-electron micrograph of a pair of permalloy squares. Striplines are deposited on top of both squares I and II. (b) X-ray images of the center part (400 nm ×400 nm) of a square at different time steps during one period of gyration. (c) V ortex-core trajectory in square I during harmonic excitation with 160 MHz. The black and the gray symbols mark thedeflection in xandydirection determined from the x-ray images, respectively. corresponds to the edge of the square. One immediately recognizes that the field strength significantly drops as a function of the distance, as already mentioned in the introduction. For a core deflection in ydirection and positive xdirection, |/vectorHint|is considerably larger than for a core deflection in negative xdirection which can be attributed to the stray fields of the domain walls in the Landau pattern. In the case of a core deflection in the latter direction, the effective magnetic moment of the square points in positive ydirection and one would consequently expect a stray field pointing in negative y direction. But for a distance 2 d/l<1.66 from the center of the square, the stray field of the domain walls having a component in positive ydirection exceeds the stray field due to the effective magnetic moment of the structure, compare arrows indicating the direction of the stray field in Fig. 1(b).T h e stray fields of the domain walls also lead to a non-vanishing stray field in the zero field equilibrium state, compare Fig. 3(a) in Ref. 28. Figure 1(c) reveals that the time-dependent strength of the stray field|/vectorHint|at a normalized distance of 2 d/l=2.2 is comparable to a rotational field with constant amplitude for about half a cycle of vortex gyration before the field amplitude significantly drops by 75 %. The stray field integrated over one cycle is dominated by the core deflection in ydirection which is responsible for the stray field component along the xaxis (bonding axis). One has to keep in mind that for a counterclockwise vortex gyration, the stray field rotates clockwise and vice versa. It has been shown in the literature that an in-plane rotating field only efficiently excites the gyroscopic vortex-core motion when the sense of rotation of the field coincides with the intrinsic sense of gyration given by the core polarization.34,35Since this is not the case for two neighboring elements having the same polarization, the time-dependent strength of the stray field helps to understandthe strong polarization selectivity of the stray-field coupling between ferromagnetic squares in the vortex state reported in Refs. 20and28. In the following experiments, we concentrate on the case of opposite core polarizations providing efficient coupling. To investigate the phase relation between coupled vortices via time-resolved STXM, pairs of squares with an edge length of l=2μm and a center-to-center distance of d=2.25μm( 2d/l=2.25) are prepared by electron-beam lithography and lift-off processing. Polycrystalline permalloy(Ni 80Fe20) with a thickness of 60 nm is thermally evaporated onto 150 nm thin silicon-nitride membranes. A 800 nm wide stripline of 120 nm copper and a gold cap of 5 nm is deposited on top of each square, see Fig. 2(a). Measurements are performed at the MAXYMUS beamline,20,36 BESSY II, Helmholtz-Zentrum Berlin. Magnetic contrast is provided via the x-ray magnetic cir- cular dichroism effect37at the Ni L 3-absorption edge (852.6 eV). The sample is mounted onto an interferometer-controlled high precision piezo stage perpendicular to the beam axis to allow directimaging of the out-of-plane magnetization component by measuring the x-ray intensity transmitted through the sample. The vortex core in element I is harmonically excited via the magnetic field generated by a sinusoidal current driven through the corresponding stripline. 38Figure 2(b) shows Downloaded 09 Jan 2013 to 152.14.136.96. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/042180-4 Vogel et al. AIP Advances 2, 042180 (2012) FIG. 3. Vector diagrams indicating the phase relation between the vortex cores in a harmonically excited pair of permalloy squares for different frequencies of the excitation signal. Only the core in the first element (I) is directly excited via analternating rf field of a strip line. The time-dependent trajectories (red symbols for element I and blue symbols for elementII) are determined via scanning-transmission x-ray microscopy (STXM). The two elements have same chiralities (see STXMimage in the top row) and opposite core polarizations. A red vector points to the reference position of the core in element Iand a blue vector points to the position of the core in element II at the same time step. exemplary x-ray images of the center part of square I at four different time steps during one period of gyration at an excitation frequency of 160 MHz. The vortex core with the polarization p=+ 1 appearing as a black spot gyrates counterclockwise following a right-hand grip rule. In the case of opposite core polarizations, vortex gyrations can be observed in both elements (see below). In ourprevious works, 28,29it has been shown that the vortex gyration in element II can be attributed to the stray-field mediated interaction and a direct excitation via the Oersted field of the stripline on top of element I can be ruled out. To determine the frequency-dependent phase relation between thevortices, the core trajectories, as shown in Fig. 2(c) for square I, are extracted from the x-ray data for different excitation frequencies between 150 MHz and 240 MHz in steps of 10 MHz. The excitation power at the function generator is kept constant. Downloaded 09 Jan 2013 to 152.14.136.96. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/042180-5 Vogel et al. AIP Advances 2, 042180 (2012) FIG. 4. Vector diagrams corresponding to Fig. 3but for elements that have opposite chiralities (see STXM image in the top row). No trajectories are shown for f=200 MHz since the time resolution of the setup is highly limited at this frequency. The definition of a phase in the case of opposite core polarizations is not straightforward since the vortices gyrate in opposite directions. According to the micromagnetic simulations, the stray field ata core deflection in ydirection dominates the magnetostatic interaction between two elements placed along the xaxis and we thus concentrate on the relative core positions along the yaxis. Figure 3 shows vector diagrams indicating the relative phase between the vortex cores in a harmonicallyexcited pair with opposite polarizations and same chiralities for different excitation frequencies. The chirality configuration is evidenced by the STXM image in the top row that reveals a bright lower part and a dark upper part of the Landau pattern in both squares. The position of the core in elementII is determined at the moment of the maximum displacement in positive ydirection of the core in element I. The x-ray data are available as movie S1 in the supplementary material. 39An in-phase motion in ydirection is observed at a lower frequency - the vectors indicating the core positions in element I and II are pointing in the same direction at 160 MHz. An antiphase motion occurs at a higher frequency (210 MHz) where the vectors are pointing in the opposite direction. Note that the amplitude of the current generating the excitation field of element I may be affected by the Downloaded 09 Jan 2013 to 152.14.136.96. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/042180-6 Vogel et al. AIP Advances 2, 042180 (2012) FIG. 5. Schematic illustration of the relative orientations of the effective magnetic moments during coupled vortex gyrations in a pair of ferromagnetic squares with opposite core polarizations at the low-frequency and at the high-frequency mode for thecase of (a) same chiralities and (b) opposite chiralities. Black arrows mark the magnetization directions in the correspondingdomains of the Landau pattern. Red and blue arrows show the effective magnetic moment of element I and element II,respectively. The dashed green arrows indicate the sense of gyration in the particular element. For each mode, the relativeconfigurations at four different time steps during one period of gyration are shown. frequency-dependent transmission properties of the printed-circuit board used to mount the sample into the x-ray microscope. However, the resulting variation of the gyration amplitude in element I has no impact on the relative phase between the vortex oscillators. Furthermore, sample impurities or structural defects may lead to the appearance of steps in the frequency-dependent amplitude of vortexgyration instead of a continuous variation. 40Forf=200 MHz the time resolution is highly limited at the chosen parameters of the electronic measurement setup which impedes a clear determination of the phase relation, see Fig. 3(f). At the first glance, the phase relation is inverted in Fig. 4which shows the case of opposite chiralities. The x-ray data for this case are available as movie S2 in the supplementary material.39An antiphase motion of the cores is observed at a lower frequency (170 MHz) and an in-phase motionoccurs at a higher frequency (220 MHz). However, one has to take into account that a core deflection in the opposite direction together with an opposite chirality in turn leads to the same effective magnetic moment of the microstructure - the magnetizations in the enlarged domains point in the same direction. Figure 5illustrates the relative orientations of the effective magnetic moments during one cycle of coupled vortex gyrations for the case of same chiralities and opposites chiralities. Thefrequency-dependent phase relation between the two vortex oscillators refers to the phase between the effective magnetic moments along the bonding axis (in xdirection) and consequently to the relative direction of the stray fields. An attractive (low-frequency) mode with the magnetic momentspointing in the same direction and a repulsive (high-frequency) mode with the magnetic moments pointing in opposite directions exist. The corresponding experiments with pairs of disks yield the same results (not shown). As stated in Ref. 29, the low-frequency mode corresponds to an in-phase oscillation of C IYIandCIIYIIand the high-frequency mode corresponds to an antiphase oscillation ofCIYIandCIIYII. In summary, we have presented micromagnetic simulations of the external stray field of a micron- sized ferromagnetic square during vortex gyration that give insight in the coupling mechanism and help to understand the distance dependence and polarization selectivity discussed in the literature. Time-resolved STXM enables to directly image the phase relation in a pair of coupled vortices whereone of the elements is harmonically excited. The normal modes of coupled vortex-core motion are selectively excited and identified as a low frequency mode for an in-phase motion and a high frequency mode for an antiphase motion of the effective magnetic moments along the bonding axis. Downloaded 09 Jan 2013 to 152.14.136.96. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/042180-7 Vogel et al. 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B 42, 7262 (1990). 38The Oersted field distribution of a comparable stripline has been calculated in the supplementary material of Ref. 29.T h e exciting rf field can be assumed almost uniform under the copper stripline. 39See supplementary material at http://dx.doi.org/10.1063/1.4771683 for movies S1 and S2. 40T. Kamionka, M. Martens, K. W. Chou, A. Drews, T. Tyliszczak, H. Stoll, B. Van Waeyenberge, and G. Meier, Phys. Rev. B83, 224422 (2011). Downloaded 09 Jan 2013 to 152.14.136.96. All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license. See: http://creativecommons.org/licenses/by/3.0/

1.4826563.pdf

Performance analysis of MgO-based perpendicularly magnetized tunnel junctions T. Devolder, , K. Garcia , G. Agnus , M. Manfrini , S. Cornelissen , and T. Min Citation: Appl. Phys. Lett. 103, 182402 (2013); doi: 10.1063/1.4826563 View online: http://dx.doi.org/10.1063/1.4826563 View Table of Contents: http://aip.scitation.org/toc/apl/103/18 Published by the American Institute of Physics Articles you may be interested in Considerations for further scaling of metal–insulator–metal DRAM capacitors Appl. Phys. Lett. 31, 01A10501A105 (2012); 10.1116/1.4767125 Roughness evolution during the atomic layer deposition of metal oxides Appl. Phys. Lett. 31, 061501061501 (2013); 10.1116/1.4812707 Graphene as anode electrode for colloidal quantum dots based light emitting diodes Appl. Phys. Lett. 103, 043124043124 (2013); 10.1063/1.4816745 Performance analysis of MgO-based perpendicularly magnetized tunnel junctions T. Devolder,1,2,a)K. Garcia,1,2G. Agnus,1,2M. Manfrini,3S. Cornelissen,3,4and T. Min3 1Institut d’Electronique Fondamentale, CNRS, UMR 8622, Orsay, France 2University Paris-Sud, 91405 Orsay, France 3IMEC, Kapeldreef 75, B-3001 Leuven, Belgium 4Dep. Electrotechniek (ESAT), KULeuven, Kasteelpark Arenberg 10, B-3001 Leuven, Belgium (Received 27 August 2013; accepted 9 October 2013; published online 29 October 2013) We studied state of the art perpendicularly magnetized tunnel junctions to identify performance improvement opportunities. The free layer has both a low damping and a large anisotropy.Conversely, the perpendicular remanence of the reference layer requires its encapsulation and its coupling with the hard layer. The weak pinning and low damping of the reference layer may make it prone to fluctuations induced by spin-torque. The combined optimization of the interface anisotropieson both sides of the MgO, together with the reproducibility of the interlayer exchange coupling are the main material challenges for our type of magnetic tunnel junctions. VC2013 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4826563 ] One of the most emblematic applications of magnetic tunnel junctions (MTJs) is the spin-torque operated magnetic random access memory (STT-MRAM).1This technology is presently undergoing a shift of interest from in-plane magne-tized 2to out-of-plane magnetized3systems because storage layers with perpendicular magnetic anisotropy (PMA) bene- fit from a larger thermal stability. In this perpendicular tech-nology, the requirements for the free layer (FL) include a high anisotropy and a low damping. In addition, reference layers (RLs) have to be insensitive to magnetic fields andspin-torques and are usually constructed in a synthetic ferri- magnet configuration to avoid the generation of stray fields that would destabilize the free layer. State-of-the-art MTJs 4rely on composite reference sys- tems with many layers, and each of the layers requires care- ful optimization. Unfortunately, the magnetic properties ofthe reference system cannot easily be determined. Indeed conventional magnetometry methods rely on magneto- transport effects, which provide pieces of information inwhich the free and reference layers properties are inter- twined, and can only be measured at the device level with potential disturbance from spin torque. Moreover, the pat-terning process may degrade 5the properties of such perpen- dicular MTJs (pMTJ), rendering6the comparison with unpatterned MTJs difficult. As a matter of fact, studies of theRL systems have remained rather scarce. In this paper, we study in detail the static and dynamic properties of the free and reference layers of a state of the artpMTJ for STT-MRAM applications. We extract the parame- ters that are needed for their modeling, e.g., the effective ani- sotropies, the interlayer coupling terms, and the Gilbertdamping parameters (Table I). From these properties and their dispersion relations, we identify optimization routes for the next generation MTJs. All layers in the pMTJ have been prepared by Canon ANELVA using a physical vapor deposition clusterC-7100EX and have been subsequently annealed at 300 /C14Cf o r half an hour in a perpendicular field of 1 T. The pMTJ stack is deposited on oxidized silicon as follows: seed/FL/MgO/ RL/Ru/HL/Cap, with the compositions and thicknesses asdepicted in Fig. 1(a)(numbers in parentheses indicate nominal thicknesses in nm). The studied pMTJ is essentially similar to that used in Ref. 7. The free layer is CoFeB and is perpendicu- larly magnetized due to interface-anisotropy induced by the MgO layer. The reference layer is a composite system, com- bining a spin polarizing section Fe/CoFeB/Ta, covered by a 5-repeat Co/Pd multilayer to strengthen its anisotropy (Fig. 1). The RL is coupled by interlayer exchange coupling to a hard layer (HL) composed of 14 repeats of Co/Pd. The transportmeasurements performed in current in-plane tunneling (CIPT) 8configuration indicate a tunnel magneto resistance of 64% and a Resistance-Area product of 8 :7X/C1lm2.I no r d e r to determine unambiguously the magnetic properties of each layer, we have fabricated 2 additional types of samples by removing (i) the hard layers only, or (ii) both the hard and thereference layers, in order to suppress, respectively, the inter- layer exchange coupling contributions, and the possible cou- pling through the MgO layer. The top layers were removedusing low energy Ar þions ( <500 eV) while rotating the sam- ple at 45/C14angle incidence. The etch stop criteria were based, respectively, on (i) the fine calibration of the etch rate of theCo/Pd hard layer and (ii) the detection of the sputtered Mg atoms by Secondary Ion Mass Spectrometry. The static magnetic properties were determined by Alternative Gradient Force Magnetometry (AGFM) (Fig. 1(b)), complemented with Polar Magneto-Optical Kerr Effects (PMOKE) to obtain a higher sensitivity. Because of TABLE I. Material properties of the reference and free layers. Layer Reference Free l0ðHk/C0MSÞ(T) /C00.0860.05 1.0 60.1 l0HJ(mT) /C0125657 61 a 0.0160.0005 0.012 60.003 a)thibaut.devolder@u-psud.fr 0003-6951/2013/103(18)/182402/4/$30.00 VC2013 AIP Publishing LLC 103, 182402-1APPLIED PHYSICS LETTERS 103, 182402 (2013) the optical absorption of the layers, the magneto-optical sig- nals of the bottommost layers were attenuated in PMOKE measurements, such that all of the observed features in the hysteresis loop could be unambiguously attributed to theirlayer of origin. The comparison of the AGFM loops (Fig. 1(b)) to the PMOKE loops (Fig. 3(a)) allows for instance to assign the highest coercivity to the HL at the top of our MTJ. The quasi-static hysteresis loops (Fig. 1(b)) indicate the switching of the layers is sequential: there is no signature of any spin-flop behavior of the synthetic antiferromagnet,which indicates qualitatively 9that the anisotropy field of ei- ther the RL or the HL system is substantially greater than the interlayer exchange coupling field. The low coercivity layeris the FL, with l 0Hc FL¼25 mT. Minor loops on the full pMTJ compared to AGFM loops of single FL indicated that the FL is slightly coupled to the RL layer by 7 mT, favoringparallel alignment. The layer switching at intermediate nega- tive field l 0Hswitch RL ¼/C0106 mT is the RL. This negative switching field is the signature of the antiferromagnetic cou-pling with the hard layer, owing to the Ru spacer layer. Minor loops (Fig. 3(a)) indicate that the reference layer switches around an offset field of /C0130 mT, with a coercivity of 22 mT. This coercivity remains constant when the HL is removed by etching, but the offset field vanishes (Fig. 1(c)), indicating that the offset was due to an interlayer exchangecoupling with the HL through Ru. Note that the saturation of the RL system is gradual. The full remanence of the minor loops suggests that this slow saturation arises from some lat-eral dispersion of the switching field. In addition, we do not observe distinct switching of the FL and the RF in Fig. 2(c), as if they were switching at the same field. Indeed after theremoval of the RL by etching, the FL coercivity is 25 mT. This indicates that the RL and FL can switch together because they have similar intrinsic coercivity in combinationwith a parallel coupling of 7 mT, as mentioned above. The hard layer finally switches at l 0Hswitch HL ¼237 mT. Unfortunately, these switching fields cannot be used in thegeneral case to deduce quantitatively the anisotropy and interlayer exchange energies. We shall thus rely on measure- ments of the dynamical properties of our pMTJs.The dynamical magnetization properties were deter- mined by Vector Network Analyzer FerroMagnetic Resonance (VNA-FMR 10) in the open-circuit total reflection configuration.11To analyze the characteristic features in the susceptibility spectra, we consider films with uniaxial anisot- ropy along z. The out-of-phase susceptibility peaks at the FMR frequency xFMR=ð2pÞ. For fields saturating the mag- netization along 6z, we have FIG. 1. (a) pMTJ stack composition with thicknesses in nm. Hysteresis loops measured by AGFM. The top loop (b) was measured on the fullstack, while the bottom loop (c) was measured on the sole FL and RL- systems, once the HL is etched away with an etch stopping in the Ru spacer layer. FIG. 2. (a) Half Width at Half Maximum (i.e., linewidth) versus resonance frequency of the most intense resonance mode, assigned to the RL layer. The bold line is linear fit with slope a¼0:01. (b) Field dependence of the resonance frequencies. The highest frequency mode corresponds to the free layer resonance, with a high field slope of cFL 0¼222 kHz /C1m/C1A/C01. For this mode, the crosses are recorded on the full pMTJ, while the squares were recorded after removing the hard layer. The lowest frequency corresponds to the reference layer, with a high field slope of cFL 0¼226 kHz /C1m/C1A/C01. Inset: typical curves of the imaginary part of the transverse susceptibility of the sample for field conditions leading to resonances at 50 GHz. A smoothingwith a convolution of 400 MHz width was applied. The free layer signal has been multiplied by 10.182402-2 Devolder et al. Appl. Phys. Lett. 103, 182402 (2013) xFMR¼c0ð6Hz/C0HkþMS6HJÞ; (1) where c0is the gyromagnetic ratio, Hzthe applied field, and Hk,MS, and HJare the uniaxial anisotropy field, the magnet- ization, and the interlayer exchange field acting on the stud- ied layer, respectively. The 6symbols are positive when the magnetization of the studied layer is parallel to the consid- ered effective field term. We will see below (Fig. 2(b)) that the frequencies of the different modes are sufficiently dis-tinct that mode hybridization 12does not occur and Eq. (1)is valid. In the full pMTJ, two ferromagnetic resonance modes are measured. These two modes are not modes of the hard layers since they are still present once the hard layer has been etched away. The weakest mode exhibits the highestfrequency, which is 28–31 GHz at zero field, depending on sample. The frequency versus field dependence of this mode is essentially a symmetric V shape, with a tiny frequencyreduction of at most 2.4 GHz at low fields, i.e., in the [/C0100 mT, 100 mT] interval. The explanation for this fre- quency decrease exceeds the scope of our study, but it recallsthat the macrospin description of the FL fails near its switch- ing field. The likely origin of the frequency dip near zero field is the existence of non uniform eigen-fluctuations thatare coupled by dipolar fields near the switching field (see the so-called nucleation instability in Ref. 13). The lowest frequency mode is roughly 10 times more intense (see Fig. 2(b), inset). Its frequency is also essentially V-shaped (Fig. 2(b)), but with a marked hysteresis in the low field region. The hysteresis almost disappears once the HL isremoved. This mode disappears when the RL is removed: it is thus a reference layer mode. Before analyzing the hyste- retic field zone, let us comment on the damping of the RLand FL layers. To extract it, we only use the high field data, i.e., jH zj>Hswitch HL, where the magnetization is unambigu- ously uniform, and the FMR linewidth is only influenced14by Gilbert damping aand long-range inhomogeneities of the internal field (so-called inhomogeneous broadening). The full width at half maximum of the resonance peak is15 Df¼2afFMRsuch that a linear fit (Fig. 2(a))o fDfversus 2fFMRyields aRL¼0:0160:0005 and aFL¼0:01260:003. Let us emphasize that the unexpected low damping of the RL might be detrimental to the MTJ performance, because itprovides to the RL an undesired substantial susceptibility to spin torque. Let us discuss the FMR frequency of the RL in the hys- teretic field interval (Fig. 3). The RL mode exhibits specific features that separate 3 field intervals. The frequency versusfield slope changes sign when the RL switches. A shallow frequency minimum happens at /C0150630 mT, where H zþHk/C0MS/C25HJ, as described below. In positive field, the frequency undergoes a sudden decrease dx=ð2pÞwhen the hard layer switches at Hsw HL. From sample to sample, there is some scattering of Hsw HLand dxas illustrated for two cases in Fig. 3(b). However, there is a linear correlation (not shown) between Hsw HLanddx. This correlation is indicative that the scattering comes from a dis-persion of the interlayer exchange coupling term H J, rather than the anisotropy of the hard multilayer that should in prin- ciple affect Hsw HLbutnotthe resonance frequency of the refer- ence layer. Note that despite this sample-to-sample variation of the low field behavior, the frequency versus field slopes at highfields were very reproducible, allowing to extrapolate these curves to zero field. The zero field frequency is f 1¼/C03.960.1 GHz when extrapolated from high field data of the full stack. It reduces to only f0 1¼0:260:1G H z f o r the sample without the hard layer and its interlayer exchange field companion. When the zero field frequency isnot extrapolated but measured on the full stack (Fig. 3(b)), it c a nb eu pt o f 2¼þ3.660.1 GHz in the strong coupling case of Fig. 3(b) (i.e., square symbols). From these numbers and Eq. (1), we can deduce that the exchange field l0ðf2/C0f1Þ=ð4pc0Þacting on the RL is /C012565m T i n t h e strong coupling case, in agreement with the RL loop offsetof/C0130 mT previously observed. We also deduce from f 1þf2and Eq. (1)that the RL layer effective anisotropy field is l0ðHk/C0MSÞ¼/C0 80650 mT. We are aware that this number is not consistent with f0 1, and we shall discuss this point later. The sign of the effective anisotropy field Hk–MSof the RL in the full stack is important because it indicates that in the full pMTJ, the reference layer system presents full per- pendicular remanence only due to interlayer exchange cou-pling with the hard system. This indicates that the interface anisotropy K FL sof the MgO/Fe interface of the reference layer is way below KRL sthat of the CoFeB/MgO interface of the free layer. We remind that the crystallization quality is16 the key factor for Tunnel Magneto-Resistance, while it isthe abruptness of the interface with the correct degree of ox-idation that determines 17the interfacial anisotropy. These independent structural evolutions occur at different anneal- ing temperatures and timescales in these two interfaces, ren-dering difficult their joint optimization in the same device a c c o r d i n gt ot h e3c r i t e r i a K FL s;KRL s,a n dT M R .T h i sf a c t argues in favor of simpler systems with two similarFIG. 3. (a) Hysteresis loop measured by Polar Magneto-Optical Kerr Effect. The minor loops of the free layer and of the reference layers are superim- posed. The arrow denotes the field sweeping direction in the next panel. (b) Frequency of the reference layer resonance for two typical samples, includ- ing that measured in panel (a) (square symbols) for the increasing field partof the hysteresis loop. In each branch the stiffness fields acting on the refer- ence layer are written. The arrows sketch the magnetization of the hard layer (gray arrow), the reference layer (blue arrow), and the applied field (orange arrow).182402-3 Devolder et al. Appl. Phys. Lett. 103, 182402 (2013) interfaces, for instance CoFeB/MgO/CoFeB. There the combined optimization of interface anisotropies and TMR is probably easier because there is only one type of KSto optimize. Finally, when the HL is removed, the magnetization of the reference layer should fall in the plane if its properties were unchanged by the etching process. This does not arisefrom the weak (7 mT) coupling between the FL and the RL. This unexpected fact might be related to higher order anisot- ropy terms that cannot be neglected when H k/C0MS/C250. Another possible contribution to this effect is the occurrence of some strain relaxation after the removal of the hard layer.This may affect the magneto-elastic (ME) contributions to the anisotropy of the reference layer. Let us estimate the order of magnitude of the maximum magneto-elastic contri-butions to the total anisotropy. It is reasonable to assume that the layers relax during the annealing step. When cooling the system to room temperature, the silicon substrate shouldshrink by 0.073%, transferring its deformation to the MTJ. The thermal expansion coefficient of most metals being 3–5 times larger than that of silicon, the metallic layers undergoa (tensile) strain of typical value /C15¼0:3% accompanied by the corresponding ME anisotropy field of 3 kY/C15=ðl 0MSÞ. Taking Y/C25200 GPa as a plausible Young’s modulus for the RL, the change in ME anisotropy is bounded by þ108 mT that would be the case of pure cobalt (magnetostriction coef- ficient k¼/C060 ppm). The real ME anisotropy is somewhere below because of the composite and polycrystalline charac- ters of the RL. However even if the exact determination of the ME contributions to the RL anisotropy exceeds the scopeof our study, its order of magnitude indicate that it may con- tribute to the difference between the capped and stressed RL and the uncapped and relaxed RL. In conclusion, we have measured the properties of pMTJ at wafer level. Their properties (Table I) indicate that the free layer has both low damping and a very large anisot-ropy, which holds promise for STT application. In addition, we evidenced that the interlayer exchange coupling of the reference layer by the hard layer is necessary to obtain afully perpendicular remanence of the reference layer, because the interface anisotropy of the Fe/MgO interface is difficult to optimize without compromising other parameters.We anticipate that the relatively low pinning of the reference layer added to its low damping will make it prone to fluctua- tions that might need to be taken into account for STT appli-cations. Strengthening the damping and the anisotropy of the reference layer or increasing the number of Co/Pd repeats might be useful to stabilize the reference system providedthis can be done without sacrificing the transport properties. Finally, the interlayer exchange coupling between the refer- ence and the hard layer seems to be subject to wafer-to-wafer dispersion. This is consistent with the high sensitivity of theinterlayer exchange coupling to the Ru thickness and to the state of the interfaces. A possible route to solve this disper- sion might be to implement a more efficient texture breakinglayer maintaining a high interlayer exchange coupling within the Reference layer, and to use for instance FeTa alloys 18for this purpose. This work was supported by the PPF SPINEL program of the “Universit /C19e Paris-Sud.” S.C. acknowledges financial support from F.W.O. Flanders. 1T. Min, Q. Chen, R. Beach, G. Jan, C. Horng, W. Kula, T. Torng, R. Tong, T. Zhong, D. Tang, P. Wang, M.-M. Chen, J. Sun, J. Debrosse, D. Worledge, T. Maffitt, and W. Gallagher, IEEE Trans. Magn. 46, 2322 (2010). 2A. V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii, R. S. Beach, A. Ong, X. Tang, A. Driskill-Smith, W. H. Butler, P. B. Visscher, D. Lottis, E. Chen, V. Nikitin, and M. Krounbi, J. Phys. D: Appl. Phys. 46, 074001 (2013). 3S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S.Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nature Mater. 9, 721 (2010). 4J. Z. Sun, R. P. Robertazzi, J. 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1.4953600.pdf

Observation of anisotropic energy transfer in magnetically coupled magnetic vortex pair N. Hasegawa , S. Sugimoto , D. Kumar , S. Barman , A. Barman , K. Kondou , and Y. Otani, Citation: Appl. Phys. Lett. 108, 242402 (2016); doi: 10.1063/1.4953600 View online: http://dx.doi.org/10.1063/1.4953600 View Table of Contents: http://aip.scitation.org/toc/apl/108/24 Published by the American Institute of Physics Articles you may be interested in Important role of magnetization precession angle measurement in inverse spin Hall effect induced by spin pumping Applied Physics Letters 110, 022404 (2017); 10.1063/1.4973704 Spin-current-driven thermoelectric generation based on interfacial spin-orbit coupling Applied Physics Letters 108, 242409 (2016); 10.1063/1.4953879 Tunable eigenmodes of coupled magnetic vortex oscillators Applied Physics Letters 104, 182405 (2014); 10.1063/1.4875618 Stochastic dynamics of strongly-bound magnetic vortex pairs AIP Advances 7, 056007 (2017); 10.1063/1.4974066 Bias field tunable magnetic configuration and magnetization dynamics in Ni 80Fe20 nano-cross structures with varying arm length Journal of Applied Physics 121, 043909 (2017); 10.1063/1.4974886 Voltage induced mechanical/spin wave propagation over long distances Applied Physics Letters 110, 072401 (2017); 10.1063/1.4975828Observation of anisotropic energy transfer in magnetically coupled magnetic vortex pair N.Hasegawa,1S.Sugimoto,1D.Kumar,2S.Barman,2A.Barman,2K.Kondou,3 and Y . Otani1,3,a) 1Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwa-no-ha, Kashiwa, Chiba 277-8581, Japan 2Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India 3CEMS, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan (Received 28 March 2016; accepted 27 May 2016; published online 13 June 2016) We have experimentally investigated the energy transfer and storage in the magnetostatically coupled vortices in a pair of disks. By measuring the frequency dependence of the rectified dc voltage, we observed a specific gyrating motion due to anomalous energy storage at the off-resonant frequencyfor anti-parallel polarities. Micromagnetic simulations based on the Landau-Lifshitz-Gilbert equation qualitatively reproduce the experimental results and reveal that the behavior arises from the aniso- tropic energy transfer, i.e., the modulation of effective damping constant of the pair disks, originatingfrom the phase difference between coupled vortex cores. These findings can be of use in magnetic vortex based logic operations. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4953600 ] The magnetic vortex has been drawing much attention as one of the fundamental structures confined in a nanometerscaled ferromagnetic disk. 1,2It is characterized by two degrees of freedom, the in-plane curling magnetization direc- tion, “chirality” ðc¼61), and the out of core magnetization, “polarity” ðp¼61). The core of a magnetic vortex is known to behave as a quasiparticle in a harmonic potential.3In the low energy excitation (or gyration) mode, the core continu-ously gyrates along an equipotential line. The resonant fre-quency for a single vortex can be determined by the potential shape which depends on the aspect ratio (thickness/ radius) of the magnetic disk 4and not on polarity or chirality. When two magnetic vortices are placed sufficiently close to each other, the dynamic dipolar interaction serves as a bind- ing force. In the two-coupled vortices system, the resonant fre-quency splits into two branches and the split-width depends on the strength of the dipolar interaction. 5,6By arranging mag- netic vortices in a two-dimensional array, it forms a so-calledmagnonic crystal, where the band structure can be tuned by polarity. 7Thus, its band structure has been intensively studied for several cases such as a one-dimensional chain,8–10at w o - dimensional array7,11a n dac a r b o nfl a k e .12In these systems, properties of the energy transfer and storage in each disk are also important for actual applications.13,14The interesting properties of the energy transfer and storage within a group ofmagnetic vortices have recently been examined in a theoreti- cal study; 15however, no experimental observation has been reported. Here, we observed the specific gyrating motion dueto anomalous energy storage, for a coupled vortices system by using the electrical detection method. Samples were fabricated on a silicon substrate by means of electron beam lithography on polymethyl-methacrylate resist and a subsequent lift-off process after electron beam deposition. We fabricated two 30 nm thick disks 500 nm inradius of Permalloy (Py: Ni 80Fe20) aligned along the xaxiswith a 100 nm gap between their nearest edges. By attaching Cu electrodes in each Py disk, we can apply different ac cur-rents I acLandIacR, where the subscripts “L” and “R” repre- sent the left and right Py disks, respectively. The Cu electrodes were patterned at the same distance from the diskcenter and deposited by thermal evaporation. Before the Cu deposition, a careful Ar ion beam etching (600 V beam volt- age) was carried out for 30 s in order to clean the Py surface.In Fig. 1(a), we show a scanning electron microscope (SEM) image of the two Py disks as well as a schematic image of the measurement circuit. Here, anti-parallel polarities were pre-pared by applying relatively large ac current 16and confirmed FIG. 1. (a) Schematic diagram of the measurement circuit and an SEM image of the sample. (b) Normalized dc voltage spectrum measured at the left disk (black dots) and the right disk (red dots), independently.a)Electronic mail: yotani@issp.u-tokyo.ac.jp 0003-6951/2016/108(24)/242402/4/$30.00 Published by AIP Publishing. 108, 242402-1APPLIED PHYSICS LETTERS 108, 242402 (2016) it by measuring the splitting intensity of resonant frequencies as reported in Ref. 6. The gyrating motion of coupled vortices can be excited by an ac current and detected as a dc voltage because the ac current is rectified via the sample resistance oscillation due to the anisotropic magnetoresistance (AMR) effect.6,17–20 The resistance oscillation associates with the core gyration amplitude and the phase of the gyrating motion as follows.When the core position r¼(x,y) is shifted from the disk center r¼(x c,yc), the resistance of the Py disk can be expressed as19,20 R¼R0þaðx/C0xcÞ2/C0aðy/C0ycÞ2; (1) where ais proportional to the sample resistance change due to AMR effect. When the ac current I¼Iaceixtis applied to the disk, the core gyrates in the steady orbit. The time evolution of the core position is expressed as: x;yðÞ ¼/C0 x0þXeixt/C0pp 2ðÞ; y0þYeixt/C1 ,w i t h ðX;YÞ¼ðX0þiX00;Y0þiY00Þ. Here, pis the po- larity, ðx0;y0Þis the core position before the gyrating motion starts and X0(X00)a n d Y0(Y00)a r et h e xandycomponent of the real (imaginary) part of the gyration amplitude. By substituting the above relation into Eq. (1), one can obtain the oscillating re- sistance. Therefore, the applied ac current is rectified by the oscillating resistance and a dc voltage Vdcnormalized by Iacis given by19,20 Vdc=Iac¼/C0paðx0/C0xcÞX00þaðy0/C0ycÞY0: (2) From the analysis based on the Thiele’s equation, it is known that X00shows the anti-Lorentzian, while Y0shows Lorentzian19,21,22spectrum shape. Eq. (2)implies that the nor- malized dc voltage Vdc/Iacis proportional to the shift in the core position from the disk center, ( x0-xc) and ( y0-yc). Figure 1(b) shows the normalized dc voltage spectra measured with Iac¼2:3 mA for the left disk (black dots) and the right disk (red dots). The normalized dc voltage spectra show almost the same peak structures, which ensure that the disks are not so different. We can also find that the spectra show Lorentzian- like shape, which implies that the value of x0L(x0R) is equal to xcL(xcR). In that case, the ratio of gyration amplitude in each vortex core can be approximated by using only the ycompo- nent, i.e., Y0 R=Y0 L¼ðVdcR=IacRÞ=ðVdcL=IacLÞ. In order to estimate the gyration amplitude in each disk, we injected two ac currents with different amplitudes and same frequency f, i.e., IL¼IacLsinð2pfÞfor the left disk and IR¼IacRsinð2pfþDIacÞfor the right disk, where DIac is the phase difference between the two ac currents. The DIac can be monitored by using an oscilloscope. We applied a large current to the left disk ðIacL¼2:3m A Þto excite and detect the gyrating motion and another current to the right disk ðIacR¼0:23 mA Þwhich is small enough not to affect the induced gyrating motion but large enough for detection. Figure 2(a) shows the normalized dc voltages as a func- tion of the phase difference DIacat 230 MHz, which is the middle frequency of the double resonant peaks at 215 and 245 MHz. At the left disk (black dots), the normalized dc voltage shows almost a constant value of 0.30 m X,, which is proportional to gyration amplitude in left disk. This implies that the collective dynamics is not perturbed by IacR. At theright disk (red dots), the DIacdependence shows the sinusoi- dal curve originating from the phase difference between the core oscillation and the IacR. When the core and IacRoscillate in-phase, the Vdc/Iactakes the maximum value of 0.69 m X, which is proportional to gyration amplitude in the right disk. Surprisingly, at this frequency, the gyration amplitude in right disk is 2.3 times larger than that of left disk, i.e., Y0R=Y0L¼2:3. From this result, we can estimate the ratio of the stored energy cin each disk by using the for- mula ;cL¼1 1þðY0R=Y0LÞ2;cR¼ðY0R=Y0LÞ2 1þðY0R=Y0LÞ2. In this frequency, the value of cin right (left) disk can be estimated to be 0.85 (0.15) of stored energy in coupled vortices system. Figure 2(b) shows the obtained values of cas a function of the input frequency. For both resonant frequencies of 215 and 245 MHz, the values are about 0.5. It means that for both cores the gyration amplitude is almost same, and the same amount of energy was stored in both disks. Interestingly on the other hand, in the off-resonant frequency range from 215 to 245 MHz, the cin the right disk is larger than that of the left disk. The maximum difference in c between left and right disks appears at around the middle frequency between two resonant peaks. This anisotropic energy storage results in the amplification effect is discussed in Ref. 15. In order to understand the physical mechanism of the frequency dependence of c, we performed a micromagnetic simulation23by solving numerically the Landau-Lifshitz- Gilbert equation with spin transfer torque terms. The materialparameters for Py in our simulations are: saturation magnet- ization M S¼0.93 T, the exchange stiffness constant A¼1.05 /C21011J/m, the spin polarization P¼0.4, and the damping coefficient a¼0.01. For the simulation, the disk is divided FIG. 2. (a) Normalized dc voltage as a function of the DIacat 230 MHz for the left (black dots) and right disk (red dots). Solid lines show the fitting with constant value (black line) and sinusoidal curve (red line). (b) Frequency dependence of the cfor both disks.242402-2 Hasegawa et al. Appl. Phys. Lett. 108, 242402 (2016)into rectangular prism like cells of 5 /C25/C230 nm3and the time step of 0.25 ps. For simplicity, non-adiabatic torque term has been ignored. The dimensions of the disks are the same as the actual sample setup, and only the ILwas applied to the left disk. Figure 3(a) shows the simulation result for frequency dependence of ccalculated from gyration amplitudes at 150 ns after the beginning of the current injection. The fre- quency is also swept between the resonant frequencies (215 MHz and 245 MHz). In general, our micromagnetic simula- tion reproduces the experimental result in a good approxima- tion. Note that even if the ac current was applied to the rightdisk, almost the same result was obtained, i.e., the gyration amplitude in the left disk is larger than that in the right disk. The magnetic interaction ene rgy between two vortex cores due to dipole interaction is analytically approximated as 5 Uint¼cLcRgxxLxR/C0gyyLyR ðÞ R2: (3) When cores gyrate in the steady orbital, the long-time average of Uintcan be written asUint¼cLcRgxjxLjjxRjcosDcore x/C0gyjyLjjyRjcosDcore y/C0/C1 2R;(4) where c,g,a n d Rare the chirality, the coefficient of the dipole interaction, and radius of magnetic disk, respectively. Thus, the amplitude of Uintis determined by the phase dif- ference of vortex core gyration Dcore.F i g u r e 3(b) shows cal- culated values of Dcoreby Landau-Lifshitz-Gilbert equation. We found that the Dcoreofxandycomponents changed smoothly from pto 2pand from 0 to p, respectively. When the input frequencies match the resonant frequencies of 215 MHz and 245 MHz, the values of Uint, respectively, take minimum and maximum value,9so that the torque sdipdue toUintarises along the azimuthal direction which is anti- parallel to the gyro torque sgyroat 215 MHz and parallel at 245 MHz as shown in schematics in Fig. 3(c).A sar e s u l t , the resonant frequency in the coupled vortex core system changes from the resonant frequency (230 MHz) in an iso- lated magnetic disk.6 On the other hand, in the off-resonant region from 215 MHz to 245 MHz, the direction of sdipchanges to the ra- dial direction, meaning that the effective damping constant of vortex core can be modulated by the sdip. Fig. 3(c) shows the frequency dependence of the radial component of the sdip obtained by assuming that gyration amplitudes of disks are the same. The positive (negative) sign of sdipcorresponds to parallel (anti-parallel) direction to the damping torque sdamp. Thus, the effective damping constant for the left (right) core increases (decreases) at the off-resonant frequency. In such case, it is easy to store the magnetic energy in the right diskrather than the left disk, such as represented in Fig. 3(a). By using this mechanism, we can control the gyrating motion and amount of stored energy in each disk in mag-nonic crystals. Note that the above discussion is also applicable to the case of parallel core polarities. In this case, there is also a fre- quency at which the relative phase difference of cores is p/2 (/C0p/2), 6i.e., the damping constant is modified by the dipole interaction. However, the dipole interaction for parallel polar- ities is smaller than that for anti-parallel polarities.5–7 Therefore, the damping constant modulation, namely, the anomalous energy storage should be suppressed. In order to confirm this trend, we have measured the same sequence for parallel core polarities and estimated cL¼0.38 and cR¼0.62 at 230 MHz. From micromagnetic simulation, we obtained cL¼0.42 and cR¼0.58 at 230 MHz. Thus, we found that the anomalous energy storage is strongly suppressed compared tothat of the anti-parallel core polarities, which agrees with the calculated result. 15 In summary, we have investigated the energy storage in coupled magnetic vortices by using the electrical detec- tion method of core gyration amplitude. The specific gyrat-ing motion due to anomalous energy storage is observed at off-resonant frequencies. Our micromagnetic simulations qualitatively reproduce the experimental results andexplain that the behavior arises from the modulation of effective damping constant. These findings about the spe- cific energy storage in coupled vortices may be importantfor magnetic vortex based signal processing and logic operations. FIG. 3. Simulated cobtained from the amplitude of steady gyration ampli- tudes at 150 ns after beginning of the current injection applied only into the left disk. (b) Phase difference between the left and right core gyration for x (black) and y(red). (c) Frequency dependence of the dipole torque working as the damping torque. Bottom schematics show the gyrating cores witheach torque. The purple, blue, and green arrows show the direction of the gyro torque, damping torque, and dipole torque, respectively.242402-3 Hasegawa et al. Appl. Phys. Lett. 108, 242402 (2016)This work was supported by Grant-in-Aid for Scientific Research on Innovative Area, “ Nano Spin Conversion Science” (Grant No. 26103002) and the RIKEN Junior Research Associate Program. 1R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye, M. E. Welland, and D. M. Tricker, Phys. Rev. Lett. 83, 1042 (1999); T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science 289, 930 (2000). 2A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Morgenstern, and R. Wiesendanger, Science 298, 577 (2002). 3S. B. Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran, J. St €ohr, and H. A. Padmore, Science 304, 420 (2004). 4V. Novosad, M. Grimsditch, K. Yu. Guslienko, P. Vavassori, Y. Otani, and S. D. Bader, Phys. Rev. B 66, 052407 (2002). 5J. Shibata, K. Shigeto, and Y. Otani, Phys. Rev. B 67, 224404 (2003). 6S. Sugimoto, Y. Fukuma, S. Kasai, T. Kimura, A. Barman, and Y. Otani, Phys. Rev. Lett. 106, 197203 (2011). 7J. Shibata and Y. Otani, Phys. Rev. B 70, 012404 (2004). 8H.-B. Jeong and S.-K. Kim, Appl. Phys. Lett. 105, 222410 (2014). 9D. S. Han, H.-B. Jeong, and S. K. Kim, Appl. Phys. Lett. 103, 112406 (2013). 10N. Hasegawa, S. Sugimoto, H. Fujimori, K. Kondou, Y. Niimi, and Y.Otani, Appl. Phys. Express 8, 063005 (2015).11A. Vogel, M. H €anze, A. Drews, and G. Meier, Phys. Rev. B 89, 104403 (2014). 12C. F. Adolff, M. H €anze, M. Pues, M. Weigand, and G. Meier, Phys. Rev. B 92, 024426 (2015). 13S. Barman, A. Barman, and Y. Otani, J. Phys. D: Appl. Phys. 43, 335001 (2010). 14H. Jung, K. S. Lee, D. E. Jeong, Y. S. Choi, Y. S. Yu, D. S. Han, A.Vogel, L. Bocklage, G. Meier, M. Y. Im, P. Fischer, and S. K. Kim, Sci. Rep. 1, 59 (2011). 15D. Kumar, S. Barman, and A. Barman, Sci. Rep. 4, 4108 (2014). 16K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville, and T. Ono, Nat. Mater. 6, 269 (2007). 17S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, and T. Ono, Phys. Rev. Lett. 97, 107204 (2006). 18R. Moriya, L. Thomas, M. Hayashi, Y. B. Bazaliy, C. Rettner, and S. S. P. Parkin, Nat. Phys. 4, 368 (2008). 19M. Goto, H. Hata, A. Yamaguchi, Y. Nakatani, T. Yamaoka, Y. Nozaki, and H. Miyajima, Phys. Rev. B 84, 064406 (2011). 20S. Sugimoto, N. Hasegawa, Y. Niimi, Y. Fukuma, and Y. Otani, Appl. Phys. Express 7, 023006 (2014). 21A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973). 22D. L. Huber, Phys. Rev. B 26, 3758 (1982). 23See http://llgmicro.home.mindspring.com for LLG Micromagnetic Simulator, 1997.242402-4 Hasegawa et al. Appl. Phys. Lett. 108, 242402 (2016)

1.4976950.pdf

Magnetization reversal processes of isotropic permanent magnets with various inter-grain exchange interactions Hiroshi Tsukahara , Kaoru Iwano , Chiharu Mitsumata , Tadashi Ishikawa , and Kanta Ono Citation: AIP Advances 7, 056224 (2017); doi: 10.1063/1.4976950 View online: http://dx.doi.org/10.1063/1.4976950 View Table of Contents: http://aip.scitation.org/toc/adv/7/5 Published by the American Institute of PhysicsAIP ADV ANCES 7, 056224 (2017) Magnetization reversal processes of isotropic permanent magnets with various inter-grain exchange interactions Hiroshi Tsukahara,1,aKaoru Iwano,1Chiharu Mitsumata,2Tadashi Ishikawa,1 and Kanta Ono1 1Institute of Materials Structure Science (IMSS), High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan 2National Institute for Materials Science (NIMS), Tsukuba 305-0047, Japan (Presented 4 November 2016; received 23 September 2016; accepted 15 November 2016; published online 15 February 2017) We performed a large-scale micromagnetics simulation on a supercomputing system to investigate the properties of isotropic nanocrystalline permanent magnets consisting of cubic grains. In the simulation, we solved the Landau–Lifshitz–Gilbert equation under a periodic boundary condition for accurate calculation of the magnetization dynamics inside the nanocrystalline isotropic magnet. We reduced the inter-grain exchange interaction perpendicular and parallel to the external field independently. Propagation of the magnetization reversal process is inhibited by reducing the inter- grain exchange interaction perpendicular to the external field, and the coercivity is enhanced by this restraint. In contrast, when we reduce the inter-grain exchange interaction parallel to the external field, the coercivity decreases because the magneti- zation reversal process propagates owing to dipole interaction. These behaviors show that the coercivity of an isotropic permanent magnet depends on the direction of the inter-grain exchange interaction. © 2017 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/1.4976950] I. INTRODUCTION The magnetization dynamics governs the properties of permanent magnets, which are applied in various devices such as the actuators in hard disk drives and high-power motors for electric vehicles.1,2 A nanocrystalline permanent magnet consists of many grains whose diameters range between nano and submicron scale.3–8The grains are separated from each other by grain boundary phases. The magnetization dynamics in nanocrystalline permanent magnets has been investigated for a long time. However, it is still not clear how the magnetization reversal process propagates in a permanent magnet. Micromagnetics simulation based on the Landau–Lifshitz–Gilbert (LLG) equation are widely used in magnetic research to investigate the magnetization dynamics because the magnetization rever- sal process can be observed directly.9–13Although micromagnetics simulations have been applied for investigating permanent magnets, it is difficult to simulate the magnetization dynamics inside the magnet accurately. In our previous studies, we implemented a new type of fast Fourier transform algorithm that enables performing micromagnetics simulation using more than 0.1 billion calculation cells in a super computing system.14–16Thus, we can accurately simulate the propagation of the magnetization reversal process inside the magnet. In this study, we performed micromagnetics simulations using a model of the isotropic nanocrystalline permanent magnet and demonstrated how the coercivity and remanent mag- netization are changed by inter-grain exchange interactions. Propagation of the magnetization aAuthor to whom correspondence should be addressed. Electronic mail: thiroshi@post.kek.jp 2158-3226/2017/7(5)/056224/4 7, 056224-1 ©Author(s) 2017 056224-2 Tsukahara et al. AIP Advances 7, 056224 (2017) reversal process depends on not only the strength but also the direction of the inter-grain exchange interaction. II. SIMULATION SYSTEM AND METHOD A schematic illustration of the isotropic nanocrystalline permanent magnet is shown in Fig. 1. The simulation system (512 nm 512 nm512 nm) consists of many cubic grains (16 nm  16 nm16 nm). We discretized the simulation model into 2.0 nm 2.0 nm2.0 nm cells. The easy axes of the grains are oriented at random. The magnetization dynamics is simulated by solving the LLG equation, which is defined as dM(x) dt= M(x)Heff(x)+MdM(x) dt, (1) where M(x)is the magnetization at x, is the gyromagnetic ratio, is the Gilbert damping constant, andHeff(x) is the effective field, which consists of anisotropy, dipole, external, and exchange fields. The exchange field Hexc(x), which is defined as Hexc(x)=X 2 M2sr
[A(x)rM(x)]ˆe, (2) where Msis the saturation magnetization and A(x) is the exchange stiffness coefficient. The exchange stiffness coefficient has different values for intra- and inter-grain exchange interactions. In this paper, the exchange stiffness constant for the inter-grain interaction is defined as A(x)=(˜A in gain g˜Aon interface, (3) where gtakes a value from 0 to 1. In particular, g ?andg krepresent gperpendicular and parallel to the external field, respectively. We consider three types of inter-grain exchange interactions called types A, B, and O. In type A (B), the inter-grain exchange interaction works on the xyplane (along thez-direction) and this interaction works along all direction in type O. We assume the following Nd–Fe–B material parameters: Ms= 1281.2 emu/cm3, the uniaxial constant K1u=4.5107erg/cm3, =1.0,j j=1.76107s1G1, and ˜A=12.5107erg/cm. We performed the micromagnetics simulation while varying the strength of the external field applied in the z-direction. III. SIMULATION RESULTS Figure 2(a) shows the hysteresis curves of type O for various values of g ?andg k. When there is no inter-grain exchange interaction ( g ?=0.0,g k=0.0), the hysteresis curve is almost the same as that of the Stoner-Wohlfarth model. In this case, the coercive force Hcis 30.4 kOe. As the strengths of the inter-grain exchange interactions increase, Hcdecreases rapidly. When the FIG. 1. Schematic of the isotropic nanocrystalline permanent magnet model. Inset shows a cubic grain.056224-3 Tsukahara et al. AIP Advances 7, 056224 (2017) FIG. 2. The hysteresis curves of (a) type O and (b) types A and B for various of g ?andg k. Inset illustrates the definition of g ?andg k. strengths of the intra- and inter-grain exchange interactions are same ( g ?=1.0,g k=1.0), Hcbecomes 9.79 kOe. This behavior of the hysteresis curve is explained by propagation of the magnetization reversal process, as argued in the past.17When there is no inter-grain exchange interaction, the magnetization tends to be reversed independently in each grain. In contrast, if the grains interact strongly through the inter-grain exchange interaction, the magnetization reversal process propagates across the grain boundary, and the magnetization is reversed in a large region. Hence, the magne- tization is gradually (quickly) damped when the inter-grain exchange interaction is weak (strong). The remanent magnetization Mrbecomes large as the strength of the inter-grain exchange interaction increases, in contrast with Hc. The reduction in Hcdepends on the direction in which the inter-grain exchange interaction works. Figure 2(b) shows the hysteresis curves of types A and B. For type A, the magnetization decreases gradually in the demagnetization process even if the strengths of the inter- and intra-grain exchange interactions are the same along the z-direction. The shape of the hysteresis curve of type A is similar to that without the inter-grain exchange interaction. The reduction in Hcis small, and Mrstill has a large value. In contrast, the hysteresis curve of type B has similar properties to that with the strong inter-grain exchange interaction, in which the domain wall moves across the grain boundary. The magnetization of type B decreases quickly when the external field approaches Hc, and Mris larger than that for type A. We performed micromagnetics simulations while changing the strength of the inter-grain exchange interaction continuously to reveal how the coercive force and remanent magnetization depend on the type of inter-grain exchange interaction. Figure 3(a) shows the Hcvalues for types A, B, and O as a function of the strength of the inter-grain exchange interaction. For type O, the reduction in Hcis largest because the magnetization reversal process is propagated easily by the strong inter-grain exchange interaction. The most remarkable property is that the reduction in Hc for type A is smaller than that for type B. This property shows that propagation of the magneti- zation reversal in the xyplane is inhibited even if the grains interact through a strong inter-grain exchange interaction in the z-direction. In contrast with that in type A, the magnetization reversal is propagated in the z-direction by the dipole interaction in type B. Thus, these properties show that it is worked for the high Hceffectively to inhibit propagation of the magnetization reversal in the xyplane. On the other hand, the strong inter-grain exchange interaction enhances Mras shown in Fig. 3(b). For types B and O, Mris larger than that for type A. FIG. 3. Coercive force Hcand remanent magnetization Mras a function of the strength of the inter-grain exchange interaction .056224-4 Tsukahara et al. AIP Advances 7, 056224 (2017) IV. SUMMARY We performed a large-scale micromagnetics simulation to investigate the magnetization dynam- ics inside an isotropic nanocrystalline permanent magnet consisting of nano meter-sized cubic grains. When we performed the micromagnetics simulations, we changed the strength of the inter-grain exchange interaction in the direction parallel and perpendicular to the external field. When the inter-grain exchange interaction worked only in the direction parallel to the external field, the magne- tization decreased gradually in the demagnetization process. In contrast, the magnetization decreased quickly even if there is no inter-grain exchange interaction in direction parallel to the external field, because the magnetization reversal process propagated in this direction owing to the dipole inter- action. Thus, the coercivity has a large value when the inter-grain exchange interaction works only along the direction parallel to the external field, although the remanent magnetization is largest, when the inter-grain exchange interaction works in all directions. Therefore, it is important for the high coercivity to inhibit the propagation of the magnetization reversal process in perpendicular to the external field. ACKNOWLEDGMENTS The authors thank the crew of the Center for Computational Materials Science of the Institute for Materials Research, Tohoku University, for their continuous support of the SR16000 supercom- puting facilities. This work was supported by the Large Scale Simulation Program No. 15/16-01 (FY2015/16) of KEK. This work was supported in part by the JST under Collaborative Research Based on Industrial Demand “High Performance Magnets: Towards Innovative Development of Next Generation Magnets.” 1K. H. J. Buschow, Mater. Sci. Rep. 1, 1 (1986). 2O. Gutfleisch, M. A. 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Phys. 33, R135 (2000). 13S.-J. Lee, S. Sato, H. Yanagihara, E. Kita, and C. Mitsumata, J. Magn. Magn. Mater. 323, 28 (2011). 14N. Inami, Y . Takeichi, C. Mitsumata, K. Iwano, T. Ishikawa, S.-J. Lee, H. Yanagihara, E. Kita, and K. Ono, IEEE. Trans. Magn. 50, 1400304 (2014). 15H. Tsukahara, S.-J. Lee, K. Iwano, N. Inami, T. Ishikawa, C. Mitsumata, H. Yanagihara, E. Kita, and K. Ono, AIP Advance 6, 056405 (2016). 16H. Tsukahara, K. Iwano, C. Mitsumata, Ishikawa, and K. Ono, Comput. Phys. Commun 207, 217 (2016). 17H. Fukushima, Y . Nakatani, and N. Hayashi, IEEE Trans. Magn. 34, 1934 (1998).

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High power all-metal spin torque oscillator using full Heusler Co2(Fe,Mn)Si Takeshi Seki, Yuya Sakuraba, Hiroko Arai, Masaki Ueda, Ryo Okura, Hiroshi Imamura, and Koki Takanashi Citation: Applied Physics Letters 105, 092406 (2014); doi: 10.1063/1.4895024 View online: http://dx.doi.org/10.1063/1.4895024 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/105/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High power radio frequency oscillation by spin transfer torque in a Co2MnSi layer: Experiment and macrospin simulation J. Appl. Phys. 113, 033907 (2013); 10.1063/1.4776719 Extensive study of giant magnetoresistance properties in half-metallic Co2(Fe,Mn)Si-based devices Appl. Phys. Lett. 101, 252408 (2012); 10.1063/1.4772546 High-power rf oscillation induced in half-metallic Co2MnSi layer by spin-transfer torque Appl. Phys. Lett. 99, 052510 (2011); 10.1063/1.3624470 Current-perpendicular-to-plane giant magnetoresistance of a spin valve using Co 2 MnSi Heusler alloy electrodes J. Appl. Phys. 105, 07E905 (2009); 10.1063/1.3068427 Current-perpendicular-to-plane giant magnetoresistance in spin-valve structures using epitaxial Co 2 FeAl 0.5 Si 0.5 / Ag / Co 2 FeAl 0.5 Si 0.5 trilayers Appl. Phys. Lett. 93, 122507 (2008); 10.1063/1.2990647 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: 149.106.210.114 On: Sun, 23 Nov 2014 21:33:13High power all-metal spin torque oscillator using full Heusler Co 2(Fe,Mn)Si Takeshi Seki,1,a)Yuya Sakuraba,1,b)Hiroko Arai,2Masaki Ueda,1Ryo Okura,1 Hiroshi Imamura,2and Koki Takanashi1 1Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan 2Spintronics Research Center, National Institute of Advanced Industrial Science and Technology, Tsukuba 305-8568, Japan (Received 25 June 2014; accepted 26 August 2014; published online 5 September 2014) We showed the high rf power ( Pout) emission from an all-metal spin torque oscillator (STO) with a Co2Fe0.4Mn 0.6Si (CFMS)/Ag/CFMS giant magnetoresistance (GMR) stack, which was attributable to the large GMR effect thanks to the highly spin-polarized CFMS. The oscillation spectra were measured by varying the magnetic field direction, and the perpendicular magnetic field was effectiveto increase P outand the Qfactor. We simultaneously achieved a high output efficiency of 0.013%, a high Qof 1124, and large frequency tunability. CFMS-based all-metal STO is promising for over- coming the difficulties that conventional STOs are confronted with. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4895024 ] The versatile magnetization dynamics in a nanomagnet triggered by torque due to the transfer of the spin angular mo-mentum, known as spin torque, 1,2provides us with a pathway for the development of spintronic devices. One of the intrigu- ing devices is the spin torque oscillator (STO), which is a pro-spective nanometer-scaled radio frequency (rf) oscillator, e.g., for on-chip communications or radar. Electric current spin-polarized by a ferromagnetic (F) layer (a spin polarizer)exerts spin torque on the local magnetic moments of another F layer (a free layer) that is separated from the spin polarizer by a nonmagnetic (N) metal or an insulator (I). When thespin torque and the intrinsic damping torque are balanced in the free layer, the magnetization steadily precesses around the effective magnetic field. For current-perpendicular-to-plane (CPP) giant magnetoresistance (GMR) devices of F jN jF or magnetic tunnel junctions (MTJs) of F jIjF, this steady magnetization precession leads to a change in the de-vice resistance through the magnetoresistance effect. 3–7Since dc electric current ( Idc) is applied to the device, the time- dependent device resistance ( R(t)) is converted into rf voltage (Vrf). Consequently, the device emits rf output power ( Pout). This is the central mechanism of the STO, and its simple architecture composed of a single nanometer-scaled elementis one of the greatest attractions of the STO. However, there are several crucial issues that need to be solved before STOs can be put into practical use, which arethe enhancement of P out, improvement of the rf oscillation quality, and increasing the frequency tunability by electric current and/or magnetic field. MTJ-based STOs, particularlywith an MgO tunnel barrier, may solve the first issue since theP outis roughly proportional to the square of the MR ra- tio.8–13However, its relatively wide oscillation linewidth (Df) is not applicable for practical applications.8,10Some so- phisticated ways for overcoming this problem have recently been proposed, e.g., the utilization of perpendicular magneticanisotropy,11,12a gyrating magnetic vortex,14and a unique device structure such as a sombrero-shape.13,15However, MTJ-based STOs have intrinsic problems, which are an im- pedance mismatch of the device to rf circuitry, and the risk of the dielectric breakdown of the tunnel barrier under largebias voltages. 9,16On the other hand, the resistance of a CPP- GMR device is easily tuned to be compatible with an rf cir- cuit because it consists of only metallic layers, and a CPP-GMR device generally has a narrow Dfcompared with the MTJ-based STOs. 6,17–24Furthermore, a CPP-GMR stack is free from the risk of dielectric breakdown in a tunnel barriermaterial. Thus, if the MR effect of a CPP-GMR device can be enhanced, it could be a candidate for use as a high per- formance STO that is beyond the capabilities of MTJ-basedSTOs. One promising way to enhance the MR ratio in a CPP-GMR device is to use highly spin-polarized F materials, that is, half metallic ferromagnets, which are ideal materialshaving the density of states at the Fermi level only in one spin channel. 25–27Here, we show high Poutin a CPP-GMR- based STO using a Co 2Fe0.4Mn 0.6Si (CFMS) layer. CFMS is a full-Heusler alloy and its high spin polarization leads to the large GMR ratio.28 Thin films were prepared on an MgO (100) single crys- tal substrate using an ultrahigh vacuum magnetron sputtering system with a base pressure below 1 /C210/C07Pa. The stacked structure is MgO subs. jCr (20) jAg (40) jCFMS (20) jAg (5)jCFMS (3) jAg (2) jAu (5) (in nanometer). The 20 nm- thick bottom CFMS layer was grown at RT by co-sputtering from Co 43.7Mn 28.0Si28.4and Co 47.5Fe24.2Si28.3targets. The bottom CFMS layer was subsequently annealed at 500/C14Ct o promote the L21ordering. A 5 nm-thick Ag spacer layer fol- lowed by a 3 nm-thick top CFMS layer was deposited at RT.The top CFMS layer was also annealed at 500 /C14C. We con- sider that there is no significant difference in the degree of L21ordering and the fundamental magnetic properties between the top and bottom CFMS layers because both layers were grown on Ag (100) and were annealed at the same temperature. In addition, our previous study28reported no significant thickness dependence of the MR effect for CFMS within the range of 3–10 nm, suggesting the high spina)Author to whom correspondence should be addressed. Electronic mail: go-sai@imr.tohoku.ac.jp b)Present affiliation: National Institute for Materials Science, Tsukuba 305-004, Japan. 0003-6951/2014/105(9)/092406/5/$30.00 VC2014 AIP Publishing LLC 105, 092406-1APPLIED PHYSICS LETTERS 105, 092406 (2014) 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: 149.106.210.114 On: Sun, 23 Nov 2014 21:33:13polarization even for the thin CFMS free layer. Electron beam lithography and Ar ion milling were used to make arectangular shaped CPP-GMR pillar. Figure 1(a) depicts a schematic illustration of the CPP-GMR device. The top CFMS layer was patterned into rectangular shapes while thebottom one remains in the extended film structure. The top and bottom CFMS layers behave as the hard and soft layers, respectively, against the external magnetic field due to thedifference in the shape anisotropy energies, and the bottom CFMS layer acts as a spin polarizer. The scanning electron microscope image of the pillar (Fig. 1(b)) shows the size of 100 nm /C2170 nm. The CPP-GMR device was connected to the rf- compatible measurement circuit using a two-terminal rfprobe. I dcwas applied to the device through a bias-Tee from a dc source meter. Positive Idcwas defined as the electric cur- rent direction that the electron flows from the top CFMS layerto the bottom one. V rfemitted from the CPP-GMR device was amplified by a preamplifier, and was fed into a spectrum analyzer. The power spectral density (PSD) value was calcu-lated by PSD ¼ðV rfðIdcÞÞ2ðZ0/C2DfRB/C2ðGampÞ2Þ/C01, where Vrfis the time-averaged value during the spectral measure- ment. Gampis the frequency-dependent gain of the preampli- fier and DfRBis the resolution bandwidth of the spectrum analyzer, which was set to 1 MHz. Z0is 50Xfor the present measurement. The rf transmission efficiency in the device ( g) was evaluated using the value of S 11measured by a vector network analyzer to calibrate the rf transmission loss. The differential resistance (d V/dI) was measured during the appli- cation of Idcusing the source meter and the lock-in amplifier. Figure 1(c) shows a MR curve when the magnetic field ( H) was applied along the in-plane easy axis ( H//x). A definite change was observed in the device resistance ( R). The high and low resistance states resulted from the antiparallel and parallel alignments of the magnetizations of twoCFMS layers, respectively. The maximum MR ratio, i.e., (R AP/C0RP)/C2100/RP, where RP¼5.88X(RAP¼7.65X) rep- resents Rin the parallel (antiparallel) configuration, was 30%. The intrinsic MR ratio was found to be 48% after sub- tracting the parasitic resistance measured by using the four- probe configuration.Figures 2(a) and2(b) compare the PSDs for H//xand H//yatH¼300 Oe, where xandycorrespond to the in- plane easy and hard magnetization axes of the rectangular pil- lar, respectively. Only the positive Idcexcited the rf oscilla- tion, in which the spin torque supported the magnetizationprecession in the top CFMS because the magnetizations of both CFMS layers were almost in parallel. The intensity of oscillation peak was low for H//x,a n d P outwas 0.03 nW. A sharp oscillation peak appeared and Poutincreased from 0.03 to 11.6 nW by varying the direction of Hfrom xtoy. In addi- tion,Dfwas reduced from 76 to 30 MHz although the peak FIG. 1. (a) Schematic illustration of the microfabricated CPP-GMR device together with the circuit for rf spectral measurement. The coordinate axes arealso shown. (b) Scanning microscopy image for the CPP-GMR pillar with the size of 100 nm /C2170 nm. (c) GMR curves measured with the magnetic field ( H) applied parallel to the long axis of the rectangular pillar ( H//x). The corresponding magnetization con-figurations are also illustrated. FIG. 2. Spin torque oscillation under in-plane magnetic field. (a) Rf spectrameasured at H//xand (b) H//y.I dcandHwere set to 5 mA and 300 Oe, respectively. (c) Mappings of normalized PSD as functions of frequency ( f) andIdcforH//xand (d) H//y.Hwas fixed at 300 Oe. (e) Resonance fre- quency ( f0) as a function of HforH//xand (f) H//y, where Idcwas 5 mA and 2.5 mA, respectively.092406-2 Seki et al. Appl. Phys. Lett. 105, 092406 (2014) 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: 149.106.210.114 On: Sun, 23 Nov 2014 21:33:13frequencies ( f0) were almost the same. Thus, the frequency purity ( Q), which is defined as f0/Df, was enhanced from 42 forH//xto 112 for H//y. Figures 2(c)and2(d)display PSD as functions of fandIdcforH//xandH//y, respectively. For H//x,f0did not depend on Idc, and the second harmonics (2f0) gradually appeared when Idc>7 mA. Less variation in f0 with Idcis different from the tendency of a typical STO that redshifts with Idc.5No remarkable Hdependence of f0was also observed for H//x(Fig. 2(e)). These facts suggest that the magnetization dynamics excited at H//xmight be far from the uniform mode due to the inhomogeneous effective magnetic field. For H//y, on the other hand, f0clearly shifted to the low fside with Idc. This is the typical STO behavior showing in-plane magnetization precession. The Hdepend- ence of f0(Fig. 2(f)) for H//ycan be interpreted by Kittel’s relationship: f0¼ðc=2pÞfHcðHcþ4pMsÞð1/C0ðH=HcÞ2Þg1=2 forH<Hc, and f0¼ðc=2pÞfðH/C0HcÞðH/C0Hcþ4pMsÞg1=2 forH/C21Hc, where cis the gyromagnetic ratio, Hcis the in- plane coercive field, and Msis the saturation magnetization. We obtained Hc¼362637 Oe and Ms¼11936306 emu cm/C03from the fitting. The obtained Hcis close to the switch- ing field of 380 Oe in the MR curve, and Msis also in agree- ment with the value obtained from the magnetization curve (1160 emu cm/C03). According to a previous experiment,29the hard axis magnetic field suppresses the inhomogeneity in demagnetizing field for an ellipsoidal or rectangular pillar, resulting in a stable uniform precession mode. It is noted thatwe obtained large P out¼11.6 nW for H//y. After correcting the loss of the emitted rf power in the rf circuit, Poutwas increased to /C2440 nW. We further optimized the magnetization precession mode of the STO by applying the perpendicular H(H//z) because it easily leads to the out-of-plane precession of mag-netization. 6,11,12,22,30The PSD mapping as functions of fand Idcis shown in Fig. 3(a)together with the differential resist- ance (d V/dI) versus Idc.A sIdcwas increased, f0shifted to the high fside (frequency blueshift). In addition, several jumps inf0appeared when the abrupt change was observed in d V/ dI. These experimental facts indicate that the excited mode was the out-of-plane precession and the mode change occurred as Idcwas increased. Figure 3(b) shows the Idcde- pendence of f0measured at various H. The frequency blue- shift was observed for each H. It is noted that the frequency tunability by Idc,df0/dIdc, became large as Hwas increased (solid lines in Fig. 3(b)). For example, d f0/dIdcwas 6.05 GHz mA/C01atH¼5 kOe. This d f0/dIdcvalue is high, suggesting that the present STO exhibits a strong nonlinearity against Idcunder the perpendicular H. As mentioned above, f0 shifted together with the discontinuous jumps. When the mode jump occurred, Dfwas remarkably broadened as shown in the inset of Fig. 3(b). Figure 3(c)summarizes Pout as a function of Idcat different perpendicular H. After the onset of oscillation at Idc/C242m A , Poutgradually increased with Idc. The maximum Poutobtained in this study was 23.7 nW for Idc¼5.6 mA and H¼3 kOe. The corresponding rf spectrum is displayed in Fig. 3(d). Together with the large Pout,Dfhad the minimum value of 10 MHz, leading to the excellent Q¼1124. However, further increasing Idcgave rise to the reduction in Pout. A striking feature here is that the present STO simultaneously achieves high Poutand high Q.The correction of the rf transmission loss gave us the increase in Poutto/C240.3lW. The 2-dimensional micromagnetic simulation was con- ducted to obtain the magnetization dynamics in the free layer by solving the Landau-Lifshitz-Gilbert (LLG) equation withSlonczewski’s spin-torque term: @m=@t¼/C0cðm/C2H effÞ þaðm/C2ð@m=@tÞÞ /C0CSTðm/C2ðm/C2pÞÞ, where cis the gyromagnetic ratio and ais the magnetic damping constant. m(p) is a unit vector representing a magnetic moment of the free (fixed) layer. The effective field, Heff, is the sum of H, the magnetocrystalline anisotropy field with a four-fold sym-metry, and the demagnetization field. C STis the spin torque term and is given by CST¼f ð glBIdcÞ=ðeMsVÞgeðm/C1pÞ, where gis the Land /C19e g-factor of the electron spin, lBis the Bohr magneton, eis the elementary charge, Msis the satura- tion magnetization, and Vis the magnetic volume of the free layer. The function of eðm/C1pÞis expressed as eðm/C1pÞ ¼½ /C0 4þð1þPÞ3ð3þm/C1pÞ=ð4P3=2Þ/C138/C01, where Pis the spin polarization of the electric current. The free layer is a rectangular-shaped magnetic thin film: 170 /C2100/C23n m3 with the edges being rounded off. We divided the film into 2/C22/C23n m3cells. The material parameters we used were as follows: Ms¼1.16/C2106A/m, anisotropy constant K1¼/C01.2/C2103J/m3, which were the measured values of the CFMS film. awas set to 0.003.31The stiffness constant A¼2.0/C210/C011J/m (Ref. 32) and P¼0.56 (Ref. 33) were brought from the Co 2MnSi (CMS) parameters. The thermal effects were neglected for simplicity. Figure 4(a) shows the simulated Idcdependence of f0at various perpendicular H. FIG. 3. Spin torque oscillation under perpendicular magnetic field. (a) Mapping of normalized PSD as functions of fandIdcforH//z. The corre- sponding differential resistance (d V/dI) versus Idcis also shown. (b) Idcde- pendence of f0and (c) Poutmeasured at H¼2 kOe (magenta squares), 3 kOe (orange squares), 4 kOe (cyan squares), and 5 kOe (black squares). The red solid lines in (b) are the results of a linear fit to the experimental data. The inset in (b) shows Dfas a function of IdcatH¼3 kOe. The maximum value ofPoutis 23.7 nW at Idc¼5.6 mA and H¼3 kOe. The corresponding rf spectrum is shown in (d).092406-3 Seki et al. Appl. Phys. Lett. 105, 092406 (2014) 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: 149.106.210.114 On: Sun, 23 Nov 2014 21:33:13The simulated results well reproduced the experimental tend- ency of the blueshift in f0with Idc. When Idcwas further increased, a gradual decrease in the peak PSD was observed just like in the experimental results (Fig. 4(b)). Snapshots of the magnetic structures are shown in Figs. 4(c) and4(d). Although non-uniform magnetic structures were observed even for Idc¼4.5 mA, the non-uniformity was more remark- able when Idcwas set at 9 mA. Thus, the PSD reduction orig- inates from the incoherent (non-uniform) magnetization precession at high Idc. Compared with the trajectory of the magnetization precession at Idc¼4.5 mA (Fig. 4(e)), which is an out-of-plane precession mode with a slightly fluctuated trajectory, Idc¼9.0 mA had a large incoherency in the pre- cession trajectory (Fig. 4(f)). This change abruptly occurred atIdc¼5.25 mA together with the jump of f0(Fig. 4(a)). Similar mode jumps were also reported in the GMR pillarwith Ni 80Fe20jCujNi80Fe2034which was due to the transi- tions between different localized nonlinear spin wave modes. Our micromagnetic simulation suggested a possible reasonfor the incoherency in the precession and a way to solve this problem. CFMS exhibits a magnetocrystalline anisotropy with a four-fold symmetry larger than the conventional ferro-magnetic layers for STO, for example, Co-Fe and Fe-Ni alloys. 35This magnetocrystalline anisotropy gives rise to the non-uniform effective magnetic field and the complicatedmagnetic structures during oscillation. Therefore, changing the pillar shape from rectangular to square, where the easy axes of CFMS align parallel to the sides of the square, is effective for suppressing the formation of complicated mag-netic structures as shown in Fig. 4(g). This improvement could lead to the further enhancement of P outowing to the coherent magnetization precession. Compared with our previous experiment, the calibrated Poutwas increased from 1.1 nW for the previous Co 2MnSi to 0.3lW for the present CFMS. In addition, the Qfactor was enhanced up to more than 1000, which might be due to the small magnetization damping of CFMS.31When we calcu- late the ratio of Poutto the input power ( Pin), corresponding to the output efficiency of STO, Pout/Pinis 0.013%. This value is enhanced up to 0.16% after calibrating the rf trans- mission loss. Figure 5plots Pout/Pinversus f0/Dffor a variety of STOs.5,6,8–16,19–21,26,27,30,36–40Surprisingly, the output ef- ficiency of our CPP-GMR-based STO is of the same order as the largest one obtained for the CoFeB jMgO jCoFeB MTJ.11We emphasize the advantages of using our CPP- GMR-based STO for practical applications since it can simultaneously provide with high Pout, high Q, and large d f0/ dIdcand is free from the risk of the dielectric breakdown for the tunnel barrier. Although we still have problems of the complicated emission properties for the present device, e.g., FIG. 4. Micromagnetic simulation for spin torque oscillation under perpen- dicular magnetic field. (a) Idcdepend- ence of f0simulated at H¼2 kOe (magenta squares), 3 kOe (orange squares), 4 kOe (cyan squares), and 5 kOe (black squares). (b) Idcdepend- ence of the peak PSD for the simula- tion (top panel) and the experiment (bottom panel). Hwas set to 3 kOe. The snapshots of magnetic structures for (c) Idc¼4.5 mA and (d) 9.0 mA at H¼3 kOe, where the xcomponent of magnetization was displayed as a col-our plot. The left (right) images were obtained at a simulation time of 93.6 ns (93.9 ns). The corresponding trajec- tories of magnetization precessions at I dc¼4.5 mA and 9.0 mA are shown in (e) and (f), respectively. (g) The snap- shots of time evolution of magneticstructures for the square-shaped pillar.092406-4 Seki et al. Appl. Phys. Lett. 105, 092406 (2014) 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: 149.106.210.114 On: Sun, 23 Nov 2014 21:33:13the dependence of Poutandf0onIdc, we believe that the all- metal STO using CFMS is a promising way for developing nanometer-scaled high performance oscillators after solvingsuch problems. This work was partially supported by the Japan Science and Technology (JST) Agency through its Strategic International Cooperative Program under the title“Advanced spintronic materials and transport phenomena (ASPIMATT),” and JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 25790011. The device fabrication waspartly performed at Cooperative Research and Development Center for Advanced Materials, IMR, Tohoku University. 1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 2L. Berger, Phys. Rev. B 54, 9353 (1996). 3A. Brataas, A. D. Kent, and H. Ohno, Nat. 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Gurney, Appl. Phys. Lett. 103, 232407 (2013). FIG. 5. Plot of output efficiency and quality of spin torque oscillation. Pout/ Pinversus f0/Dfwas summarized for a variety of STOs. The solid green and purple outlined stars denote the present values detected and calibrated, respectively. This plot includes the values reported for CoFeB (CFB) jMgO jCFB,8–11,13,15,16,36CFB jMgO jFeB,12CFB jMgO jPy,14FeCo jFeCo- AlO jFeCo,37,39CojCujCo,5CoFe jCujCoFe,40CoFe jCujCoFe j Py,20CoFe jCujPy,6CojCujPy,30PyjCujPy,19,21,38CMS jAgj CMS,26and Co 2Fe(Ge, Ga) (CFGG) jAgjCFGG.27The blue and red circles denote the data of CPP-GMR-based and MTJ-based STOs,respectively.092406-5 Seki et al. Appl. Phys. Lett. 105, 092406 (2014) 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: 149.106.210.114 On: Sun, 23 Nov 2014 21:33:13

1.403779.pdf

An adaptive controller for the active absorption of sound F. Ordufia-Bustamante a) and P.A. Nelson Institute of Sound and Vibration Research, University of Southampton $02 5NH, England (Received 17 September 1991; accepted for publication 7 January 1992 ) An adaptive system is presented that allows a secondary acoustic source to become an active absorber of sound at the end of a closed duct. The system can also be generalized in order to achieve other termination impedances. The system consists of a loudspeaker, two microphones, and signal processing hardware including a digital signal microprocessor. The signals from the microphones are processed to obtain an error signal that represents the difference between the actual and the desired acoustic impedance of the termination. An absorbing termination requires, for example, that the microphone pair acts as a unidirectional probe picking up the sound reflected from the active termination only. This signal is used as the error signal that the digital controller is required to minimize. A simple analysis shows that this can be done adaptively using the "filtered-X" LMS algorithm. A simple experimental setup is used to obtain an absorbing termination which is shown to work with periodic, random, and transient input signals. PACS numbers: 43.50. Ki, 43.50.Gf, 43.55.Ev INTRODUCTION It is well known that an acoustic source can become a sink of acoustic energy under certain conditions. 1;_ The ab- sorption cross section associated with a free-field point mon- opole source is A 2/4rr, where A is the acoustic wavelength. At low frequencies therefore, at a frequency of 100 Hz for example, a single compact loudspeaker could be made to provide a perfectly absorbing area equivalent to 0.93 m 2. This could be very beneficial in room acoustics at low fre- quencies where passive methods of damping room reso- nances are bulky, expensive, and inefficient. In the work pre- sented here, we assume that we have access to a signal which is linearly related to the pressure fluctuation that impinges on the secondary source. This would be the case, for exam- ple, in a sound reproduction system where the primary source of sound is itself a loudspeaker. The general problem is to operate on this "primary" signal in order to produce and maintain a desired acoustic impedance in front of the secondary source. Again, in the context of sound reproduc- tion systems, such an approach could be used to improve the listening conditions of the room in which the recorded mate- rial is being replayed. Another interesting application is the active control of reverberation times. Existing electroacous- tic techniques can extend the reverberation time; 3 reduction of reverberation times could be achieved using a set of auxil- iary sources actively controlled to absorb energy from the sound field. The system proposed for the adaptive control of acous- tic impedance is shown in Fig. 1. The aim of the adaptive digital controller is to change the acoustical impedance pre- sented by the secondary source to a desired specification. An error signal indicating the difference between the actual and the desired impedance is obtained by processing the signals from a two-microphone probe, the task imposed on the con- troller being to reduce this error signal in a least-squares sense. The availability of the primary signal avoids dealing with secondary source feedback problems that would arise if the primary signal is obtained directly from the sound field (from a reference microphone close to the primary source for example). Previous work on this problem (without the adaptive feature) shows successful results for purely resistive desired impedances at one end of a narrow tube, allowing control of the reflection coefficient between 0 and 1.5 for periodic sig- nals with frequencies from 100 to 800 Hz. 4-6 (A reflection coefficient greater than unity meaning that the active termi- Primary Secondary source source 2.Mie probe p(,) p ACOUSTIC IMPEDANCE CONTROLLER This work was conducted whilst this author was on temporary leave from Secci6n de Acfistica, CI-UNAM, Apdo. Postal 70-186, Mxico DF, Mexico. FIG. 1. Diagram of the acoustic impedance controller. The aim is to drive the acoustic impedance in front of the secondary source to a desired specifi- cation. The irregular line represents the boundary of the acoustic environ- ment being controlled. 2740 J. Acoust. Soc. Am. 91 (5), May 1992 0001-4966/92/052740-08500.80 ¸ 1992 Acoustical Society of America 2740 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Sat, 22 Nov 2014 02:56:56nation is actually supplying energy to the sound field. ) The main difficulty of the previous approach, a consequence of being nonadaptive, is the complicated design of the control filter which must be repeated carefully for each new acoustic system (and for each new position of the controller within the same system) where the control is to be applied. Other work where adaptive signal processing tech- niques are employed in active sound field control, although not dealing directly with the acoustic impedance control, is that described for example by Elliott et al. 7 and Nelson and Elliott. 8 In particular, Curtis et al. 9 discuss the active sup- pression of resonances in one-dimensional acoustic systems and deal with the effect of various control strategies, includ- ing two different terminations (an acoustical virtual earth or pressure release boundary and a perfectly absorbing termi- nation) to reduce the acoustic potential energy in a finite narrow duct. A perfectly absorbing termination is shown in this paper to provide significant reductions of the total po- tential energy in the duct. I. THE ERROR SIGNAL AND THE CONTROL ALGORITHM A. The error signal An error signal can be synthesized from the signals of a pair of microphones located in front of the secondary source. This error signal should have the property that after being minimized by the action of the controller, the acoustic im- pedance in front of the secondary source matches the desired specification. As this specification is very likely to be made in the frequency domain, a later transformation to the time domain will be required. Using z transforms, the error signal can be defined as 1ø E(z) = H12 (z)P 1 (z) -- P2 (z), ( 1 ) where Pl (z) and P2 (z) are the z transforms of the signals from the microphones and H12 (z) is the desired transfer function between them (see Fig. 2). [Note that when the error signal is identically zero, E(z) =0, then H12 (z) --P2 (z)/P1 (z).] In order to synthesize the error signal in the time domain H12 (z) must be specified as a reali- zable (i.e., causal and stable) digital filter. The desired transfer function between the outputs from the microphones H12 (z) can be related to the desired acous- tic impedance at the center of the microphone pair. For p(k) Acoustic System , H12(z) e(k) FIG. 2. Definition of the error signal specifying the desired digital transfer function between the signals of the two-microphone probe. sound consisting only of plane waves traveling along the axis of a narrow tube this (analog) transfer function is '2 X a ( j )/pc -- j tan (co-/2) Hi2 (jo) -- , (2) Xa ( jco ) /pc + j tan ( w'/2 ) where ' = s/c is the acoustic time delay between the two microphones (s being the spacing between them) and Xa (jco) is the desired (analog) acoustic impedance. (The use of Xinstead of the usual Z to denote acoustic impedance is introduced to avoid confusion with the z-transform vari- able. ) In the particular case when the desired acoustic imped- ance is Xa (jco) = pc, corresponding to a perfectly absorbing termination, Eq. (2) simplifies to a time delay, Hi2 (jco) -- exp(--jo'r). (3) This can be readily transformed to a digital transfer function if the sampling period is chosen as an integer fraction N of the acoustic time delay ( Ts = 'r/N), in this case the digital transfer function reduces simply to a delay of N samples, H12(z) =z -N. (4) In the time domain, the error signal is then calculated as e(k) ----Pl (k- N) --P2 (k), (5) where k is the digital time index and lowercase letters denote the time signals corresponding to the z transforms in cap-. itals. This simple form is amenable to a highly efficient real time implementation in a microprocessor. B. The optimal controller and its adaptive FIR approximation The general form of the impedance controller is shown in Fig. 3. The matrix of transfer functions of the electroa- coustic system, including the acoustic transfer functions and those of the loudspeakers and microphones, relates the pri- mary and secondary inputs Sl (z) and S2 (z) to the signals of the microphones by, [Pl (z)] [Gll (z)G21 (z) ] IS1 (z) ] P2 (z) = (12 (Z) (22 (z) S2 (z) ' (6) The error signal is obtained as discussed in the previous sec- tion and the control filter H(z) is applied to the primary signal to obtain the signal applied to the secondary source in the form, S2 (z) -- H(z)Sl (z). (7) Combining Eqs. (6) and (7) the z transform of the error signal can be written as, E(z) = [E 1 (Z) + E 2 (z)H(z) ]S 1 (Z), (8) where the transfer functions between the primary and sec- ondary signals and the error signal are E1 (z) and E2 (z), respectively. These transfer functions can be written in terms of H12 (z) [ from Eqs. ( 1 ), (6), (7) and (8) ] as El (z) = H12 (2)Gll (2) -- G12 (2), (9a) E2 (z) = Hl (z)G21 (z) -- G22 (z). (9b) A reduced diagram of the controller can be drawn using these transfer functions as shown in Fig. 4 (a). The structure 2741 J. Acoust. Soc. Am., Vol. 91, No. 5, May 1992 F. Ordura-Bustamante and P. A. Nelson: Adaptive controller 2741 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Sat, 22 Nov 2014 02:56:56an(z) e(k) FIG. 3. General form of the acoustic impedance controller with the error signal given by a combination of the signals from the two-microphone probe. obtained after interchanging the order in which the transfer functions E2 (z) and H(z) are applied is shown in Fig. 4(b). If H(z) is implemented as an adaptive FIR filter the struc- ture in Fig. 4(b) leads directly to the filtered-X LMS algo- rithm to update the coefficients of the control filter. 13 q(k) . E() .{z, I (a) e(k) I / rgz) '- mz) + e(k) I (b) FIG. 4. Reduced diagram of the acoustic impedance controller in terms of the transfer functions from the primary and secondary signals to the error signal. (a) Direct reduction. (b) After interchanging the secondary error filter with the control filter. The optimal unconstrained control filter that makes the error identically zero is easily seen to be, Hop t (z) = -- g 1 (z)/g 2 (j). (10) This represents a stable filter provided the zeros of E2 (z) are all within the complex unit circle, in a practical implementa- tion (especially with fixed-point arithmetic) the optimal so- lution could lead to instability even if E2 (z) has a zero inside but close to the complex unit circle. The causality require- ment is satisfied if the minimum delay in E1 (z) is greater than, or at least equal to, the minimum delay in E2 (z) in such a way that the series expansion of Hop t (z) contains only negative powers of z, and thus represents a physically reali- zable system which introduces positive delays only. This condition can be satisfied if the two-microphone probe is located close to the secondary source and far enough from the primary source (on the assumption that the primary sig- nal is available to the controller with a negligibly small de- lay). When the control filter is constrained to be FIR, it can be designed to approach the optimal filter Hopt (z) in a least- squares sense. The FIR filter structure has the advantage of being both stable and causal. Figure 5 shows a complete diagram of the controller as used in the experiments described in the next section. It shows, particularly, how the control filter H(z) is updated using information from the filtered input signal and the error signal (i.e., the filtered-X LMS algorithm). The secondary (11(Z) G21(z) Pt (k) () I + e(k) s(n) + e(n) FIG. 5. Diagram of the acoustic impedance controller showing the adapta- tion of the H(z) filter with the filtered-X LMS algorithm and the adaptive identification of the secondary error filter E2 (z) [ r(k) = random se- quence ]. 2742 J. Acoust. Soc. Am., Vol. 91, No. 5, May 1992 F. Ordura-Bustamante and P. A. Nelson: Adaptive controller 2742 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Sat, 22 Nov 2014 02:56:56Primary source " 1o5 m Microphones 1 and 2 Foam Secondary source FIG. 6. Acoustical system used during the experimental work. error filter E2 (z) must be identified previously by another adaptive algorithm (the LMS algorithm in the "system iden- tification" form). The input to this identification stage is provided by an auxiliary pseudorandom sequence denoted by r(k). During the experiments described below, the sec- ondary error filter E2 (z) was identified before adapting the control filter H(z), with E2 (z) remaining fixed afterward. However, the identification of E2 (z) can be used even dur- ing the adaptation of the control filter H(z) so that the con- troller is able to adapt continuously to changes in the acous- tic system. TM The Appendix provides a formulation that takes into account a general form for the desired acoustic impedance and thus allows the discussion presented here to be extended to other types of termination impedances. II. EXPERIMENTS WITH AN ABSORBING TERMINATION A. Hardware implementation of the controller The adaptive impedance controller described above was implemented in hardware using a Texas Instruments TMS320C25 digital signal microprocessor working with 16- bit fixed-point data. Signals were processed via 12-bit D/A and A/D converters and using analog low-pass filters for anti-aliasing and reconstruction purposes. The experiments were aimed to achieve an absorbing termination at one end of a narrow PVC duct terminated by loudspeakers. Details of the acoustical system used are shown in Fig. 6. The duct has a circular cross section which can propagate higher-or- der modes, other than the plane-wave propagating mode, for kr> 1.84 approximately (see, for example, Kinsler et al. 5 ). With r = 0.05 m this gives a value around 2000 Hz for the lowest cut-on frequency. All the experiments were per- formed at frequencies below 1000 Hz which is well in the region of purely plane-wave propagation. The loudspeakers were enclosed in small cavities filled with open-cell foam as shown. Prior to actually adapting the main control filter, a training procedure was carried out to obtain an FIR approx- imation of the transfer function between the secondary source and the error signal E2 (z) using the LMS algorithm in a standard "system identification" configuration and the pseudorandom input r(k) as shown in Fig. 5. After identify- ing this transfer function, the control filter H(z) was updat- ed using the filtered-X LMS algorithm. The filter length was set to 256 coefficients and the sampling frequency to 1700 Hz so that the sampling period equals the time delay between Y, 400mV + X, BOOmV 400m 300m 200m lOOm -lOOm -200m -300m -400m FIG. 7. Lissajous figures of microphone 1 (Xaxis) versus microphone 2 (Yaxis) after adaptation using a sinusoidal input at 308 Hz. The thin elongated ellipse shows the same signals without control. --500m -400m -300m -200m --lOOm 0 lOOm 200m 300m 400m 500m 2743 J. Acoust. Soc. Am., Vol. 91, No. 5, May 1992 F. Orduda-Bustamante and P. A. Nelson: Adaptive controller 2743 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Sat, 22 Nov 2014 02:56:56the microphones for a microphone separation of 0.2 m and a speed of sound of c = 340 m/s. In this way, Eq. (4) is valid with N = 1 ( Ts = ') and reduces to a single delay H2 (z) = z- . ( 11 ) A Briiel&Kjaer dual channel signal analyzer type 2032 was used to perform the measurements reported below. B. Experimental results Using sinusoidal periodic signals the effect of the con- troller is shown by the Lissajous figures obtained from the two microphone signals (i.e., plotting œ vs œ2 ). These are shown in Fig. 7, where the frequency of the input signal (308 Hz) is close to a resonance of the duct (see the frequency response of the uncontrolled duct in Fig. 9), and in Fig. 8 where the frequency of the input signal (250 Hz) is far from resonance. When no control is applied, a standing wave is produced. This implies that at different points along the tube the sound pressure varies with time almost in phase or al- most in antiphase ("almost" because of the damping present in the system). This is revealed in the Lissajous figures as a very elongated ellipse (with no damping it would be a straight line crossing the origin). Also, as the sound pressure varies along the tube, the amplitude of each signal is ob- served to be different, especially at resonance. An extreme case is shown in Fig. 7 where the first microphone happens to be located close to an anti-node at this particular frequency and the corresponding signal has an amplitude considerably smaller than that coming from the second microphone. This results in an almost horizontal ellipse. The other ellipses shown in Figs. 8 and 9 (the "fat" ones) were obtained after adapting the system to achieve an absorbing termination. It is clear that the signals are forced to move out of phase and have a similar amplitude. This is precisely the behavior that should be expected if a plane wave were traveling along the duct. This shows that the active termination is effectively absorbing sound. Experiments with random and transient inputs lead to the results shown in Figs. 9-11. Figure 9 shows the frequen- cy response function between a reference microphone close to the primary source (not microphone 1) and the signal from microphone 2 in the error probe. This response is shown without control and when the control is applied try- ing to achieve an absorbing termination. The success of the controller can be judged from the extinction of the resonant behavior and the onset of a unit gain and a linear phase re- sponse in almost the whole 0- to 400-Hz range. This frequen- cy response is characteristic of a pure time delay which is the response that should be expected between the near field of the primary source and the second microphone in an infinite lossless duct. Figure 10 shows the signal captured by the second microphone in response to a pulse applied to the pri- mary loudspeaker. Note that the duration of the pulse (broadened by the low-pass filter) is of the order of magni- tude of the acoustic transit time along the duct so it is diffi- cult to distinguish individual reflections. However, the over- all effect of these reflections can still be noted clearly in the uncontrolled response. The trace corresponding to the con- trolled duct shows that the reflections are suppressed to a large extent when the control is applied. This is a feature to be ex, pected from an absorbing termination. Figure 11 shows the frequency response between microphones 1 and 2. The uncontrolled response shows clearly the influence of the im- pedance presented by the (uncontrolled) secondary loud- speaker at the end of the duct. However, when the control is applied the response shows again the unit gain and linear phase corresponding to a pure time delay as in Fig. 9. Y 400mV # X=, EIOOmV 4DOm 300m ZOOm lOOm --lOOm -ZOOm -300m -400m -',, ................. , .... , ......... - .................... . . . --500m --400m --300m --ZOOm lOOm 0 lOOm ZOOm 00m 400m 500m FIG. 8. Lissajous figures of microphone 1 (Xaxis) versus microphone 2 (Yaxis) after adaptation using a sinusoidal input at 250 Hz. The thin elongated ellipse shows the same signals without control. 2744 J. Acoust. Soc. Am., Vol. 91, No. 5, May 1992 F. Orduda-Bustamante and P. A. Nelson: Adaptive controller 2744 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Sat, 22 Nov 2014 02:56:56Y, O. Od8 40d8 X, OHz + 400Hz LIN 3O ZO 10 0 -10 oo ], oo I -:zoo I -''-u -00 0 50 1 oo 150 FIG. 9. Frequency response function be- - . . , . . . aoo 350 4oo tween a reference microphone close to the primary source and microphone 2. With- out control (solid line) and after adapta- tion (dashed line). :00 :50 300 :350 4.00 Ya -00 TO 00 OEG X, 0Hz 400Hz LIN 1.13 13.5 -13. 5 -1.13 1.13 0.5 -0. 5 -1.13 Ya 1. OOV X, O. OOmo .4. 230mo IA, AA AA .A ........... , Vv.V.,v-v , 13 213m 413m 8Om 813m 11313m 1213m 1413m 1813m 1813m Z1313m 220m 2413m 13 213m 413m 6Om 813m 1013m 120m 1413m 1613m 1813m 2013m 2213m 2413m Y, 1. OOV X, O. OOmm -' Z$Omm FIG. 10. Response at microphone 2 to a pulse applied to the primary loudspeaker. Without control (above) and after adap- tation (below). 2745 J. Acoust. Soc. Am., Vol. 91, No. 5, May 1992 F. Orduda-Bustamante and P. A. Nelson: Adaptive controller 2745 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Sat, 22 Nov 2014 02:56:562O 10 -10 oo loo -lOO -oo Y, 30. OdB 4OdB X, OHz 4OOHz LIN 0 50 ! O0 150 00 50 :900 50 400 I .... .... , .... ! .... , .... , .. . . . i .... , .... 0 5 100 150 00 :50 00 50 400 Y: -200 TO 2OO DEG X: OHz 4OOHz LIN FIG. 11. Frequency response function be- tween microphones 1 and 2. Without con- trol (solid line) and after adaptation (dashed line). III. CONCLUSIONS The analysis presented here shows that adaptive control of acoustic impedance can be achieved using the well-known filtered-X LMS algorithm. Experiments with periodic, ran- dom, and transient inputs reveal that an active absorbing termination can be achieved in practice following a simple and very efficient digital signal processing technique. ACKNOWLEDGMENTS This work was presented as a part of the M.S. Research Project of one of us (FOB) at the Institute of Sound and Vibration Research and sponsored by the Universidad Na- cional Aut6noma de M(xico. We are most grateful with members of the Signal Processing and Control Group within ISVR for their technical support during this work. APPENDIX: THE ERROR SIGNAL FOR AN ARBITRARY DESIRED IMPEDANCE The (digital) acoustic impedance is the ratio of the z transforms of the sound pressure and particle velocity sig- nals, X(z) -- P(z)/U(z). (A1) Introducing the standard finite difference approximation used for sound intensity measurements, P(z) and U(z) can be written in terms of the z transforms of the sound-pressure signals picked up by the pair of microphones as follows: P(z) = «[P, (z) + P2 (z) ], U(z) = (1/,os)I(z)IF, (z) -- P2 (z)]. (A2a) (A2b) Equation (A2b) is a digital realization of the momentum equation jcopu - - Op/Ox, ,o is the mass density of air, s is the separation of the microphones and I(z) is a digital real- ization of an analog integrator (with frequency response function 1/jco); a possible realization (using the bilinear transformation 16 ) being, I(z) = (Ts/2)(1 -I-z-)/(1 (A3) where Ts is the sampling period. Let Xd (z) represent a digital realization of the desired acoustic impedance at the center of the microphone probe. Then it is clear [ from Eqs. (A 1 ), (A2a), and (A2b) ] that the requirement X(z) = X, (z) implies a restriction on the ratio of the pressure signals Hi2 (z) -- P2 (z)/Pl (z) only. Thus an error signal can be defined more generally as, E(z) = H (z)P (z) + H2 (z)P2 (z), (A4) with Hi (z), H2 (z) chosen to satisfy the required ratio H (z)/H2 (z) = - H2 (z) when E(z) = 0. This freedom in choosing H (z) and H2 (z) can be used to improve the design of the controller. Note that in order to synthesize the error signal in the time domain H (z) and H2 (z) must again be specified as realizable (i.e., causal and stable) digital filters. When the error signal is defined by Eq. (1) (as in the main text of the paper), the implementation requires a digi- tal approximation ofH2 (z) itself. A possibility is as follows. At low frequencies, the analog transfer function between the microphones [ Eq. (2) in the paper ] can be approximated as, 2746 J. Acoust. Soc. Am., Vol. 91, No. 5, May 1992 F. Ordufia-Bustamante and P. A. Nelson: Adaptive controller 2746 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Sat, 22 Nov 2014 02:56:56( H2 (jco) = , co',l. (ASa) Xa (jw)/pc + jw/2 The desired digital transfer function derived from this can be written as, (2/r)I(z)Xa(z)/pc-- 1 H2 (z) = , (A5b) (2/)I(Z)Xd(Z)/pC + 1 where the digital integrator I(z) is defined in Eq. (A3). The desired acoustic impedance X d (z) can be inserted in this equation in order to find a particular filter structure for H2 (Z). Finally, Eq. (A5b) suggests the use of H (z) = 1 -- (2/)I(z)Xa(z)/pc, (A6a) H: (z)= 1 + (A6) in Eq. (A4) which then can be written as, E(z = P: + (z) - (z) ]. (A7) This provides an alternative definition of the error signal that can be easily implemented in a digital microprocessor. P. A. Nelson and S. J. Elliott, "Active Minimization of Acoustic Fields," J. Theoret. Appl. Mech. Suppl. 6, 39-98 (1987). S. D. Snyder and C. H. Hansen, "Active noise control in ducts: Some physical insights," J. Acoust. Soc. Am. 86, 184-194 (1989). 3 H. Kuttruff, Room Acoustics (Applied Science, United Kingdom, 1979), 2nd ed. 4 K. Karcher, "Active Modification of the Acoustic Wall Impedance for Normal Incidence," Ph.D. thesis, Drittes Physikalisches Institut, Univ. of Gottingen, 1982 [Abstract in J. Acoust. Soc. Am. 78, 810 (1985) ]. M. Rollwage, "Free Field Investigation on Coherent Active Control of the Acoustic Wall Impedance," (Ph.D. thesis), Drittes Physicalisches In- stitut., Univ. of Gottingen, 1984 [Abstract in J. Acoust. Soc. Am. 78, 810 (1985)]. 6 D. Guicking, K. Karcher, and M. Rollwage, "Coherent Active Methods for Applications in Room Acoustics," J. Acoust. Soc. Am. 78, 1426-1434 (1985). ? S. J. Elliott, I. M. Stothers, and P. A. Nelson, "A multiple error LMS algorithm and its application to the active control of sound and vibra- tion," IEEE Trans. ASSP-3$, 1423-1434 (1987). 8 p. A. Nelson and S. J. Elliott, Active Control of Sound (Academic, New York, 1992). 9 A. R. D. Curtis, P. A. Nelson, S. J. Elliott, and A. J. Bullmore, "Active Suppression of Acoustic Resonance," J. Acoust. Soc. Am. 81, 624-631 (1987). ,o Felipe Ordufia-Bustamante, "The Adaptive Control of Acoustic Imped- ance," M.S. thesis, Southampton University, 1989. "J. Y. Chung and D. A. Blaser, "Transfer Function Method of Measuring In-Duct Acoustic Properties. I. Theory," J. Acoust. Soc. Am. 68, 907- 913 (1980). ,2 j. y. Chung and D. A. Blaser, "Transfer Function Method of Measuring In-Duct Acoustic Properties. II. Experiment," J. Acoust. Soc. Am. 68, 914-921 (1980). ,3 B. Widrow and S. D. Stearns, Adaptive Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1985). ,4 L. J. Eriksson, M. C. Allie, C. D. Bremigan, and J. A. Gilbert, "Active Noise Control on Systems with Time-Varying Sources and Parameters," Sound Vib. (1989). ,5 L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamen- tals of Acoustics (Wiley, New York, 1982), p. 222, 3rd ed. ,6 R. A. Roberts and C. T. Mullis, Digital Signal Processing (Addisson- Wesley, Reading, MA, 1987). 2747 J. Acoust. Soc. Am., Vol. 91, No. 5, May 1992 F. Ordufia-Bustamante and P. A. Nelson: Adaptive controller 2747 Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Sat, 22 Nov 2014 02:56:56

1.4773370.pdf

Stochastic switching asymmetry in magnetoresistive stacks due to adjacent nanowire stray field M. T. Bryan, N. A. Porter, J. S. Claydon, M. A. Bashir, G. Burnell et al. Citation: Appl. Phys. Lett. 101, 262404 (2012); doi: 10.1063/1.4773370 View online: http://dx.doi.org/10.1063/1.4773370 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v101/i26 Published by the American Institute of Physics. Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 24 Apr 2013 to 142.51.1.212. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsStochastic switching asymmetry in magnetoresistive stacks due to adjacent nanowire stray field M. T. Bryan,1N. A. Porter,2J. S. Claydon,2,a)M. A. Bashir,1G. Burnell,2C. H. Marrows,2 T. Schrefl,3and D. A. Allwood1 1Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom 2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United kingdom 3St. P €olten University of Applied Sciences, St. P €olten, Austria (Received 2 November 2012; accepted 11 December 2012; published online 27 December 2012) Giant-magnetoresistance was used to measure the switching of patterned multilayer stacks either close to or removed from a ferromagnetic nanowire. Stray fields from the nanowire greatly changed the stack’s free layer hysteresis characteristics. Four distinct switching modes were observed when the applied field opposed the pinned layer magnetization, but reproducibleswitching occurred otherwise. Micromagnetic modeling suggested that the asymmetry was due to interlayer stray field interactions and the Oersted field from the measuring current, while the switching modes were due to transverse components from the nanowire stray field. The resultsdemonstrate the feasibility of remote electrical detection of nanowire magnetization. VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4773370 ] Stray field interactions between magnetic elements are important for several proposed applications, from data proc- essing1–3to atomistic4and biological5–8manipulation and sensing. Coupling produces behavior that would be energeti-cally unfavorable in isolated elements. 9–16Furthermore, the interaction modifies the switching process of the individual elements.17The switching field of each element depends on the element separation and also the shape and magnetization of its neighbor. Recently, we proposed using stray field cou- pling to monitor domain wall characteristics in a nanowirewith an adjacent magnetoresistive (MR) stack. 18This scheme would provide a localized, electrically readable probe for nanowire devices that was compatible with semiconductortechnology. Here, we demonstrate that nanowire stray fields can be detected by giant-magnetoresistive (GMR) stacks adja- cent to the wire end. This design improves on previous“racetrack”-style read-out schemes, where the nanowire forms part of the MR stack, 19–21as it is more adaptable to complex circuits and simplifies the fabrication requirements. Unexpect-edly, we found that the stochastic switching behavior of the GMR stacks did not respond symmetrically with reversal of the nanowire magnetization. Although we use GMR stacks,the results are equally applicable to magnetic tunnel junctions (MTJs), as the magnetization behavior of the free layer is not dependent on the composition of the spacer layer in the stack. Elliptical GMR stacks were fabricated using electron beam lithography (EBL) and lift-off, either as isolated ele- ments or adjacent to Permalloy (Ni 80Fe20) nanowires, with an average separation of g¼140 nm (Fig. 1). The stacks were designed with the same diameter in one (y) direction, meas- ured as dy¼640 nm, but with an orthogonal diameter varied to achieve aspect ratios a¼dx/dy¼0.90, 1.00, and 1.05. Pre- vious modeling had suggested that sensors are most sensitive for near circular geometries.18Each stack was sputtered under a 150 Oe magnetic field, Hpin, directed along the – ydirection(Fig. 1) with the composition Ta(5.3 nm)-Ni 80Fe20(3.9 nm)- Co90Fe10(0.5 nm)-Cu(2.6 nm)-Co 90Fe10(2.2 nm)-Ru(0.79 nm)- Co90Fe10(2.2 nm)-IrMn(10 nm)-Ru(2.5 nm). The free layer, used to detect the nanowire stray field, was composed of theneighboring Ni 80Fe20and Co 90Fe10layers positioned at the bottom of the stack. Throughout the experiment, the reference layer (the Co 90Fe10layer between the Cu and Ru layers) mag- netization was aligned with the field applied during growth. Thin film measurements of this composition yielded a GMR ratio ( Rp-Rap)/Rap¼5.14%, where Rpand Rapare the stack resistance with the free and reference layers parallel and anti- parallel. Following an in situ ion mill to clean the sample sur- face, Ti(20 nm)-Au(100 nm) electrodes patterned in a furtherEBL and lift-off stage were sputtered on to the stacks. The electrodes overlapped the stacks by a distance of s¼150 nm on each side, to enable electrical connection. The thermallyevaporated Permalloy nanowire, also patterned by EBL and lift-off, was 400 nm wide, 20 nm thick, and 10 lm long, termi- nating in a nucleation pad (side 2 lm, pitch 45 /C14,F i g . 1(b)). FIG. 1. (a) Schematic showing the geometry of the wire, GMR stack and contact electrodes, and indicating the direction of the pinning field during growth, Hpin. (b) False-color SEM image of a working device. CE ¼coupled ellipse, N ¼nanowire, and E1, E2, and E3 ¼electrodes. Note electrode E3 is not connected to the ellipse shown.a)Present address: Nanometrics (UK) Ltd., York YO26 6RU, United Kingdom. 0003-6951/2012/101(26)/262404/4/$30.00 VC2012 American Institute of Physics 101, 262404-1APPLIED PHYSICS LETTERS 101, 262404 (2012) Downloaded 24 Apr 2013 to 142.51.1.212. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsNanowire switching was measured using a magneto-optical Kerr effect (MOKE) magnetometer with a laser spot focused to a 4 lmd i a m e t e r .22Separately, two-wire MR measurements (current 0.2 lA) were used to detect stack switching. In all measurements, the magnetic field was applied parallel to the wire length (the y-axis). A hybrid finite element/bounda ry element method that sol- ves the Landau-Lifshitz-Gilbert and quasi-static Maxwell equa- tions was used to calculate the magnetization dynamics and MR of the system.18,23The modeled structures were similar to Fig. 1(a), with g¼100 nm and dy¼dx¼600 nm. Due to computational limitations, a 10 nm tetrahedral mesh was usedthroughout. Also, the modeled wire was 400 nm wide, 5 nm thick, and 1.6 lm long, with a tapered end instead of a nuclea- tion pad and no edge roughness. The stack composition wassimplified to Ta(6 nm)-Ni 80Fe20(5.5 nm)-Cu(2.5 nm)-Co 90Fe10 (3 nm)-Ru(0.8 nm)-Co 90Fe10(3 nm)-IrMn(10 nm)-Ta(5 nm), but the maximum GMR ratio matched the experimental thinfilm value of 5.14%. Standard magnetization constants of Perm- alloy were used (exchange stiffness A¼13 pJ m /C01, saturation magnetization Ms¼800 kA m/C01, zero magnetocrystalline ani- sotropy, and Gilbert damping constant a¼0.02). GMR calcula- tions included the current shunting effect due to the different conductivities of the layers in the stack24and were performed using a 1 mA current passing between the 50 nm thick, 424 nm wide Au electrodes overlapping the stack by s¼88 nm. Except where stated, the reference layer was initialized along the /C0y direction and the pinned (Co 90Fe10-IrMn) layer along the þy direction, to achieve a high resistance state under a positive field. Due to the effects of nanopatterning and the contact re- sistance, the experimentally measured saturation GMR ratio for each ellipse, MR S, varied between structures and was smaller than the thin film GMR ratio. For isolated ellipses MR S¼1.17%, 1.33%, and 1.76% for a¼0.90, 1.00, and 1.05. The corresponding MR Sfor the ellipses with adjacent nanowires (“coupled ellipses”) appeared to display a system- atic reduction by a factor of 0.3, so that for a¼0.90, 1.00, and 1.05, MR S¼0.86%, 0.92%, and 1.23%. These values are similar to the GMR ratios expected from micromagnetic models of ellipses adjacent to domain walls.18 Figures 2(a)and2(c)show the MR response of the ellip- ses, averaged over 2–24 field cycles and normalized to MR S. Orange peel coupling between the free and reference layers caused an offset in all of the hysteresis loops. The isolatedellipses had a coercivity of 1–2 Oe, with hard-axis switching behavior seen when a¼1.05, but easy-axis switching behav- ior occurring otherwise. In each of the coupled ellipses, thecoercivity increased to 13 Oe, due to the nanowire stray field. The nanowire coercivity was 65 Oe (Fig. 2(d)), so the ellipse switching preceded the nanowire reversal in each direction.Therefore, the nanowire stray field always acted to stabilize the ellipse magnetization. The 11 Oe increase in the coupled ellipse coercivity (Figs. 2(a)–2(c)) indicates the average stray field over the nucleation volumes. Switching in the isolated ellipses, while different for each aspect ratio, was reproducible over consecutive field cycles.The a¼1.05 coupled ellipse also displayed reproducible switching, although the switching transition was less linear than in the isolated case, showing that in addition to shiftingthe ellipse coercivity, the nanowire stray field also modified the ellipse switching mechanism. Stochastic changes wereobserved in the switching of the a¼0.90 coupled ellipse over 24 single-cycle hysteresis loops (Fig. 3). While switching was reproducible during the increasing field ramp, four distinctswitching modes occur during the decreasing field ramp. These consisted of either a single-step process at low ( /C06O e , mode A) or high ( /C020 Oe, mode B) fields, or a two-step pro- cess with one step at negative ( /C03 Oe, mode C) or positive (4 Oe, mode D) fields, and the other step at /C021 Oe. The switching mechanisms of the four modes may be interrelated.For example, mode C appears to begin similarly to mode A, but later change to mode B. Similarly, the final mode D switch also bears a strong resemblance to mode B, although the ini-tial switch is distinct from either of the single-step switching modes. Mode D accounts for 46% of the observed reversals, with more than twice the number of counts as the next com-mon modes (B and C). However, using a chi-square test, we FIG. 2. Averaged normalized MR hysteresis loops of ellipses with aspect ra- tio (a) a¼0.90, (b) a¼1.00, and (c) a¼1.05, measured for isolated ellipses, E, and for ellipses with adjacent nanowires, E þW. (d) Normalized MOKE hysteresis loop from a nanowire. FIG. 3. MR hysteresis loops from the a¼0.90 ellipse (adjacent to a nano- wire) showing stochastic switching. 24 single-cycle loops have been grouped into four data sets (A–D), corresponding to different switchingmodes. The number of cycles averaged in each set is displayed as a fraction of the total measured.262404-2 Bryan et al. Appl. Phys. Lett. 101, 262404 (2012) Downloaded 24 Apr 2013 to 142.51.1.212. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionscannot discount the null hypothesis that the four modes are equally likely (5% significance level), so the observed distri- bution could be due to random statistical fluctuations, rather than an indication that mode D is energetically favored. Micromagnetic modeling gives an insight into the origin of the asymmetric switching behavior. Figure 4shows the modeled switching of isolated and coupled 600 nm diametercircular stacks. The coupled wires are magnetized along the þy (“Wire þM y”) or – y(“Wire –M y”) directions with end domains magnetized along the þxdirection. In the coupled ellipses, switching occurs via domain wall nucleation and the formation of a C-shaped magnetization state (Figs.4(b)–4(i)). Supplementary material SM1 shows the transi- tional states of the isolated ellipse are broadly similar to those of the coupled ellipses. 28However, in addition to off- setting the applied field, the wire stray field pins domain wall motion, so the hysteresis loop of the coupled stack has fea- tures that do not occur in isolated ellipses. In the experimental data (Fig. 2), the nanowire reversal caused different stray field conditions to act on the ellipse during the increasing and decreasing field ramps. We willreproduce the nanowire reversal in the modeled hysteresis by considering the “Wire –M y” data during the increasing field ramp, but the “Wire þMy” data during the field decrease. In this way, the ellipse switching path follows Figs. 4(b)–4(i).The coercivity of this “composite” hysteresis path is 19 Oe, while the coercivity of isolated ellipse is 8 Oe, which agrees qualitatively with experimental observations (Fig. 2). All of the modeled stacks displayed asymmetry between switching during the increasing and the decreasing field ramps, including the isolated stack. This suggests that the ex- perimental asymmetric switching may originate from withinthe stack rather than the interaction with the wire. One source of asymmetry comes from an interaction between the free layer and the stray field from the other layers in the stack. Models with reversed reference and pinned layers (not shown) display a negatively biased free layer. However, thebias favors some of the intermediate magnetization states more than others, shifting the field required to switch between the states by between 7 Oe and 23 Oe. The switch-ing field variation shows that interlayer coupling is sensitive to the alignment of the free poles in the free and reference layers. Further asymmetry (not shown) may be caused by theOersted field generated by the current used to measure the stack resistance. Compared to an isolated stack without a current applied, the current introduces a bias to the fieldrequired to switch between states of between 4 Oe and 9 Oe. The small variation in the bias reflects the non-uniform cur- rent distribution through the structure, which interacts withthe magnetization structure in the free layer. Further field non-uniformity occurs when the stacks are coupled to a wire. Figure 5shows the stray field profile from wires with opposite end domain magnetizations (at rema- nence). The simulations were performed without a stack, but the footprint of a 600 nm diameter stack is indicated for refer-ence. In each case, the stray field varies between 3 Oe and 45 Oe over the area of the free layer, averaging 9 Oe; close to the experimental shift in coercivity for coupled ellipses (Figs.2(a)–2(c)). The mirror symmetry axis of the stray field is not co-incident with the structural symmetry about the y-axis, but instead is offset by 680 nm, depending on the end domain magnetization. When the end domain is magnetized along the þxdirection (Fig. 5(a)), the angle of the stray field within the stack footprint varies between 63 /C14and 136/C14from the x-axis, such that the average field angle, 99/C14, has a component in the –xdirection, opposing the end domain magnetization. When FIG. 4. (a) Micromagnetically calculated MR hysteresis loops of an isolated 600 nm diameter circular GMR stack (No wire) and similar GMR stacks with adjacent nanowires magnetized in the þy(Wire þMy) or the – y (Wire-M y) directions. (b)–(i) show the magnetic configuration of the free layer and the nanowire at the points indicated in (a). The gray rectangles show the contact electrode overlap. FIG. 5. Micromagnetic models of possible magnetization configurations atthe nanowire end and the corresponding stray field. Arrows indicate the magnetization (stray field) direction; the color shows the x-magnetizationcomponent (stray field strength). The circles indicate where a 600 nm diame- ter circular GMR stack would sit in relation to the wire edge.262404-3 Bryan et al. Appl. Phys. Lett. 101, 262404 (2012) Downloaded 24 Apr 2013 to 142.51.1.212. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissionsthe end domain is in the – xdirection (Fig. 5(b)), the field is mirrored about the y-axis, so the stray field angle varies between 44/C14and 117/C14, averaging to 81/C14. Experimentally, the end domain configuration is likely to vary stochastically over different field cycles, introducing a mechanism for the modal switching seen in Fig. 3. Previ- ous investigations have shown that modal switching canresult from different initial magnetization configurations, 25 or can be induced by a transverse field,26which enhances (or destabilizes) the domain components that are parallel (or anti-parallel) to it. A similar mechanism is likely to gen- erate switching modes here, as Fig. 4shows that the magnet- ization curls at the side of the ellipse where switching is most likely to begin (Figs. 4(e)–4(g)). Four interaction configurations are possible, with the magnetization of theellipse at the nucleation point curling either clockwise or counter-clockwise to form a domain wall and the wire end domain aligned along either the þxor –xdirection. The four observed switching modes seen in Fig. 3are consistent with each interaction configuration having an unique switching path. Such a mechanism of modal switching does notexclude the possibility that the interaction between the stack and the wire may cause a bias towards particular configura- tions. Provided that the unfavored configurations existed asmetastable states that were energetically accessible, for example, via thermal activation during switching, a configu- rational bias would only affect the probability of each modeoccurring, not the number of modes. To examine the effect of the stray field symmetry, the free layer hysteresis was modeled with the end domain mag-netized in the – xdirection. This caused the magnetization of the free layer during the reversal process to be mirrored about the y-axis (compared to Fig. 4). For a perfect system, no energy difference occurs when both the free layer mag- netization and the field are reflected about the same axis. The mirroring of the reversal process did not, therefore, affectthe MR hysteresis loops, which were identical to those in Fig. 4(a). In experiment, the symmetry of the reflection is broken by factors, such as shape definition and edge rough-ness, which may modify the local domain wall nucleation field, 27and misalignment of the stack and wire, which would alter the stray field profile across the stack upon reversal ofthe end domain (Fig. 5). In conclusion, we have investigated the switching of GMR stacks interacting with the ends of nearby nanowires.Stray field from the wires biases the free layer reversal, pro- viding a remote method of reading nanowire device magnet- ization. Experimentally, asymmetric stochastic behavior wasobserved, with reproducible switching of the free layer tak- ing place during the increasing field ramp but up to four modes of reversal occurring during the decreasing fieldramp. Micromagnetic modeling suggested that the asymme- try may be related to stray fields from within the stack and the Oersted field of the measuring current. Modal switchingwas attributed to the transverse stray field component from the end domain of the wire. Interaction with the end domaininfluences the transverse magnetization of the free layer, which may become metastable if the end domain magnetiza- tion reverses. Fabrication factors that occur experimentally are likely to break the mirror symmetry of the stray field pro-file and separate the switching behavior of the four possible modes of reversal. The system displays a high degree of field non-uniformity, highlighting the need to understand thelocalized nature of the initial switching process. EPSRC Grants EP/F069359/1 and EP/F068573/1 funded this work. 1R. P. Cowburn and M. E. Welland, Science 287, 1466 (2000). 2A. Imre, G. Csaba, L. Ji, A. Orlov, G. H. Bernstein, and W. Porod, Science 311, 205 (2006). 3A. Lyle, A. Klemm, J. Harms, Y. Zhang, H. Zhao, and J.-P. Wang, Appl. Phys. Lett. 98, 092502 (2011). 4T. J. Hayward, A. D. West, K. J. Weatherill, P. J. Curran, P. W. Fry, P. M. Fundi, M. R. J. Gibbs, T. Schrefl, C. S. Adams, I. G. Hughes, S. J.Bending, and D. A. Allwood, J. Appl. Phys. 108, 043906 (2010). 5J. Llandro, T. J. Hayward, D. Morecroft, J. A. C. Bland, F. J. Castano, I. A. Colin, and C. A. Ross, Appl. Phys. Lett. 91, 203904 (2007). 6G. Vieira, T. Henighan, A. Chen, A. J. Hauser, F. Y. Yang, J. J. Chalmers, and R. Sooryakumar, Phys. Rev. Lett. 103, 128101 (2009). 7M. T. Bryan, K. H. Smith, M. E. Real, M. A. Bashir, P. W. Fry, P. Fischer, M. Y. Im, T. Schrefl, D. A. Allwood, and J. W. Haycock, IEEE Magn. Lett. 1, 1500104 (2010). 8P. Vavassori, M. Gobbi, M. Donolato, M. Cantoni, R. Bertacco, V. Metlushko, and B. Ilic, J. Appl. Phys. 107, 09B301 (2010). 9P. Vavassori, D. Bisero, V. Bonanni, A. Busato, M. Grimsditch, K. M. Lebecki, V. Metlushko, and B. Ilic, Phys. Rev. B 78, 174403 (2008). 10L. O’Brien, D. Petit, H. T. Zeng, E. R. Lewis, J. Sampaio, A. V. Jausovec, D. E. Read, and R. P. Cowburn, Phys. Rev. Lett. 103, 077206 (2009). 11T. J. Hayward, M. T. Bryan, P. W. Fry, P. M. Fundi, M. R. J. Gibbs, M. Y. Im, P. Fischer, and D. A. Allwood, Appl. Phys. Lett. 96, 052502 (2010). 12T. J. Hayward, M. T. Bryan, P. W. Fry, P. M. Fundi, M. R. J. Gibbs, D. A. Allwood, M. Y. Im, and P. Fischer, Phys. Rev. B 81, 020410(R) (2010). 13M. T. Bryan, J. Dean, T. Schrefl, F. E. Thompson, J. Haycock, and D. A. Allwood, Appl. Phys. Lett. 96, 192503 (2010). 14L. O’Brien, D. Petit, E. R. Lewis, R. P. Cowburn, D. E. Read, J. Sampaio, H. T. Zeng, and A. V. Jausovec, Phys. Rev. Lett. 106, 087204 (2011). 15E. Rapoport and G. S. D. Beach, J. Appl. Phys. 111, 07B310 (2012). 16E. Rapoport and G. S. D. Beach, Appl. Phys. Lett. 100, 082401 (2012). 17S. Basu, P. W. Fry, M. R. J. Gibbs, T. Schrefl, and D. A. Allwood, J. Appl. Phys. 105, 083901 (2009). 18M. A. Bashir, M. T. Bryan, D. A. Allwood, T. Schrefl, J. S. Claydon, G. Burnell, and C. H. Marrows, J. Appl. Phys. 110, 123912 (2011). 19S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 20S. Glathe, R. Mattheis, and D. V. Berkov, Appl. Phys. Lett. 93, 072508 (2008). 21T. Uemura, K. Sawada, K. Matsuda, and M. Yamamoto, Appl. Phys. Lett. 95, 012502 (2009). 22U. J. Gibson, L. F. Holiday, D. A. Allwood, S. Basu, and P. W. Fry, IEEE Trans. Magn. 43, 2740 (2007). 23O. Ertl, G. Hrkac, D. Suess, M. Kirschner, F. Dorfbauer, J. Fidler, and T. Schrefl, J. Appl. Phys. 99, 08S303 (2006). 24A. T. McCallum and S. E. Russek, Appl. Phys. Lett. 84, 3340 (2004). 25K. Shigeto, K. Miyake, T. Okuno, K. Mibu, T. Ono, Y. Yokoyama, T. Kawagoe, Y. Suzuki, and T. Shinjo, J. Magn. Magn. Mater. 240, 301 (2002). 26M. T. Bryan, P. W. Fry, T. Schrefl, M. R. J. Gibbs, D. A. Allwood, M. Y. Im, and P. Fischer, IEEE Trans. Magn. 46, 963 (2010). 27M. T. Bryan, D. Atkinson, and R. P. Cowburn, J. Phys.: Conf. Ser. 17,4 0 (2005). 28See supplementary material at http://dx.doi.org/10.1063/1.4773370 for the magnetic configuration of the isolated ellipse free layer during the “No wire” hysteresis loop in Fig. 4(a).262404-4 Bryan et al. Appl. Phys. Lett. 101, 262404 (2012) Downloaded 24 Apr 2013 to 142.51.1.212. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://apl.aip.org/about/rights_and_permissions

1.3383044.pdf

Effect of light rare earth element Nd doping on magnetization dynamics in Co–Nb films Zhengmei Zhang, Min Lin, Jingyi Zhu, Dangwei Guo, Guozhi Chai, Xiaolong Fan, Yuancai Yang, and Desheng Xuea/H20850 Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, People’ s Republic of China /H20849Received 28 November 2009; accepted 9 March 2010; published online 23 April 2010 /H20850 /H20849Co0.85Nb0.15/H20850100−xNdx/H20849x=0, 1.1, 3.4, and 6.4 /H20850soft magnetic thin films have been prepared on Si substrates by oblique sputtering /H1101116°. The dynamic properties of the films were systematically investigated in a wide frequency range from 0.1 to 7 GHz. Strong enhancement of the dampingparameter which is one key materials parameter that controls the dynamic response and the fullwidth at half maximum of the imaginary permeability spectra were observed when Nd element wasdoped. The fitted value of the damping parameter for /H20849Co 0.85Nb0.15/H2085093.6Nd6.4film is around 0.1, which is almost one order larger than that 0.015 of /H20849Co0.85Nb0.15/H20850100film. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3383044 /H20852 I. INTRODUCTION The dynamic response of magnetic materials is of fun- damental interest and is essential for various applications inmodern magnetic storage technology. 1The applications of soft magnetic film materials are usually based on the analysisof the dynamic magnetic properties or the process of magne-tization subjected to an effective magnetic anisotropy fieldH effas given by the Landau–Lifshitz–Gilbert /H20849LLG /H20850 equation2 d dtM=−/H9253M/H11003Heff+/H9251 4/H9266MS/H20873M/H11003d dtM/H20874, /H208491/H20850 where /H9253is the gyromagnetic ratio, Mis the instantaneous magnetization vector, and 4 /H9266Msis the saturation magnetiza- tion of the film. The first term in Eq. /H208491/H20850describes the gyro- scopic precession of Mwith a characteristic precession or resonance frequency frproportional to Heff.3The second term in Eq. /H208491/H20850describes its dissipation. The dissipation, or magnetic damping, is described by the dimensionless con-stant /H9251. Therefore, the key materials parameters which de- scribe the dynamic response of soft magnetic thin films arethe resonance frequency and the damping parameter. For application it is desirable that the damping parameter and the resonance frequency of magnetic materials can betuned independently. While the resonance frequency can becontrolled relatively easily base on the bianisotropic picture 4 by, e.g., controlling the in-plane uniaxial magneticanisotropy, 5,6the ways to change the damping parameter are doping ferromagnetic /H20849FM/H20850thin films with transition metal elements,7,8depositing multilayer,9,10or diluting the FM material.11–13Previously most studies on this topic were fo- cused on NiFe-based films and heavy rare earth elementdoping. 7–9,14–16 Amorphous Co–Ti,17Co–Nb,6and Co–Zr /H20849Ref. 18/H20850thin films have attracted much attention since they have attractivesoft magnetic properties, which are required in many high frequency applications.19In this work, we investigate the properties of Co–Nb thin films with doping low light rareearth element Nd. The soft magnetic properties of thesamples are retained by doping Nd, and a tuning of thedamping parameter /H9251is achieved. II. EXPERIMENT /H20849Co0.85Nb0.15/H20850100−xNdx/H20849x=0, 1.1, 3.4, and 6.4, respec- tively /H20850soft magnetic thin films with thickness /H11011160 nm were prepared by radio frequency sputtering on Si /H20849111/H20850sub- strates attached to a water cooling system. A Co plate of 70mm in diameter and 3 mm in thickness was used as the targeton which Nb and Nd chips were placed. The composition ofthe deposited films was adjusted by controlling the numberof the Nd chips, at the same time, the number and the loca-tion of the Nb chips remained unchanged. Films were depos-ited at an oblique angle of 16° to enhance the anisotropyfield. 6,20The oblique angle is defined as the angle between the normal direction of target and the line from the targetcenter to the substrate center, as shown in Fig. 1 of Ref. 6. The background pressure was less than 2 /H1100310 −5Pa, and the working Ar pressure was 0.15 Pa with an Ar flow rate of 20SCCM /H20849SCCM denotes cubic centimeter per minute at STP /H20850, and the radio frequency power density was 1.7 W /cm 2. The compositions were measured by energy dispersive x-rayspectroscopy. The static magnetic measurements were per-formed using a vibrating sample magnetometer /H20849Lakeshore model 7304 /H20850. The microwave permeability measurements of the films were carried out with a PNA E8363B vector net-work analyzer using the microstrip method from 100 MHz to7 GHz. 21All the above measurements were carried out at room temperature. III. RESULTS AND DISCUSSION Figure 1shows the in-plane magnetic hysteresis loops of the /H20849Co0.85Nb0.15/H20850100−xNdxthin films. The static magnetica/H20850Author to whom correspondence should be addressed. Electronic mail: xueds@lzu.edu.cn.JOURNAL OF APPLIED PHYSICS 107, 083912 /H208492010 /H20850 0021-8979/2010/107 /H208498/H20850/083912/4/$30.00 © 2010 American Institute of Physics 107 , 083912-1properties approximate ideal easy axis /H20849EA/H20850and hard axis /H20849HA/H20850loop shapes, as evidenced by high coercive squareness along the EA and near-zero remanence along the HA. Differ-ence in the hysteresis loops measured along EA and HAexhibits an in-plane uniaxial magnetic anisotropy. The coer-civity of all samples with Nd doping keeps low values,which looks insensitive to the Nd concentration. The depen-dence of coercivity and static magnetic anisotropy field onNd composition for /H20849Co 0.85Nb0.15/H20850100−xNdxfilms is shown in Fig. 2. The coercivity along EA Hceis slightly reduced from /H110114 Oe for CoNb film to less than /H110111 Oe within the Nd- doped films. From the M-Hloops, the static magnetic aniso- tropy field Hk-stavarying from 70 to 140 Oe is roughly esti- mated, which shows no coherent dependence on Ndconcentration. The permeability spectra of the films are shown in Fig. 3, where /H9262/H11032and/H9262/H11033represent the real and imaginary parts of complex permeability, respectively. Compared to the staticmeasurements of M-Hloops shown in Fig. 1, the permeabil- ity spectra have more obvious dependence on the Nd con-centration, indicated by shifting of resonance frequency andbroadening of the peak width of the imaginary complex per-meability In order to determine the values of 4 /H9266Msand dynamic anisotropy Hk-dyn, we measured the resonance frequency fr,at which the /H9262/H11033show maximum, as a function of an applied magnetic field Happlalong in-plane EA. As shown in Fig. 4, the square value of frshows a linear relationship with Happl, which indicates that the resonance mechanism of/H20849Co 0.85Nb0.15/H20850100−xNdxfilms is natural resonance. The perme- ability spectrum of a thin film corresponding to the uniform gyromagnetic resonance can be calculated from the LLGequation. In the limit H k-dyn/H112704/H9266Ms, the resonance fre- quency can be simplified as8 fr2=/H925324/H9266MS/H20849Hk-dyn+Happl/H20850, /H208492/H20850 where /H9253=2.8 GHz /kOe. Plotting fr2as a function of Happl then allows calculation of 4 /H9266Msfrom the slope and of Hk-dyn from the intersection with the abscissa. The saturated FIG. 1. /H20849Color online /H20850EA and HA magnetization loops of /H20849Co0.85Nb0.15/H20850100−xNdxfilms with varying Nd concentration. FIG. 2. /H20849Color online /H20850Dependence of EA coercivity Hceand static magnetic anisotropy field Hk-staon the Nd concentration. FIG. 3. /H20849Color online /H20850Frequency dependence of the real /H9262/H11032/H20849a/H20850and imagi- nary/H9262/H11033/H20849b/H20850parts of complex permeability of /H20849Co0.85Nb0.15/H20850100−xNdxfilms, respectively. FIG. 4. /H20849Color online /H20850Dependence of the square of resonance frequency fr2 on the applied field Happlof/H20849Co0.85Nb0.15/H20850100−xNdxfilms.083912-2 Zhang et al. J. Appl. Phys. 107 , 083912 /H208492010 /H20850magnetizations 4 /H9266Ms=14.6 kOe and 12.7 kOe for /H20849Co0.85Nb0.15/H2085098.9Nd1.1and /H20849Co0.85Nb0.15/H2085096.6Nd3.4films can be obtained, respectively. Figure 5shows the values of Hk-dynandfrat zero exter- nal field as a function of Nd concentration. The varying ten-dency of f ris similar to Hk-dyn, but not entirely consistent since fr2/H11008Hk-dyn4/H9266MsatHappl=0, and 4 /H9266Msof Co–Nb film has been varied with the increase in Nd concentration. It canalso be found that the dynamic anisotropy field obtainedfrom the dynamic measurement is consistent with that fromstatic measurement. The slight discrepancy between bothvalues was also observed in other material systems. 22 Based on the LLG equation, the permeability spectrum of an in-plane magnetized thin film can be expressed as23,24 /H9262=1+fm/H20849f0+fm+i/H9251f/H20850 fr2−f2+if/H9004fr, /H208493/H20850 where fm=/H92534/H9266Ms/2/H9266,f0=/H9253Hk-dyn /2/H9266,fr2=f02+fmf0, and /H9004fr=/H9251/H208492f0+fm/H20850. All the experimental results of permeability spectra in Fig. 3can be fitted with Eq. /H208493/H20850.4/H9266MsandHk-dyn take the values from Fig. 4. The fitted results of /H9251is shown in Fig. 6. A linear relationship between /H9251and Nd concentra- tion is found. From the slope of the linear increase wedetermine the contribution to the total effective damping pa-rameter of Nd concentration by the formula, /H9251=/H9251CoNb +CNd/H9251Nd,14where CNdis the Nd atomic concentration in percent. The value for /H9251Ndis 0.014, and /H9251CoNb /H110150.013 forCoNb film, which are considerably larger than the bulk value /H9251Co/H110150.005 in Co.25,26In Fig. 3, the most obvious depen- dences of the spectra on the Nd concentration are that theresonance peak of the imaginary part grows graduallybroader, and meanwhile the maximum peak value /H9262/H11033de- creases. By means of the imaginary part of the frequency- dependent permeability, the full width at half maximum/H20849FWHM /H20850/H9004fcan be obtained with the results shown in Fig. 6. The pure Co–Nb film has a linewidth of about 0.55 GHz and the value increases in Nd-doped film and reaches 2.22GHz for the Nd concentration of 6.4. Based on the relation-ship between /H9004fand /H9251expressed as27/H9004f/H11008/H208494/H9266Ms+Hk-sta/H20850/H9251 for/H9251/H112701,/H9004fshould increase linearly with /H9251when 4 /H9266Ms andHk-stakeep constant. But in this case, 4 /H9266MsandHk-staof the Co–Nb–Nd films are variables due to the introduction ofNd element. These give rise to the different dependence /H9004f and /H9251on Nd concentration. IV. CONCLUSION In summary, we have developed an effective method to tune the magnetization dynamics in soft materials retainingthe important soft magnetic properties. Although it is diffi-cult to get an accurate dependence of the static anisotropy onNd concentration, the resonance frequency is consistent withthat of static anisotropy. This work also showed that the dy-namic damping effect in the CoNb-based soft magnetic filmscould be greatly strengthened by doping light rare earth ele-ment Nd. The FWHM of imaginary permeability is broad-ened correspondingly. ACKNOWLEDGMENTS The authors should thank Professor Yungui Ma for Eng- lish revision. Financial supports by National Natural ScienceFoundation of China /H20849NSFC /H20850/H20849Grant No. 10774062 /H20850, the Keygrant Project of Chinese Ministry of Education /H20849Grant No. 309027 /H20850, and National Science Fund for Distinguished Young Scholars /H20849Grant No. 50925103 /H20850are gratefully ac- knowledged. 1R. W. Wood, J. Miles, and T. Olson, IEEE Trans. Magn. 38, 1711 /H208492002 /H20850. 2T. L. Gilbert, IEEE Trans. Magn. 40, 3443 /H208492004 /H20850. 3C. Kittel, Phys. Rev. 73, 155 /H208491948 /H20850. 4X. De-Sheng, L. Fa-Shen, F. Xiao-Long, and W. Fu-Sheng, Chin. Phys. Lett. 25, 4120 /H208492008 /H20850. 5S. D. Li, Z. G. Huang, J. Duh, and M. Yamaguchi, Appl. Phys. Lett. 92, 092501 /H208492008 /H20850. 6X. L. Fan, D. S. Xue, M. Lin, Z. M. Zhang, D. W. Guo, C. J. Jiang, and J. Q. Wei, Appl. Phys. Lett. 92, 222505 /H208492008 /H20850. 7J. O. Rantschler, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F. Egelhoff, Jr., B. B. Maranville, D. Pulugurtha, A. P. Chen, and L. M.Connors, J. Appl. Phys. 101, 033911 /H208492007 /H20850. 8S. G. Reidy, L. Cheng, and W. E. Bailey, Appl. Phys. Lett. 82,1 2 5 4 /H208492003 /H20850. 9J. McCord, R. Kaltofen, O. G. Schmidt, and L. Schultz, Appl. Phys. Lett. 92, 162506 /H208492008 /H20850. 10B. K. Kuanr, S. Maat, S. Chandrashekariaih, V . Veerakumar, R. E. Cam- ley, and Z. Celinshi, J. Appl. Phys. 103, 07C107 /H208492008 /H20850. 11J. Fassbender and J. McCord, Appl. Phys. Lett. 88, 252501 /H208492006 /H20850. 12Y . Guan, and W. E. Bailey, J. Appl. Phys. 101, 09D104 /H208492007 /H20850. 13Y . Liu, Z. W. Liu, C. Y . Tan, and C. K. Ong, J. Appl. Phys. 100, 093912 /H208492006 /H20850. 14G. Woltersdorf, M. Kiessling, G. Meyer, J. U. Thiele, and C. H. Back, FIG. 5. /H20849Color online /H20850Dependence of EA dynamic anisotropy Hk-dynand resonance frequency fron the Nd concentration. FIG. 6. /H20849Color online /H20850Dependence of the FWHM /H9004fand damping param- eter/H9251on the Nd concentration.083912-3 Zhang et al. J. Appl. Phys. 107 , 083912 /H208492010 /H20850Phys. Rev. Lett. 102, 257602 /H208492009 /H20850. 15G. D. Fuchs, J. C. Sankey, V . S. Pribiag, L. Qian, P. M. Braganca, A. G. F. Garcia, E. M. Ryan, Z. Li, O. Ozatay, D. C. Ralph, and R. A. Buhrman,Appl. Phys. Lett. 91, 062507 /H208492007 /H20850. 16W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE Trans. Magn. 37, 1749 /H208492001 /H20850. 17K. Hayashi, M. Hayakawa, Y . Gchiai, H. Matsuda, W. Ishikawa, Y . Iwasaki, and K. Aso, J. Appl. Phys. 61, 2983 /H208491987 /H20850. 18C. J. Jiang, D. S. Xue, D. W. Guo, and X. L. Fan, J. Appl. Phys. 106, 103910 /H208492009 /H20850. 19Y . Luo, M. Esseling, K. Zhang, G. Güantherodt, and K. Samwer, Euro- phys. Lett. 73, 415 /H208492006 /H20850. 20Z. M. Zhang, X. L. Fan, M. Lin, D. W. Guo, G. Z. Chai, and D. S. Xue, J. Phys. D 43, 085002 /H208492010 /H20850.21Y . Liu, L. F. Chen, C. Y . Tan, H. J. Liu, and C. K. Ong, Rev. Sci. Instrum. 76, 063911 /H208492005 /H20850. 22R. Lopusnik, J. P. Nibarger, T. J. Silva, and Z. Celinski, Appl. Phys. Lett. 83,9 6 /H208492003 /H20850. 23N. X. Sun, S. X. Wang, and T. J. Silva, IEEE Trans. Magn. 38,1 4 6 /H208492002 /H20850. 24C. B. Craus, A. R. Chezan, D. O. Boerma, and L. Niesen, J. Phys.: Con- dens. Matter 16, 9227 /H208492004 /H20850. 25J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 /H208492000 /H20850. 26F. Schreiber, J. Pflaum, Z. Frait, T. Mühge, and J. Pelzl, Solid State Com- mun. 93, 965 /H208491995 /H20850. 27K. Seemann, H. Leiste, and C. Klever, J. Magn. Magn. Mater. 321, 3149 /H208492009 /H20850.083912-4 Zhang et al. J. Appl. Phys. 107 , 083912 /H208492010 /H20850Journal of Applied Physics is copyrighted by the American Institute of Physics (AIP). Redistribution of journal material is subject to the AIP online journal license and/or AIP copyright. For more information, see http://ojps.aip.org/japo/japcr/jspJournal of Applied Physics is copyrighted by the American Institute of Physics (AIP). Redistribution of journal material is subject to the AIP online journal license and/or AIP copyright. For more information, see http://ojps.aip.org/japo/japcr/jsp

1.1846945.pdf

Spin-current effect on ferromagnetic resonance in patterned magnetic thin film structures Haiwen Xi, Yiming Shi, and Kai-Zhong Gao Citation: Journal of Applied Physics 97, 033904 (2005); doi: 10.1063/1.1846945 View online: http://dx.doi.org/10.1063/1.1846945 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Evolution of ferromagnetic and spin-wave resonances with crystalline order in thin films of full-Heusler alloy Co2MnSi J. Appl. Phys. 111, 023912 (2012); 10.1063/1.3677996 Effect of localized magnetic field on the uniform ferromagnetic resonance mode in a thin film Appl. Phys. Lett. 94, 172508 (2009); 10.1063/1.3123264 Ferromagnetic resonance and spin momentum exchange in crystalline magnetic ultrathin films in noncollinear configuration J. Appl. Phys. 103, 07B505 (2008); 10.1063/1.2830645 Effect of the classical ampere field in micromagnetic computations of spin polarized current-driven magnetization processes J. Appl. Phys. 97, 10C713 (2005); 10.1063/1.1853291 Ferromagnetic resonance of amorphous Fe–Hf thin films J. Appl. Phys. 87, 5251 (2000); 10.1063/1.373311 [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: 158.42.28.33 On: Mon, 24 Nov 2014 11:06:01Spin-current effect on ferromagnetic resonance in patterned magnetic thin film structures Haiwen Xi, Yiming Shi, and Kai-Zhong Gao Phoenix Spintronics Group (PSG), 3019 Bobcat Trail NW, The Wilds South, Prior Lake, Minnesota 55372 (Received 4 August 2004; accepted 11 November 2004; published online 6 January 2005 ) We have theoretically investigated the ferromagnetic resonance in the magnetic thin film structures undertheinfluenceofspin-transfertorqueusingamodifiedLandau–Lifshitz–Gilbertequationinthelinearization regime. The study shows that spin currents do not shift the resonance field but ratherchange both the resonance amplitude and the linewidth. Ferromagnetic resonance under thiscircumstance can be characterized by an effective damping constant. Depending upon its direction,thespincurrentcanpumpenergyintoordissipateenergyfromthemagneticsystem.Inaddition,thequality factor of the resonance can be tuned by changing the current intensity. Ferromagneticresonance excited by ac electrical currents is also theoretically demonstrated and discussed in thisarticle. © 2005 American Institute of Physics .[DOI: 10.1063/1.1846945 ] I. INTRODUCTION The recent experimental observations of spin-wave excitations1,2and irreversible magnetization reversals3–7in magnetic hybrid structures have proved the original idea thatwas proposed by Berger 8and Slonczewski9independently in the late 1990s. The interplay of spin-polarized electrical cur-rents and local magnetic moments, called the spin-transfertorque effect, is of considerable research interest because ofits potential use in magnetoresistive random access memory(MRAM )for the current-switching write process 10and in nanodevices for generating microwave signals, of which thefrequency is tunable by a dc current. 9,11Although there has been intensive study toward understanding the microscopicorigins 12and the magnetization response13to spin currents in magnetic structures, the spin-transfer torque effect remainsintriguing to researchers. Ferromagnetic resonance (FMR )is a powerful experi- mental technique for the study of magnetic properties, suchas relaxation of the magnetization, of single-layer andmultilayer films. 14In light of the recent experimental study.15 of the spin-pumping effect by FMR measurements in mag-netic trilayers, we have theoretically investigated the spin-current effect on FMR. Using the linearization approach andassuming spatially uniform magnetization precession, wehave obtained analytical results of the resonance field andlinewidth as functions of the spin current. In addition, wepropose to use an ac spin-polarized current as the excitationsource for FMR in the article. The FMR result based on thisscenario is also presented. II. MAGNETIC STRUCTURE AND SPIN-TRANSFER TORQUE Magnetic trilayers with two ferromagnetic (FM)metallic layers separated by a nonmagnetic (NM)metallic spacer shown in Fig. 1 are considered in the study. An electricalcurrent is applied perpendicularly into the trilayers. Whenthe current leaves the first FM layer, FM 1, it is polarized with the majority of electron spins pointing along the direction ofthe FM 1magnetization. The spin polarization remains when the current passes through the NM layer, of which the thick-ness is much less than the spin diffusion length, and flowsinto the second ferromagnet, FM 2. In real FMR measure- ments, the electrical current should be confined and then thetrilayers have to be patterned. When the spin-polarized current is incident on FM 2, the magnetic moments of FM 2will experience a torque exerted by the spin current due to the s-dexchange interaction. Re- gardless of the microscopic origin of the spin-transfer torque,which is well discussed elsewhere, 12the Landau–Lifshitz– Gilbert (LLG )equation is employed in this study of the mag- netization dynamics. For the torque applied by the spin cur-rent, only the component perpendicular to the magnetizationand in the plane made by the magnetization and the spinpolarization direction is taken into account. 5,13Therefore the spin-transfer torque can be expressed as the second term ofthe LLG equation, FIG. 1. The configuration of a patterned FM1/NM/FM2trilayer and the coordinate system for FMR measurement. The graphic arrow shows thedirection of the electrical current.JOURNAL OF APPLIED PHYSICS 97, 033904 (2005 ) 0021-8979/2005/97 (3)/033904/5/$22.50 © 2005 American Institute of Physics 97, 033904-1 [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: 158.42.28.33 On: Mon, 24 Nov 2014 11:06:01dM dt=−gMˆHeff+gM Ms3sMˆIsd+aM MsˆdM dt, s1d where gis the gyromagnetic ratio, ais the Gilbert damping constant, Mis the magnetization vector of FM 2,Msis its saturation magnetization, Heffis the effective magnetic field including the applied field, the demagnetizing field, the an-isotropy field, the exchange field, and the dipolar interaction,andI sis the spin current vector that is normalized to the total moment of FM 2.The magnetic field induced by the current is ignored. The spin-current vector, Is, has the same direction as of the FM 1magnetization and a magnitude that is written as9,13,16 Is=" 2ehJ MstFM, s2d whereJis the electrical current density, his the polarization ratio, and tFMis the FM 2layer thickness. Equation (2)indi- cates that the spin-transfer torque is an interface effect.WhenFM 2is very thin, we assume that the spin torque is applied on the FM 2moments through the thickness. In this article, Is, which has the dimension of the mag- netic field, is used in the discussion. Iscan be positive if the electrical current flows from FM 1to FM2or negative if the current is applied in the opposite direction. It has been re-ported that the polarization ratio for Co is about 0.35. For acurrent density of 1.0 310 7A/cm2injected int o a 3 nm-thick Co layer with a saturation magnetization of 1420 emu/cm3, Is, is about 60 Oe. It is orders smaller in magnitude than the typical resonance field in FMR measurements. III. FEROMAGNETIC RESONANCE EXCITATED BY AC FIELDS Figure 1 shows the magnetization and the field configu- ration for a patterned trilayer in the FMR measurement. Thetrilayer lies in the x−yplane with a uniform electrical current flowing through in the zdirection. A static magnetic field H is applied along the positive xdirection while an ac field hstd with a magnitude in the order of mOe and a frequency in the microwave region is applied usually normal to the staticfield. The FMR study is isolated on the FMR of the FM 2 magnetization while FM 1is considered simply as a “spin polarizer.” Ferromagnetic resonances for both ferromagneticlayers would appear. As we will show, the resonance field isrelated to the magnetic properties such as anisotropies andmagnetization.The FMR should be able to be separated fromeach other from the difference in magnetic properties of thetwo FM layers. In order to illustrate the essence of the spin-current effect on FMR, the simplest case where there is noexchange coupling, magnetocrystalline anisotropy, or eventhe demagnetizing field on the FM 2magnetization is studied first and the assumptions are justified later in the text. There-fore,H eff=H+hstd, s3d where we introduce the excitation field as a complex vari- able, hstd=hexpsivtd. s4d vis the angular frequency of the ac excitation field. The nonlinearity of Eq. (1)implies that the magnetiza- tion dynamics be sophisticated. However, in FMR measure-ments, the magnetization precesses around the steady fielddirection with a small angle. Therefore, the magnetizationcan be written as M=M 0+mstd, s5d whereM0is the magnetization at equilibrium in the absence of the microwave field and mstdis the magnetization pertur- bation by the microwave field. SinceM0is time independent without the influence of the excitation field, it can be readily obtained from Eq. (1), mˆ0ˆH−mˆ0ˆsmˆ0ˆIsd=0, s6d wheremˆ0=M0/Msis the unit vector of M0. The direction of Isis defined by the FM 1magnetization vector. It is reason- able to assume that the FM 1magnetization is aligned by the large in-plane static field in our study for the trilayers com-posed of soft transition metal materials. Therefore, I s//H and Eq. (6)implies that M0has to be parallel or antiparallel toH. We know that the antiparallel configuration is not a stable solution when the current torque is sufficiently small.Therefore, M 0is considered in the discussion to lie along the positivexdirection with a magnitude of Ms. By neglecting the high-order terms of mstdand knowing thatM0//Is//H, we can derive from Eq. (1)that dmstd dt=−gmstdˆH−gM0ˆhstd +gmˆ0ˆsmstdˆIsd+amˆ0ˆdmstd dt. s7d This linearization approach is valid when mstd!Msand hstd!Hand is well described in many textbooks such as Ref. 17. Responding to the oscillating field hstdexpressed in Eq.(4),mstdcan be written as mstd=mexpsivtd. s8d Then the differential Eq. (7)can be transformed into an al- gebraic equation, ivm+iavmˆmˆ0+gmˆH−gsmˆIsdˆmˆ0 =−gMsmˆ0ˆh. s9d With the knowledge that the xaxis coincides with the direc- tion ofM0,Is, andH, we can write Eq. (9)as the compo- nents in a Cartesian system and obtain ivmx=0, s10ad siv−gIsdmy+sgH+iavdmz=gMshz, s10bd033904-2 Xi, Shi, and Gao J. Appl. Phys. 97, 033904 (2005) [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: 158.42.28.33 On: Mon, 24 Nov 2014 11:06:01siv−gIsdmz−sgH+iavdmy=−gMshz. s10cd Equations (10)indicate that the ac field component along the static field does not play a role in exciting the magnetizationas it is well known in FMR study. In addition, the magneti-zation dynamics remains in the plane perpendicular to thestatic field. By doing some algebra, the solution of Eqs. (10) can be written in a tensor form, m=XJ·h, s11ad where the magnetic susceptibility tensor XJis XJ= 100 0 0 xaixb 0−ixb xa2. s11bd The tensor components are all complex numbers. Taking xa=xa8−ixa9andxb=xb8−ixb9, we obtain xa8=1 DMsfH3−s1−a2dH02H+Is2H−2aIsH02g, s12ad xa9=1 DMsfaH0H2+as1+a2dH03−aIs2H0−2IsH0Hg, s12bd xb8=1 DMsfH0H2−s1+a2dH03−Is2H0−2aIsH0Hg,s12cd xb9=1 DMsf2aH02H−s1−a2dIsH02−IsH02−Is3g, s12dd D=fH2−s1+a2dH02+Is2g2+4saH−Isd2H02. s12ed Here, the notation, H0=v/g, is used. Figure 2 shows the response of the magnetic suscpetibil- ity tensor components to the static field calculated from Eqs.(12). The frequency used in calculation is an X-band 9.53 GHz and the gryomagnetic ration gis 1.76 3107Hz/Oe. The peaks of the imaginary parts, xa9andxb9, represent the ferromagnetic resonance, which is the resonantabsorption of magnetic energy in the thin film. The peakposition does not shift with the spin current and is simplydetermined by the microwave frequency, H res=H0=v/g. s13d Thereby, the resonance amplitude is xa,res9=xb,res9=2H02 4H02+saH0+Isd2Ms aH0−Is. s14d Equation (14)suggests that the resonance amplitude diverges when the spin current Isapproaches aH0.As a matter of fact, when IsøaH0, s15d the parallel configuration of the magnetization with respect to the static field is no longer stable.18Any small perturba- tion will result in a reversal of the magnetization to the op-posite direction. Our discussion of the ferromagnetic reso-nance within the linearization region should be constrained under the condition, I s,aH0. Figure 3 shows the depen- dence of the resonance amplitude on the spin current. When the spin current is smaller than the resonance field by orders of magnitude, Eq. (14)can be approximated to be xa,res9=xb,res9=Ms 2aeffH0, s16d where the effective damping constant is aeff=aS1−Is aH0D. s17d The resonance linewidth, which is defined as the interval between the field values at the points when xa9is half of the resonance amplitude, is FIG. 2. The dependence of real and imaginary parts of the magnetic sus- ceptibility components vs on the static field from Eqs. (12)forIs=−50, 0, and 50 Oe. The frequency of the microwave field vis 9.53 GHz. The damping constant ais 0.02. The components are in units of Ms/aH0. FIG. 3. The imaginary part of the magnetic susceptibility at resonance as a function of the spin-current parameter Isfora=0.01, 0.02, 0.03, and 0.04. It is in units of Ms/aH0.033904-3 Xi, Shi, and Gao J. Appl. Phys. 97, 033904 (2005) [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: 158.42.28.33 On: Mon, 24 Nov 2014 11:06:01DH=2aeffH0. s18d Figure 4 shows the spin-current dependence of the resonance linewidth with the dots calculated from Eqs. (12b)and(12d) well fit to the lines from Eq. (18). Implied by Eq. (17), the spin-current effect on the damping constant8,19is derived from the phenomenological term in the LLG equation in thesituation with uniform magnetization precession and no ex-citation of spin waves skÞ0d. Energy is pumped in the mag- netic thin film when I s.0, i.e., the electrical current flows from FM 1into FM 2. It is reflected in the high resonance amplitude and in the narrow linewidth, which is similar tothe situation of a small damping constant. When the electri-cal current flows out of FM 2toward FM 1, it helps the energy dissipation, resulting in a large effective damping constant.The quality factor Qof the resonance can be tuned by chang- ing the current intensity and direction, Q=DH 2H0=H0 2saH0−Isd. s19d In reality, the magnetization in magnetic multilayers is subject to the demagnetizing field HD, the magnetocrystal- line anisotropy HK, and the interlayer coupling Hex. There- fore, the effective field in the LLG equation should be Heff=HD+HK+Hex+H+hstd. s20d When the static field is along the in-plane anisotropy east axis and the spin current Isis small, the magnetization stays in parallel to the static field at equilibrium. As demonstratedabove, the spin torque does not shift the resonance field.Therefore, the resonance field is determined by H 02=sHres+HK+HexdsHres+HK+Hex+4pMsd,s21d which is well known for ferromagnetic resonance.17Note that we assume collinear interlayer exchange coupling with-out the biquadratic coupling and no perpendicular magneticanisotropy. Equation (21)is valid when only FM 1is consid- ered in the study and the exchange coupling behaves like anexternal magnetic field. In general cases where a variety ofinterlayer exchange coupling play a role, there is no analyti- cal formula for the resonance modes. 20–22 The effect of spin current on FMR is to change the reso- nance amplitude and linewidth. The calculation to derive thesusceptibility tensor components is tedious for the real caseof magnetic multilayers with in-plane coupling and aniso-tropy. The expressions are sophisticated but they neither helpto illustrate the role of the spin current nor render usefulfeatures for FMR study. For real thin film structures, Eq. (16) for the susceptibility at resonance and Eq. (17)for the effec- tive damping constant are valid with the replacements H 0!Hres, and H0!2pMs+HK+Hex+Hres, respectively. The first one is made because the spin current does not shift the resonance field. The spin-current effect isin the damping of the system. The second replacement ismade because of the fact that when the spin current reaches acritical value, I sscd=as2pMs+HK+Hex+Had, the parallel state of the magnetization is no longer stable in thin-film structures.18Hais the in-plane applied field. Under the influence of the spin current, the magnetization rotatesaway from the equilibrium in a spiral motion and reverses tothe opposite direction. IV. FERROMAGNETIC RESONANCE EXCITATED BY AC CURRENTS Instead of ac magnetic fields, ac currents can be used to excite the magnetization. Let us go back to the simple casewhere there is no anisotropy, exchange coupling, and demag-netizing field. In the presence of a steady magnetic field, themagnetization at equilibrium, M 0, satisfies M0ˆH=0, s22d which simply means that M0is parallel to the steady field. Using the linearization approach on the master LLG equation(1), we obtain dmstd dt=−gmstdˆH+gMsmˆ0ˆmˆ0ˆisstd +amˆ0ˆdmstd dt, s23d whereisstddenotes the ac spin current. With the ac spin current written as isstd=isexpsivtd, s24d and Eq. (8)for a magnetization excitation at an angular fre- quency of v, the linearized equation of motion is therefore transformed to be ivm+iavmˆmˆ0+gmˆH=gMsmˆ0ˆsmˆ0ˆisd. s25d FIG. 4. Resonance linewidth DHas a function of the spin-current parameter Isfora=0.01, 0.02, 0.03, and 0.04.033904-4 Xi, Shi, and Gao J. Appl. Phys. 97, 033904 (2005) [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: 158.42.28.33 On: Mon, 24 Nov 2014 11:06:01Equation (25)is then solved for min a Cartesian coor- dinate system. M0andHare in the positive xdirection as shown in Fig. 1 while the spin current is allowed to be in anarbitrary direction in the three-dimension space.The solutionis written in a tensor form, m=XJ c·i, s26d whereXJcis the “susceptibility” tensor of the magnetization to the spin current. Comparing Eq. (25)to Eq. (9)in the situation of no dc spin current, the relationship between thesusceptibility tensors can be readily obtained to be XJ c=XJIs=0·1100 00 1 0− 10 2 =−i10 00 0 xbixa 0−ixa xb2 Is=0. s27d It shows that ac spin currents function the same way as ac magnetic fields in achieving ferromagnetic resonance. Onlythe transverse components of the ac spin current play a rolein magnetic precession, which is also a right-hand rotationrelative to the direction of the static magnetic field. The dif-ference of current excitation from field excitation is a 90° lagin phase. The similarity between ac current excitation and ac field excitation suggests that the results of ac field excitation forthe general situations where exchange coupling and anisotro-pies are involved apply to ac current excitation. One of theadvantages of current excitation lies on that a wide range ofexcitation frequency can be easily achieved. Since the mag-nitude of the spin current could be as large as several tens ofOersteds, the linearization approach is no longer valid. Non-linear magnetization dynamics such as frequency doublingand chaotic behavior will appear. 23 V. SUMMARY AND CONCLUSIONS We have studied the spin-current effect on the ferromag- netic resonance in patterned magnetic thin film structures.Starting from the Landau-Lifshitz-Gilbert equation with aspin-transfer torque term, we conduct the calculations of themagnetic susceptibility using the linearization approach withthe assumption of uniform magnetization precession. Thestudy reveals that the spin current does not change the reso-nance field. It plays a role in the resonant absorption of elec-tromagnetic energy as it is a factor on the damping constantfor the magnetic thin films. Analytical expressions of the ferromagnetic resonance dependence on the spin currenthave been derived. In addition, we consider the possibility ofusing ac current to excite ferromagnetic resonance. While itrenders similar results as in the case of ac magnetic fieldexcitation, spin-current excitation can be large in magnitudeand would be used in the study of the magnetization dynam-ics beyond the linearization regime. 1M. Tsoi, A. G. M. Jansen, J. Baas, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett. 80, 4281 (1998 ); Nature (London )406,6 7 9 1 (2000 ); M. Tsoi, V. Tsoi, J. Baas, A. G. M. Jansen, and P. Wyder, Phys. Rev. Lett. 89, 246803 (2002 ). 2Y. Ji, C. L. Chien, and M. D. Stiles, Phys. Rev. Lett. 90, 106601 (2003 ). 3E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science285, 867 (1999 ); F. J. Albert, N. C. Emley, E. B. Myers, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 89, 226802 (2002 ). 4J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000 ). 5J. E. Wegrowe, X. Hoffer, Ph. Guittenne, A. Fábián, L. Gravier, T. Wade, and J. Ph. Ansermet, J.Appl. Phys. 91, 6806 (2002 ); J. E. 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D. Stiles and A. Zangwill, J. Appl. Phys. 91,6 8 1 2 (2002 ). 13For example, see J. Miltat, G. Albuquerque, A. Thiaville, and C. Vouille, J. Appl. Phys. 89, 6982 (2001 ). 14For example, see M. Farle, Rep. Prog. Phys. 61, 755 (1998 ). 15K. Lenz, T. Tolinski, J. Linder, E. Kosubek, and K. Baberschke, Phys. Rev. B69, 144422 (2004 ). 16X. Zhu, J.-G. Zhu, and R. M. White, J.Appl. Phys. 95, 6630 (2004 ); J.-G. Zhu and X. Zhu, IEEE Trans. Magn. 40, 182 (2004 ). 17A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves (CRC Press, Boca Baton, FL, 1996 ),C h a p .1 . 18J. Z. Sun, Phys. Rev. B 62, 570 (2000 );Z .L ia n dS .Z h a n g , ibid.68, 024404 (2003 ). 19Y. Tserkovnyak, A. Brataas, and G. E.W. Bauer, Phys. Rev. B 66, 224403 (2002 ); Phys. Rev. Lett. 88, 117601 (2002 ). 20B. Heinrich, in Ultrathin Magnetic Structure II , edited by B. Heinrich and J. A. C. Bland (Springer-Verlag, Berlin, 1994 ), p. 195. 21S. M. Rezende, C. Chesman, M. A. Lucena, A. Azevedo, F. M. deAguiar, and S. S. P. Parkin, J. Appl. Phys. 84, 958 (1998 ). 22A. Layadi, Phys. Rev. B 65, 104422 (2002 );69, 144431 (2004 ). 23H. Xi, K. Z. Gao, and Y. Shi (unpublished ).033904-5 Xi, Shi, and Gao J. Appl. Phys. 97, 033904 (2005) [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: 158.42.28.33 On: Mon, 24 Nov 2014 11:06:01

1.5049749.pdf

Enhanced magnon spin transport in NiFe 2O4 thin films on a lattice-matched substrate J. Shan , A. V. Singh , L. Liang , L. J. Cornelissen , Z. Galazka , A. Gupta , B. J. van Wees , and T. Kuschel Citation: Appl. Phys. Lett. 113, 162403 (2018); doi: 10.1063/1.5049749 View online: https://doi.org/10.1063/1.5049749 View Table of Contents: http://aip.scitation.org/toc/apl/113/16 Published by the American Institute of PhysicsEnhanced magnon spin transport in NiFe 2O4thin films on a lattice-matched substrate J.Shan,1,a)A. V. Singh,2L.Liang,1L. J. Cornelissen,1Z.Galazka,3A.Gupta,2 B. J. van Wees,1and T. Kuschel1,4 1Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands 2Center for Materials for Information Technology, The University of Alabama, Tuscaloosa, Alabama 35487, USA 3Leibniz Institute for Crystal Growth, Max-Born-Str. 2, 12489 Berlin, Germany 4Center for Spinelectronic Materials and Devices, Department of Physics, Bielefeld University, Universit €atsstraße 25, 33615 Bielefeld, Germany (Received 25 July 2018; accepted 29 September 2018; published online 16 October 2018) We investigate magnon spin transport in epitaxial nickel ferrite (NiFe 2O4, NFO) films grown on magnesium gallate spinel (MgGa 2O4, MGO) substrates, which have a lattice mismatch with NFO as small as 0.78%, resulting in the reduction of antiphase boundary defects and thus in improved magnetic properties in the NFO films. In nonlocal transport experiments where platinum (Pt) stripsfunction as magnon spin injectors and detectors, enhanced signals are observed for both electrically and thermally excited magnons, and the magnon relaxation length ( k m) of NFO is found to be around 2.5 lm at room temperature. Moreover, at both room and low temperatures, we present distinct features from the nonlocal spin Seebeck signals which arise from magnon-polaron forma- tion. Our results demonstrate excellent magnon transport properties (magnon spin conductivity, km, and spin mixing conductance at the Pt/NFO interface) of NFO films grown on a lattice-matchedsubstrate which are comparable with those of yttrium iron garnet. Published by AIP Publishing. https://doi.org/10.1063/1.5049749 Magnons, the collective excitation of spins, are playing the central role in the field of insulator spintronics. 1 Magnons in magnetic materials can interact with conductionelectrons in adjacent heavy metals, transferring spin angular momentum and thus allowing for magnonic spin current excitation and detection using electrical methods. 2–9 Besides, magnons can be driven thermally, known as the spin Seebeck effect (SSE).10–12Both magnons generated by a spin voltage bias and a temperature gradient can be trans- ported for a certain distance on the order of a few to tens ofmicrometers, as reported recently in ferrimagnetic 2,13and even in antiferromagnetic materials,14making magnons promising candidates as information carriers. Nickel ferrite (NFO) is a ferrimagnetic insulator with an inverse spinel structure. It is widely used in high-frequencysystems in conventional applications. 15Recently, NFO and other spinel ferrites were explored for spintronic applications, where effects such as spin Hall magnetoresistance (SMR),16–20 SSE,21–28and nonlocal magnon spin transport13were reported. In most of these studies, large magnetic fields of a few teslasare required to align the magnetization of the ferrites, possiblydue to the presence of antiphase boundaries. 29 However, it was recently shown that the NFO films grown on nearly lattice-matched substrates with similar spi-nel structures, such as MgGa 2O4and CoGa 2O4, exhibited superior magnetic properties due to the elimination of anti- phase boundaries, leading to, for instance, a larger saturation magnetization ( MS), smaller coercive fields, and a lower Gilbert damping constant, compared to the NFO films grownon the typically used MgAl 2O4(MAO) substrate.30An enhanced longitudinal SSE effect was reported on such NFO films.31It can be expected that the nonlocal transport proper- ties of magnon spin are also elevated in these NFO films, aswe discuss in this paper. We studied two NFO films on MGO (100) substrates, with thicknesses of 40 nm and 450 nm, respectively. NFO films were grown by pulsed laser deposition, in the same way as described in Refs. 30and31. Prior to further pro- cesses, the 450-nm-thick sample was characterized by super- conducting quantum interference device (SQUID) magnetometry, exhibiting an in-plane coercive field lowerthan 5 mT [see Fig. 1(b)]. Afterwards, multiple devices were fabricated on both samples. Figure 1(a)shows schematically the typical geometry of a device, where two identical Ptstrips serving as magnon spin injectors and detectors are pat- terned in parallel with a center-to-center spacing d, ranging from 0.3 to 25 lm for all devices. The lengths and widths of the Pt strips are designed to be different for shorter- andlonger- ddevices, as summarized in Table I. In Geometry I, Pt strips are 100 nm in width, allowing for fabrication of devices with narrow spacings. In Geometry II, Pt strips arewider and longer, permitting larger injection currents which yield a larger signal-to-noise ratio, so that small signals can be resolved. For all devices, Pt is sputtered with a thicknessof 8 nm, showing a conductivity of around 3 /C210 6S/m. Contacts consisting of Ti (5 nm)/Au (60 nm) were patterned in the final step of device fabrication. Electrical measurements were performed with a stan- dard lock-in technique, where a low-frequency ac current,I¼ffiffiffi 2p I 0sinð2pftÞ, was used as the input to the device, anda)j.shan@rug.nl 0003-6951/2018/113(16)/162403/5/$30.00 Published by AIP Publishing. 113, 162403-1APPLIED PHYSICS LETTERS 113, 162403 (2018) voltage outputs were detected at the same (1 f) or double fre- quency (2 f), representing the linear and quadratic effects, respectively. In this study, typically I0is 100 lA and fis set to be around 13 Hz. For the local detection VL, as shown in Fig. 1(a),V1f Ldetects the resistance and magnetoresistance (MR) effect of the Pt strip, and V2f Lincorporates the current- induced local SSE.32,33For the nonlocal detection VNL;V1f NL represents the nonlocal signals from magnons that are injected electrically via SHE,2,3andV2f NLstands for the non- local SSE.2,13,34–38The conductance of the NFO thin films was checked by measuring resistances between random pairs of electrically detached contacts, which yielded values over GX, confirming the insulating nature of the NFO films. We first perform angular-dependent measurements at room temperature for both local and nonlocal configurations, with results plotted in Figs. 1(c)–1(f) . The sample was rotated in-plane with a constant magnetic field applied. Thestrength of the field is 300 mT, large enough to saturate the NFO magnetization along the field direction. A strong MR effect, with DR/R/C250.1%, was observed from the local V 1f L signal [see Fig. 1(c)]. This MR effect was checked to be magnetic-field independent in the range from 100 to 400 mT, indicating that the observed MR effect is the SMR effect which is sensitive to the NFO magnetization that is saturated in this range, instead of the Hanle MR effect39which depends on the external magnetic field. This is in marked contrast to the previous observations from sputtered NFO thin films grown on MAO, where only the Hanle MR effectwas observed at fields above 1 T. 13The SMR ratios for both 40- and 450-nm thick samples exhibit similar values, ranging between 0.07% and 0.1%, around 3 to 4 times larger than those for Pt/yttrium iron garnet (YIG) systems with a similar Pt thickness.6,34,40It is also more than twice as large as the SMR reported from Pt/NFO systems with the NFO layer grown by chemical vapor deposition on MAO substrates.17 Using the average SMR ratio of 0.08% and the spin Hallangle of Pt of 0.11,6,34we estimated the real part of the spin mixing conductance ( Gr) for Pt/NFO systems to be 5.7/C21014S/m2with the SMR equation,41being more than 3 times larger than that of the Pt/YIG systems determined with the same method.6 Figures 1(e)and1(f)plot typical results from the nonlo- cal measurements in V1f NLand V2f NL, showing cos2ðaÞand cosðaÞdependences, respectively, the same as observed pre- viously in YIG or NFO films with Pt or Ta electro- des.2,13,34,42,43For the magnon transport process represented byV1f NL, both the magnon excitation and detection efficien- cies are governed by cos ðaÞ, which in total yields a cos2ðaÞ behavior. For V2f NL, on the other hand, the thermal magnon excitation is independent of abut the detection process is, thus showing a cos ðaÞdependence. Their amplitudes, denoted as VEIandVTG, respectively, can be obtained from sinusoidal fittings. Next, we present VEIandVTGfor all devices as a func- tion of don both the 40- and 450-nm-thick samples to inves- tigate the magnon relaxation properties, as shown in Fig. 2. For both VEIand VTG, discontinuities are found between Geometries I ( d/C202lm, filled with yellow color) and II (d/C212lm), even though the data from Geometry II are care- fully normalized to Geometry I as was done for Pt/YIG non- local devices to link the data between the two geometries.34 However, this normalization method is based on the assump- tion of noninvasive contacts and does not account for the additional spin absorption that was induced by widening the Pt contact width. This normalization method works well for Pt/YIG systems but becomes less satisfactory for Pt/NFOsystems as we study here, which is expected in view of a larger G rvalue. For VEI, the datapoints at d>15lm(d>12lm for 450 nm NFO) are not plotted as the signal amplitudes become much smaller than the noise level. For shorter dis-tances ( d<1lm), the signals on both samples are even com- parable to those measured on thin YIG films with similar device geometry, 2,34although a fairer comparison should be made with the same thickness of the magnetic insulators. We can also make a comparison between the VEIsignals from the 40-nm-thick NFO film studied here and the 44-nm-thick sputtered NFO film on the MAO substrate studied in Ref. 13. We found that for the same device geometry ( d¼350 nm)FIG. 1. (a) Schematic geometry of local and nonlocal measurements. An electric current Iis applied at one Pt strip, and voltages can be detected at the same strip (locally) or at the other one (nonlocally).An in-plane magnetic field is applied at an angle denoted by a. (b) In-plane mag- netization of the 450 nm-thick NFO film obtained from SQUID at room tempera- ture. (c)–(f) Room-temperature local and nonlocal measurements shown in first and second harmonic signals, withI¼100lA. They are measured on the 40-nm-thick NFO sample with an exter- nal magnetic field of 300 mT under angu- lar sweep. Only for (e), a background of 910 nV is subtracted. TABLE I. Sample details of Geometry I and II. Geometry Pt length ( lm) Pt width ( lm) distances ( lm) I 10 0.1 0.3–2 II 20 0.5 2–25162403-2 Shan et al. Appl. Phys. Lett. 113, 162403 (2018)and Pt thickness, the VEIsignal amplitudes obtained here are around 100 times larger than found in Ref. 13, showing the superior quality of the NFO films studied in this paper. To extract kmfor these NFO samples at room tempera- ture, we performed exponential fittings as shown in Fig. 2(a) by the dashed lines. We limit the fit to the datapoints in the exponential regime where d>2lm. Both datasets yield km /C252.5lm for the two NFO samples with different thick- nesses. It is noteworthy that the VEIsignals for the 450 nm NFO are in general smaller than those for the 40 nm NFOsample, except for one datapoint at the shortest distance. However, one would expect the opposite, as increasing the NFO thickness from 40 to 450 nm enlarges the magnon con- ductance without introducing an extra relaxation channel vertically, given that 450 nm is still much smaller than k m /C252.5lm. This puzzle is similar to that for Pt/YIG systems,34 and the reason is not yet clear to us. Now, we move to the thermally generated nonlocal SSE signals VTGas shown in Fig. 2(b). According to the bulk- generated SSE picture,6,34,38,44at a certain distance ( drev), VTGshould reverse sign, where in short distances VTGhas the same sign as the local SSE signal, and further away, thesign alters. d revis influenced by the thickness of the magnetic insulator and interfacial spin transparency at the contacts.34With our measurement configuration [the polarities of local and nonlocal measurement configurations are opposite as shown in Fig. 1(a)], the VTGsignals measured from all devi- ces are in fact opposite in sign compared to the local SSE signals [see Fig. 1(d)], meaning that drevis positioned closer than the shortest dwe investigated. Only an upturn is observ- able for VTGof the 450 nm NFO sample in a short- drange. Compared to Pt/YIG systems, where drevis about 1.6 times of the YIG thickness, for Pt/NFO systems, the sign-reversal takes place much closer to the heater, possibly because of the Pt/NFO interface being more transparent for a larger Gr. Exponential fittings can also be carried out for VTGon both samples. Note that only the datapoints in the exponen- tial regime can be used to extract km, which typical starts at d¼kmand extends to a few km.36Further than the exponen- tial regime, VTGstarts to decay geometrically as 1/ d2, domi- nated by the temperature gradient present near the detector. Based on kmthat we extracted from the decay of the electri- cally injected magnon signals, we identify 2 /C20d/C208lma s the exponential regime and obtain kmto be around 2.2 or 2.3lm from the decay of VTG. The consistency between km found from magnon signals excited electrically and ther- mally illustrates again the same transport nature of the mag- nons generated in both methods. Owing to the excellent quality of the NFO films, we are able to study their magnetoelastic coupling by means of the nonlocal SSE. It was observed in YIG that for both the local and nonlocal SSE signals, spike structures arose at certain magnetic fields, at which the magnon and phonon dispersions became tangent to each other, resulting in a maximal magne- toelastic interaction and the formation of magnon-polar- ons.38,45–48At these conditions, the spin Seebeck signals have extra contributions from the magnon-polarons, provided that the magnon and phonon impurity scattering potentials are dif- ferent.45,46It was found that for YIG films, the acoustic qual- ity is higher than the magnetic one, with peaks observed in local SSE and nonlocal SSE ( d<drev) measurements and dips observed for nonlocal SSE where d>drev.38,45This effect is explained as several parameters such as km, the bulk spin Seebeck coefficient, the magnon spin, and heat conductivities are all modified by the emergence of magnon-polarons.38,46 So far, this resonant enhancement/suppression of SSE caused by magnetoelastic coupling has only been clearly observed in YIG; besides, a bimodal structure was found in the SSE of a Ni0.65Zn0.35Al0.8Fe1.2O4thin film and was speculated to be related to magnon-phonon interactions.49 Here, we present distinctive magnon-polaron features in the nonlocal SSE measurements on our NFO films. Figure 3 shows field-sweep data of VTGperformed on one device of the 450-nm-thick NFO sample at T¼150 K and 293 K. At both temperatures, asymmetric dip structures of VTGare clearly visible, around 64.2 T at T¼150 K, and shift to 64.0 T at T¼293 K. The change in the characteristic mag- netic field of 0.2 T for a temperature decrease of about 150 K is comparable to Pt/YIG systems.38The sign of the anoma- lies is in accordance with the previous observation reported in Ref. 38, considering that the spacing between the Pt strips (d¼1lm) is further than drev. This implies that the studied NFO film may also have a higher acoustic than magneticFIG. 2. Distance dependence of (a) VEIand (b) VTGmeasured at B¼200 mT on both NFO samples at room temperature, normalized to I¼100lA. The datapoints filled with yellow color are obtained from devi- ces in Geometry I, while the rest belongs to Geometry II. The datapoints from Geometry II are normalized to Geometry I as described in Ref. 34for better comparison. Dashed lines are exponential fittings with the formula V¼Aexpð/C0d=kmÞ, with the coefficient Abeing different for each fitting. The extracted kmfrom each fitting is indicated nearby. The dotted orange lines in (b) are 1/ d2fittings for long- dresults.162403-3 Shan et al. Appl. Phys. Lett. 113, 162403 (2018)quality like YIG, although a careful study which measures the anomalies from the local SSE is needed. The magnetic fields where the anomalies occur can be evaluated by the phonon and magnon dispersions. In our experiments, limited by the maximal applied magnetic field(l 0H/C257 T), we could only probe the first anomaly which involves transverse acoustic (TA) phonons with a lower sound velocity. The TA phonons follow the dispersion rela- tionx¼vTk, where vTis the TA phonon sound velocity. vT is related to the elastic constant C44and material density q byC44¼qv2 T49,50and is determined to be 3968 m/s for NFO using C44¼82.3 GPa and q¼5230 kg/m3.31,51 We assume that magnons in NFO can also be described by a parabolic dispersion relation like for YIG (x¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDexk2þcl0HÞðDexk2þcðl0HþMSÞÞp ), where Dex is the exchange stiffness, l0the vacuum permeability, and c the gyromagnetic ratio. From Fig. 1(b), we obtain MSof our NFO sample to be 160 emu/cm3at room temperature, which equals 201 mT. The Dexof NFO is not yet experimentally reported. In our experiment, from the peak positions observed at room temperature ( l0HTA¼64.0 T), we can determine the only unknown parameter Dexto be 5.5 /C210/C06 m2/s with both phonon and magnon dispersions. This value is close to Dexwhich can be estimated from the exchange integrals among Ni2þ,F e3þ(octahedral site), and Fe3þ (tetrahedral site).52,53Using the parameters given in Ref. 54,theDexof NFO can be estimated to be 6.4 /C210/C06m2/s, within 17% difference of our experimental value. Anomalies were also observed in the 450 nm NFO sample for electrically excited magnons in the field-sweep measure- ments of VEIatT¼150 K, albeit with a lower signal-to-noise ratio. For the 40 nm NFO sample, however, no clear anomalieswere identified in the measured range ( l 0H/C206.6 T) for VEI orVTG. In summary, we have studied the magnon spin transport properties of epitaxial NFO films grown on MGO substrates in a nonlocal geometry. We obtained large nonlocal signals for both electrically and thermally excited magnons at shortcontact spacings, comparable to that of YIG. From the relaxa- tion regime, k mwas found to be around 2.5 lm. Furthermore, we observed anomalous features as a result of magnon-polarons formation in the field-dependent SSE measurements at both 150 and 293 K, from which the exchange stiffness con- stant of NFO can be determined. Our results demonstrate theimproved quality of NFO grown on a lattice-matched sub- strate, showing NFO to be a potential alternative to YIG for spintronic applications. Specifically, both A and B sites ofspinels can be versatilely substituted with other atoms at diverse composition ratios, allowing for more adaption and optimization of the material properties compared to garnets. We thank Gerrit Bauer, Matthias Althammer, and Koichi Oyanagi for helpful discussions and would like to acknowledge M. de Roosz, H. Adema, T. Schouten, and J. G. Holstein for technical assistance. This work was supported by the researchprograms “Magnon Spintronics (Nr. 159)” and “Skyrmionics(Nr. 170)” of the Netherlands Organisation for Scientific Research (NWO), the NWO Spinoza prize awarded to Professor B. J. van Wees, DFG Priority Programme 1538 “SpinCaloric Transport” (KU 3271/1-1), NanoLab NL, EU FP7 ICT Grant No. InSpin 612759, and the Zernike Institute for Advanced Materials. The work at Alabama was supported byNSF Grant No. ECCS-1509875. 1A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). 2L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat. Phys. 11, 1022 (2015). 3S. T. B. Goennenwein, R. Schlitz, M. Pernpeintner, K. Ganzhorn, M. Althammer, R. Gross, and H. Huebl, Appl. Phys. Lett. 107, 172405 (2015). 4J. Li, Y. Xu, M. Aldosary, C. Tang, Z. Lin, S. Zhang, R. Lake, and J. Shi,Nat. Commun. 7, 10858 (2016). 5H. Wu, C. H. Wan, X. Zhang, Z. H. Yuan, Q. T. Zhang, J. Y. 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1.4953231.pdf

Composite media for high density heat assisted magnetic recording Zengyuan Liu , Yipeng Jiao , and R. H. Victora Citation: Appl. Phys. Lett. 108, 232402 (2016); doi: 10.1063/1.4953231 View online: http://dx.doi.org/10.1063/1.4953231 View Table of Contents: http://aip.scitation.org/toc/apl/108/23 Published by the American Institute of Physics Composite media for high density heat assisted magnetic recording Zengyuan Liu,1,2Yipeng Jiao,1,2and R. H. Victora1,2 1Center for Micromagnetics and Information Technologies, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Electrical and Computer Engineering Department, University of Minnesota, Minneapolis, Minnesota 55455, USA (Received 19 April 2016; accepted 23 May 2016; published online 6 June 2016) A heat assisted magnetic recording composite media with a superparamagnetic writing layer is proposed. The recording process is initiated in the write layer that is magnetically softer than the longterm storage layer. Upon cooling, the composite structure copies the information from the writing layer to the lower Curie temperature (Tc) storage layer, e.g., doped FePt. The advantages include insensitivity to Tc variance in the storage layer, and thus the opportunity to significantly lower theFePt Tc without the resulting Tc distribution adversely affecting the performance. The composite structure has a small jitter within 0.1 nm of the grain size limit owing to the sharp transition width of the optimized superparamagnetic writing layer. The user density of the composite structure can reach4.7 Tb =in: 2for a Gaussian heat spot with a full-width-at-half-maximum of 30 nm, a 12 nm reader width, and an optimized bit length of 6 nm. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4953231 ] Heat Assisted Magnetic Recording (HAMR) is often con- sidered to be the next-generation technology of the informa-tion storage industry. Unlike perpendicular magnetic recording, HAMR recording media is not only recorded by a magnetic field, but also assisted by heat. HAMR is greatlyaffected by the Curie temperature (Tc) distribution becausewriting occurs at temperatures near the Tc of FePt media. 1–3 To increase the areal density capacity (ADC) of HAMR, noise mitigation is crucial. Different composite structures have been proposed to control the noise caused by the Tc variance. Theidea is to utilize phenomena that can provide narrower transi-tion width than that generated by typical Tc variance. For example, FeRh/FePt exchange coupled media 4benefits from a sharp FeRh phase transition. The FePt/Cr/X/FePt compositestructures 5make use of the sharp AFC (antiferromagnetic coupling) transition provided by the Cr layer. Simulations5,6 showed these structures do yield smaller jitter compared to FePt media with varying Tc. However, owing to intrinsic ex- perimental variations during fabrication, neither of these twostructures will likely have narrower transition width. A design idea is to decouple the recording problem from the long-term storage issue. This idea relies on a composite structure that has a superparamagnetic writing layer and a (doped) FePt storage layer. The decoupling process requiresthe composite structure to later copy information (byexchange coupling) to the storage layer from the writing layer during the recording process. To avoid the effects of Tc variance in the FePt storage layer, the recording processmust happen at a higher temperature than the writing temper-ature of the storage layer. To achieve the goal, the Curie tem-perature of the writing layer should be higher than that of the storage layer. 7,8At the same time, the anisotropy of the writ- ing layer should be substantially large, and the magneticmoments of the write layer must exceed that of the storagelayer until writing is completed. In addition, the temperaturedependence for the magnetic properties of the writing layer should be optimized to provide a sharp transition width. Thermal switching probability distribution (SPD) has beeninvestigated for FePt media. 7The results in this paper will examine the potential of the composite structure to provide a narrower transition width than that found in FePt media with typical Tc variance. The design of the writing layer with substantially large anisotropy and high Curie temperature is one aspect that dis- tinguishes the bilayer structure from the other bilayer structure designs with different Curie temperatures.9The other aspect is that the writing process explicitly uses superparamagneticwriting, which means writing with thermal fluctuations. Thewriting layer in the composite structure offers another direc- tion of optimization to reduce the noise in the recording media, and reduces the demands on the FePt layer. These dis-tinctive features potentially allow higher user areal densitywith lower write temperatures than previously demonstrated. Micromagnetic simulation 11,12based on the Landau- Lifshitz-Gilbert (LLG) equation is implemented. The renor-malized cell size is 1 :5n m /C21:5n m /C21:5 nm. The compos- ite structure has two layers, e.g., a 3 nm superparamagnetic writing layer and 6 nm FePt storage layer. The Curie temper- ature of the writing layer is 900 K. The Curie temperature ofthe FePt layer is 700 K. The Tc variance within the compos-ite structure distributes as rT c;wl¼0% and rTc;sl¼3% (“sl” stands for storage layer and “wl” stands for writing layer). Hk variance is assumed to be zero. The Gilbert damping con- stant of the composite structure is 0.02 at 350 K. Ms;slð350KÞ ¼922 :3 emu =cm3;Ku;slð350K Þ¼4:11/C2107ergs =cm3and Aex;slð350K Þ¼1:1/C210/C06ergs =cm. The magnetic profiles of the writing layer are given by Ms;wlT Tc;wl/C18/C19 Ms;slT Tc;sl/C18/C19 ¼Ms;wl350KðÞ =Ms;sl350KðÞ ;/C30 (1) Ku;wlT Tc;wl/C18/C19 Ku;slT Tc;sl/C18/C19 ¼Ku;wl350KðÞ =Ku;sl350KðÞ ;/C30 (2) 0003-6951/2016/108(23)/232402/5/$30.00 Published by AIP Publishing. 108, 232402-1APPLIED PHYSICS LETTERS 108, 232402 (2016) Aex;wlT Tc;wl/C18/C19 Aex;slT Tc;sl/C18/C19 ¼Tc;wl=Tc;sl:/C30 (3) The exchange coupling between these two layers is taken to beffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiAex;wl/C1Aex;slp. Different magnetic profiles of the writing layer are produced by various combinations of Ms;wlð350 K Þ andKu;wlð350 K Þ. Therefore, by changing the magnetization and anisotropy of the writing layer, different writing temper-atures ( T w) and FWHMs are obtained from the calculated SPD.8The results for average grain diameter of 5.6 nm with 20% standard deviation of diameter are shown in Fig. 1. The error bars represents the FWHM ( Tw6FWHM =2). Based on the results shown in Fig. 1, the writing temperature of the composite structure decreases with increasing magnetizationof the writing layer. The composite structure has higher writ-ing temperature when the writing layer has larger anisotropy.The writing temperature is (at most) 710 K when T c;wl¼ 900K and Ku;wl¼1:0/C2107erg=cm3. Therefore, Tpeak¼ 850K is sufficiently high to ensure good recording. Protecting the a-C (amorphous carbon) overcoat13constrains the Tc choices of the writing layer. The black lines are basedon the SPD of the single FePt layer with 6 nm thickness and3% Tc variance. The solid line shows the writing temperature of the FePt layer is 655 K. The FWHM of the FePt layer is 60 K. Thedashed lines represent 655 K þFWHM/2 and 655 K-FWHM/ 2. In Fig. 1, the composite structure has narrower FWHM (small transition jitter 7,8) if the writing layer has higher ani- sotropy. The FWHM of the composite structure decreases with increasing magnetization of the writing layer and becomes larger when Tw is lower than 685 K. These resultscontrast the use of a soft layer, 9such as Fe, where the anisot- ropy is zero and its moment (magnetization multiplied bythickness) must be higher. Considering the anisotropy andFWHM, the optimized composite structure chosen has 6 nmFePt storage layer and 3 nm writing layer with M s;wl ¼550 emu =cm3,Ku;wl¼7/C2106erg=cm3, and Tc;wl¼900 K. Potential materials for the superparamagnetic writing layer include variations on conventional perpendicular media. Thewriting temperature of the chosen composite structure is692 K. The FWHM of it is 37 K, which is 38% less than 60 K of the FePt media. The transition jitter 6,14of the functional composite media is calculated by implementing a previously developed heat- assisted-magnetic-recording simulation11that uses CUDA/C based on NVIDIA graphics to accelerate the computation ofthe magnetic recording process. The write head field is 8000 Oe. The angle between the applied field and the z direc- tion is 22 :5 /C14. The thermal field11,15,16delivered by the near field transducer on the media is modeled by a 2-D Gaussian distribution. The FWHM of the thermal profile is 40 nm in thecross-track and down-track directions. The equivalent thermal gradient is about 15 K/nm. The bit length (BL) is 21 nm, and the recording pattern is a single tone. The peak temperature(T peak) of thermal field is 850 K. The head velocity is 20 m/s. TherHkdistribution and rTcdistribution are the same that were used in the SPD calculation. The transition jitter of the composite structure is calculated based on 200 zero crossings of the play back signal. The nonmagnetic grain boundary FIG. 1. Relationship between Tw of the composite structure and Ms;wlfor Tc,wl¼900 K, cooling rate 100 K/ns. FIG. 2. (a) The relationship between the transition jitter and grain pitch. The curves with different colors correspond to different structures. The black dashed line represents the grain size limited jitter values under differentgrain pitches (based on Equation (4)). (b) Relationship between the calcu- lated transition jitter and Tc variance in FePt layer.232402-2 Liu, Jiao, and Victora Appl. Phys. Lett. 108, 232402 (2016) (Bnd) is assumed to be 1 nm. The reader width WR ¼20 nm. The magnetic flying height is 6 nm. The shield to shield dis- tance is 11 nm. The noise from the reader is assumed to be zero. The results are shown in Fig. 2(a). The blue curve repre- sents the jitter of the composite structure optimized by SPDcalculation. For comparison, the transition jitter of 6 nm FePt media and 9 nm FePt media is also calculated. The dashed curve is the grain-size theoretical limit, 16which follows r2 jitter ;theory ¼1 WR =D/C1D2 12: (4) In Equation (4), D stands for the grain pitch, which is hDiþ Bnd ( hDi: grain diameter). The jitter of single FePt media shows that 3% Tc variance is detrimental to the recording process. For example, the gap between single FePt layers and the theoretical value is 0.4–0.5 nm. The blue curve shows that the jitter of the media is greatly reduced by intro-ducing the superparamagnetic writing layer. The difference with theoretical limit is only 0.2 nm. However, the jitter deviates from the expected value when the recording grain pitch is smaller than 4.8 nm. This is due to thermal fluctua- tions of the small grains causing DC noise. The DC noisecan be mitigated by increasing the thickness of both layers. At the same time, the ratio of the layer thickness remains unchanged (3 nm/6 nm ¼0.5). This ensures that the relation- ship between magnetic moment of two layers is identical to that of 3 nm–6 nm structure (3 nm: writing layer, 6 nm: FePtlayer) during the recording process. Two composite struc- tures with 4.5 nm–9 nm configuration and 6 nm–12 nmconfiguration are examined. With bigger total volume, these two composite media have even smaller jitter. 8The 4.5 nm–9 nm configuration has only 0.1 nm bigger jitter thanthe theoretical value, and it removes the DC noise existing in the 3 nm–6 nm structure. The 6 nm–12 nm composite struc- ture approaches the theoretical limit under all grain pitchesused. The 4.5 nm–9 nm composite structure is taken to be theoptimized media owing to its reasonable total thickness (13.5 nm) and smaller jitter when the grain pitch is 4.8 nm. Using the Voronoi grains that have average diameter of 5.5 nm and nonmagnetic grain boundary of 1.0 nm, theresults for the transition jitter under different Tc variance in the FePt layer are shown in Fig. 2(b). The FePt traditional HAMR media has larger jitter (worse recording density)when rT cis bigger. The transition jitter of the two composite structures is much smaller than that of single layer FePt media. The jitter is only 0.1 nm–0.2 nm bigger than the theo- retical limit. Furthermore, the transition jitter of the compos-ite structures remains almost unchanged even under differentTc variance in FePt layer. This is due to the superparamag- netic writing process: the composite structure copies infor- mation from the writing layer, which makes the wholesystem be impervious to the rT cin the FePt storage layer. Significant doping of FePt can be used to reduce FePt Curie temperature at the cost of reduced archival stabilityand, typically, increased Tc distribution. However, evenreduced anisotropy will still be archival at grain sizes likely to be commercially accessible, and our proposed composite structure removes the jitter dependence on the storage layerTc distribution. The black solid curve in Fig. 2(b) representsTABLE I. SNR, BER, and EBR under different BLs and FWHMs. BL (nm) FWHM (nm) SNR(dB) BER C/BL (nm/C01) 4 30 8.6( 61:1Þ 0.1492 0.098 4 40 9.1( 60:7Þ 0.1254 0.114 6 30 11.2( 60:7Þ 0.0077 0.156 6 40 12.3( 60:5Þ 0.0058 0.158 8 30 13.7( 60:4Þ 0.0052 0.119 8 40 13.5( 60:8Þ 0.0049 0.119 10 30 14.6( 60:7Þ 0.0044 0.096 10 40 15.2( 61:0Þ 0.0039 0.096 FIG. 3. Illustration of overlapping adjacent tracks to maximize the user den- sity. The extent of squeezing is specified by the CTCD (center to center dis- tance). The red spot represents the heat profile with 30 nm FWHM. Thehead direction is along the down track. The recording patterns of the central track are used for playing back.TABLE II. User density under different CTCDs using 15 nm reader width. CTCD (nm) SNR (dB) BER C/BL (/nm) UD (Tb =in2) 30 11.2( 60:7) 0.0077 0.1558 3.4 27 11.1( 60:5) 0.0083 0.1551 3.7 24 11.1( 60:6) 0.0113 0.1518 4.1 21 11.1( 60:7) 0.0106 0.1525 4.7 18 10.2( 60:7) 0.0521 0.1175 4.2 FIG. 4. The effective bit ratio (C/BL, dashed) and user density (solid) under different CTCD. Different colored curves correspond to three different reader widths (WR).232402-3 Liu, Jiao, and Victora Appl. Phys. Lett. 108, 232402 (2016) the composite media in which the Curie temperatures of both layers and the peak temperature are scaled down by 100 K. The magnetic properties of the composite structure are scaled based on the temperature dependence for the anisot-ropy field of FePt with different Curie temperature 17(by doping Ni-content). Here, the anisotropy is assumed to be proportional to the Curie temperature. When Tpeak¼750K, the structure still works well with even smaller jitter. Inconclusion, besides being unaffected by rT cin FePt layer, the composite structure can also decrease the writing tem- perature. Thus, two major HAMR media concerns areaddressed. The proposed composite structure increases the storage capacity due to the decreased transition jitter. To achieve higher storage capacity, a small grain pitch, e.g., recording grains with 3.8 nm diameter and 1 nm nonmagnetic grainboundary, should be used. The functional composite struc- ture chosen is 4.5 nm–9 nm owing to the good jitter perform- ance even using very small grains. The user density (UD) isdefined as an evaluation parameter to optimize the whole HAMR system, including the recording media design, re- cording process, and reader configuration. The recording pa-rameters that affect the user density are FWHM of the heat profile and the recording bit length (BL). The Shannon Capacity (C) 10,18is calculated based on bit error rate (BER). The EBR (effective bit ratio10) is defined by C=BL. The user density (UD) can be defined as EBR/FWHM. To calculate the bit error rate (BER), a pseudo random bit sequence10(PRBS) is recorded. The noise from the reader is assumed to be zero. The SNR is calculated based on SNR ¼Ð VðxÞ2dx=Ð dVðxÞ2dx. The V(x) stands for the play back signal voltage along the down track direction. The BER is calculated using a 1-D MMSE (minimum mean square error) equalizer and improved Viterbi detector.19,20The tar- get function G¼½g0;g1;…;gL/C01/C138Tðg0¼1;L¼3Þand the equalizer function F¼½f/C0K;f/C0Kþ1;…;f0;…;fK/C138Tare deter- mined by minimizing the difference between the equalizedsignal and the ideal signal. The total equalizer length is (2 K þ1)BL, which should remain constant (around 80 nm) for different BL. The Viterbi detector and decoder 21are used to determine the states (“0” or “1”) of the signals. The resultsof BER, SNR, and EBR under different recording parameters are shown in Table I. The reader width is taken to be 1=2of the FWHM. Table Ishows that the composite structure can reach a maximum user density when FWHM of the heat pro- file is 30 nm and the recording bit length is 6 nm. Based on EBR/FWHM, the maximum user density is 3 :4T b =in:2.In order to continue to maxim ize the user density of the composite structure, the adjacent tracks are allowed to overlapeach other. This idea is illustrated in Fig. 3.T h eF W H Mo ft h e heat profile and the recording b it length are set to be the optimal case in Table I.T h et h r e et r a c k su s et h es a m eh e a tp r o fi l e ,b u t different pseudo random bit seque nce, to mimic recording on ad- jacent tracks. The CTCD stands fo r center to center distance of the two adjacent tracks. The sm aller CTCD means larger overlap of two adjacent tracks. Using different CTCD values, SNR, BER, and effective bit ratio (EBR) are calculated in order to findthe optimal CTCD that maximize s the user density. The results for reader width of 15 nm are shown in Table II. To calculate the user density, the FWHM should be changed to CTCD. The over- lap of two adjacent tracks can be as large as 9 nm for the com-posite structure before the use r density starts to decrease. To optimize the reader configuration under different CTCD, the effective bit ratio and the user density are calcu-lated as Table IIunder different reader widths, including 12 nm, 15 nm, and 20 nm. The results are shown in Fig. 4.T h e dashed lines show the results for an effective bit ratio. The ab- rupt decrease of all three dashed curves when CTCD is smallerthan 21 nm is due to the recording patterns of the central trackbeing distorted by the adjacent tracks, as shown in Fig. 5. This also results in the decreased user density. The solid lines show the results for user density. It continues to increase with decreasing CTCD due to almost unchanged effective bit ratio, when CTCD is bigger than 21 nm. Boththe user density and the effective bit ratio show that theresults saturate when the reader width continues to decrease. For example, the results under the reader widths of 15 nm and 12 nm are almost the same. The maximum user densitycan reach 4 :7T b =in: 2. Thus, using this technique, the user density can be increased by 40% without making any changes to the original HAMR recording head configuration. In conclusion, a composite structure with a 4.5 nm superparamagnetic writing layer and a 9 nm FePt storage layer is proposed. The transition jitter of the proposed com-posite structure approaches the grain-size theoretical limitand is impervious to the Tc variance in FePt layer. A reduced writing temperature is demonstrated. Under a 30 nm FWHM heat profile, 6 nm bit length, 21 nm CTCD, and 12 nm readerwidth, the user density of the proposed composite structurecan reach 4 :7T b =in: 2. The authors acknowledge useful discussion with Pin- Wei Huang and Tao Qu. The authors also thank Seagate for funding this project. FIG. 5. Recording patterns of central track under no overlap (top one) and CTCD ¼18 nm (bottom one). These two cases use the same PRBS pattern for central track. Regions enclosed by red rectangles illustrate the distortion of the patterns on the central track.232402-4 Liu, Jiao, and Victora Appl. Phys. Lett. 108, 232402 (2016) 1O. Hovorka, S. Devos, Q. Coopman, W. J. Fan, C. J. Aas, R. F. L. Evans, X. Chen, G. Ju, and R. W. Chantrell, Appl. Phys. Lett. 101, 052406 (2012). 2A. Chernyshov, T. Le, B. Livshitz, O. Mryasov, C. Miller, R. Acharya,and D. Treves, J. Appl. Phys. 117, 17D111 (2015). 3A. Chernyshov, D. Treves, T. Le, F. Zong, A. Ajan, and R. Acharya, J. Appl. Phys. 115, 17B735 (2014). 4J. U. Thiele, S. Maat, and E. Fullerton, Appl. Phys. Lett. 82, 2859 (2003). 5R. H. Victora, S. Wang, P. Huang, and A. Ghoreyshi, IEEE Trans. Magn. 51(4), 3200307 (2015). 6P. Huang and R. H. Victora, IEEE Trans. Magn. 50(11), 3203304 (2014). 7S. Wang, M. Mallary, and R. H. Victora, IEEE Trans. Magn. 50(11), 3202304 (2014). 8Z. Liu and R. H. Victora, IEEE Trans. Magn. PP(99), 1 (2016). 9D. Suess and T. Schrefl, Appl. Phys. Lett. 102, 162405 (2013). 10Y. Jiao, Y. Wang, and R. H. Victora, IEEE Trans. Magn. 51(11), 3002304 (2015).11R. H. Victora and P. Huang, IEEE Trans. Magn. 49(2), 751 (2013). 12Y. Jiao, Z. Liu, and R. H. Victora, J. Appl. Phys. 117, 17E317 (2015). 13N. Wang, K. Komvopoulos, F. Rose, and B. Marchon, J. Appl. Phys. 113, 083517 (2013). 14P. Huang and R. H. Victora, J. Appl. Phys. 115, 17B710 (2014). 15M. H. Kryder, E. C. Gage, T. W. Mcdaniel, W. A. Challener, R. E. Rottmayer, G. Ju, Y.-T. Hsia, and M. F. Erden, Proc. IEEE 96(11), 1810–1835 (2008). 16B. Valcu and N.-H. Yeh, IEEE Trans. Magn. 46(6), 2160 (2010). 17J.-U. Thiele, K. R. Coffey, M. F. Toney, J. A. Hedstrom, and A. J. Kellock, J. Appl. Phys. 91, 6595 (2002). 18C. E. Shannon, Bell Syst. Tech. J. 27(3), 379–423 (1948). 19J. Moon and W. Zeng, IEEE Trans. Magn. 31(2), 1083–1088 (1995). 20Y. Wang, M. F. Erden, and R. H. Victora, IEEE Magn. Lett. 3, 4500304 (2012). 21J. Caroselli, S. A. Altekar, P. McEwen, and J. K. Wolf, IEEE Trans. Magn. 33(5), 2779–2781 (1997).232402-5 Liu, Jiao, and Victora Appl. Phys. Lett. 108, 232402 (2016)

1.1944902.pdf

Domain-wall displacement triggered by an ac current below threshold Gen Tatara, Eiji Saitoh, Masahiko Ichimura, and Hiroshi Kohno Citation: Applied Physics Letters 86, 232504 (2005); doi: 10.1063/1.1944902 View online: http://dx.doi.org/10.1063/1.1944902 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/86/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Influence of the free-layer domain structure on domain-wall displacing type field sensors J. Appl. Phys. 109, 07E523 (2011); 10.1063/1.3561810 Effect of ac on current-induced domain wall motion J. Appl. Phys. 101, 09A504 (2007); 10.1063/1.2713211 Current-driven domain-wall depinning J. Appl. Phys. 98, 016108 (2005); 10.1063/1.1957122 Domain wall displacement induced by subnanosecond pulsed current Appl. Phys. Lett. 84, 2820 (2004); 10.1063/1.1711168 Magnetic domain wall motion triggered by an electric current Appl. Phys. Lett. 83, 2617 (2003); 10.1063/1.1578165 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: 128.237.130.127 On: Sun, 14 Dec 2014 19:51:17Domain-wall displacement triggered by an ac current below threshold Gen Tataraa! PRESTO, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan, and Graduate School of Science, Tokyo Metropolitan University, 1-1 Minamiosawa, Hachioji, Tokyo 192-0397, Japan Eiji Saitoh Department of Physics, Keio University, Yokohama 223-8522, Japan Masahiko Ichimura Advanced Research Laboratory, Hitachi, Ltd., Hatoyama, Saitama, 350-0395, Japan Hiroshi Kohno Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan sReceived 17 November 2004; accepted 30 April 2005; published online 2 June 2005 d It is theoretically demonstrated that a displacement of a pinned domain wall, typically of order of mm, can be driven by use of an ac current which is below threshold value. The point here is that finite motion around the pinning center by a low current is enhanced significantly by the resonanceif the frequency is tuned close to the pinning frequency as demonstrated by recent experiment. ©2005 American Institute of Physics .fDOI: 10.1063/1.1944902 g Domain-wall motion driven by electric current is of spe- cial interest recently. From the viewpoint of application tomagnetic memories, the most urgent issue is to reduce thethreshold current. Threshold current in experiments so far onmetallic wires under dc or slow pulse current is mostly oforder of 10 11–1012sA/m2d.1–4Smaller value of 1010sA/m2d is reported when a pulse of order of ns is applied.5 To realize lower current density, further understanding of the driving mechanism as well as extrinsic effects, such aspinning, are necessary. Current-driven domain-wall motionin an artificial pinning potential was studied quite recently. 6 Besides the pinning potential being controllable, the experi-ment has a novelty in that the applied current is ac current ofMHz range, and the resonance behavior was observed. Thedata indicated surprisingly that the domain wall was drivenmostly by the force due to charge current, in contrast to thecase of dc current, in which the spin torque due to spincurrent dominates. This was argued there to be due to thestrong enhancement of the force at the resonance and, at thesame time, the reduction of spin torque effect in slow acdynamics fcompared with microscopic spin frequency s,GHz dg. The displacement observed there is driven below the critical current and thus is finite, but still it is rather big sestimated to be ,10 mmd. This result indicates that by use of ac field, domain wall can be driven by a different mecha- nism from the dc case, and this would be useful in realizingdomain-wall displacement at low current. Motivated by this experiment, we studied the depinning of a domain wall under ac current theoretically. It is shownthat depinning under ac field occurs at a current densitywhich is lower than the dc case roughly by a factor of Gilbertdamping. Thus the use of ac current is quite promising fordevice application. We consider a wire in the zdirection. Choosing a hard axis as the ydirection, the spin Hamiltonian isH S=S2 2hJss„ud2+sin2us„fd2d +sin2usK+K’sin2fdj, s1d where the easy- and hard-axis anisotropies sKandK’din- clude the effect of demagnetizing field.We consider a case oflargeK ’or small current ssmaller than the critical current d, thus fcan be treated as small. Note that most dc experi- ments so far are in the current region larger than the criticalvalue, thus the analysis below does not apply. We consider apinning potential of a harmonic type with a range of j, VsXd=1 2MwV02sX2−j2dusj−uXud, s2d whereMw;"2N/K’l2is the mass of wall, l;˛J/Kis the wall thickness, and the oscillation frequency at the bottom isdenoted by V 0. We consider a current with frequency v0. The equation of motion for the domain wall at z=Xstdis written as fwe neglect terms of Osa2d, where ais the Gilbert damping parameter g X¨+1 tX˙+V02X=fjstd, s3d where 1/ t;saK’/"ds1+s"V0/K’d2d.fjis the total force due to ac current divided by Mw, which is given in terms of charge current jand spin current jsas6,7 fjstd;a3 2SeFK’lAn "e2Rw "Sj+a" K’] ]tjD+g] ]tjsG,s4d where gis the adiabaticity parameter sg!1 for thick wall, lkF@1d,Rwis resistance due to wall, and nandAare elec- tron density and cross sectional area, respectively. We con-sider a current with an angular frequency of v0and the am- plitudej0switched at t=0;jstd=j0e−iv0tustd. By using parameter brepresenting the spin polarization of current sjs;bjd, we can write fjstd=f0sv0de−iv0tustd+f1dstd, whereadElectronic mail: tatara@ess.sci.osaka-u.ac.jpAPPLIED PHYSICS LETTERS 86, 232504 s2005 d 0003-6951/2005/86 ~23!/232504/2/$22.50 © 2005 American Institute of Physics 86, 232504-1 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: 128.237.130.127 On: Sun, 14 Dec 2014 19:51:17f0sv0d;a3 2SeFK’ "R˜S1−ia"v0 K’D−ibgv0Gj0, s5d andR˜;lAn e2Rw/". The d-function part is given by f1;a3 2SesaR˜+bgdj0. s6d We here neglect f1, focusing on the resonating behavior due to oscillating force, f0. The oscillating component of the so- lution is then obtained as Xstd=f0 2Ve−iv0t11−ef−1 2t+isv0+Vdgt Sv0+V+i1 2tD −1−ef−1 2t+isv0−Vdgt Sv0−V+i1 2tD2, s7d where V;˛V02−1/4 t2. After a long time, t*t, the ampli- tude ofXreduces to uXstdu!uf0u1 ˛sv02−V02d2+Sv0 tD2. s8d This is enhanced at resonance, v0=V0,t ob e uXstdures=uf0u1 1 tV0="uf0u aK’V0S1+S"V0 K’D2D. s9d Depinning occurs when uXstdu*j. At resonance, the condi- tion is given by uf0u=1/tV0j=ajK’V0s1+s"V0/K’d2d/" ;fcac. The threshold current is thus given by jcac=2Se a3ajV0S1+S"V0 K’D2D ˛R˜2+S"V0 K’D2 sgb+aR˜d2. s10d In the case of dc current, threshold condition is given by f0 =V02j;fcdc, i.e.,fcac=s1/tV0dfcdc. Thus, the threshold cur- rent in the ac case at resonance is written in terms of dc case asjcac=ejcdc, where e;1 tV0=aK’ "V0S1+S"V0 K’D2D. s11d Since ais generally small s,0.01d, we see that depinning threshold is much lower when an ac current is applied than the dc case unless K’/"V0is very large or very small. sIn most metallic wires, the pinning angular frequency V0is small compared with oscillation frequency scale of a singlespin,K ’/".dnote, we Here, however, that even in the case of very weak pinning created by external magnetic field sV0 ,2p325 MHz and K’,0.1 K din Ref. 6, the ac enhance- ment by a factor of 2 is expected; e.0.5. The displacementthere is estimated to be DX,sa3j0/2edsR˜/alV0d,10mm. The dominant driving mechanism under ac current is determined by the ratio fsee Eq. s10dg h=bg"V0 R˜K’, s12d where the spin transfer effect dominates over the momentum transfer if h.1. In the case of Ref. 6, V0.2p325 MHz, l=70 nm, A=70345 nm2.K’is estimated to be 0.1 K, and a,2.5 Å. Thus, spin transfer dominates if Rw&0.5bg 310−5sVd. The spectrum obtained there suggests a larger resistance, Rw,3310−4sVd, i.e., momentum transfer is dominant, in sharp contrast to the case of dc current. This experimental finding for a thick wall with thickness ,70 nm is quite surprising. At the same time, this fact has an impor-tant implication for applications. In fact, domain-wall resis-tance decays quite rapidly for a thickness larger than Fermiwavelength, 8,9typically exponentially Rw~e−2pzkFlif the wall profile is the form of tanh, where zis a number of order of unity. In the case of a linear wall, where Szchanges lin- early as function of x, it decays slower as Rw ~sin2zkFl/skFld2. Even in this modest case, we will have Rwlarger by an order of magnitude if we can reduce the wall thickess by a factor of 3 and then the driving force would belarger by order of magnitude. For the tanh type of wall, thechange would be more drastic. It is critically important thatfor the ac case, where momentum transfer contributes, R wis another parameter to control the threshold. For the application, switching speed is crucially impor- tant. The speed based on resonance switching consideredabove is determined by the time scale t. In the case of Ref. 6, sK’=0.1 K and a=0.01 d, the time scale corresponds to 20 MHz. Faster operation would be possible by using larger K’. To implement the ac current-driven domain-wall motion into device, one would need to fabricate two pinning centers,between which the wall hops each time the ac pulse is ap-plied. One of the authors sG.T.dthanks Monka-shou, Japan and The Mitsubishi Foundation for financial support. 1J. Grollier, D. Lacour, V. Cros, A. Hamzic, A. Vaurés, A. Fert, D. Adam, and G. Faini, J. Appl. Phys. 92, 4825 s2002 d, J. Grollier, P. Boulenc, V. Cros, A. Hamzic, A. Vaurés, A. Fert, and G. Faini, Appl. Phys. Lett. 83, 509s2003 d. 2N. Vernier, D. A. Allwood, D. Atkinson, M. D. Cooke, and R. P. Cowburn, Europhys. Lett. 65, 526 s2004 d. 3M. Kläui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, and L. J. Heyderman, Appl. Phys. Lett. 83, 105 s2003 d. 4A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 s2004 d. 5C. K. Lim, T. Devolder, C. Chappert, J. Grollier, V. Cros, A. Vaurès, A. Fert, and G. Faini, Appl. Phys. Lett. 84, 2820 s2004 d. 6E. Saitoh, H. Miyajima, T. Yamaoka and G. Tatara, Nature sLondon d432, 203s2004 d. 7G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 s2004 d. 8G. G. Cabrera and L. M. Falicov, Phys. Status Solidi B 61,5 3 9 s1974 d. 9G. Tatara, Y.-W. Zhao, M. Munoz, and N. Garcia, Phys. Rev. Lett. 83, 2030 s1999 d.232504-2 Tatara et al. Appl. Phys. Lett. 86, 232504 ~2005 ! This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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1.4818326.pdf

Perpendicular magnetic anisotropy in Ta/Co2FeAl/MgO multilayers M. S. Gabor, T. Petrisor, C. Tiusan, and T. Petrisor Citation: J. Appl. Phys. 114, 063905 (2013); doi: 10.1063/1.4818326 View online: http://dx.doi.org/10.1063/1.4818326 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i6 Published by the AIP Publishing LLC. 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 27 Sep 2013 to 131.91.169.193. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsPerpendicular magnetic anisotropy in Ta/Co 2FeAl/MgO multilayers M. S. Gabor,1,a)T. Petrisor, Jr.,1C. Tiusan,1,2,b)and T. Petrisor1 1Center for Superconductivity, Spintronics and Surface Science, Technical University of Cluj-Napoca, Cluj-Napoca, Romania 2Institut Jean Lamour, P2M, CNRS-Nancy University, Nancy, France (Received 6 June 2013; accepted 29 July 2013; published online 12 August 2013) In this paper, we demonstrate the stabilization of perpendicular magnetic anisotropy (PMA) in Ta/ Co2FeAl/MgO multilayers sputtered on thermally oxidized Si(100) substrates. The magnetic analysis points out that these films show significant interfacial anisotropy even in the as-depositedstate, K S¼0:67 erg =cm2, enough to provide PMA for the as-deposited films with thicknesses below 1.5 nm. Moreover, the interfacial anisotropy is enhanced by thermal annealing up to 300/C14C. The presence of a magnetic dead layer, whose thickness increases with annealing temperature, wasalso identified. VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4818326 ] I. INTRODUCTION Magnetic tunnel junctions (MTJs) with ferromagnetic electrodes showing perpendicular magnetic anisotropy(PMA) are of great interest since they exhibit several advan- tages with respect to the MTJs with ferromagnetic electrodes displaying in-plane magnetic anisotropy. They allow thereduction of the spin transfer torque (STT) switching current density (J c0)1,2due to the absence of the demagnetization term, and show higher stability against thermal fluctuationsas a result of their larger anisotropy energy compared to the established in-plane magnetized materials. The conventional PMA materials explored so far include the L1 0ordered alloys (FePt and CoPt), Co-based multilayers (Co/Pd and Co/Pt), and rare-earth and transition metal alloys.3–9 However, these materials suffer from some limitations including insufficient chemical and/or thermal stability and low spin polarization. Recently, it was shown that CoFeB- MgO interfacial magnetic anisotropy can be used to manu-facture MTJs with thin CoFeB layers ( <1.5 nm) showing perpendicular magnetic easy axis and which maintain a large magnetoresistive ratio. 10However, the Gilbert damping con- stant ( a) of CoFeB films was found to increase with decreas- ing the films thickness10and since J c0is proportional to a, this might constitute a disadvantage for STT switching. A material that seems to be an ideal candidate as a ferro- magnetic electrode for perpendicular magnetic tunnel junc- tions (p-MTJs) is the full Heusler alloy Co 2FeAl (CFA). It was demonstrated to provide giant tunneling magnetoresist- ance (GTMR) in MgO based MTJs.11–14Moreover, this ma- terial is of special importance since it was shown to possessa low Gilbert damping, 15–17essential for applications con- cerning STT switching. CFA is a material with a relative low cubic magnetic anisotropy,18which was found to provide PMA in Pt/CFA/MgO and Cr/CFA/MgO trilayer struc- tures.19,20In this paper, we extend these studies and demon- strate the presence of PMA in relatively thick CFA films inTa/CFA/MgO trilayers. Furthermore, we will show that theCFA films thinner than 1.5 nm exhibit perpendicular mag- netic easy axis even before annealing. II. EXPERIMENTAL The Si/SiO 2/Ta(6 nm)/CFA(0.9–4.8 nm)/MgO(0.65 nm)/ Ta(1.2 nm) multilayer stacks were elaborated using a magne- tron sputtering system having a base pressure lower than4/C210 /C09Torr. The metallic films were deposited at room tem- perature (RT) by DC sputtering under an Ar pressure of 1 mTorr. For the growth of the CFA layer, a 2-in. Co 2FeAl stoichiometric target was used. The MgO film was grown at RT by rf sputtering from a MgO polycrystalline target in an Ar pressure of 15 mTorr. After deposition, the multilayerswere ex situ annealed for 1 h at temperatures up to 350 /C14Ci na vacuum better than 3 /C210/C08Torr in the absence of an applied magnetic field. The crystalline structure of the multilayers wasanalyzed by x-ray diffraction (XRD) using a four-circle dif- fractometer. The magnetic properties of the films were studied at RT using a Vibrating Sample Magnetometer (VSM). III. RESULTS AND DISCUSSIONS In order to determine the crystal structure of the multi- layers, we performed x-ray diffraction experiments in graz- ing incidence geometry (GIXRD). For getting the maximumdiffracted signal, our analysis has been performed on the sample with the thicker CFA layer: Ta(6 nm)/CFA(4.8 nm). Figure 1shows the diffraction pattern recorded for this sam- ple, annealed at 200 /C14C. In spite of the relative low diffracted signal, the XRD pattern clearly shows peaks corresponding to the Ta (110) and CFA (220), (400), and (422) reflections,respectively. This indicates that at least the CFA film has a polycrystalline structure. After the background subtraction, the diffraction peaks were fitted using pseudo-Voigt func-tions. This allowed us to evaluate their full width at half maximum (FWHM) and to determine the mean structural co- herence length along the grazing direction by means of thewell-known Scherrer equation. 21The obtained values are 1.35 nm for the Ta film and 5.83 nm for the CFA film, respectively. For the samples with thinner CFA films, theXRD analysis is difficult to perform due to the relative lowa)Electronic mail: Mihai.Gabor@phys.utcluj.ro b)Electronic mail: Coriolan.Tiusan@phys.utcluj.ro 0021-8979/2013/114(6)/063905/4/$30.00 VC2013 AIP Publishing LLC 114, 063905-1JOURNAL OF APPLIED PHYSICS 114, 063905 (2013) Downloaded 27 Sep 2013 to 131.91.169.193. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsdiffracted signal. However, extrapolating the XRD results obtained for the 4.8 nm thick CFA film sample, it is reasona- ble to assume that the thinner CFA films are also polycrystal- line. This represents a quite different situation with respectto the (100) textured thin films: Fe/MgO(100), CoFeB/ MgO(100), and Co 2FeAl/MgO(100) in which PMA has been reported.10,20,22 Figure 2shows in-plane and out-of-plane magnetization curves for the as-deposited and 200/C14C annealed samples with CFA thickness of 1.4 and 1.7 nm, respectively. One canobserve that the 1.4 nm thick CFA film shows PMA even in the as-deposited state and that the PMA is maintained after annealing. The squareness ratio (M R/MS) of the out-of-plane magnetization loop increases from 0.7, before annealing, to 0.9, after annealing. On the other hand, the as-deposited 1.7 nm thick film shows no apparent PMA, while the PMA isclearly achieved after annealing, with an out-of-plane mag- netization loop squareness ratio of about 0.7. The origin of PMA at Fe-MgO and Co-MgO interfaces is commonlyattributed to the hybridization of the O 2 pand Co or Fe 3 dorbitals. 22,23In contrast with previous reports,19,20in the case of our samples, the 0.65 nm thick MgO layer seems to provide adequate Co-O and Fe-O bonding at the interface topromote the PMA even before annealing. To get deeper insight of the role of the annealing on the magnetic properties of the samples, we investigated the evo-lution of the saturation magnetization (M S) for CFA films with different thicknesses with respect to the annealing tem- perature (see Fig. 3). The M Swas determined by taking into account the CFA layers nominal thicknesses. In the case of the 4.8 nm thick film, the Ms shows first a slight increase up to the annealing temperature of 250/C14C, and then a small decrease for higher annealing temperatures. In the case of the thinner films, the Ms shows no obvious variation up to anannealing temperature of 150 /C14C. However, a rather strong decrease of Ms is observed for higher annealing tempera- tures. Moreover, even for the as-deposited samples, the Msshows a monotonous reduction with the decrease of the CFA film thickness. These findings give evidence towards a mag- netically dead layer present in the CFA films, whose thick-ness increases with increasing the annealing temperature. In order to quantify the extent of the magnetic dead layer, we have plotted the M St vs. t (CFA layer nominal thickness), as shown in Fig. 4(a) for the case of the samples annealed at 150/C14C. In this representation, the intercept with the thickness axis of the linear fit of the data gives the magni-tude of the dead layer, whereas the slope gives the saturation magnetization (M o S) of the CFA film. Our analysis indicates a thickness of the magnetic dead layer (t DL) of around 0.5 nm for annealing temperatures up to 200/C14C, which increases up to 1.4 nm for the samples annealed at 350/C14C (Fig. 4(b)). This trend is most likely due to the interdiffusion at theCFA-Ta interface, which becomes more pronounced with increasing the annealing temperature. At the same time, the saturation magnetization of the films shows an increase withthe annealing temperature, which can be attributed to improvement of the structural quality and of the chemical order with the annealing (Fig. 4(b)). To quantify the magnitude of the PMA in CFA films, the effective perpendicular magnetic anisotropy constant K eff was determined from the saturation field, H S, using the rela- tion Kef f¼HSMo S=2. In the case of perpendicular magnetic easy axis, the H Swas considered to be positive and was FIG. 1. X-ray diffraction pattern measured in grazing incidence geometry (GIXRD) for the Si/SiO 2/Ta(6 nm)/CFA(4.8 nm)/MgO(0.65 nm)/Ta(1.2 nm) multilayer stack annealed at 200/C14C. The symbols represent experimental data, while the lines are the result of the theoretical fit. The vertical dashed lines mark the positions of the Ta(110) and CFA(220), (400), and (422) reflections, respectively. FIG. 2. In-plane and out-of-plane magnetization curves for the as-deposited and 200/C14C annealed samples with CFA thickness of 1.4 and 1.7 nm. FIG. 3. Evolution of the saturation magnetization (Ms) with the annealingtemperature for multilayer stacks with different CFA thicknesses.063905-2 Gabor et al. J. Appl. Phys. 114, 063905 (2013) Downloaded 27 Sep 2013 to 131.91.169.193. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsdetermined from the in-plane magnetization curve. On the other hand, in the case of in-plane magnetic easy axis, the H S was considered to be negative and was determined from theout-of-plane hysteresis loop. The K effcan be described by the phenomenological relation24 Kef f¼Kv/C02pMo2 SþKS=tCFA ef f; where Kvdescribes the bulk magnetocrystalline anisotropy, 2pMo2 Sthe shape anisotropy, and KSthe interface anisotropy. By fitting the tCFA ef f(CFA layer effective thickness) dependence of the product Kef ftCFA ef fwith the above equation, the different anisotropy contributions can be evaluated. For the filmsannealed at 150 /C14C (see Fig. 5), the bulk anisotropy ( Kv) was found to be negligible and KSof about 0.77 erg/cm2. This value increases to about 1.14 erg/cm2if, when deriving theKS, the nominal thickness of the CFA layer is used instead of the effective one. The KSfor the as-deposited samples was estimated to about 0.67 erg/cm2. An increase in 0.77–0.95 erg/cm2range was observed with increasing the annealing temperature up to 300/C14C. This can be related to an improvement of the CFA-MgO interface quality with anneal-ing. At an annealing temperature of 350 /C14C, the KSshows a sharp decrease down to about 0.48 erg/cm2, most likely due to the CFA-Ta enhanced interdiffusion and to the degradation ofthe thin MgO layer at the upper CFA interface. Figure 6shows a contour plot, deduced from the experi- mental data points, of the K ef ftCFA nom:product vs. the nominal thickness of the CFA layer and the annealing temperature. The critical CFA layer thickness, which separates the out-of-planeof the in-plane anisotropy, is situated around 1.5 nm for the as- deposited samples and for the samples annealed at 150/C14C. This critical thickness increases to about 1.9 nm for the samples annealed up to 300/C14C. This increase is the result of two phe- nomena: first, the enhancement of the KS, and second the reduction of the tCFA ef fwith increasing the annealing tempera- ture. For the samples annealed at 350/C14C due to the decrease of KSand to the increase of the magnetic dead layer thickness (Fig. 4(b)) the CFA thickness range in which one can obtain films with PMA becomes extremely narrow, not suitable for practical purposes related to potential applications of CFA. IV. CONCLUSION We have experimentally demonstrated the stabilization of PMA in Ta/CFA/MgO multilayer stacks grown by sputter- ing on thermally oxidized Si(100) substrates. Our CFA layers show significant interfacial anisotropy, KS¼0:67 erg =cm2, even in the as-deposited state. This is sufficient to provide PMA for as deposited magnetic films with thickness below 1.5 nm. An increase of KSin the 0.77-0.95 erg/cm2range was observed with increasing the annealing temperature up to 300/C14C. The presence of a magnetic dead layer, whose thick- ness increases with annealing temperature was also identified.The occurrence of PMA in the as-deposited samples renders the Ta/Co 2FeAl/MgO system promising from an application point of view. FIG. 4. (a) M St vs. CFA nominal layer thickness for the samples annealed at 150/C14C. The straight line is the result of the linear fit. (b) Dependence of the magnetic dead layer thickness and sat- uration magnetization on the annealingtemperature. FIG. 5. Dependence of the Kef ftCFA ef fproduct on the tCFA ef ffor the samples annealed at 150/C14C. The straight line is the result of a linear fit. FIG. 6. Contour plot of the Kef ftCFA nom :product vs. the nominal thickness of the CFA layer and the annealing temperature. The dashed line depicts the boundary between the in-plane and out-of-plane anisotropy. The gray area is the region where the clear ferromagnetic signal of the CFA films is lost.063905-3 Gabor et al. J. Appl. Phys. 114, 063905 (2013) Downloaded 27 Sep 2013 to 131.91.169.193. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsACKNOWLEDGMENTS This work has been partially supported by CNCSIS UEFISCSU, Project No. PNII IDEI No.4/2010, code ID-106 and by POS CCE ID. 574, code SMIS-CSNR 12467. 1S. Yakata, H. Kubota, Y. Suzuki, K. Yakushiji, A. Fukushima, S. Yuasa, and K. Ando, J. Appl. Phys. 105, 07D131 (2009). 2S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E. E. Fullerton, Nature Mater. 5, 210–215 (2006). 3N. Nishimura, T. Hirai, A. Koganei, T. Ikeda, K. Okano, Y. Sekiguchi, and Y. Osada, J. Appl. Phys. 91, 5246–5249 (2002). 4H. Ohmori, T. Hatori, and S. Nakagawa, J. Appl. Phys. 103, 07A911 (2008). 5G. Kim, Y. Sakuraba, M. Oogane, Y. Ando, and T. Miyazaki, Appl. Phys. Lett. 92, 172502 (2008). 6K. Yakushiji, T. Saruya, H. Kubota, A. Fukushima, T. Nagahama, S. Yuasa, and K. Ando, Appl. Phys. Lett. 97, 232508 (2010). 7J.-H. Park, C. Park, T. Jeong, M. T. Moneck, N. T. Nufer, and J.-G. Zhu, J. Appl. Phys. 103, 07A917 (2008). 8B. Carvello, C. Ducruet, B. Rodmacq, S. Auffret, E. Gautier, G. Gaudin, and B. Dieny, Appl. Phys. Lett. 92, 102508 (2008). 9A. Barman, S. Wang, O. Hellwig, A. Berger, E. E. Fullerton, and H. Schmidt, J. Appl. Phys. 101, 09D102 (2007). 10S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nature Mater. 9, 721–724 (2010).11W. Wang, H. Sukegawa, R. Shan, S. Mitani, and K. Inomata, Appl. Phys. Lett. 95, 182502 (2009). 12W. Wang, E. Liu, M. Kodzuka, H. Sukegawa, M. Wojcik, E. Jedryka, G. H. Wu, K. Inomata, S. Mitani, and K. Hono, Phys. Rev. B 81, 140402 (2010). 13W. Wang, H. Sukegawa, and K. Inomata, Phys. Rev. B 82, 092402 (2010). 14Z. Wen, H. Sukegawa, S. Mitani, and K. Inomata, Appl. Phys. Lett. 98, 192505 (2011). 15S. Mizukami, D. Watanabe, M. Oogane, Y. Ando, Y. Miura, M. Shirai, and T. Miyazaki, J. Appl. Phys. 105, 07D306 (2009). 16G. Ortiz, M. S. Gabor, T. Petrisor, Jr., F. Boust, F. Issac, C. Tiusan, M. Hehn, and J. F. Bobo, J. Appl. Phys. 109, 07D324 (2011). 17M. Belmeguenai, H. Tuzcuoglu, M. S. Gabor, T. Petrisor, C. Tiusan, D. Berling, F. Zighem, T. Chauveau, S. M. Ch /C19erif, and P. Moch, Phys. Rev. B87, 184431 (2013). 18M. S. Gabor, T. Petrisor, Jr., C. Tiusan, M. Hehn, and T. Petrisor, Phys. Rev. B 84, 134413 (2011). 19X. Li, S. Yin, Y. Liu, D. Zhang, X. Xu, J. Miao, and Y. Jiang, Appl. Phys. Express 4, 043006 (2011). 20Z. Wen, H. Sukegawa, S. Mitani, and K. Inomata, Appl. Phys. Lett. 98, 242507 (2011). 21A. L. Patterson, Phys. Rev. 56, 978–982 (1939). 22H. X. Yang, M. Chshiev, B. Dieny, J. H. Lee, A. Manchon, and K. H. Shin, Phys. Rev. B 84, 054401 (2011). 23A. Manchon, C. Ducruet, L. Lombard, S. Auffret, B. Rodmacq, B. Dieny, S. Pizzini, J. Vogel, V. Uhl /C19ı/C20r, M. Hochstrasser, and G. Panaccione, J. Appl. Phys. 104, 043914 (2008). 24C. A. F. Vaz, J. A. C. Bland, and G. Lauhoff, Rep. Prog. Phys. 71, 056501 (2008).063905-4 Gabor et al. J. Appl. Phys. 114, 063905 (2013) Downloaded 27 Sep 2013 to 131.91.169.193. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions

1.2822407.pdf

Magnetization dynamics in planar spin transfer devices and stabilization by repulsion in a spin-flip transistor Ya. B. Bazaliy Citation: Applied Physics Letters 91, 262510 (2007); doi: 10.1063/1.2822407 View online: http://dx.doi.org/10.1063/1.2822407 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/91/26?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Micromagnetic simulations of spin-wave normal modes and the spin-transfer-torque driven magnetization dynamics of a ferromagnetic cross J. Appl. Phys. 115, 17D123 (2014); 10.1063/1.4863384 Spin transistor based on T-shaped graphene junctions J. Appl. Phys. 110, 033701 (2011); 10.1063/1.3615951 Effect of quantum confinement on spin transport and magnetization dynamics in dual barrier spin transfer torque magnetic tunnel junctions J. Appl. Phys. 108, 104306 (2010); 10.1063/1.3503882 Optically controlled spin polarization in a spin transistor Appl. Phys. Lett. 94, 162109 (2009); 10.1063/1.3111442 Datta–Das transistor with enhanced spin control Appl. Phys. Lett. 82, 2658 (2003); 10.1063/1.1564867 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: 131.230.73.202 On: Sun, 21 Dec 2014 18:15:02Magnetization dynamics in planar spin transfer devices and stabilization by repulsion in a spin-flip transistor Ya. B. Bazaliya/H20850 Instituut-Lorentz, Universiteit Leiden, P .O. Box 9506, 2300 RA Leiden, The Netherlands, Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA,and Institute of Magnetism, National Academy of Science of Ukraine, 36-b V ernadsky Blvd.,Kyiv 03142, Ukraine /H20849Received 11 October 2007; accepted 16 November 2007; published online 27 December 2007 /H20850 In systems with dominating easy-plane anisotropy, magnetization dynamics is governed by effective one dimensional equation for the in-plane angle. Rederiving this equation in the presence of spintorques, we obtain a convenient and intuitive description of spin transfer devices. In the case of aspin-flip transistor, the method provides a surprising prediction: the device can be stabilized in the/H20849normally unstable /H20850energy saddle point by a spin torque repelling from that point. Stabilization by repulsion happens due to the presence of dissipative environment and requires a Gilbert dampingconstant that is large enough to ensure overdamped dynamics at zero current. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2822407 /H20852 The spin transfer effect is a nonequilibrium interaction that arises when a current of electrons flows through a non-collinear magnetic texture. 1–3This interaction can become significant in nanoscopic magnets and is nowadays studied ina variety of systems. Current induced magnetic switching 4 and magnetic domain wall motion5based on spin transfer effect serve as an underlying mechanism for a number ofmemory and logic applications currently in development. Inthose applications, spin transfer torque plays a key role cre-ating dynamic regimes that are not present in conventionalmagnetic devices. However, full dynamic study is notstraightforward even for simple spin transfer devices due to the nonlinearity of the Landau-Lifshitz-Gilbert /H20849LLG /H20850equa- tion governing the magnetization motion. A simple deviceconsists of a dynamic magnet /H20849free layer /H20850influenced by the current coming from a stationary spin polarizer /H20849fixed layer /H20850. 2When magnetic anisotropy of the free layer is an easy axis parallel to the spin polarizer direction, the problemcan be solved exactly. 2,6However, if in addition the easy plane anisotropy of actual devices7is accounted for, and/or the polarizer direction is different, the calculations becomemuch more involved. 6,8–12As the anisotropy gets more com- plicated, additional dynamic features appear: “cantedstates,” 6,10,12multiple precession states,8–10“magnetic fan” regimes,13–16etc. Experimentally, the easy plane anisotropy energy is usually much larger than the easy axis energy, i.e.,the system is in the regime of a planar spintronic device 17 /H20849Fig.1/H20850. Fortunately, the limit of dominating easy plane en- ergy is characterized by a simplification of the dynamicequations, 18,19which mathematically arises from the exis- tence of a small parameter: the ratio of the energy modula-tion within the plane to the easy plane energy. Physically, themagnetization moves with little deviations from the easyplane and thus can be well described by the in-plane anglegoverned by an effective planar equation. In this paper, we present the effective planar equation that takes into account spin transfer torques in general formand does not suffer from the limitations of the previouslyemployed linear expansion in current magnitude. 19The power of the approach is then demonstrated in the case of thespin-flip transistor.20In that device, spin transfer attracts the system to one of the two energy saddle points which even-tually becomes stable.21Here we show that spin transfer can also stabilize the other saddle point even though it repels thesystem from it. Although highly counterintuitive, this predic-tion is later verified by the no-approximation calculations. In the macrospin model, the magnetization of the free layer has a constant absolute value Mand a direction given by a unit vector n/H20849t/H20850. The LLG equation 2,6reads n˙=/H9253 M/H20875−/H9254E /H9254n/H11003n/H20876+u/H20849n/H20850/H20851n/H11003/H20851s/H11003n/H20852/H20852+/H9251/H20851n/H11003n˙/H20852./H208491/H20850 Here, /H9253is the gyromagnetic ratio, E/H20849n/H20850is the magnetic en- ergy of the free layer, /H9251is the Gilbert damping constant, sis a unit vector along the direction of the polarizer, and the spintransfer strength u/H20849n/H20850is proportional to the electric current I.6,19In general, u/H20849n/H20850=f/H20851/H20849n·s/H20850/H20852I, with the function f/H20851/H20849n·s/H20850/H20852 being material and device specific. Equation /H208491/H20850can be writ- ten in polar angles /H20851/H9258/H20849t/H20850,/H9278/H20849t/H20850/H20852, /H9258˙+/H9251/H9278˙sin/H9258=−/H9253 Msin/H9258/H11509E /H11509/H9278+u/H20849s·e/H9258/H20850/H11013F/H9258, /H9278˙sin/H9258−/H9251/H9258˙=/H9253 M/H11509E /H11509/H9258+u/H20849s·e/H9278/H20850/H11013F/H9278, /H208492/H20850 with tangent vectors e/H9278=/H20851zˆ/H11003n/H20852/sin/H9258ande/H9258=/H20851e/H9278/H11003n/H20852. The easy plane is chosen at /H9258=/H9266/2, and the magnetic energy has the form E=/H20849K/H11036/2/H20850cos2/H9258+Er/H20849/H9258,/H9278/H20850, where Eris the “re- sidual” energy. In the planar limit, K/H11036→/H11009, Eq. /H208492/H20850can be expanded in small parameters /H20841Er/H20841/K/H11036/H112701,/H20841u/H20849n/H20850/H20841/K/H11036/H112701. a/H20850Electronic mail: yar@mailaps.org. FIG. 1. /H20849Color online /H20850Planar spin transfer devices. Hashed parts of the devices are ferromagnetic and white parts are made from a nonmagneticmetal.APPLIED PHYSICS LETTERS 91, 262510 /H208492007 /H20850 0003-6951/2007/91 /H2084926/H20850/262510/3/$23.00 © 2007 American Institute of Physics 91, 262510-1 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: 131.230.73.202 On: Sun, 21 Dec 2014 18:15:02Assuming time-independent uands, we obtain an effective equation of the in-plane motion, 1 /H9275/H11036/H9278¨+/H9251eff/H9278˙=−/H9253 M/H11509Eeff /H11509/H9278, /H208493/H20850 which has the form of Newton’s equation of motion for a particle in external potential Eeff/H20849/H9278/H20850with a variable viscous friction coefficient /H9251eff/H20849/H9278/H20850. The expressions for the effective friction and energy are /H9251eff/H20849/H9278/H20850=/H9251−/H20849/H9003/H9278+/H9003/H9258/H20850//H9275/H11036, /H9003/H9278=/H20849/H11509F/H9278//H11509/H9278/H20850/H9258=/H9266/2,/H9003/H9258=/H20849/H11509F/H9258//H11509/H9258/H20850/H9258=/H9266/2, /H208494/H20850 and Eeff/H20849/H9278/H20850=Er/H20849/H9266/2,/H9278/H20850+/H9004E/H20849/H9278/H20850, /H9004E=−M /H9253/H20885/H9278/H20875u/H20849n/H20850/H20849s·e/H9258/H20850−/H9003/H9258 /H9275/H11036F/H9278/H20876 /H9258=/H9266/2d/H9278/H11032. /H208495/H20850 Equation /H208493/H20850with definitions /H208494/H20850and /H208495/H20850gives a general de- scription of a planar device in the presence of spin transfertorque. At nonzero current, the effective friction can becomenegative /H20849see below /H20850, and the effective energy is not neces- sarily periodic in /H9278/H20849e.g., in the case of “magnetic fan”19/H20850. Physically, this reflects the possibility of extracting energyfrom the current source which amounts to “negative dissipa-tion.” In many planar devices, the polarizer direction slies in the easy plane, /H9258s=/H9266/2, with a direction defined by the azi- muthal angle /H9278s. At the same time, the residual energy has a property /H20849/H11509Er//H11509/H9258/H20850/H9258=/H9266/2=0, i.e., does not shift the energy minima away from the plane. We will also use the simplest form f/H20851/H20849n·s/H20850/H20852=const for the spin transfer strength. A more realistic function will not change the result qualitatively and can be easily used if needed. With these restrictions the ef-fective friction and the energy correction get the form /H9251eff=/H9251+2ucos/H20849/H9278s−/H9278/H20850 /H9275/H11036, /H9004E=−Mu2 2/H9253/H9275/H11036cos2/H20849/H9278s−/H9278/H20850. /H208496/H20850 In a spin-flip transistor, the polarizer direction is given by/H9278s=/H9266/2. Following Ref. 20, we consider an easy axis residual energy Er/H20849/H9258,/H9278/H20850=−/H20849K"/2/H20850sin2/H9258cos2/H9278. Stable equilib- ria are found as the minimum points of Eeffat which /H9251effis positive. The results are summarized in a switching diagram/H20849Fig.2/H20850plotted on the plane of the material characteristic /H9251 and the experimental parameter u/H11011I. We will discuss the u/H110220 region. The effect of the opposite current is completely symmetric. First, one observes stabilization of the /H9278=/H9266/2 /H20849parallel /H20850equilibrium for u/H11022u1/H11013/H20881/H9275"/H9275/H11036,/H9275"=/H9253K"/M. This is in accord with the results of Ref. 21and can be explained by noticing that spin torque attracts the magnetization of thefree layer to the parallel direction. In addition, when the damping constant is larger than the critical value /H9251*=2/H20881/H9275"//H9275/H11036, a window of stability u1/H11021u /H11021u2/H11013/H9251/H9275/H11036/2 of the /H9278=−/H9266/2/H20849antiparallel /H20850equilibrium opens on the diagram. Note that a sufficiently large easyplane energy can always ensure /H9251*/H11021/H9251.If one thinks about the stability of the antiparallel equi- librium /H20849/H9258,/H9278/H20850=/H20849/H9266/2,−/H9266/2/H20850atu/H110220 in terms of Eq. /H208491/H20850, its stabilization seems completely unexpected. The anisotropy torques do not stabilize this equilibrium because it is a saddlepoint of the total magnetic energy E, and the added spin transfer torque repels nfrom this point as well. The whole phenomena may be called “stabilization by repulsion.” Tomake sure that it is not an artifact of approximation /H208493/H20850, the result was verified using the LLG equations /H20851Eq. /H208492/H20850/H20852with no approximations for the axis-and-plane energy E =/H20849K /H11036/2/H20850cos2/H9258−/H20849K"/2/H20850sin2/H9258cos2/H9278. Calculating the eigenval- ues of the linearized dynamic matrices6at the equilibrium points /H20849/H9266/2,±/H9266/2/H20850, we obtained the same switching dia- gram and confirmed the stabilization of the antiparallel di- rection. Typical trajectories n/H20849t/H20850calculated numerically from the LLG equation with no approximations illustrate the sta- bilization of the /H9278=−/H9266/2 equilibrium in Fig. 3. In accord with the predictions of Eqs. /H208493/H20850and /H208496/H20850, the wedge of stabil- ity consists of two regions /H20849b/H20850and /H20849c/H20850characterized by over- damped and underdamped dynamics during the approach tothe equilibrium. The dividing dashed line is given by u 3 =/H9275"//H9251+/H9251/H9275/H11036/4. It was also checked numerically that small deviations of sfrom the /H20849/H9266/2,/H9266/2/H20850direction do not change the behavior qualitatively. When current is further increased to u/H11022/H9251/H9275/H11036/2, the an- tiparallel state becomes unstable again. Numeric calculationshows that above this threshold the trajectories approach astable precession cycle /H20851Fig.3/H20849d/H20850/H20852. Precession states can be FIG. 2. /H20849Color online /H20850Switching diagram of the spin-flip transistor. In each zone, thin arrows show the possible stable directions of the free layer mag-netization. Spin polarizer is shown by a thick arrow. Angular dependenciesof /H9251effand Eeffare given in insets. Stable subregions “b” and “c” differ in overdamped vs underdamped approach to the equilibrium. FIG. 3. Typical trajectories of n/H20849t/H20850for/H9275"//H9275/H11036=0.01 and /H9251=1.5/H9251*. The plot labels correspond to the regions in Fig. 2; current magnitudes are marked on the panels. The /H9278=−/H9266/2 equilibrium is /H20849a/H20850unstable /H20849b/H20850stable with over- damped approach /H20849c/H20850stable with oscillatory approach /H20849d/H20850unstable with a stable cycle surrounding it.262510-2 Ya. B. Bazaliy Appl. Phys. Lett. 91, 262510 /H208492007 /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: 131.230.73.202 On: Sun, 21 Dec 2014 18:15:02efficiently studied in the framework of planar approximation as well.22 The fact that /H9251/H11022/H9251*condition is required for the stabi- lization means that dissipation terms play a crucial role en-tangling two types of repulsion to produce a net attraction tothe reversed direction. Note that an interplay of a strongeasy plane anisotropy and dissipation terms produces unex-pected effects already in conventional /H20849u=0/H20850magnetic sys- tems. There the same threshold /H9251*represents a boundary between the oscillatory and overdamped approaches theequilibrium.18 In summary, we derived a general form of the effective planar equation /H20851Eq. /H208493/H20850/H20852for a macrospin free layer in the presence of spin transfer torque produced by a time-independent spin polarized current. Importantly, the behaviorof the effective equation’s solutions can be qualitatively un-derstood using the analogy with mechanical motion of a par-ticle in an external potential with variable friction. The pre-dictive power of the approach is illustrated by the discoveryof the “stabilization by repulsion” phenomena in the spin-flipdevice. We suggest that the latter can be experimentallyfound by observing magnetization jumps at the threshold current u= /H20881/H9275"/H9275/H11036. For/H9251/H11021/H9251*, deterministic jumps into the parallel state are predicted, while for /H9251/H11022/H9251*, one should see stochastic jumps into either parallel or antiparallel states. Thedirection of the jump can be determined from the magnetore-sistive signal. In addition, one can initially set the free layerinto the parallel or antiparallel state by external magneticfield, turn the current on, and switch off the field after that.Both states should be stabilized by a moderate current /H20881/H9275"/H9275/H11036/H11021u/H11021/H9251/H9275/H11036/2. Furthermore, as the current is in- creased above the /H9251/H9275/H11036/2 threshold, the parallel state should remain a stable equilibrium, but the antiparallel state shouldtransform into a precession cycle with an oscillating compo-nent of magnetoresistance. The antiparallel stable state andthe corresponding precession cycle can be used to engineermemory or logic devices, and microwave nanogeneratorswith tunable frequency. The author wishes to thank C. W. J. Beenakker, G. E. W. Bauer, and Yu. V. Nazarov for illuminating discussions. Re-search at Leiden University was supported by the Dutch Sci-ence Foundation NWO/FOM. 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1.3626593.pdf

Suppression of the spin pumping in Pd/Ni81Fe19 bilayers with nano-oxide layer Duck-Ho Kim, Hong-Hyoun Kim, and Chun-Yeol You Citation: Appl. Phys. Lett. 99, 072502 (2011); doi: 10.1063/1.3626593 View online: http://dx.doi.org/10.1063/1.3626593 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v99/i7 Published by the American Institute of Physics. Related Articles Tensile and fatigue behaviors of printed Ag thin films on flexible substrates Appl. Phys. Lett. 101, 191907 (2012) Making a continuous metal film transparent via scattering cancellations Appl. Phys. Lett. 101, 181110 (2012) Quantum spin transport through magnetic superatom dimer (Cs8V-Cs8V) J. Chem. Phys. 137, 164311 (2012) Narrow-band optical transmission of metallic nanoslit arrays Appl. Phys. Lett. 101, 171106 (2012) Spatially resolving variations in giant magnetoresistance, undetectable with four-point probe measurements, using infrared microspectroscopy Appl. Phys. Lett. 101, 162402 (2012) Additional information on Appl. Phys. Lett. Journal Homepage: http://apl.aip.org/ Journal Information: http://apl.aip.org/about/about_the_journal Top downloads: http://apl.aip.org/features/most_downloaded Information for Authors: http://apl.aip.org/authors Downloaded 12 Nov 2012 to 147.226.7.162. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsSuppression of the spin pumping in Pd/Ni 81Fe19bilayers with nano-oxide layer Duck-Ho Kim, Hong-Hyoun Kim, and Chun-Y eol Y oua) Department of Physics, Inha University, Namgu Incheon 402-751, Republic of Korea (Received 17 February 2011; accepted 30 July 2011; published online 17 August 2011) We demonstrate that the spin pumping effect can be effectively suppressed with a nano-oxide layer. Spin pumping effect manifests itself by an enhancement of the Gilbert damping parameter innormal metal/ferromagnetic hetero-structures, while many spintronics devices prefer smaller damping parameter. Since the spin pumping effect is directly related with the spin dependent interface conductance, we can modify the spin pumping by altering the interface conductance withthe nano-oxide layer. We prepared series of Pd/Ni 81Fe19bilayers with different pausing time between Pd and Ni 81Fe19depositions in order to control the interface conductance. The Gilbert damping parameters are determined from the line-width measurements in the ferromagneticresonance spectra for each pausing time sample. They are 0.0490, 0.0296, 0.0278, and 0.0251 for 0, 6, 30, and 60 s pausing time, respectively. We find that the damping parameter of Pd/Ni 81Fe19is almost recovered to one of the Cu/Ni 81Fe19bilayer with 60 s pausing time, while the static magnetic properties are not noticeably changed. VC2011 American Institute of Physics . [doi:10.1063/1.3626593 ] Spin pumping effect is an important physical phenom- enon because it alters details of spin dynamics in the magnetic multilayers by enhancing the Gilbert damping parameters.1–3 Spin pumping effect provides extra damping mechanism in the normal metal/ferromagnetic (NM/FM) hetero-structures, so that the spins in the FM layer have larger damping parame- ters. The damping parameter is an important physical propertyof the material which governs the details of spin dynamics in many modern spintronics devices, including spin transfer tor- que (STT) magnetoresistive random access memory(MRAM), 4race-track memory,5,6spin torque nano-oscillator,7 non-uniform magnetic field induced domain wall memory,8 and domain wall MRAM.9Since the role of the damping is energy dissipation, it determines the critical current density in S T T - M R A M ,d o m a i nw a l lv e l o c i t yi nd o m a i nw a l ld e v i c e s , etc. For example, the critical current density of the STT-MRAM is proportional to the damping parameter, 10,11so that the reducing damping parameter is a critical issue in the real- ization of the STT-MRAM. The free FM layer in the STT-MRAM must contact with a top NM electrode in order to make electric circuits, therefore, the spin pumping effect is inevitable. Since the spin pumping make an additional damp-ing process, the damping parameters of the free layers always increase with the top NM electrode. In this letter, we will show that the spin pumping effect is effectively suppressed by forming nano-oxide layer (NOL) between the NM and FM layers. Since it is known that the spin pumping is linked to the spin dependent spin conductiv-ities, 1,2we tailor the interface conductance by forming NOL in order to suppress the spin pumping. The physical role of the NOL has been actively studied in the giant magnetoresist-ance (GMR) spin valve structures. 12–14It acts as an additional scattering source and provides multiple scatterings for the conducting spins. In this study, we employed NOL in order tocontrol the interface conductance without altering other mag- netic properties such as coercivity, anisotropy, and magnetic moments. In order to achieve our goal, we prepare a series of Pd/Ni 81Fe19(Permalloy; Py) hetero-structures with NOL (0, 6, 30, and 60 s pausing time). The Gilbert damping parameters of them are extracted from the line-width of the ferromagnetic resonance (FMR) spectra by the vector network analyzer(VNA) with the varying external magnetic field. 15It is con- firmed that the spin pumping effect is effectively suppressed when the pausing time increases, as we expected. We find thatthe Gilbert damping parameter of Pd/Py is almost recovered to one of the Cu/Py bilayer with 60 s pausing time, while the static magnetic properties are not noticeably changed. We prepare a series of Pd/Py bilayers with various NOL and a Cu/Py bilayer as a control sample, which has small or negligible spin pumping effect. 16All samples are prepared under the base pressure of low 10/C08Torr with 1.5 mTorr Ar working gas pressure by the dc-magnetron sputtering. The deposition rates are 0.586, 0.514, and 0.842 A ˚/s for Pd, Py, and Cu, respectively. Before the deposition process, Si sub- strates are prepared in cleaning processes, with acetone, chloroform, and isopropyl alcohol. First, we deposit 10-nmPd layer, as a NM, on the top of the substrate at the room temperature. According to the previous study, 16,17it is known that Pt and Pd provide large spin pumping effect dueto their large atomic numbers. After deposition of Pd layer, we pause the deposition process for t p(¼0, 6, 30, and 60 s). During the pausing time, NOL has been formed at the top ofthe Pd layer. The formation of the NOL must be bound up with the pausing time t p. We do not know exact relation between the thickness of NOL and tp, but we can conjecture that the thicker NOL will be formed for the longer tp. After that, we deposit 5-nm Py layer and additional 5-nm Cu layers are deposited as capping layers. Here, we takeCu as a capping layer because Cu is poor sink of the injected spins. Therefore, the full structures of our samples are Si(sub.)/Pd(10 nm)/NOL( t p)/Py(5 nm)/Cu(5 nm).a)Author to whom correspondence should be addressed. Electronic address: cyyou@inha.ac.kr. 0003-6951/2011/99(7)/072502/3/$30.00 VC2011 American Institute of Physics 99, 072502-1APPLIED PHYSICS LETTERS 99, 072502 (2011) Downloaded 12 Nov 2012 to 147.226.7.162. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsThe schematic structures of the samples are sketched in Fig. 1(a)without NOL and (b) with NOL. The white arrows indi- cate spin current and scattering process between NM and FM layers and their width means spin conductance’s amount. Weshow the role of NOL in Fig. 1(b), it reduces transmission and reflection spin current at the interface between Pd and Py. So due to the NOL, the interface conductance will be suppressed.Even though the NOL is not detectable by x-ray diffraction, such extra scattering at the NOL has been evidenced by the improved GMR in spin valve systems. 12We also perform the x-ray diffraction (not shown here), however, there is no noticeable difference between the samples with different NOLs. The hysteresis loops are measured using vibrating sam-p l em a g n e t o m e t e ra n da r ed e p i c t e di nF i g . 2for Pd/NOL( t p)/ Py and Cu/Py samples. Since the coercivity of the FM thin film is very sensitive on its microstructure in addition to theanisotropy field, there is small change of the coercivity for dif- ferent t psamples, but noticeable differences are not found. It implies that the effect of NOL in the static magnetic propertiesis negligible. And it is what we want to achieve in this study. We aim to tailor the Gilbert damping parameters only without altering other magnetic properties. The saturation magnetization M sand Gilbert damping parameters aare extracted from the resonance frequency, fres, and line-width Dfof the FMR spectra measurement. We used VNA-FMR to measure imaginary parts of the suscepti- bility of the samples.18–20The measured imaginary parts of the susceptibility raw data are calibrated with the careful cal-ibration procedures. 18–20The calibrated imaginary parts of the susceptibility are shown in Fig. 3(a)for a tp¼30 s sam-ple. The peaks of the spectra are moving with the magnetic field from 41 to 106 mT, and the resonance peaks are well fitted with the Kittel’s equation18,21 fres¼cl0 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðHextþHuniÞðHextþHuniþMsÞp ; (1) where, c,l0,Hext, and Huniare the gyromagnetic ratio, mag- netic permeability, external magnetic field, and uni-axial ani- sotropy field, respectively. Fig. 3(b) shows the resonance peaks and fitted results together and we obtained Msof 0.89660.002 T. Msoftp¼0, 6, and 60 s are 0.828 60.001, 0.82860.001, and 0.896 60.002 T, respectively. The varia- tion of Msis also small as we desire. Now, let us discuss about the determination of the Gilbert damping parameter a.T h em e a s u r e d aincludes not only the intrinsic damping, but also extrinsic contributions such as the surface roughness, magnetic inhomogeneity, inhomogeneous magnetic field, and extra damping processes such as two-mag-non scattering process, spin pumping, scattering from impur- ities, etc. Furthermore, VNA-FMR spectra have extra broadening source, non-unifor m spin wave excitation in copla- nar wave guide. According to the Bilzer’s studies, 18the experi- mentally measured line-widths of the VNA-FMR spectra are Dfapp/C25Dfffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ1 f2 resCs ; (2) where Dfappis the apparent line-width and Cis determined by one of the following two expressions: Cspinwave ¼f4 eff 16Df2ðkmaxdÞ2andCinhomo ¼c2l2 0 16p2DH2 inhomo a2, where kmax,d, and DHinhomo are the highest wave vector, film thickness, and fi- nite zero-frequency filed line-width, respectively. Eq. (2) takes into account the effects of the non-uniform spin waveexcitation in coplanar wave guide and inhomogeneous mag- netic properties of the samples. They lead 1/ f 2 rescontribution in Eq. (2). In order to exclude the 1/ f2 resterm, we used only FIG. 1. (Color online) Schematics of the sample structures of Si/Pd/Py/Cu, (a) without NOL and (b) with NOL ( tp¼6, 30, and 60 s) between Py and Pd. Bold lines explain about transmission spin current, Ipumpdetermines addi- tional damping, from the ferromagnetic layer into normal metal and reflec- tion spin current, Iback, back to the ferromagnetic layer. FIG. 2. (Color online) Hysteresis loop in Si/Pd(10 nm)/NOL( tp¼0, 6, 30, and 60 s)/Py(5 nm)/Cu(5 nm) and Si/Cu(10 nm)/Py(5 nm)/Cu(5 nm). FIG. 3. (Color online) (a) Imaginary parts of the susceptibility as a function of frequency with varied Hext. (b) Resonance frequency as a function of field Hext: symbols mean experimental data and solid lines are fitted results with the Kittel’s equation. (c) The line-width as a function of Hext: symbols mean experimental data and solid lines are linear fitting for Si/Pd(10 nm)/NOL(30 s)/Py(5 nm)/Cu(5 nm).072502-2 Kim, Kim, and Y ou Appl. Phys. Lett. 99, 072502 (2011) Downloaded 12 Nov 2012 to 147.226.7.162. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionslarge field (large fres) region to find aas shown in Fig. 3(c). The line-width is related with aas follows: Dfapp¼acl0 2p/C16 2ðHextþHuniÞþMs/C17 þDfex; (3) where Dfeximplies the extrinsic line-width contributions. Therefore, the slope of Dfappversus Hextplot gives acl0 p. In Fig. 4, the shifted apparent line-widths, DfappðHextÞ /C0DfappðHext¼80 mT Þ, are plotted as a function of Hextfor each sample. We shifted DfappðHextÞto get better compari- son. From the slopes of each data, the Gilbert damping pa- rameter aare obtained. We depict the results in Fig. 5with the Cu/Py/Cu result. When tp¼0 s, enhancement of ais almost 113% compared with Cu/Py/Cu: it certifies the exis- tence of spin pumping effect in FM/NM interface. For tp¼6 s, the enhancement of ais sharply decreased up to 29%. With tp¼60 s, ais almost recovered to the Cu/Py/Cu value. We also calculate spin-mixing conductance for each case: 4.92/C21019, 1.25 /C21019, 9.87 /C21018, and 4.32 /C21018m/C02 for 0, 6, 30, and 60 s pausing time, respectively.22,23 The origin of the suppression can be easily understood. The enhancement of the damping is due to the extra spin flipprocess in the adjacent NM. The NOL acts as an additional scattering center at the interface and it blocks the conduction channel. However, the scattering at NOL is spin conservedprocess, which has been reported in the spin valve structure with NOL. 12 It must be noted that the formation of NOL affects to the conductance at the interface, but not to the tunneling magne- toresistance (TMR). Since TMR structure has an insulator barrier, where the conductance is much poorer. Furthermore,TMR is sensitive only at the interface between the insulator and FM, not the interface between FM and NM. In conclusion, we perform VNA-FMR measurement to find spin pumping effect on Pd/Py and Pd/NOL/Py series. We show that the spin pumping effect is effectively sup- pressed by the forming NOL, without altering the static mag-netic properties. We believe that the NOL will be helpful to reduce the damping parameter of the free layer, and the effect of the NOL on the STT must be studied to reduce thecritical current density for STT-MRAM applications.This work is supported by a 2011 Inha University research grant. 1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002). 2Y. Tserkovnyak and A. Brataas, Phys. Rev. Lett. 88, 117601 (2002). 3O. Mosendz, J. E. Pearson, F. Y. Fradin, S. D. Bader, and A. Hoffmann, Appl. Phys. Lett. 96, 022502 (2010). 4R. Beach, T. Min, C. Horng, Q. Chen, P. Sherman, S. Le, S. Young, K. Yang, H. Yu, X. Lu, W. Kula, T. Zhong, R. Xiao, A. Zhong, G. Liu, J. Kan, J. Yuan, J. Chen, R. Tong, J. Chien, T. Torng, D. Tang, P. Wang, M. Chen, S. Assefa, M. Qazi, J. DeBrosse, M. Gaidis, S. Kanakasabapa-thy, Y. Lu, J. Nowak, E. O’Sullivan, T. Maffitt, J. Z. Sun, and W. J. Gal- lagher, Tech. Dig. - Int. Electron Devices Meet. 2008 , 305. 5S. S. P. Parkin, U.S. patent 6834005 Dec. 21 (2004). 6S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008). 7S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, and J. A. Katine, Nature 437, 389 (2005). 8C.-Y. You, J. Magn. Magn. Mater. 321, 888 (2009); Appl. Phys. Lett. 92, 192514 (2008) ibid. 92, 152507 (2008). 9IEICE Tech. Rep. 109, SDM2009-114, 91 (2009). 10J. Z. Sun, Phys. Rev. B 62, 570 (2000). 11Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L.-C. Wang, and Y. Huai, J. Phys.: Condens. Matter. 19, 165209 (2007). 12M. F. Gillies and A. E. T. Kuiper, J. Appl. Phys. 88, 5894 (2000). 13A. Gupta, S. Mohanan, M. Kinyanjui, A. Chuvilin, U. Kaiser, and U. Herr, J. Appl. Phys. 107, 093910 (2010). 14Y. Kamiguchi, H. Yuasa, H. Fukuzawa, K. Koui, H. Iwasaki, and M. Sahashi, Dig. Intermag 1999, paper DB-01. 15C. Bilzer, T. Devolder, J.-V. Kim, G. Counil, C. Chappert, S. Cardoso, andP. P. Freitas, J. Appl. Phys. 100, 053903 (2006). 16S. Mizukami, Y. Ando, and T. Miyazaki, Jpn. J. Appl. Phys. 40, 580 (2001). 17O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann, Phys. Rev. Lett. 104, 046601 (2010). 18C. Bilzer, PhD thesis, Universite Paris-sud 11, France, 2007. 19C. Bilzer, T. Devolder, P. Crozat, and C. Chappert, J. Appl. Phys. 101, 074505 (2007). 20C. Bilzer, T. Devolder, P. Crozat, and C. Chappert, IEEE Trans. Magn. 44, 3265 (2008). 21H. Suhl, Phys. Rev. 97, 555 (1955). 22T. Yoshino, K. Ando, K. Harii, H. Nakayama, Y. Kajiwara, and E. Saitoh, J. Phys.: Conf. Ser. 266, 012115 (2011). 23In order to extract correct spin mixing conductance, a series of various thicknesses of ferromagnetic layers is required. However, we focus our attention to the reduction of the spin pumping effect, so we prepared only one thickness of ferromagnetic layer with different oxidation time. There-fore, the obtained spin mixing conductance is somewhat larger than others because of the inaccurate determination methods. FIG. 4. (Color online) The shifted line-widths as a function of Hextfor Si/ Pd(10 nm)/NOL(0, 6, 30, and 60 s)/Py(5 nm)/Cu(5 nm). The solid symbols are experimental values from each of spectra, and the solid lines are linear fit results. The line-widths are shifted to better comparison of the slopes. FIG. 5. (Color online) Gilbert damping parameters aas a function of tpfor Si/Pd(10 nm)/NOL( tp)/Py(5 nm)/Cu(5 nm). The solid line is for eye guide and the dash line is damping parameter of Si/Cu(10 nm)/Py(5 nm)/Cu(5 nm) for the comparison.072502-3 Kim, Kim, and Y ou Appl. Phys. Lett. 99, 072502 (2011) Downloaded 12 Nov 2012 to 147.226.7.162. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

1.420353.pdf

Effects of three parameters on speaking fundamental frequency Harry Hollien Institute for Advanced Studies of the Communication Processes, University of Florida, Gainesville, Florida 32611 Patricia A. Hollien Forensic Communication Associates, Gainesville, Florida 32601 Gea de Jong Institute for Advanced Studies of the Communication Processes, University of Florida, Gainesville,Florida 32611 ~Received 19 December 1996; accepted for publication 16 July 1997 ! Speaking fundamental frequency levels and usage ~SFF,F0!are of interest to many investigators who study human speech and voice. Substantial research in the area has been carried out; commonfociincludeSFFasrelatedtoinfantcry,age,gender,adolescentvoicechange,language,race,voicepathology,andsoon.Yettherestillareanumberofrelationshipswhicharenotwellunderstoodandthree of them will be addressed in this project. They involve the long-held notions that ~1!a secular trend exists with SFF being lowered over time, ~2!the use of university students in research of this type will create bias because they are physically different from average individuals, and ~3!SFF can vary systematically for different types of speech ~especially for oral reading and extemporaneous speaking !. Experiments assessing these questions were carried out, but only certain of the postulates were supported. That is, while some evidence of a secular trend was found, it appearedinconsequential during the past quarter of this century; second, although university students werefound to be slightly larger than a cohort approaching the average population, only minor vocaldifferences were found. Finally, it was observed that, in general, oral reading resulted in highermean SFF’s than those for spontaneous speech. However, this difference was not robust and, due toreversals, the resulting metric did not appear to be of good predictive value for individual speakers.©1997 Acoustical Society of America. @S0001-4966 ~97!00911-9 # PACS numbers: 43.70.Gr @AL# INTRODUCTION Speaking fundamental frequency usage ~SFF orF0!has been of particular interest to phoneticians and related profes-sionals for many years. Indeed, fairly sophisticated investi-gations began some decades ago ~Cowan, 1936; Lynch, 1934; Murray and Tiffin, 1934; Weaver, 1924 !; however, most of the relevant research has been carried out only sincethe middle of this century. Much has been accomplished butconfusion yet exists about a substantial number of relation-ships. For example, a general listing of some of the moresalient would include: ~1!speaker characteristics of age, sex, health, race, education, language, temporal shifts, etc.; ~2! the type of speech used ~reading, spontaneous, extemporane- ous, etc. !; and ~3!text materials ~cold running speech, dra- matic, informative, poetry, etc. !. As would be expected, it often is difficult to generalize from the data available because of the effects of the citedparameters, plus the sharply differing research approachesand measurement techniques employed. For example, thesize of the populations studied has varied widely ~from a half dozen individuals to several hundreds !and the age range within a group has extended from a few months to manydecades. Sometimes median SFF is reported instead of themean; standard deviation ~variability !of the frequency dis- tribution is usually reported ~correctly !in tones or semitones, but sometimes it is found in Hertz. Finally, data extractionmethods vary greatly; while adequate in most cases, these values occasionally are based on little more than estimates.Thus it would appear useful if some attempt were made atdata coordination and if certain long-standing questions wereaddressed directly. A. Areas exhibiting reasonable agreement It would appear appropriate to review briefly those speaking fundamental frequency ~SFF!relationships about which there appears to be some agreement. How gender andage affect SFF are areas where many of the major factors areunderstood and the data are relatively complete. For ex-ample, there is little doubt but that SFF levels are differentfor the sexes—at least, after puberty is reached ~Fitch, 1990; Fitch and Holbrook, 1970; Higgins and Saxman, 1991;Krook, 1988; Ku ¨nzel, 1989; Snidecor, 1994 !. The general- ized curves found in Fig. 1 contrast these differences; theyare based on the best available SFF evidence drawn prima-rily from data on Americans of European origin and Europe-ans where language appears not to be a biasing factor ~see references listed below !. Of course, graphed data such as these only provide a broad perspective of the F0 patterns. To illustrate, note the closely paralleling curves for infants andchildren; there appear not to be very great ~gender !differ- ences between them ~if any at all !—at least before adolescent voice change takes place ~Bennett, 1983; Eguchi and Hirsh, 1969; Fairbanks et al., 1949a, 1994b; Hasek et al., 1980; 2984 2984 J. Acoust. Soc. Am. 102(5), Pt. 1, November 1997 0001-4966/97/102(5)/2984/9/$10.00 © 1997 Acoustical Society of America Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.105.215.146 On: Sat, 20 Dec 2014 13:43:43McGlone and McGlone, 1972; Wheat and Hudson, 1988 !.I t is during this latter period ~i.e., puberty !where the sex-based differences in SFF level develop ~Curry, 1940; Duffy, 1970; Hollien and Malcik, 1967; Hollien et al., 1965, 1994 !. Second, reconsideration of Fig. 1 will also reveal that there are enough F0-age data to provide a reasonably good perspective of how these factors generally covary over life ~Brownet al., 1991; de Pinto and Hollien, 1982; Fitch and Holbrook, 1970; Higgins and Saxman, 1991; Hollien, 1987,1995; Hollien and Malcik, 1967; Hollien and Paul, 1969;Hollien and Shipp, 1972; Horii and Ryan, 1981; Krook,1988; Linke, 1953/73; McGlone and Hollien, 1963; Mysak,1959; Philhour, 1948; Pronovost, 1942; Ringel andChodzko-Zajko, 1987; Saxman and Burk, 1967; Shipp et al., 1992; Snidecor, 1951 !. However, many of the specifics about SFF and age are still lacking and some controversies areextant. For example, it is not yet known if F0 patterns for the elderly are random in nature ~Chodzko-Zajko et al., 1995; Ringel and Chodzko-Zajko, 1987 !or follow orderly patterns such as those associated with the male–female coalescencetheory ~Hollien, 1987, 1995 !—a model which suggests that older males and females tend to exhibit parameter shifts to-ward each other. The possibility that this theory may hold isreflected in the lower of the two F0 curves for older females ~see again Fig. 1 !. It can be inferred by comparing the SFF levels reported by Honjo and Isshiki ~1980!with those for younger Japanese women ~Terasawa et al., 1984; Tsuge et al., 1987 !, and the suggestion of such a trend also is found in other studies ~Krook, 1988; Mack, 1984; Stoicheff, 1981 !.On the other hand, the flat trend line for older women is based on the McGlone and Hollien ~1963!data~especially when compared to Saxman and Burk, 1967 !, and the abso- lute values for older women reported by Krook and by Sto-icheff. In any event, it was considered expedient to includeboth possibilities in the figure. B. Areas exhibiting some confusion Other SFF-linked relationships are even less well under- stood and the resulting confusion can impair comprehensionof many specific SFF relationships. Moreover, they also tendto bias understanding in related areas. Examples include theSFF interface with race ~Hollien and Malcik, 1967; Hudson and Holbrook, 1981; Mack, 1984; Morris, 1997; Wheat andHudson, 1988; Yamazawa and Hollien, 1992 !, with language ~Boe¨et al., 1975; Hanley and Snidecor, 1967; Hanley et al., 1966; Honjo and Isshiki, 1980; Majewski et al., 1972; Terasawa et al., 1984; Tsuge et al., 1987; Yamazawa and Hollien, 1992 !, and with special behaviors such as smoking ~Braun, 1994; Gilbert and Weismer, 1974; Sorenson and Horii, 1982; Stoicheff, 1981 !. Even more important are the long-standing SFF contro- versies involving the effects of secular trends, subject class,and the types of speech used in research. If bias results frominadequate control of any of these factors, seemingly estab-lished relationships could be distorted. ~1!Secular trends. This term may be a misnomer; neverthe- less, it has developed a certain currency. In this instance,it refers to a commonly held opinion that SFF will shiftover time due to systematic changes in laryngeal struc-ture or phonatory behavior. More specifically, it is ar-gued that SFF for adults has been gradually lowered overrecent years—and continues to drop. Yet, not very muchdata can be found to support this notion. The issue willbe discussed in detail and, then, evaluated in the firstsection to follow. ~2!Subject class. A long-held opinion appears to be that measurement of certain vocal factors—in this instanceSFF—will be biased by the type of subject studied. Forexample, SFF results are sometimes challenged becauseuniversity students were used as subjects. It is contendedthat they do not adequately represent the overall popula-tion and, hence, they somehow bias the resultingdata—in this case, that their SFF levels will be differentfrom those found in the general population. While thereare some data which suggest this postulate may be true,they are not very robust. Evaluation of this issue will befound in the second section of this report. ~3!Type of speech material. While it is without question that the use of certain types of speech material can pro-duce SFF shifts ~the contrasts among various emotion- ally laden samples constitute examples !, the issue as to whether or not oral reading will result in different SFFlevels than will extemporaneous speech has not yet beenresolved. Again, data are sparse and variable. Hence, re-search attempting to resolve this issue was carried out~Sec. III !. FIG. 1. Graphic portrayal of generalized data for male/female speaking fundamental frequency levels as a function of age. The data are plotted inHertz; variation around the means is indicated by the hatched areas. Notethat there now is a question as to whether female SFF remains constant or islowered in older women. 2985 2985 J. Acoust. Soc. Am., Vol. 102, No. 5, Pt. 1, November 1997 Hollien et al.: Speaking fundamental frequency Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.105.215.146 On: Sat, 20 Dec 2014 13:43:43As may be seen, the ideas about these three factors may be due more to subjective thought than to the results pro-vided by appropriate research. Nevertheless, it appears thatthey can be tested; they form the basis of the research tofollow. Specifically, it is held that: ~1!secular trends exist for speaking fundamental frequency usage; ~2!the use of univer- sity students will bias SFF data as they can be expected tobehave differently than normal healthy subjects; and ~3!dif- ferences in SFF can result from the type of speech chosen forthe research. Multiple experiments have been carried out totest these postulates; as stated, they will be considered seri-ally and in the order cited above. I. SECULAR TRENDS The first issue involves the possibility that secular trends in speaking fundamental frequency level ~SFF orF0!have occurred over the past half-century. Questions here are con-tinually raised and available research can be used to supportor contradict this position. For example, an SFF reduction ofabout 1.7 st. can be observed if Pronovost’s 1942 data foryoung males are compared to those reported thirty years laterby Hollien and Shipp ~1972!, i.e., 36.2 st. vs 34.5 st. Note also that his means ~i.e., Pronovost’s !are over 2.6 st. higher than are those reported by Krook in 1988 ~36.2 st. vs 33.5 st.!. A second example: a 4.7 st. difference separates the mean SFF data for 14-year-olds when Curry’s 1940 value of242 Hz ~7!is compared to the 184-Hz level reported 25 yearslater by Hollien and Malcik ~1967!. However, when taken as a whole, the trends among the available data are not particu-larly compelling and the cited contrasts depend on selectivecomparisons. For example, little difference is found if Prono-vost’s 1942 data are compared to those reported in 1973 byHollien and Jackson ~i.e., 132 Hz vs 129 Hz !rather than Hollien and Shipp ~1972!. Nor can much difference be found if the mean data on middle aged women, reported in 1967 bySaxman and Burke ~1967!, are contrasted to those published 21 years later by Krook ~1985!~i.e., 189 Hz vs 191 Hz !. In short, it is not at all clear if secular trends actually exist and ~assuming they do !if they are robust. Indeed, if methodogical differences are ignored, relationships such asthose found in Fig. 2 can be plotted. In this case, they consistof those mean data, reported between 1940 and 1995, which we consider to be about as valid as any available—evenwhen methodology and population size are taken into ac-count. As may be seen, there is only a slight suggestion ofoverall SFF reduction for either the men or the women. Nev-ertheless, common opinion holds that a more robust SFFlowering trend exists. New data are provided in an effort toclarify the issue. A. Data selection/generation Several problems must be addressed if a reasonable set of comparisons are to be made over the 50-year period ofinterest. For example, the procedures which led to data col- FIG. 2. Plot of mean speaking fundamental frequency data from 17 studies ~American/European subjects only, drawn from de Pinto and Hollien, 1982; Fitch, 1990; Fitch and Holbrook, 1970; Hanley, 1951; Hanley and Snidecor, 1966; Higgins and Saxman, 1991; Hollien and Jackson, 1973; Hollien and Paul, 1969;Hollien and Shipp, 1972; Hollien et al., 1982; Krook, 1988; Ku ¨nzel, 1989; Linke, 1953/1973; Philhour, 1948; Pronovost, 1942; Snidecor, 1951; Yamazawa and Hollien, 1992 !. Note that group size is classified as ‘‘small’’ ~squares !, ‘‘medium’’ ~circles !, or ‘‘large’’ ~triangles !. 2986 2986 J. Acoust. Soc. Am., Vol. 102, No. 5, Pt. 1, November 1997 Hollien et al.: Speaking fundamental frequency Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.105.215.146 On: Sat, 20 Dec 2014 13:43:43lection should be both valid and compatible for those studies considered and the analysis approaches should be similar~see McKinney, 1965, and Hess, 1982, for general discus- sions about such equipment !. Population size should be large enough to permit predictions and be of roughly the sametype and size. Young healthy adults, 18–30 years of ageattending a university were targeted as feasible groups, aswere cohorts of 15–35 subjects. Further, it is known that thePhonellograph ~see Cowan, 1936 !and the Fundamental Fre- quency Indicator ~Hollien and Tamburrino, 1965; Hollien, 1990!produce virtually identical data ~Hollienet al., 1994 !. Accordingly, studies where the investigator utilized one orthe other of these devices were sought. It was found that data from two early studies on males appeared compatible enough to permit reasonable contrast;they were Philhour ~1948!and Hollien and Shipp ~1972!. Data were then collected in 1995 for a similar group of sub-jects in order to complete the triad ~i.e., the current data also were obtained on 25 healthy university students !. While it was a little more difficult to identify parallel materials forwomen, it was judged that the data collected by Linke in1953 ~but published in 1973 !could be contrasted with a parallel group established for this purpose 42 years later. It isnoteworthy also that all five groups were of reasonably simi-lar size ~range: 18–27 subjects !, the speech materials were common to all and both equipment and measurement proce-dures were compatible. B. Results The basic data for the five groups cited may be found in Tables I and II. The males will be considered first. Whileonly a slight lowering of SFF appeared to have occurred overthe past 45–50 years, the shift proved significant at the 5%level ~F510.87; df 52,71,F 0.553.98!. Possible variation within the trend slope was evaluated by application of aTukey’s post hoc test at the alpha 50.05 level. It was found that the Philhour data were significantly different from theothers ~required for significance .1.357; obtained 51.692 !, whereas the 1972–1995 contrasts were not(obtained 50.927). These relationships suggest that, if a secular trend does exist, it is confined primarily to the periodprior to the 1970’s.Are there other factors—particularly external ones— which could explain the cited patterns? One source might bethat of equipment fidelity. For example, early transducersand tape recorders were not of as high quality as those em-ployed in the 1970s ~and afterward !; hence, low frequencies might have been damped in the Philhour research. On theother hand, the Phonellograph tends to detect more of thevery low frequencies produced ~vocal fry especially !than do processing systems such as FFI. Thus equipment limitationsprobably were not to blame as small restrictions in systemfrequency response undoubtedly were counterbalanced bythose created by the processing equipment. About the onlyvariable which could have artificially resulted in a downward secular shift is that of subject age. The Hollien–Shipp andpresent subjects ~mean age: 24.4 and 23.4 years respectively ! might be, on average, slightly older than Philhour’s ‘‘youngadults’’ and, hence, exhibit slightly lower SFF. Moreover,certain ~additional !data are available which might be of rel- evance to this issue. They result from two studies that some-what paralleled those reported but were not used as they didnot meet the group size or speech sample criteria. The firstwas of a cohort of young males who served as controls foranother project completed just after the Hollien and Shippresearch ~Hollienet al., 1982 !. The mean age for this cohort of 41 subjects was 20.9 years and mean SFF was 125 Hz—avalue more like Philhour’s than the others. Further, theHollien–Jackson ~1973!SFF data were at a level nearly that of Philhour’s and, with a mean of 20.3 years, that group wasquite young also. In short, it would appear that the data do not fullysup- port the hypotheses that a secular trend exists; i.e., one whereadult SFF levels have been systematically lowering over thepast 50 years. Indeed, a reduction in SFF level of less than0.06 st. per year over the period studied is simply too smalla shift to support the notion of a robust secular trend. The data for the female subjects ~see again Table II !are not consistent at all with the hypothesis as stated. In thiscase, it may be seen that the two groups of young healthyuniversity females, studied 42 years apart, exhibit almostidentical SFF levels. There are so few differences ~excepting time of data collection !extant between Linke’s study and these new data, that this finding is rather compelling. Cer-TABLE I. Summary table of the characteristics of the three groups of male subjects and the results of the SFF comparisons. ParameterPhilhour 1948Hollien/Shipp 1972Present 1995 Number of subjects 24 25 25 Ages ~years ! Mean YAa24.4 23.4 Range {{{ 20–29 21–30 Mean SFF Hertz 132 120 114Semitone level 36.2 34.5 33.6 Standard deviation~tones ! 1.7 1.3 1.3 aYA5young adults.TABLE II. Summary table of the characteristics of two groups of female subjects and the results of the SFF comparisons. ParameterLinke 1953Present 1995 Number 27 18 Ages ~years ! Mean YAa22.5 Range {{{ 21–30 Mean SFF Hertz 200 198Semitone level 43.3 43.2 Standard deviation~tones ! 1.5 1.2 aYA5young adults. 2987 2987 J. Acoust. Soc. Am., Vol. 102, No. 5, Pt. 1, November 1997 Hollien et al.: Speaking fundamental frequency Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.105.215.146 On: Sat, 20 Dec 2014 13:43:43tainly there is no reason to reject a hypothesis that both groups are at the same level ( P50.73). C. Discussion Certain relationships emerged when the data were fur- ther evaluated and contrasted with those reported in the earlyliterature. First, it is now obvious that, if a powerful seculartrend actually exists, much of it occurred prior to 1940 ~wit- ness Weaver, who in 1924 reported a mean F0 level of 312 Hz for females !. Quite clearly, the restricted frequency re- sponse of the equipment used biased these very early studies.Yet a legitimate shift may have occurred, at least, when par-allel data are considered. Second, another dividing point ap-pears to have occurred, one which separates the two parts ofthe current half-century. Thus even if some sort of seculartrend was operant during the first half of the period, there islittle evidence that it still exists. The SFF differences some-times reported during these more recent years appear relatedto~1!the types of subjects studied ~for example, singers, nonsmokers !,~2!problems with the subjects themselves ~voice pathologies !, and/or ~3!behavioral states ~subject stress, intoxication !. II. SUBJECT CLASS The second issue involves the very common use of uni- versity students as subjects in phonetics research in generaland studies of speaking fundamental frequency in particular.Concern is often expressed that individuals of this class ex-hibit characteristics different from those found in ordinarypopulations. The logic behind this opinion is that universitystudents probably are larger, healthier, and smarter than in-dividuals in otherwise similar cohorts. While this concern israrely stated formally, it often surfaces at professional meet-ings, in reviews of proposals/publications and the like. Ofcourse, it must be conceded that there may be a higher pro-portion of individuals who are ill or marginally competentamong nonstudent populations and that students also may bemore facile with tasks involving high levels of language skilland/or intellectual ability. On the other hand, these possibledifferences may not exist for other types of ~behavioral ! tasks; for example, those involving motor speech productionand the rather specific characteristic of speaking fundamentalfrequency ~SFF!usage. A. Subjects Speech materials for two young healthy male popula- tions were gathered in the early 1970s. These studies werepart of a much larger project designed to provide baselinespeech/voice data ~including SFF !for adults of all types. Some of the data from the first of these substudies have beenpublished ~Hollien and Jackson, 1973 !; however, no data from the other experiments have been reported as yet. Subjects for the first study ~i.e., Hollien and Jackson, 1973!were 157 male University of Florida students, aged 18–26 years ~mean age 20.3 years !, who were demonstrably healthy and exhibited no speech and hearing disorders.Nearly all of them had grown up in the Southeastern portionof the United States but none exhibited a marked regional dialect. The second cohort ~unreported data !consisted of 142 healthy young men drawn from the enlisted ranks availableat the Pensacola Naval Air Station. Their mean age andrange was almost identical to the first group. All had beenscreened for speech, hearing, and voice disorders, and exhib-ited none. While, as with the university group, these subjectswere laryngeally healthy, they included a slightly greaternumber of smokers than did the students. Those selected hadcompleted high school but not attended college. Except forvery few subjects in the Northeast group, none spoke with anobservable regional dialect. These speakers were selected be-cause they were thought to represent a normal, nonuniversitypopulation—but a healthy one. While it is conceded that theyprobably were a little ‘‘above average’’ in some respects,their characteristics should be much closer to those for allyoung U.S. males than were those for the individuals in theuniversity cohort. In short, the two groups studied were quitesimilar in most respects—that is, except for educational sta-tus. Note, also, that the second group was established toroughly parallel the demography of the United States. Theywere classified on the basis of the urban–rural populationand by the four largest census regions; the relevant demo-graphic data maybe found in Table III. As can be seen, thesesubjects exhibited about a normal urban–rural balance butwere biased toward the Northeast, and away from the NorthCentral region. However, it was judged that they reasonablyrepresented the U.S. population and that they roughlymatched the students except for the factor being tested. Fi-nally, the two groups can be considered contemporary as therecordings for both were made over the same 14 month pe-riod. B. Procedure While speaking fundamental frequency was obtained for both oral reading and extemporaneous speaking, only thefirst of these two sets of data will be considered here. Thematerial for the reading sample was a modified version ofR.L. Stevenson’s prose passage ‘‘An Apology for Idlers.’’Subjects in both groups were provided ample practice timebefore recordings were made of this approximately 3-minpassage. Height and weight were obtained just prior to theTABLE III. Demographic data for the 142 subjects making up an average but healthy ~military !group. Because these individuals were studied in the early 1970’s, the demographic information is based on the 1960 U.S. census found in the 1968 Statistical Abstract. All values are in percent. AreaU.S. populationMilitary subjects ~A!Region Northeast 24.9 36.6North Central 30.7 19.0South 28.8 21.8West 15.6 14.4Unclassified a{{{ 8.5 ~B!Type Urban 69.9 68.4Rural 30.1 31.6 aPrimarily former military dependents. 2988 2988 J. Acoust. Soc. Am., Vol. 102, No. 5, Pt. 1, November 1997 Hollien et al.: Speaking fundamental frequency Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.105.215.146 On: Sat, 20 Dec 2014 13:43:43speech trials. Recordings were made in an IAC sound-treated booth using a high quality Altec condenser microphonecoupled to a calibrated Ampex 350 tape recorder. The result-ing tapes were processed by IASCP’s Fundamental Fre-quency Indicator ~FFI-8 !coupled to a dedicated computer ~Hollien and Tamburrino, 1965; Hollien, 1990 !. C. Results and discussion Data for the two groups may be found in Table IV. As will be noted, they are almost identical in age and roughlysimilar in physical size, with the university students bothslightly larger ~on average by a half-inch and three pounds ! and somewhat more variable relative to these factors. Thesedata are consistent with the notion that university studentswill be larger than other healthy but otherwise average popu-lations. The data for speaking fundamental frequency level ~see again Table IV !reveal over a semitone difference between the groups and this value is of statistical significance ~t 54.45;t 0.0551.65 for df .120!. However, it can be said that, while the two groups appear different with respect to(F0) level, the shift is in a somewhat different direction than might be predicted on the basis of physical size. That is,while SFF does not appear to correlate very well with asubject’s physical dimensions ~Hollienet al., 1994; Ku ¨nzel, 1989; Majewski et al., 1972 !, there are no data at all which suggest that smaller people will tend to exhibit lower F0 than will larger individuals. Thus even if a size–voice cor-relation could be argued, the trend here is in an unexpecteddirection—and, perhaps, in an inappropriate one. The onlyfactor which might account for this shift could be a differen-tial in the number of cigarette smokers in each group. Whilethis variable was not controlled, the subjective impressionwas that there was only a slightly greater proportion ofsmokers in the military group and, in any case, subjects wereyoung and could not have been long-term habitual smokers. Nevertheless, the possible effects of this factor cannot beignored. The values for the standard deviation of the distribution ~s.d.!reflect the variability in F0 usage by the individual speakers. The virtually identical s.d. found for each of thetwo groups suggest that they are very much alike with re-spect to this parameter. Admittedly, the range is slightlylower and a little more restricted for the university studentsbut this difference appears to be of little import. III. ORAL READING VERSUS EXTEMPORANEOUS SPEAKING Subjective impressions have led to the opinion that a person generally will exhibit a lower F0 when speaking ex- temporaneously than when engaged in the more formal ac-tivity of reading written material aloud. Sometimes resultsare questioned on the basis of which of these approaches wasemployed; in any event, the relationships cited here do notappear to be very well understood. Some research has beencarried out ~Fitch, 1990; Hanley, 1951; Hanley et al., 1966; Higgins and Saxman, 1991; Hollien and Jackson, 1973;Snidecor, 1944 !, but no clear patterns have emerged. In some instances, the differences were quite small ~Fitch, 1990; Han- leyet al., 1966; Morris, 1997 !; in others reversals were ob- served ~Fitch, 1990; Hanley, 1951 !. There is also the possi- bility that different experimental procedures were employedeven within a study. In one case, anyway, the subjects ap-parently sat when reading but stood when producing the ex-temporaneous speech. In short, it would appear reasonable tostudy the issue directly and possibly provide somewhat moredefinitive information about these relationships. A. Procedure Published data again were combined with new material to assess the issue of interest. However, all of the studiescited followed the same research protocols ~i.e., those estab- lished by Hollien and Jackson, 1973 !. Not only were the same procedures used for both tasks but the ~oral!reading passage was specifically balanced by an equally long extem-poraneous speaking task ~presentation order was counterbal- anced !. Four different experiments met these criteria; they were: ~1!the 157 ‘‘university’’ males studied by Hollien and Jackson ~1973!;~2!the 142 ‘‘military’’ males cited above; ~3!the 25 males from the secular trends study; and ~4!18 females also from that study. The contrasts were between‘‘The Apology for Idlers’’ and a 2-min segment taken from 3min of extemporaneous speech ~responses to questions such as, ‘‘What did you do on your last vacation?,’’ ‘‘What TVprogram is your favorite?’’ !in the first two investigations and between ‘‘The Rainbow Passage’’ and 40-s samplesdrawn from subjects’ 2-min response to the same ~neutral ! questions in the second two. As stated above, subjects wereyoung, healthy individuals between the ages of 18 and 30years who stood or sat ~depending upon which study !in fixed relationship to a microphone. All data reduction wasaccomplished by FFI processing.TABLE IV. Basic data for the two relatively large populations of young adult males. The students were drawn from the University of Florida; themilitary cohort were young healthy adults who roughly parallel the popula-tion distribution of the United States. Parameter University Military Number 157 142 Age~years ! Mean 20.3 20.5Range 17.9–25.8 17.8–25.5 Height ~inches ! Mean 69.9 69.3 a Range 63.3–77.5 62.0–75.5 Weight ~pounds ! Mean 164.4 160.6a Range 116.0–260.0 112.0–222.5 SFF~Hz! Mean 129.4 121.5Range 92.6–178.1 95.0–159.4Mean SFF ~STL! 35.7 34.6 Standard deviation ~T! Mean 1.6 1.6Range 0.5–2.5 0.8–3.3 aHeight and weight were not available for one subject. 2989 2989 J. Acoust. Soc. Am., Vol. 102, No. 5, Pt. 1, November 1997 Hollien et al.: Speaking fundamental frequency Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.105.215.146 On: Sat, 20 Dec 2014 13:43:43B. Results and discussion The contrasts between the speaking materials are best understood by consideration of Table V. First, it can be seenthat the mean SFF level for the oral reading task was alwayshigher than that for the more spontaneous speaking condi-tions. Moreover, the actual differences, while a little lessthan a semitone on average, are rather uniform—rangingonly from 0.6 st. to 0.9 st. Thus it is tempting to concludethat the hypothesis, as stated, is supported by this research,especially since these differences were consistent for fourseparate studies, involving 341 subjects, carried out underreasonably precise conditions. This position is furtherstrengthened by the results of an ANOVA ~F574.74, df 51,3,F 0.05510.13 !. On the other hand, about one subject in five exhibited either no shift at all or a higher F0 for the extemporaneous speaking condition than for oral reading.This finding was not unexpected, especially since prior dataon the issue were somewhat mixed. However, it precludes any strong predictions for individual subjects. Additionally, it must be conceded that an uncontrolled variable could have biased the data for the spontaneousspeech procedure; it involved the stress levels that mighthave been experienced by some subjects. That is, while theindividuals studied responded to questions carefully struc-tured for their neutrality, some of them appeared to become alittle excited during their response ~for example, they had ‘‘absolutely loved’’ the motion picture they were describ-ing!. Since it is reasonably well accepted that F0 can rise ~for some subjects anyway !with increases in stress or certain types of emotions ~Hollien, 1990 !, the possibility exists that such bias could have caused at least some of the reversals. Inany event, it appears that higher group SFF can be expectedfor oral reading and there now is rather substantial supportfor this relationship.IV. SUMMARY AND CONCLUSIONS The three postulates evaluated were that: ~1!a secular trend exists for SFF; one which is manifested as a loweringof talker F0 over time: ~2!university students will exhibit somewhat different speaking behaviors than will average in-dividuals; and ~3!there are systematic differences in SFF level between reading aloud and speaking extemporaneously.Based on the obtained data, it must be concluded that noneof them ~except perhaps the third !can be supported at any but a minimal level. To be specific: ~1!While a statistically significant trend was noted for the SFF lowering of parallel groups over the past 50 years,there was no firm evidence that much ~if any !of the shift occurred during the latter half of the period. Of course,the data presented were generated only on young adults.However, these findings were reasonably consistent withthose from contemporary studies for individuals of otherages—especially when compensation was made for dif-ferences in population size/type, measurement proce-dure, data processing approaches and the like. If a physi-ologically based secular trend did occur it probablyshould be linked to periods prior to the 1970’s. More-over, it can be argued that recent data now are suffi-ciently extensive that direct comparisons with the earlierstudies are no longer necessary ~except for historical pur- poses !. ~2!As expected, the university students studied were slightly larger than an otherwise comparable group ofaverage young males. They also exhibited a slightlyhigher SFF. The problem here is that the vocal differ-ences were not particularly compelling, appeared to be inan inappropriate direction and, although the SFF contrastbetween groups was statistically significant, both levelswere well within the expected F0 range for young adults. Thus the second hypothesis hardly appears towarrant support. As a matter of fact, it can be argued thatuniversity students probably will make good subjects forstudies of motor speech and voice even though theirgroupF0 may be slightly higher than those for average subjects. They are less likely to exhibit minor deficits orpathologies and they certainly should be able to easilyunderstand and carry out the tasks required of them. ~3!The third postulate was supported to some extent; how- ever, with certain reservations. It appears that, under oth-erwise parallel conditions, most individuals will producespeech at a mean fundamental frequency which is higherfor oral reading than for extemporaneous speech. Thisdifference can be considered minimal for most applica-tions and corrections should be easy to make ~on a group basis, that is !. It, also, must be stressed that, due to po- tential for reversals, these relationships do not result ingood predictive metrics for individual speakers. ACKNOWLEDGMENTS The research upon which this report is based was sup- ported by grants from the National Institutes of Health, theDreyfus Foundation and the Office of Naval Research. TheTABLE V. Summary table contrasting speaking fundamental frequency measures for oral reading and extemporaneous speech. Study NReading ExtemporaneousDiff. ~ST!Reversals ~percent ! University males 157 21 Mean SFF ~Hz! 129 123 Mean SFF ~STL!a35.7 34.8 0.9 s.d.~T! 1.6 1.6 Military males 142 23 Mean SFF ~Hz! 122 116 Mean SFF ~STL! 34.6 33.8 0.8 s.d.~T! 1.6 1.7 Current study: Men 25 28 Mean SFF ~Hz! 114 110 Mean SFF ~STL! 33.6 33.0 0.6 s.d.~T! 1.3 1.3 Current study: Women 18 17 Mean SFF ~Hz! 198 190 Mean SFF ~STL! 43.2 42.4 0.9 s.d.~T! 1.2 1.1 Means 0.8 22 aSTL5semitone level; T 5tones; ST 5semitones. 2990 2990 J. Acoust. Soc. Am., Vol. 102, No. 5, Pt. 1, November 1997 Hollien et al.: Speaking fundamental frequency Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.105.215.146 On: Sat, 20 Dec 2014 13:43:43authors also wish to thank Reva Schwartz, James Greene, Patricia Phillips, Allan Alderman and personnel at the U.S.Naval Air Station, Pensacola, Florida for their kind assis-tance and support. 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1.2711702.pdf

Ultrafast optical modification of magnetic anisotropy and stimulated precession in an epitaxial Co 2 Mn Al thin film Y. Liu, L. R. Shelford, V. V. Kruglyak, R. J. Hicken, Y. Sakuraba, M. Oogane, Y. Ando, and T. Miyazaki Citation: Journal of Applied Physics 101, 09C106 (2007); doi: 10.1063/1.2711702 View online: http://dx.doi.org/10.1063/1.2711702 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Magnetic properties of Fe0.4Mn0.6/Co2FeAl bilayers grown on GaAs by molecular-beam epitaxy J. Appl. Phys. 110, 093904 (2011); 10.1063/1.3657780 Quadratic magneto-optical Kerr effect in Co2MnSi J. Appl. Phys. 110, 043904 (2011); 10.1063/1.3622512 Magnetic properties of full-Heusler alloy Co 2 Fe 1 − x Mn x Al films grown by molecular-beam epitaxy Appl. Phys. Lett. 97, 232506 (2010); 10.1063/1.3525376 Structural and magnetic properties of epitaxial L 2 1 -structured Co 2 ( Cr , Fe ) Al films grown on GaAs(001) substrates J. Appl. Phys. 97, 103714 (2005); 10.1063/1.1888050 Epitaxial structure and magnetic anisotropies of metastable single crystal Co 0.70 Mn 0.30 film J. Appl. Phys. 81, 2036 (1997); 10.1063/1.364010 [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: 129.24.51.181 On: Mon, 01 Dec 2014 18:37:46Ultrafast optical modification of magnetic anisotropy and stimulated precession in an epitaxial Co 2MnAl thin film Y . Liu,a/H20850L. R. Shelford, V. V. Kruglyak, and R. J. Hicken School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom Y . Sakuraba, M. Oogane, Y . Ando, and T . Miyazaki Department of Applied Physics, Graduate School of Engineering, Tohoku University, Sendai 980–8579, Japan /H20849Presented on 11 January 2007; received 1 November 2006; accepted 13 December 2006; published online 2 May 2007 /H20850 An all-optical pump-probe method was used to study magnetization precession in an epitaxial Co2MnAl Heusler alloy thin film. The frequency and amplitude of precession showed a clear fourfold variation as the orientation of the static field was applied in different directions within theplane of the film, revealing that the precession is induced by an ultrafast modification of themagnetocrystalline anisotropy field. The effective fields acting upon the magnetization have beendetermined and the damping parameter is found to decrease rapidly as the strength of the appliedfield is increased. © 2007 American Institute of Physics ./H20851DOI: 10.1063/1.2711702 /H20852 Heusler alloys show great promise for use in spintronic devices due to their predicted half-metallic band structure.Recently, large room temperature tunnel magnetoresistance/H20849TMR /H20850values have been observed in magnetic tunnel junc- tions /H20849MTJs /H20850with Co 2MnAl /H20849Ref. 1/H20850and other Heusler alloy2,3electrodes. It is expected that the half-metallic band structure may suppress spin-flip processes and hence reducethe damping of the magnetization precession. Ferromagneticresonance /H20849FMR /H20850has previously been used to study the in- trinsic Gilbert damping parameter of thin Co 2MnAl films.4,5 However, time domain measurements of the damping param- eter over a continuous range of frequencies are needed toachieve a comprehensive understanding of the dynamicproperties while measurements within the low field regimeare of particular technological relevance. All-optical pump-probe techniques have been demonstrated in the last decadethat enable ultrafast demagnetization, coherent magnetiza-tion rotation, and hot electron relaxation in magnetic thinfilms 6–11to be studied over an extended range of frequencies. In this paper, we present all-optical pump-probe mea- surements of precessional magnetization dynamics in an ep-itaxial Co 2MnAl thin film. By varying the strength and ori- entation of the static magnetic field and fitting the resultingtransient Kerr rotation signals to a macrospin solution of theLandau-Lifshitz-Gilbert /H20849LLG /H20850equation, we have studied the variation of the frequency, amplitude, phase, and damping ofthe precession. We demonstrate that precession is induced byan ultrafast modification of the magnetocrystalline aniso-tropy and determine the dependence of the damping uponboth the strength and orientation of the magnetic field. A MgO /H20849001 /H20850/Cr /H2084940 nm /H20850/Co 2MnAl /H2084930 nm /H20850/MgO /H2084910 nm /H20850 structure was grown by magnetron sputtering at room tem- perature on a MgO /H20849001 /H20850substrate.12The sample was post- annealed at 300 °C so that epitaxial Co 2MnAl /H20849001 /H20850was obtained with the B2 lattice structure.1,12Optical pump-probe measurements were made upon the samples with 130 fspulses of 800 nm wavelength from a Ti:sapphire regenera- tive amplifier. The p-polarized pump pulse had energy of up to 0.8 /H9262J and was directed onto the sample surface at near normal incidence. The response of the sample magnetizationwas determined from a time resolved magneto-optical Kerreffect /H20849TRMOKE /H20850rotation measurement made with an s-polarized 400 nm wavelength probe pulse, with energy of 4 nJ, incident at an angle of 40°. The pump and probe beamswere focused onto the sample surface with spot sizes of 140and 80 /H9262m, respectively, and overlapped as they were viewed with a high-magnification charge-coupled device/H20849CCD /H20850camera. Following excitation by the pump beam, the magnetiza- tion and magnetic anisotropy of the film undergo an ultrafastmodification. As a result, the effective magnetic field actingupon the magnetization may no longer be parallel to themagnetization. The resulting torque stimulates precession ofthe magnetization which may be described by the LLG equa-tion, /H11509M /H11509t=− /H20841/H9253/H20841M/H11003Heff+/H9251 M/H20873M/H11003/H11509M /H11509t/H20874, /H208491/H20850 where /H9251is the Gilbert damping parameter, /H9253=2.80 /H11003/H9266 /H11003106/H11003gHz/Oe is the gyromagnetic ratio of the electron, andgis the spectroscopic splitting factor. The effective mag- netic field Heffincludes the applied static magnetic field, the demagnetizing field, and contributions due to the magneto-crystalline anisotropy. For the case of a small amplitude uni-form precession, an algebraic expression for the time depen-dent magnetization may be obtained if the magneto-crystalline anisotropy is assumed to be instantaneously re-duced by the optical pumping. When the static field is ap-plied parallel to the plane of incidence of the probe, the Kerrrotation signal contains an oscillatory component due to theout-of-plane component of the dynamic magnetization. Al-ternatively, if the static field is applied perpendicular to theplane of incidence, the Kerr signal results from the sum oftwo oscillatory terms that are linearly proportional to the a/H20850Electronic mail: yanwei.liu@exeter.ac.ukJOURNAL OF APPLIED PHYSICS 101, 09C106 /H208492007 /H20850 0021-8979/2007/101 /H208499/H20850/09C106/3/$23.00 © 2007 American Institute of Physics 101 , 09C106-1 [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: 129.24.51.181 On: Mon, 01 Dec 2014 18:37:46components of magnetization that lie within the plane of incidence.13In either case the oscillatory Kerr rotation has the form /H9278K/H11008/H9004/H9278cos /H208492/H9266f0t+/H9272/H20850exp /H20849−t//H9270/H20850, /H208492/H20850 where /H9004/H9278,f0,/H9270, and/H9272are the amplitude, frequency, decay constant, and initial phase of the signal, respectively. Thevalue of /H9272is dependent on the experimental geometry, while in the limit /H9251/H112701,/H9004/H9278,f0, and /H9270can be expressed algebra- ically as /H9004/H9278=−/H208491/2 /H20850sin /H208514/H20849/H9278−/H92784/H20850/H20852/H9004/H20849K1/Ms/H20850 Hcos /H20849/H9278/H20850+/H20849K1/2Ms/H20850cos /H208514/H20849/H9278−/H92784/H20850/H20852, /H208493/H20850 f0=1 2/H9266/H20841/H9253/H20841/H20849H/H9251H/H9252/H208501/2, /H208494/H20850 /H9270=2 /H20841/H9253/H20841/H20849H/H9251+H/H9252/H20850/H9251, /H208495/H20850 H/H9251=Hcos /H20849/H9278/H20850+K1 2Ms/H208533 + cos /H208514/H20849/H9278−/H92784/H20850/H20852/H20854+4/H9266Ms, /H208496/H20850 H/H9252=Hcos /H20849/H9278/H20850+2K1 Mscos /H208514/H20849/H9278−/H92784/H20850/H20852. /H208497/H20850 Here/H9278and/H92784are the angles that the magnetization and /H20851100 /H20852axis of the Co 2MnAl describe with the applied static field H.MSandK1are the saturation magnetization and cu- bic magnetocrystalline anisotropy constant, respectively,while the value of /H9278is determined from the static equilib- rium condition, MHsin /H20849/H9278/H20850+K1 2sin /H208514/H20849/H9278−/H92784/H20850/H20852=0 . /H208498/H20850 Measurements were first performed with the magnetic field /H20849HL/H20850applied in the plane of the sample and parallel to the plane of incidence, as shown in Fig. 1/H20849a/H20850. A typical time- resolved rotation Kerr signal is shown in Fig. 1/H20849b/H20850, for which the magnetic field of 852 Oe was applied at 20° relative tothe /H20851110 /H20852axis of the Co 2MnAl. The signal consists of an oscillatory component superimposed upon a component dueto an ultrafast demagnetization. By comparing fine scans ofthe initial rise of the signal with the longitudinal hysteresisloop /H20849not shown /H20850, the peak demagnetization was found to be 5%. It has been demonstrated that the amplitude and phase ofthe magnetization precession may be strongly dependent onthe helicity of the pump beam for garnet films studied in theFaraday geometry. 14However, the amplitude and phase of the demagnetization and precessional components werefound to be independent of the helicity of the pump in ourinvestigation, while the precession amplitude was linearlydependent on the pulse energy, as shown in the inset of Fig.1/H20849b/H20850. In order to eliminate the contribution from the demagne- tization signal, the experiments were mainly conducted withthe static field /H20849H T/H20850applied perpendicular to the plane of incidence, as shown in Fig. 1/H20849a/H20850. Figure 2/H20849a/H20850shows the tran- sient Kerr rotation signals obtained in this transverse fieldgeometry as a static field of 211 Oe, sufficient to saturate the sample, was applied in different directions relative to the/H20851110 /H20852axis of the Co 2MnAl. The transient Kerr rotation sig- nals show that the precession amplitude depends strongly onthe orientation of the static magnetic field. The raw time-resolved signals were fitted to the sum of a damped sinusoid/H20851Eq. /H208492/H20850/H20852and to a small exponentially decaying background that accounts for the slow recovery of the magnetic aniso-tropy. The saturation magnetization was set to the bulk valueof 730 emu/cm 3.4,12Equations /H208493/H20850,/H208494/H20850, and /H208496/H20850–/H208498/H20850were used to fit the amplitude and frequency of the precession ofthe magnetization, shown in Figs. 2/H20849b/H20850and2/H20849c/H20850, respectively. Figure 2/H20849b/H20850shows that the precession amplitude has a clear fourfold variation with the orientation of the static field. No FIG. 1. /H20849a/H20850Schematic diagram of the experimental configuration with a longitudinal /H20849HL/H20850or a transverse /H20849HT/H20850magnetic field. /H20849b/H20850Typical time re- solved Kerr rotation signal obtained with Hparallel to the plane of inci- dence. The magnetic field of 852 Oe was applied at an angle of 20° relativeto the /H20851110 /H20852axis of the Co 2MnAl. The inset shows the dependence of the precession amplitude on the pulse energy, where the solid line is a guide tothe eye. FIG. 2. /H20849a/H20850Transient Kerr rotation signals obtained with a magnetic field of 211 Oe applied at difference angles to the /H20851110 /H20852axis of the Co2MnAl and perpendicular to the plane of incidence. /H20849b/H20850The dependence of the preces- sion amplitude, /H20849c/H20850the precession frequency, and /H20849d/H20850the damping parameter on the orientation of the magnetic field are shown.09C106-2 Liu et al. J. Appl. Phys. 101 , 09C106 /H208492007 /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: 129.24.51.181 On: Mon, 01 Dec 2014 18:37:46precession was observed with the field applied parallel to /H20849easy axis /H20850or 45° from /H20849hard axis /H20850the /H20851110 /H20852direction, while the maximum amplitude was found for the field applied mid-way between these orientations. The precession signal is alsoseen to undergo a 180° phase change as the static fieldcrosses the easy or hard axis. From Fig. 2/H20849c/H20850, the precession frequency is seen to exhibit a fourfold variation which hasbeen fitted to Eq. /H208494/H20850. The fourfold variation of the preces- sion amplitude and frequency with the orientation of themagnetic field confirms that the magnetization precession isinduced by an ultrafast optical modification of the magneto-crystalline anisotropy, 9,15leading to a reorientation of the effective field acting upon the magnetization. When the staticfield is applied along either the easy or hard axis, no preces-sion is induced because the effective field remains parallel tothe magnetization after excitation. In all other orientationsH effrotates suddenly from alignment parallel to M, produc- ing a torque on M. The damping parameter /H9251calculated using Eq. /H208495/H20850shows a weak fourfold variation to which a sine curve has been added as a guide to the eye. However,we note that the extracted values of /H9251are slightly modified if the frequency of the fitted function is allowed to vary withtime so as to account for the slow recovery of the magneto-crystalline anisotropy, and so further measurements are re-quired to determine whether the observed fourfold variationis an intrinsic property of the sample or an artifact of thefitting procedure. Figure 3/H20849a/H20850shows the dependence of the transient Kerr rotation signal upon the strength of the magnetic field. Herethe static field was applied at 20° relative to the Co 2MnAl /H20851110 /H20852easy axis. The transient precession signals were again fitted using Eq. /H208492/H20850, and the fitted parameter values are sum- marized in Figs. 3/H20849b/H20850–3/H20849d/H20850. As the magnetic field is in- creased, the precession frequency increases while the preces-sion amplitude and the damping parameter /H9251decrease. The solid lines in Figs. 3/H20849b/H20850and3/H20849c/H20850are fits to Eqs. /H208493/H20850and /H208494/H20850, respectively. By fitting both the field and angle dependencedata simultaneously, best fit values of g=2.10 and K 1= −4572 erg/cm3were determined. A value of /H9251 =0.015±0.003 is obtained at a frequency of 10 GHz, whichis twice the value obtained in FMR measurements at the same frequency.4A number of authors have reported a field dependent damping parameter and suggested a variety of un-derlying mechanisms. 16–20The damping parameter extracted from the fits reported here should be regarded as a phenom-enological parameter that accounts for the combined effect ofintrinsic damping, inhomogeneous broadening, two magnonscattering, any higher order spin waves processes, and propa-gation effects resulting from the nonuniform spatial profile ofthe precession. Further work is now required to understandwhich of these mechanisms make a significant contributionto the damping observed in the present study. We have used an all-optical pump-probe method to study the magnetization dynamics of a thin Co 2MnAl Heusler alloy film. The frequency and amplitude of the observed preces-sional oscillations showed a clear fourfold variation, reveal-ing that the magnetization precession is induced by an ul- trafast modification of the magnetocrystalline anisotropyfield. By fitting the Kerr signal to a macrospin model, wehave determined the values of the effective fields acting uponthe magnetization and shown that the value of the dampingparameter decreases rapidly as the strength of the appliedfield is increased. The authors would like to acknowledge the support of the New Energy and Industrial Technology Development Or-ganization /H20849NEDO /H20850and the Engineering and Physical Sci- ences Research Council /H20849EPSRC /H20850. 1Y. Sakuraba, J. Nakata, M. Oogane, Y. Ando, H. Kato, A. Sakuma, T. Miyazaki, and H. Kubota, Appl. Phys. Lett. 88, 022503 /H208492006 /H20850. 2S. Kämmerer, A. Thomas, A. Hütten, and G. Reiss, Appl. Phys. Lett. 85, 79 /H208492004 /H20850. 3Y. Sakuraba, M. Hattori, M. Oogane, Y. Ando, H. Kato, A. Sakuma, T. Miyazaki, and H. Kubota, Appl. Phys. Lett. 88, 192508 /H208492006 /H20850. 4R. Yilgin, M. Oogane, S. Yakata, Y. Ando, and T. Miyazaki, IEEE Trans. Magn. 41, 2799 /H208492005 /H20850. 5M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys., Part 1 45,3 8 8 9 /H208492006 /H20850. 6E. Beaurepaire, J. C. Merle, A. Daunois, and J. Y. Bigot, Phys. Rev. Lett. 76, 4250 /H208491996 /H20850. 7J. Hohlfeld, E. Matthias, R. Knorren, and K. H. Bennemann, Phys. Rev. Lett. 78, 4861 /H208491997 /H20850. 8M. 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. 9Q. Zhang, A. V. Nurmikko, A. Anguelouch, G. Xiao, and A. Gupta, Phys. Rev. Lett. 89, 177402 /H208492002 /H20850. 10A. Kimel, A. Kirilyuk, A. Tsvetkov, R. V. Pisarev, and T. Rasing, Nature /H20849London /H20850429, 850 /H208492004 /H20850. 11V. V. Kruglyak, R. J. Hicken, M. Ali, B. J. Hickey, A. T. G. Pym, and B. K. Tanner, Phys. Rev. B 71, 233104 /H208492005 /H20850. 12H. Kubota, J. Nakata, M. Oogane, Y. Ando, H. Kato, A. Sakuma, and T. Miyazaki, J. Appl. Phys. 97, 10C913 /H208492005 /H20850. 13J. Wu, J. R. Moore, and R. J. Hicken, J. Magn. Magn. Mater. 222,1 8 9 /H208492000 /H20850. 14F. Hansteen, A. Kimel, A. Kirilyuk, and T. Rasing, Phys. Rev. B 73, 014421 /H208492006 /H20850. 15H. B. Zhao et al. , Appl. Phys. Lett. 86, 152512 /H208492005 /H20850. 16W. Platow, A. N. Anisimov, G. L. Dunifer, M. Farle, and K. Baberschke, Phys. Rev. B 58, 5611 /H208491998 /H20850. 17J. P. Nibarger, R. Lopusnik, and T. J. Silva, Appl. Phys. Lett. 82,2 1 1 2 /H208492003 /H20850. 18M. L. Schneider, T. Gerrits, A. B. Kos, and T. J. Silva, Appl. Phys. Lett. 87, 072509 /H208492005 /H20850. 19M. Djordjevic, G. Eilers, A. Parge, M. Münzenberg, and J. S. Moodera, J. Appl. Phys. 99, 08F308 /H208492006 /H20850. 20J. Wu, N. D. Hughes, J. R. Moore, and R. J. Hicken, J. Magn. Magn. Mater. 241,9 6 /H208492002 /H20850. FIG. 3. /H20849a/H20850Transient Kerr rotation signals with Hperpendicular to the plane of incidence and at 20° to the Co2MnAl /H20851110 /H20852axis. /H20849b/H20850The dependence of the precession amplitude, /H20849c/H20850the precession frequency, and /H20849d/H20850the damping parameter on the strength of the magnetic field /H20849the solid line is a guide to the eye, while /H20849b/H20850and /H20849c/H20850show fits to the macrospin model /H20850.09C106-3 Liu et al. J. Appl. Phys. 101 , 09C106 /H208492007 /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: 129.24.51.181 On: Mon, 01 Dec 2014 18:37:46

1.372855.pdf

Thermal field fluctuations in a magnetic tip / implications for magnetic resonance force microscopy J. D. Hannay and R. W. Chantrell D. Rugar Citation: Journal of Applied Physics 87, 6827 (2000); doi: 10.1063/1.372855 View online: http://dx.doi.org/10.1063/1.372855 View Table of Contents: http://aip.scitation.org/toc/jap/87/9 Published by the American Institute of PhysicsMagnetic Imaging and Modeling R. D. Gomez, Chairman Thermal field fluctuations in a magnetic tip Õimplications for magnetic resonance force microscopy J. D. Hannaya)and R. W. Chantrell Computational Magnetism Group, School of Electronic Engineering and Computer Systems, University of Wales Bangor, Dean Street, Bangor, LL57 1UT, United Kingdom D. Rugar IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, California 95120 Thermally excited magnetic fluctuations are fundamental to the behavior of small ferromagnetic particlesandhavepracticalconsequencesfortheproposeddetectionofindividualspinsbymagneticresonance force microscopy ~MRFM !. In particular, fluctuating fields from a nearby magnetic tip can increase the relaxation rate of spins in a sample if there is significant spectral density of fieldfluctuation at the Larmor frequency of the target spin. As an initial step toward understanding thisissue, magnetic field fluctuations have been simulated which emanate from a magnetic tip withdimensions 60nm 360nm 32 mm. It was found that the fluctuations in a cobalt magnetic tip were toostrongforMRFMexperimentsaimedatdetectingindividualelectronspins.However,theresultsobtained for a PrFeB tip fell within the tolerance required. © 2000 American Institute of Physics. @S0021-8979 ~00!81308-0 # I. INTRODUCTION Magnetic resonance force microscopy ~MRFM !is a rela- tively new scanning probe microscope technique that com-bines aspects of magnetic resonance imaging ~MRI!and atomic force microscopy ~AFM !. MRFM uses a small ferro- magnetic tip to generate a strong magnetic field gradient soas to generate a magnetic force on a nearby sample. Theforce is detected using a sensitive micromechanical cantile-ver while the magnetization of the sample is manipulatedusing magnetic resonance techniques. MRFM was originallyproposed by Sidles as a means to determine the three-dimensional atomic structure of biological molecules, suchas proteins. 1Although this ambitious goal is still out of reach, MRFM has been shown to be a sensitive technique fordetecting magnetic resonance in micron-size samples. 2,3 More recently, experiments aimed at detecting individualelectron spins in paramagnetic samples have been initiated.For these experiments to be successful, one requirement thatmust be met is that the random flip rate of the target spinmust be slow, preferably less than 1 flip per second. Several influences contribute to the random flip rate. First, there is an intrinsic spin-lattice relaxation time T 1.T1 is usually quite fast for electron spins ~often in the nanosec- ond or microsecond range !, but for certain systems, such as E8centers in SiO 2at low temperatures, T1can be signifi- cantly longer than 1 s.4 Another possible contributor to the random flip rate is magnetic noise emanating from the ferromagnetic tip. Forexample, a randomly fluctuating transverse magnetic fieldH x(t) will induce spin flips if there is any significant spectral density of Hx(t) at the Larmor frequency of the target spin. The flip rate caused by Hx(t) is given by Wflip51 8g2SHx~vLarmor !, ~1! whereWflipis the mean flip rate ~flips per second !,gis the gyromagnetic ratio of the target spin ~21.76 3107radOe21s21for an electron spin !andSHx(vLarmor)i s the~single-sided !spectral density of Hx(t)~with units of Oe2/Hz!. Equation ~1!shows that a key quantity is the spectral density of the field noise. This will depend on many param-eters, including temperature, magnetic anisotropy, spin wavemodes, etc. In this paper we shall make initial calculations ofthe power spectral density of the transverse fluctuating fieldemanating from a magnetic tip. II. COMPUTATIONAL MODEL To make the problem concrete, we first consider a rect- angular bar of cobalt with a square x-ycross-section of 60 nm by 60 nm and length along the zaxis of 2 mm. This bar is modeled by 900 single domain cubes of 20 nm side eachof which is represented by a single spin obeying coherentrotation. The total magnetic energy of the ith cube includes the Zeeman energy from the applied field, uniaxial crystallineanisotropy energy, inter-granular exchange coupling energyand the magnetostatic energy. The total energy density forcubeiis then given by a!Electronic mail: hannay@sees.bangor.ac.ukJOURNAL OF APPLIED PHYSICS VOLUME 87, NUMBER 9 1 MAY 2000 6827 0021-8979/2000/87(9)/6827/3/$17.00 © 2000 American Institute of PhysicsEtot,i52MiHapp1Ksin2ui22A* D2Ms2( jÞiNN MiMj 2( jÞiTotalF3~Mirij!~Mjrij! rij5 2MiMj rij3G, ~2! whereKis the uniaxial anisotropy energy constant, uiis the angle between the easy axis orientation and the magnetiza-tion direction, A *is the effective exchange energy constant, Dis the cube side, Msthe saturation magnetization and rijis the vector which connects the centers of cubes iandj. The exchange energy is analogous to that from spin–spin ex-change and only nearest neighbor cubes are included in thecalculation. The magnetostatic interaction energy for onecubeiis approximated as a sum of dipolar contributions from all other cubes within the magnetic tip. The reduced effective field acting on the ith cube is given by h eff,i521 HK]Etot,i ]Mi, ~3! whereHK52K/Msis the magnetocrystalline anisotropy field. If we combine Eqs. ~2!and~3!we have heff,i5happ1~mˆieˆi!eˆi1C*( jÞiNN mˆj 1Ms2D3 2K( jÞiTotal1 rij3@3~mˆjrˆij!rˆij2mˆj#, ~4! wherehappis the reduced externally applied field vector, mˆi5Mi/Msis the magnetization unit vector of cube i,eˆiis the easy axis unit vector of cube i, andC*5A*/(D2K)i s the reduced effective exchange coupling parameter. The time evolution for the magnetization vector of each cube is governed by the phenomenological stochastic Gilbertequation. 5This particular form for the equation of motion is used in preference to that of Landau and Lifshitz6because it can be directly derived from a Lagrangian function and aRayleigh dissipation function. 5However, the Gilbert equa- tion is usually cast in the Landau–Lifshitz form to aid sim-plicity, resulting in the so called stochastic Landau–Lifshitz–Gilbert ~LLG!equation which, by using a reduced time variable t5tuguHK/(11a2), can be written as dmˆi dt52mˆiˆ@heff,i1hth,i~t!# 2amˆiˆ$mˆiˆ@heff,i1hth,i~t!#%, ~5! where arepresents the dissipation or damping constant. Equation ~5!is the Langevin equation of this particular system because the reduced effective field that is acting oneach cube ihas been augmented by a fluctuation field h th,i(t). This stochastic field accounts for the interaction be- tween the simple magnetic spin and the many degrees offreedom associated with the surrounding complex dissipativesystem which causes random fluctuations of the magnetiza-tion. The resultant relationship between cause and effect isexpressed in the fluctuation–dissipation theorem. 7In order todefine the statistical properties of hth,i(t), assumptions fol- lowing those made in Brownian motion8~also known as the Wiener process !theory are made.9Therefore we assume, by virtue of the Central Limit Theorem ~SWAP !, that the ran- dom field has the form of Gaussian white noise and has theproperties h th,i~t!5xˆhth,i1~t!1yˆhth,i2~t!1zˆhth,i3~t!, ~6! ^hth,ix~t!&50, ~7! ^hth,ix~t!hth,iy~t8!&5s2dxyd~t2t8!, ~8! where dxyis the Kronecker delta and d(t2t8) is the Dirac delta function. Equation ~7!states that the ensemble mean of hth,i(t) is zero and it is also assumed that the stochastic random field is different for each cube i. Equation ~8!states that different components of hth,i(t) are uncorrelated and identical components are uncorrelated for t8Þt. Brown9 formulated an expression for the variance of Eq. ~8!which for this model is given by s25kBTa KVi~11a2!Dt, ~9! wherekBis Boltzmann’s constant, Tthe temperature and Dt is the reduced time integration step. III. RESULTS AND DISCUSSION After magnetizing the bar along its anisotropic long axis we reduce the applied field to 3000 Oe. The transverse ran-dom field is sampled at a point 20 nm directly below thecenter of the tip at a temperature of 5° K and the root powerspectral density of these field fluctuations is calculated usingFast Fourier Transform ~FFT!techniques. Figure 1 shows the results for a reduced exchange coupling parameter C * 50.4. There are two dominant features apparent in Fig. 1 and initial interpretations suggest that the lowest frequencyfeature may be associated with the magnetostatic magneticmode of the tip with the ~broader !higher frequency feature resulting from a superposition of the exchange spin wavemodes. If we assume that the target spin has a Larmor pre- FIG. 1. Root power spectral density of field fluctuations for a strongly exchange coupled cobalt tip.6828 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Hannay, Chantrell, and Rugarcession frequency ’131010Hz then we can see that the strongly exchange coupled cobalt tip gives SHx1/2(vLarmor)55 31026Oe/AHz. Using Eq. ~1!we see that this gives a flip rateWflip’1000s21which is three orders of magnitude faster than the experiment can tolerate. Therefore a cobalt tipwould not be suitable for single spin MRFM experiments. Inorder to study the dependence of the spectrum on the ex-change coupling value, the calculations are repeated for acobalt tip with a weaker intergranular exchange couplingC *50.1~Fig. 2 !. The peak associated with the exchange spin wave modes has been shifted to a lower frequencywhich, qualitatively agrees with previous results, indicatingthat strong exchange coupling speeds up the precession ofthe magnetic moments. 10 The fluctuations should decrease significantly for a more anisotropic magnet. One alternative tip material that is underconsideration for MRFM experiments is PrFeB, which has avery large anisotropy constant K52.3310 8erg/cm3at low temperatures.11We keep the geometry of the cobalt tip to aid the comparison of results. Figure 3 shows the root powerspectral density of the fluctuation field for a PrFeB tip. ThepeaksofFigs.1and2cannolongerbeseenatthisfrequencyrange because they will now have been shifted to a muchhigher frequency due to the much larger effective dc fieldacting on the tip. It is immediately clear that the PrFeB tipreduces the fluctuations dramatically. The value of S Hx1/2(vLarmor)’0.631027Oe/AHz gives us a flip rate which is on the order of 1 second, thus satisfying the requirements of a single spin MRFM experiment. IV. CONCLUSIONS A dynamic model of thermally activated field fluctua- tions emanating from a magnetic tip has been described. Wehave shown that cobalt is not a suitable tip material for singlespin MRFM experiments due to the relatively large randomfield fluctuations below the tip. We have found also that high exchange coupling leads to a more rapid relaxation withinthe bar. The spectrum is found to be sensitive to the precisevalue of the exchange coupling strength. For cobalt the spec-trum shows features which can be tentatively associated withexchange and magnetostatic spin-wave modes which will bediscussed elsewhere. The large anisotropy of the PrFeB tipshifts any such features to frequencies outside the experi-mental range of interest. As expected, the field fluctuationsare dramatically reduced in high anisotropy materials such asPrFeB. Since the reduced fluctuations result in a muchslower random flip rate for nearby target spins, high anisot-ropy magnetic tips are likely to be key elements in futuresingle spin MRFM experiments. ACKNOWLEDGMENTS The financial support of the United Kingdom Engineer- ing and Physical Sciences Research Council ~UK EPSRC !is gratefully acknowledged. J.D.H. wishes to thank Seagate~Fremont, USA !for their support of a Co-operative Awards in Science and Engineering ~CASE !studentship. D.R. also thanks H. J. Mamin, B. Stipe, C. S. Yannoni, and J. Sidlesfor stimulating discussions and the U.S. Office of Naval Re-search for partial financial support. 1J. A. Sidles, Phys. Rev. Lett. 68, 1124 ~1992!. 2D. Rugar, O. Zuger, S. Hoen, C. S. Yannoni, H. M. Vieth, and R. D. Kendrick, Science 264, 1560 ~1994!. 3J. A. Sidles, J. L. Garbini, K. J. Bruland, D. Rugar, O. Zuger, S. Hoen, and C. S. Yannoni, Rev. Mod. Phys. 67, 249 ~1995!. 4J. G. Castle, Jr. and D. W. Feldman, J. Appl. Phys. 36,1 2 4 ~1963!. 5T. L. Gilbert, Phys. Rev. 100, 1243 ~1955!. 6L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8,1 5 3 ~1935!. 7H. B. Callen and T. A. Welton, Phys. Rev. 83,3 4~1951!. 8M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17,3 2 3 ~1945!. 9W. F. Brown, Phys. Rev. 130, 1677 ~1963!. 10J. D. Hannay, R. W. Chantrell, and H. J. Richter, J. Appl. Phys. 85, 5012 ~1999!. 11S. Hirosawa, Y. Matsuura, H. Yamamoto, S. Fujimura, M. Sagawa, andH. Yamauchi, J. Appl. Phys. 59, 873 ~1986!. FIG. 3. Root power spectral density of field fluctuations for a PrFeB tip. FIG. 2. Root power spectral density of field fluctuations for a weakly ex- change coupled cobalt tip.6829 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Hannay, Chantrell, and Rugar

1.4870711.pdf

Electric field control of multiferroic domain wall motion Hong-Bo Chen, Ye-Hua Liu, and You-Quan Li Citation: Journal of Applied Physics 115, 133913 (2014); doi: 10.1063/1.4870711 View online: http://dx.doi.org/10.1063/1.4870711 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/13?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Strain-mediated electric-field control of multiferroic domain structures in ferromagnetic films Appl. Phys. Lett. 102, 112407 (2013); 10.1063/1.4795938 Stress-based control of magnetic nanowire domain walls in artificial multiferroic systems J. Appl. Phys. 109, 023915 (2011); 10.1063/1.3532041 Electric field induced magnetization of multiferroic horizontal heterostructures J. Appl. Phys. 108, 084901 (2010); 10.1063/1.3496626 Electrically controlled magnetization switching in a multiferroic heterostructure Appl. Phys. Lett. 97, 052502 (2010); 10.1063/1.3475417 Reversible magnetic domain-wall motion under an electric field in a magnetoelectric thin film Appl. Phys. Lett. 92, 112509 (2008); 10.1063/1.2900886 [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: 147.226.7.162 On: Thu, 20 Nov 2014 09:21:26Electric field control of multiferroic domain wall motion Hong-Bo Chen, Y e-Hua Liu, and Y ou-Quan Li Zhejiang Institute of Modern Physics and Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of China (Received 10 March 2014; accepted 26 March 2014; published online 4 April 2014) The dynamics of a multiferroic domain wall in which an electric field can couple to the magnetization via inhomogeneous magnetoelectric interaction is investigated by the collective-coordinateframework. We show how the electric field is capable of delaying the onset of the Walker breakdown of the domain wall motion, leading to a significant enhancement of the maximum wall velocity. Moreover, we show that in the stationary regime the chirality of the domain wall can be efficientlyreversed when the electric field is applied along the direction of the magnetic field. These characteristics suggest that the multiferroic domain wall may provide a new prospective means to design faster and low-power-consumption domain wall devices. VC2014 AIP Publishing LLC . [http://dx.doi.org/10.1063/1.4870711 ] I. INTRODUCTION Manipulation of magnetic properties by an external elec- tric field has long been a big challenge in the quest for novel spintronic devices. In conventional ferroelectric or ferromag-netic materials, the controlled motion of ferroic domain walls (DWs) is essential to achieve the desired functional- ities. 1,2Usually, the motion of DW in magnetic materials are driven by a magnetic field3–5or spin-polarized current.6,7A key concept in the context of DW motion is the so-called Walker breakdown,3which distinguishes the two regimes with high- and low-mobility and sets a limit to the DW velocity. To achieve fast and efficient control of DW motion, various attempts have been made to prevent this breakdownprocess, such as applying a transverse field 8–11or consider- ing a perpendicular magnetic anisotropy,12–14and the spin-orbit coupling effect.15–19In this context, a way to manipulate the dynamics of the DWs by an electric field, which is critical for developing low-power-consumption spintronic devices, would be extremely appealing. Recently,it has been shown that modulating the magnetic anisotropy by an applied electric field is possible, 20,21and thus will allow for the electric field control of DW dynamics in ultra-thin metallic ferromagnets. 22–28Nevertheless, the search for alternative schemes allowing fast and energy-efficient DW propagation is of great relevance in advanced spintronicsresearch. Multiferroic materials, 29which exhibit simultaneously ferroelectric and magnetic orders, may provide a promisingarena to realize electric control of magnetization and even for DW motion. Multiferroics display a particularly rich variety of magnetoelectric (ME) cross-coupling effects. Anintriguing scenario of ME coupling is that spiral spin orders can by themselves produce electric polarization, which is called the spin-current mechanism, 30–33or equivalently the inhomogeneous magnetoelectric interaction .34Therefore, a nonzero electric polarization can be induced not only just in bulk but also within local magnetic textures, like magneticDWs and vortices. The possibility of such a magnetoelectric- ity in ferromagnetic N /C19eel walls has been anticipatedtheoretically 34,35and recently demonstrated experimen- tally.36,37Especially, this ME coupling also enables the electric field couple to the magnetization with its spatial gra-dients, which necessarily presents in metallic as well as insu- lating ferromagnets. 38The influence of this coupling on the spin waves of the multiferroics has recently been explored inRef. 39. However, the relevance of the electric field to the motion of a multiferroic DW, even though it is crucial to future ME multiferroic devices based on DW control, stillremains unclear. In this paper, we identify the dynamical nature of a prototypical type of multiferroic DW, that is, a magneticDW simultaneously displaying an electric polarization. This is a good basis for studying the dynamical properties of the multiferroic DWs in which the electric field can couple tothe magnetization via the inhomogeneous magnetoelectric interaction. We derive the equations of motion for electric field controlled DW dynamics in such a multiferroic DW.We report two main findings. The first one is that the mag- netic DW velocity can be considerably enhanced due to the delay of the occurrence of Walker breakdown by an appliedelectric field. This electric-field-modulated higher DW speed implies faster device operation, which is one of the main aim of the conventional DW device applications. The secondfinding is that the electric field can be used to control the switching of the DW chirality. This control of the chirality could provide an additional degree of freedom, which can beuseful in future magnetoelectric logic devices. The paper is organized as follows: In Sec. II, we present the model for a multiferroic DW. We obtain the equations ofmotion for the multiferroic DW dynamics using the collec- tive coordinate description. In Sec. III, we discuss how the electric field influences the DW velocity and the chiralityswitching. At the end, in Sec. IV, we present a brief sum- mary of the results obtained in this work. II. THEORETICAL MODEL The system under consideration is schematically depicted in Fig. 1. We will focus on a case of one-dimensional 0021-8979/2014/115(13)/133913/6/$30.00 VC2014 AIP Publishing LLC 115, 133913-1JOURNAL OF APPLIED PHYSICS 115, 133913 (2014) [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: 147.226.7.162 On: Thu, 20 Nov 2014 09:21:26insulating N /C19eel-type DW when magnetic easy (hard) axis is taken to be along the ^zð^yÞdirection with a voltage applied along the zdirection. The theoretical model we employ con- sists of three distinct contributions H¼H SþH ZþH E: (1) The first contribution HSdescribes the Hamiltonian of local spins in a magnetic DW, with an easy axis and a hard axis,chosen as ^zand ^ydirections, respectively. In the continuum limit, this Hamiltonian takes the form 40 HS¼ðd3x a3J 2rSðÞ2/C0K 2ðSzÞ2þK? 2ðSyÞ2/C20/C21 : (2) Here, Sis the local spin vector, ais the lattice constant, Jis the exchange coupling between local spins, while KandK? are anisotropy energies associated with the easy and hard axes of the spins, respectively. Here, we consider a homoge- neous system, i.e., a system without pinning potential. Interms of spherical coordinates ðh;/Þ, the energy functional in Eq. (2)has a stationary DW as a classical solution hðxÞ¼2 arctan ½e ðx/C0XÞ=kDW/C138;/ðxÞ¼/0; (3) which gives the x-dependent spin configuration S¼Sðsinhcos/0;sinhsin/0;coshÞ; (4) with Sis the magnitude of spin. In Eq. (3),Xis the position of the DW center, kDW¼ffiffiffiffiffiffiffiffi ffi J=Kp is the DW width. /0is the tilting angle between spins at the DW center and theeasy plane, which is spatially homogeneous. In particular, if / 0¼0o r p, the domain wall considered above is called pure N /C19eel wall (as shown in Fig. 1) with opposite chirality: clockwise (CW) or counterclockwise (CCW), respectively, while if /0¼6p=2 the wall becomes a pure Bloch wall. Such a description of the DW assumes its rigidity, that is, the DW can only move or rotate. The second ingredient of our model is the Zeeman energy, which is given by HZ¼/C0ð d3xM/C1B; (5) where M¼/C0g/C22h a3Sis the magnetic moment per unit volume, g(>0) is the gyromagnetic ratio, and Bis a uniform external magnetic field pointing along the negative zdirection, i.e., B¼/C0B^z;B>0.The third contribution HEmodels the coupling between the electric field and the spin degrees of freedom of the mul- tiferroic DW. According to the inhomogeneous magneto-electric interaction, 34we take the following form: HE¼/C0E/C1P; (6) with the electric polarization Pinduced within the DW given as34 P¼c0ðd3x a3½Sðr /C1SÞ/C0ðS/C1r ÞS/C138; (7) where c0is the magnetoelectric coupling coefficient. One can learn immediately from Eq. (7)that inhomogeneous magnetoelectric interaction induces electric polarization P within the N /C19eel wall so that it is actually multiferroic. To study the dynamics of a rigid planar DW, we employ a well-known collective coordinate description.40In this approach, the position Xand angle /0in Eq. (3)of the wall is regarded as time-dependent collective coordinates fXðtÞ;/0ðtÞg. The chirality of the DW is determined by the tilt angle /0ðtÞ. The domain wall is described by a Lagrangian of local spins given by L¼ðd3x a3/C22hS_/0ðcosh/C01Þ/C0H : (8) The first term represents the spin Berry phase. Inserting the DW ansatz (Eq. (3)) into Eq. (8)and integrating the space coordinate, the DW Lagrangian in terms of X(t) and /0ðtÞ can be written as L¼/C0/C22hNS kDWX_/þv?sin2/0/C0gBX/C0cEcos/0/C0/C1 :(9) Here, N¼2AkDW=a3is the number of spins in the wall region, with Abeing the cross-sectional area of the system, v?¼kDWK?S=2/C22h, and c¼pSc0=2/C22h. To derive the equations of motion of a DW, one further needs to introduce the dissi- pation function Wto incorporate the Gilbert damping, which is written as40 W¼ðd3x a3/C22ha 2S_S2¼a/C22hNS 2_X=kDW/C0/C12þ_/2 0hi ; (10) where ais the Gilbert damping parameter. We then utilize the generalized Euler-Lagrangian equation40 d dtdL d_q/C0dL dq¼/C0dW d_q; (11) where qrepresents X(t) and /0ðtÞand the last term describes the energy dissipated. The equations of motion for the col- lective coordinates, derived from the Eqs. (9)–(11), are given as follows: _X kDW/C0a_/0¼v? kDWsin 2/0þcE kDWsin/0; (12a) _/0þa_X kDW¼gB: (12b) FIG. 1. Schematic of the one-dimensional multiferroic DW structure con- sisting of a N /C19eel-type magnetic DW which induces an electric polarization (blue arrows) within the DW. A voltage is applied along the zdirection, while the external magnetic field Bis kept in /C0^zdirection. The white broad arrows denote the local spins in the wall.133913-2 Chen, Liu, and Li J. Appl. Phys. 115, 133913 (2014) [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: 147.226.7.162 On: Thu, 20 Nov 2014 09:21:26These equations provide a basic description of the multifer- roic DW dynamics under the magnetic and electric fields. Thus, the application of the electric field on the DW introdu-ces an additional spin torque proportional to sin / 0into the equations of motion. This new term will act as a chirality sta- bilizer, influencing significantly the DW dynamics. In whatfollows, we solve Eqs. (12) numerically and calculate the values for Xand/ 0after a sufficiently long time. The aver- age terminal velocity of the DW is defined as vDW¼h _Xi, where the angular brackets refer to a long-time average. To do the numerical simulation, we take a fixed value for the Gilbert damping parameter a¼0.02. The initial DW tilt angle is set to /0i¼/0ðt¼0Þ¼0 throughout, so the initial chirality of the DW is clockwise. III. RESULTS AND DISCUSSION A. Electric field mediated DW velocity Let us discuss, how the electric field affects the field- driven DW motion based on the equations of motion we derived previously. The numerical simulation results of Eqs. (12) are presented in Figs. 2–4. It is important to note that when the electric field is switched off, the equations of motion in Eqs. (12) are reduced to those of a DW purely driven by a magnetic field, whose behaviours are wellknown. 3,4,40In that case, the DW motion is characterized by the existence of two dynamic regimes, separated by a thresh- old field called Walker field.3That is, for an external field smaller than the Walker field BW¼aK?S=2g/C22h, the DW moves with a constant velocity which increases linearly with the external magnetic field up to BW. In this regime, the DW chirality which describes the sense of rotation of the spins in the wall is preserved during the motion. For fields B>BW, the Walker breakdown occurs and the DW undergoes oscil-latory motion, which makes the DW velocity decrease rap- idly. Such a behaviour was originally predicted by Schryer and Walker 3and was observed experimentally, for example, by Beach et al.5 We first show in Fig. 2the time-averaged precession velocity h_/0i, as a function of magnetic field Bfor various values of electric fields applied along þ^zdirection. We find thath_/0i¼0 up to a threshold applied field, even when theexternal electric field is switched on. This zero precession velocity means that the wall angle /0tilts out of the plane until it reaches a certain angle. From then on, it no longerchanges. In the regime, where h_/ 0i¼0 the wall moves at a constant velocity. As h_/0ibecomes finite, the wall tilt angle /0starts precessing, causing an oscillatory motion that slows down the domain wall. In Fig. 2, we clearly see that the zero h_/0iregime (stationary regime) is significantly extended by the application of an electric field. Figure 3(a) shows the time-averaged DW velocity vDW as a function of Bfor several applied electric fields. When the electric field is switched on, the vDW(B) curves show similar behavior to that of the conventional magnetic field driven model. For each applied E,vDWreaches a maximum velocity, namely, Walker velocity ( vW). The corresponding threshold magnetic field is Walker field ( BW), and above BW thevDWdrops abruptly. More specifically, the Walker field BWincreases with Eand there is no change of DW mobility. It seems that the presence of an electric field surely acts as a chirality stabilizer and plays a pivotal role to delay the onset of the Walker breakdown and allows for higher attainableDW velocities. Figures 3(b) and3(c)summarize the increase of both the Walker field B Wand the Walker velocity vWwith the magnitude of E. It is clearly shown that both BWandvW exhibit a nearly linear behavior, so we have an scenario where the maximum velocity of the wall is substantially enhanced by the application of an electric field. To elucidate the role of the electric field in the suppres- sion of Walker breakdown, we further examine analytically the DW dynamics for Bsmaller than BW, since in such a sta- tionary regime @/0=@t¼0a s t!1 . From Eqs. (12),w e obtain B aB0¼sin 2/0þDsin/0; (13)FIG. 2. The behavior of the time-averaged precession velocity h_/0ias a function of magnetic field Bat several applied electric field E.h_/0iis given in units of K?S=2/C22h, the electric field Eis given in units of E0¼v?=c, and units B0¼K?S=2g/C22h. The regime with h_/0i¼0 represents the stationary re- gime of the DW motion.FIG. 3. (a) The velocity of the DW vDWas a function of Bunder the applica- tion of various Eapplied along the þ^zdirection. Eis given in units of E0¼v?=c. (b) and (c) are the dependence of the Walker field BWand the Walker velocity vWwith the electric field, respectively. Both show a quasi-linear increase with E.133913-3 Chen, Liu, and Li J. Appl. Phys. 115, 133913 (2014) [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: 147.226.7.162 On: Thu, 20 Nov 2014 09:21:26where B0¼K?S=2g/C22handD¼E=E0, with E0¼v?=c.A s long as this equation is satisfied, the DW will propagatewithout oscillatory motion. The Walker field B Wis deter- mined from the maximum of the r.h.s. of Eq. (13). Equation (13) shows that BWdepends not only on the sign of D(orE) but also on the initial tilt angle /0ið¼/0ðt¼0ÞÞ(either 0 or p). The Walker field BWis found to be associated with the tilt angle /0W¼arccos ½ð/C0Dþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2þ32p Þ=8/C138. Then, we can obtain the Walker field as BWð/0i¼0;E>0Þ¼BW1with BW1¼ð16/C0D2þjDj~DÞ1=2ð3jDjþ~DÞaB0 16ffiffi ffi 2p ; (14) where ~D¼ðD2þ32Þ1=2. For a large Eone has simply BW/C24aE, and further, the Walker velocity vW/C24E. The ana- lytical results of BWandvWversus Eare shown as red solid lines in Figs. 3(b) and3(c), respectively, which are both con- sistent with the numerical simulations. B. Electric field induced DW chirality switching In this section, we investigate the effect of the applied electric field on the chirality of DW. We show that a reliablecontrol of the chirality switching of a moving DW will be achieved by the application of an electric field along thedirection of the magnetic field. Each DW has two possible chiralities: CW and CCW. This DW chirality can be used asan information unit. 2Therefore, a controllable switching of the DW chirality is desirable. We now change the direction of the applied electric field to align the /C0^zdirection, i.e., parallel to the magnetic field. The numerical results are shown in Fig. 4. Figure 4(a) illus- trates the vDW(B) curves for various E. We can see that a negative Ealso results in a suppression of the Walker break- down and an increase of the DW velocity, similar to the action of a positive Ediscussed in Sec. III. On the other hand, the chirality of the DW is determined by the tilt angle /0ðtÞ. It should be noted that /0ðtÞincreases from the initial tilt angle /0ibut eventually becomes saturated to a constant value in the limit t!1 below the Walker breakdown. Hence, controlling the terminal tilt angle /0can be used to switch the chirality of the DW. For a moving DW drivenpurely by a magnetic field, its initial chirality is preserved below the Walker field, 40and the chirality switching is diffi- cult to achieve in a controllable way. However, it can beshown that this picture will not hold for multiferroic DW when Eis included. Figure 4(b) shows the terminal DW tilt angle / DWð¼/0ðt!1 Þ Þ as a function of the external mag- netic field Bin the presence of electric field E. We can see that below the Walker field, the DW tilt angle /DWinitiallyFIG. 4. (a) Bdependence of the DW velocity vDWfor several choices of Eapplied along the /C0^zaxis. The color dots mark the critical fields Bcindicating the chirality switching for each case. (b) The terminal DW tilt angle /DWas a function of B. The sudden jumps of /DWdenote the chirality switching occurring, from CW to CCW. (c) Component of the spin along the x-axis (spin modulation direction) at the DW center. Below Bc,Sx>0 (red solid line) means CW chir- ality; and above Bc,Sx<0 (blue dashed line) means CCW chirality. The applied E¼/C00.5E0. (d) The dependence of Bcwith the field strength jEj.133913-4 Chen, Liu, and Li J. Appl. Phys. 115, 133913 (2014) [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: 147.226.7.162 On: Thu, 20 Nov 2014 09:21:26increases with Bfrom zero, and then suddenly jumps to a value larger than pat a critical field Bcfor each E. Therefore, the switching of the DW chirality from the ini-tially clockwise to terminally counterclockwise has clearly occurred at these critical magnetic fields. Fig. 4(c) further shows the x-component of the spin at the DW center for the applied fields Bnear the critical field B c. The applied electric field is chosen as E¼/C00.5E0. The signs of the spin compo- nent along the x-axis indicate that below Bc(i.e., B¼0.0132 B0), the DW chirality is CW, when Breaches up toBc, the DW chirality switches to CCW. The positions of Bcextracted from the /DWðBÞcurves are shown as blue spheres in Fig. 4(d). It is shown that Bcdecreases with jEj down to zero at jEj¼2E0. In order to better understand this remarkable chirality switching process, we take the following analytical analyses. For a negative E, we can obtain from Eq. (13) the Walker field BWð/0i¼0;E<0Þ¼BW2, with BW2¼ð16/C0D2/C0jDj~DÞ1=2ð/C03jDjþ~DÞaB0 16ffiffiffi 2p : (15) For smaller negative E, we have BWð/0i¼0;E<0Þ/C24/C0 aE. Interestingly, if setting the initial tilt angle /0i¼p(i.e., the initial chirality is CCW), we can also obtain the Walker fieldasB Wð/0i¼p;E<0Þ¼BW1. In this case, the terminal tilt angle /DWis larger than p. Moreover, it is easy to see that for E<0, BWð/0i¼0;E<0Þ is smaller than BWð/0i¼p;E<0Þ. Such a difference between the two Walker fields BWð/0i¼0;E<0Þand BWð/0i¼p;E<0Þ enables the chirality of a moving DW to be switched. That is,when Bincreases beyond the first Walker field B Wð/0i¼0;E<0Þ, the Walker breakdown process does not occur until Bfurther reaches the higher Walker field BWð/0i¼p;E<0Þ. In the meantime, the initial tilt angle /0iswitches from 0 to pat the first threshold field BWð/0i¼0;E<0Þand thus the DW chirality can be switched. The Edependence of BWð/0i¼0;E<0Þis shown in Fig. 4(d) as red solid line, which is in excellent agreement with the numerical results Bcextracted from the /DWðBÞ curves. However, for E>0;BWð/0i¼0;E>0Þ> BWð/0i¼p;E>0Þ, we have only one threshold field BWð/0i¼0;E>0Þand hence there is no DW chirality switching process. The switching of the DW chirality with the application of electric field can be read by a magnetic field sensor since the stray field near a DW depends on its chirality.It should be noted that the controlled chirality switching of a moving DW can also be achieved by applying an oblique magnetic field as proposed by Seo et al. 41Nevertheless, we here offer a more efficient alternative to flip the chirality of the DW with the help of an external electric field. IV. SUMMARY We considered multiferroic DW systems that exhibit both a coexistence and a coupling of electric polarization and a magnetic DW. The effects of electric field on the DWdynamics via the inhomogeneous magnetoelectric interaction have been investigated. We have revealed the dynamicalnature of a multiferroic DW and demonstrated the efficiency of an electric field control of magnetic DW motion. In partic- ular, we showed that the electric field can achieve not only anearly linear enhancement of the maximum wall velocity but also a controllable switching of DW chirality. This control of the motion of the multiferroic DWs via electric fields can beuseful for designing low-power and high-speed DW-based magnetoelectric memory and logic devices. ACKNOWLEDGMENTS This work was supported by NSF-China (Grant Nos. 11074216 and 11274272), the Fundamental Research Funds for the Central Universities in China, and China PostdoctoralScience Foundation. 1G. Catalan, J. Seidel, R. Ramesh, and J. F. Scott, Rev. Mod. Phys. 84, 119 (2012). 2D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P.Cowburn, Science 309, 1688 (2005); S. S. P. Parkin, M. Hayashi, and L. Thomas, ibid.320, 190 (2008). 3N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974). 4X. R. Wang, P. Yan, and J. Lu, Europhys. Lett. 86, 67001 (2009); B. Hu and X. R. Wang, Phys. Rev. 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Phys. 115, 133913 (2014) [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: 147.226.7.162 On: Thu, 20 Nov 2014 09:21:26

1.4914111.pdf

Spin-orbit torque induced spike-timing dependent plasticity Abhronil Sengupta, Zubair Al Azim, Xuanyao Fong, and Kaushik Roy Citation: Applied Physics Letters 106, 093704 (2015); doi: 10.1063/1.4914111 View online: http://dx.doi.org/10.1063/1.4914111 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/106/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Large voltage-induced modification of spin-orbit torques in Pt/Co/GdOx Appl. Phys. Lett. 105, 222401 (2014); 10.1063/1.4903041 Current induced domain wall dynamics in the presence of spin orbit torques J. Appl. Phys. 115, 17D502 (2014); 10.1063/1.4860946 Spin-orbit torque opposing the Oersted torque in ultrathin Co/Pt bilayers Appl. Phys. Lett. 104, 062401 (2014); 10.1063/1.4864399 Tailoring spin-orbit torque in diluted magnetic semiconductors Appl. Phys. Lett. 102, 192411 (2013); 10.1063/1.4806981 Spin torque switching in triple-quantum dot device with ferromagnetic contacts for memory applications J. Appl. Phys. 109, 07C702 (2011); 10.1063/1.3535545 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: 132.203.227.61 On: Mon, 13 Jul 2015 08:05:24Spin-orbit torque induced spike-timing dependent plasticity Abhronil Sengupta,a)Zubair Al Azim, Xuanyao Fong, and Kaushik Roy School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA (Received 23 December 2014; accepted 24 February 2015; published online 4 March 2015) Nanoelectronic devices that mimic the functionality of synapses are a crucial requirement for performing cortical simulations of the brain. In this work, we propose a ferromagnet-heavy metalheterostructure that employs spin-orbit torque to implement spike-timing dependent plasticity. The proposed device offers the advantage of decoupled spike transmission and programming current paths, thereby leading to reliable operation during online learning. Possible arrangement of suchdevices in a crosspoint architecture can pave the way for ultra-dense neural networks. Simulation studies indicate that the device has the potential of achieving pico-Joule level energy consumption (maximum 2 pJ per synaptic event) which is comparable to the energy consumption for synapticevents in biological synapses. VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4914111 ] Large scale cortical brain simulations on present day supercomputers, based on Von-Neumann model of computa- tion, have proved highly inefficient with respect to the ultra-high density and energy efficient processing capability of the human brain. For instance, the IBM Blue Gene supercom- puter consumed 1.4 MW of power to simulate 5 s of brain ac-tivity of a cat. 1On the contrary, the human brain consumes power of the order of a few Watts. In order to harness the re- markable efficacy of the human brain in cognition and per-ception related tasks, the field of neuromorphic computing attempts to develop non Von-Neumann computing models inspired by the functionality of the basic building blocks,i.e., neurons and synapses in the biological brain. The computational fabric of the brain consists of a highly interconnected structure where neurons are connectedby junctions termed as synapses. Each synapse is character- ized by a conductance and helps to transmit weighted signals in the form of spikes from the pre-neuron to the post-neuron.It is now widely accepted that synapses are the main compu- tational element involved in learning and cognition. The theory of Hebbian Learning 2postulates that the strength of synapses is modulated in accordance to the temporal rela- tionship of the spiking patterns of pre-neurons and post- neurons. In particular, Spike-Timing Dependent Plasticity(STDP) has emerged as one of the most popular approaches of Hebbian Learning. 3According to STDP, if the pre-neuron spikes before the post-neuron, the conductance of the syn-apse potentiates (increases), while it depresses (decreases) if the pre-neuron spikes after the post-neuron. The relative change in synaptic strength decreases exponentially with thetiming difference between the pre-neuron and post-neuron spikes. The timing window during which such plastic synap- tic learning occurs has been observed to be of the order/C24100 ms. The number of synapses also outnumbers the number of neurons in the mammalian cortex by a large extent. It is cru-cial to accommodate as many synapses as possible per neu- ron for efficient implementation of a neuromorphic system capable of online learning. 6Although there have beenseveral attempts to emulate synaptic functionality by CMOS transistors,4,5the area overhead and power consumption involved are quite large due to the significant mismatchbetween the CMOS transistors and the underlying neuro- science mechanisms. As a result, nanoscale devices that emulate the functionality of such programmable, plastic,Hebbian synapses have become a crucial requirement forsuch neuromorphic computing platforms. To that end, researchers have proposed several programmable devices based on phase change materials, 6,7Ag–Si memristors,8and chalcogenide memristors9that mimic the synaptic function- ality. Neuromorphic computing architectures employing such memristive devices have been also demonstrated.10–12 However, nanoscale devices attaining the ultra-high density (1011synapses per cm/C02) and low energy consumption (/C241 pJ per synaptic event) of biological synapses have still remained elusive. In order to address some of the challengesinvolved in the search for an ideal “electronic” synapse, wepropose a device structure based on a ferromagnet with oppositely polarized magnetic domains separated by a transi- tion region called domain wall . 13,14We will refer to such a ferromagnetic material as a domain-wall magnet (DWM) for the rest of this text. Recently, spin-orbit torque generated by a heavy metal (HM) has emerged as one of the promising mechanisms to manipulate the magnetization of a ferromagnet lying on top due to high spin injection efficiency.15,16In particular,17 demonstrates deterministic control of domain wall motion in a ferromagnet by orthogonal current flow in an underlying HM layer in presence of an external magnetic field. In thiswork, we exploit this functionality to propose a device struc-ture with decoupled spike transmission and learning current paths which is a crucial requirement for STDP since the learning event can take place at any time during the opera-tion of the network. Spin-orbit torque generated in our pro-posed magnetic heterostructure serves as the basic physical phenomena responsible for generating STDP. The proposed four terminal synaptic device structure is shown in Fig. 1(a). It consists of a magnetic heterostructure where a magnetic material with Perpendicular MagneticAnisotropy (PMA) is in contact with a non-magnetic HM a)Electronic mail: asengup@purdue.edu 0003-6951/2015/106(9)/093704/5/$30.00 VC2015 AIP Publishing LLC 106, 093704-1APPLIED PHYSICS LETTERS 106, 093704 (2015) 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: 132.203.227.61 On: Mon, 13 Jul 2015 08:05:24with high spin-orbit coupling. The magnetic material is a DWM where the domain wall is longitudinal, running paral- lel to the length of the DWM. The DWM is also part of a Magnetic Tunneling Junction (MTJ) structure where a tun-neling oxide barrier (MgO) separates the DWM from the Pinned Layer (PL). Fig. 1(b) depicts the side-view of the de- vice structure. The spike current from the pre-neuron passesbetween terminals A and B through the MTJ structure. The learning current required to program the synapses flows through the HM between terminals C and D to implementSTDP learning. An in-plane magnetic field His also applied during the learning stage. The spike transmission and learn- ing operations are described next.The resistance model of the device is shown in Fig. 2(a). Considering the total width of the MTJ to be W MTJand the width of the ferromagnetic domain whose magnetization is parallel to the PL to be w, the equivalent conductance of the device can be expressed as Gdev¼GAP ;max 1/C0w WMTJ/C18/C19 þGP;maxw WMTJ/C18/C19 þGDW :(1) Here, GAP,max (GP,max) represents the conductance of the de- vice when the entire DWM magnetization is oriented anti- parallel (parallel) to the PL and GDWrepresents the conduct- ance of the domain wall. Hence, the device conductancevaries linearly with the domain wall position as demon- strated in Fig. 2(b). Non-equilibrium Green’s function (NEGF) based electron transport simulation framework 18 was modified to simulate the variation of the device conduct- ance with domain wall position. The resistance of the DWM- HM heterostructure that lies in the path of the spike current between terminals A and B is negligible in comparison to the resistance of the tunneling oxide barrier. Hence, when a volt-age spike from the pre-neuron is applied between terminals A and B, the device conductance will determine the strength of the spike current transmitted which can be modulated byprogramming the domain wall position. In order to implement STDP in the device, a current is passed between terminals C and D. When a programming current flows from terminal C to terminal D through the HM in the /C0x direction, spin-Hall effect 22leads to the accumula- tion of þy directed spin-polarized electrons at the HM- DWM interface which generates spin-orbit torque on the DWM. Negligible Dzyaloshinskii-Moriiya Interaction (DMI) and shape anisotropy due to the formation of the longitudinal domain wall leads to the formation of a Bloch wall in thesample. 17The external in-plane magnetic field orients the magnetic moment of the domain wall along 6x direction. Thus, the final magnetization state of the ferromagnet isdetermined by the cross-product of the accumulated spins at the HM-DWM interface and the direction of the applied magnetic field. For a magnetic field applied along the þx direction, application of current through the HM in the /C0x direction results in domain wall motion in the /C0y direction so that þz magnetic domain in the ferromagnet starts to expand. It is worth noting here that conventional bulk spin- transfer torque does not contribute to the domain wall movement. The magnetization dynamics of the ferromagnet can be described by solving Landau-Lifshitz-Gilbert equation with additional term to account for the spin momentum torque generated by the accumulated spin current at the HM-DWMinterface 19,20 d^m dt¼/C0c^m/C2Hef f/C0/C1þa^m/C2d^m dt/C18/C19 þb^m/C2^mP/C2^m ðÞ ; (2) where ^mis the unit vector of DWM magnetization at each grid point, c¼2lBl0 /C22his the gyromagnetic ratio for electron, a is Gilbert’s damping ratio, Heffis the effective magnetic field, b¼/C22hPhJ 2l0etM s(/C22his Planck’s constant, Pis polarization of FIG. 1. (a) Proposed synaptic device structure. (b) Side-view of the device with spike transmission and learning current paths. FIG. 2. (a) Resistance model of the proposed synaptic device structure. (b)Variation of device conductance as a function of the domain wall position.093704-2 Sengupta et al. Appl. Phys. Lett. 106, 093704 (2015) 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: 132.203.227.61 On: Mon, 13 Jul 2015 08:05:24the PL, Jis input charge current density, his spin-orbit tor- que efficiency,17l0is permeability of vacuum, eis elec- tronic charge, tis FL thickness, and Msis saturation magnetization), and ^mPis direction of input spin current. Micromagnetic simulations were performed in MuMax3.21 The simulation parameters are given in Table Iand were used for the rest of this work, unless otherwise stated. The simulation framework was calibrated with experi- mental results reported for Ta (HM)–CoFeB (DWM) hetero- structure in Ref. 17. Fig. 3(a) depicts the position of the domain wall in the sample with CoFeB dimensions of600/C2200/C21n m 3as a function of time, due to the applica- tion of a current density of J¼3.5/C2106A/cm2. The domain wall positions predicted by our simulation framework are in good agreement with the experimental results (refer to Fig. 4in Ref. 17). For a given duration of the programming current, the do- main wall displacement increases linearly with the magni- tude of the current density. Fig. 3(b) illustrates the linear increase of the domain wall displacement with programmingcurrent amplitude for different time durations. Since the de- vice conductance is also a linear function of the domain wall position, the programming current follows a linear relation-ship with conductance change in the device. Reversing thedirection of programming current or the direction of the magnetic field causes the domain wall to move in oppositedirection. This enables us to implement STDP in the device as discussed in the following paragraph. Fig. 4(a) shows the proposed synaptic device with access transistors to decouple the spike transmission and learning current paths. A possible arrangement of the synap- ses in an array connecting the pre-neurons and post-neuronsis depicted in Fig. 4(b). The spike transmission signal from the pre-neuron is V SPIKE while the programming signals VPREandVPOST are used to implement STDP. When the pre- neuron spikes, the spike is transmitted using the signal VSPIKE through the MTJ structure. As long as the post- neuron does not spike, the spike transmission current pathremains activated. Assuming that the resistance offered by the access transistors in the spike transmission current path is small in comparison to the resistance of the device, thespike voltage will be modulated by the device conductance and the weighted spike current will be transmitted to the post-neuron summing amplifier. As soon as the pre-neuronspikes, it also applies an appropriate programming voltage V PREwhich extends over the time window to be used for learning. When the post-neuron spikes, the VPOST signal gets activated. The VPOST signal is a short pulse of a few ns dura- tion that essentially samples the appropriate amount of pro- gramming current from the VPRE signal. The spike transmission path gets de-activated and the appropriate pro- gramming current corresponding to the time delay between the pre-neuron and post-neuron spikes passes through theTABLE I. Simulation parameters. Parameters Value Ferromagnet dimensions 200 /C2100/C21n m3 Grid size 2 /C22/C21n m3 Heavy metal dimensions 200 /C21000/C210 nm3 Domain wall width 22 nm MTJ (PL) dimensions 120 /C2100/C21n m3 Saturation magnetization, Ms 800 KA/m Spin orbit torque efficiency, h 0.08 Gilbert damping factor, a 0.024 MgO thickness 1.2 nmExchange correlation constant 3 /C210 /C011J/m Perpendicular magnetic anisotropy 6 /C2105J/m/C03 Magnetic field, H 10 G FIG. 3. (a) Position of the domain wall in Ta-CoFeB heterostructure with CoFeB dimensions of 600 /C2200/C21n m3as a function of time, due to the application of a current density of J¼3.5/C2106A/cm2. (b) Variation of do- main wall displacement with programming current through HM for different programming time durations. FIG. 4. (a) Synaptic device with access transistors for separate spike trans- mission and learning current paths. (b) Possible arrangement of synapses in an array.093704-3 Sengupta et al. Appl. Phys. Lett. 106, 093704 (2015) 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: 132.203.227.61 On: Mon, 13 Jul 2015 08:05:24HM to move the domain wall to the appropriate location. To account for learning during the negative timing window, the post-neuron programming signal VPOST can be applied with a delay corresponding to the duration of the negative time win- dow. For this work, the VPOST signal was taken to be of dura- tion 10 ns. Once the VPOST signal is activated, programming current flows through the device in case the VPREsignal remains active. Assuming that the magnetic field requiredfor learning is generated by a local current carrying wire, the current through the wire can be turned on only when the post-neuron spikes. Hence, spin-orbit torque is the underly- ing physical phenomena involved in the learning process as conventional bulk spin-transfer torque will not have any effect on the longitudinal domain wall. It is worth noting here that some amount of spike current will flow through the PL of the MTJ and the HM. The spin current injected into the ferromagnet due to spike current flowing through the HMis much larger than the spin current injected due to the PL as the spin injection efficiency of spin-Hall effect is much greater than the polarization of the PL. However, the spike generation and post-neuron summing amplifier circuits can be appropriately designed such that the magnitude of the spike current is kept lower than the threshold current density required for domain wall movement to avoid any change in synaptic conductance. Fig.5(a)depicts the STDP characteristics (relative con- ductance change as a function of spike timing difference) implemented in our device which are in accordance to the characteristics measured in rat hippocampal glutamatergic synapses by Bi and Poo. 3Simulation of the programming circuit with access transistors was done using a commercial 45 nm transistor model. The pre-neuron signal VPRErequired to achieve the desired STDP characteristics is shown in Fig. 5(b). For a time duration of 10 ns, the amount of program- ming current required to switch the DWM from the com- pletely parallel to the anti-parallel state or vice-versa was found to be /C24200lA. Assuming that this current flows from a 1 V supply, the corresponding energy consumption is /C242 pJ( V/C2I/C2t). The desired programming current was achieved by appropriately sizing the access transistors for learning. The magnetic field along þx direction can be produced by a current flowing along þy direction through a wire located at a height hfrom the device. The magnitude of the magnetic field Bproduced by a current Ifieldis given by Biot- Savart’s Law asB¼l0If ield 2ph: (3) For instance, for a magnetic field B¼10 G, and height h¼100 nm, the current Ifieldrequired is /C24500lA. This field current can be utilized to provide the necessary magnetic field for all the synapses in a particular row of the array. The number of synapses that can be driven by the field currentwill be limited by the resistance of the wire. Hence, the aver-age energy consumption per synapse for magnetic field gen-eration will be given by 5pJ N, where Nis the number of synapses in a particular row of the array. Additionally it hasbeen shown in Ref. 23that a ferromagnet cladding region with high permeability can be used to enhance and concen-trate the magnetic field, thereby causing an increase of mag-netic field strength by almost /C2413/C2(Ref. 23) for a given magnitude of field current. Such narrow gap cladding (NGC) field enhancement techniques 23not only helps to reduce the power consumption of the field current but also helps toprovide immunity against any noise that may arise from straymagnetic fields of neighboring magnets. Hence, the energyconsumption due to magnetic field generation can belimited to sufficiently low values in comparison to theprogramming energy consumption of the synapse by appro-priate design. The major contributions of this work over state-of-the- art approaches can be summarized as follows. We proposeda device structure for STDP that enables decoupled spiketransmission and programming current paths. While thelearning current flows mainly through the HM and is respon-sible for generating STDP, the spike current gets modulatedby the MTJ conductance, i.e., the strength of the synapse.The synaptic programming scheme is also simple and intui- tive and has a direct correspondence to the learning scheme to be implemented. In most of the proposed programmableresistive synapses, 6–9significant amount of programming voltage is applied across the device during the entire timeduration of the learning window for the pre-/post-neuronresulting in large amount of redundant power consumption.Additionally they are characterized by relatively high pro-gramming threshold voltages (approximately a few volts).Our proposed programming scheme leads to current flowthrough the synapse only for a small time duration (a few ns)in case the post-neuron spikes before/after the pre-neuron during the learning time window. Programming currents needed to modulate the device conductance is also low dueto high spin injection efficiency of spin-Hall effect. Ultra-low energy consumption of the order of 2 pJ per synapticevent and the possibility of arranging such DWM-HM heter-ostructures in a crossbar fashion demonstrate the potential ofthis device as a possible candidate for an “electronic” syn-apse in brain-inspired computing platforms. The work was supported in part by the Center for Spintronic Materials, Interfaces, and Novel Architectures (C-SPIN), one of six centers of STARnet, a SemiconductorResearch Corporation program, sponsored by MARCO andDARPA, by the Semiconductor Research Corporation, theNational Science Foundation, and by the National SecurityScience and Engineering Faculty Fellowship. FIG. 5. (a) Relative conductance change of proposed device as a function of spike timing, (b) VPREprogramming signal as a function of time for pre- neuron spiking event at t¼0.093704-4 Sengupta et al. Appl. Phys. Lett. 106, 093704 (2015) 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: 132.203.227.61 On: Mon, 13 Jul 2015 08:05:241Sally Adee, IBM Unveils a New Brain Simulator, IEEE Spectrum, 2009. 2D. Hebb, The Organization of Behavior (Wiley, New York, 1949). 3G. Q. Bi and M. M. Poo, J. Neurosci. 18, 10464 (1998). 4P. Merolla, J. Arthur, F. Akopyan, N. Imam, R. Manohar, and D. S. Modha, in Proceedings of the IEEE Custom Integrated Circuits Conference (IEEE, 2011), p. 1. 5J. S. Seo, B. Brezzo, Y. Liu, B. D. Parker, S. K. Esser, R. K. Montoye, B. Rajendran, J. A. Tierno, L. Chang, D. S. Modha, and D. J. Friedman, in Proceedings of the IEEE Custom Integrated Circuits Conference (IEEE, 2011), p. 1. 6B. L. Jackson, B. Rajendran, G. S. Corrado, M. Breitwisch, G. W. Burr, R.Cheek, K. Gopalakrishnan, S. Raoux, C. T. Rettner, A. Padilla, A. G. Schrott, R. S. Shenoy, B. N. Kurdi, C. H. Lam, and D. S. Modha, ACM J. Emerg. Technol. Comput. Syst. 9, 12 (2013). 7D. Kuzum, R. G. D. Jeyasingh, B. Lee, and H.-S. P. Wong, Nano Lett. 12, 2179 (2012). 8S. H. Jo, T. Chang, I. Ebong, B. B. Bhadviya, P. Mazumder, and W. Lu,Nano Lett. 10, 1297 (2010). 9Y. Li, Y. Zhong, L. Xu, J. Zhang, X. Xu, H. Sun, and X. Miao, Sci. Rep. 3, 1619 (2013). 10G. Indiveri, B. Linares-Barranco, R. Legenstein, G. Deligeorgis, and T. Prodromakis, Nanotechnology 24, 384010 (2013). 11T. Serrano-Gotarredona, T. Prodromakis, and B. Linares-Barranco, IEEE Circuits Syst. Mag. 13, 74 (2013).12B. Rajendran, Y. Liu, J.-s. Seo, K. Gopalakrishnan, L. Chang, D. J. Friedman, and M. B. Ritter, IEEE Trans. Electron Devices 60, 246 (2013). 13X. Wang, Y. Chen, H. Xi, H. Li, and D. Dimitrov, IEEE Electron Devices Lett. 30, 294 (2009). 14X. Wang, Y. Chen, Y. Gu, and H. Li, IEEE Electron Devices Lett. 31,2 0 (2010). 15J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999). 16L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). 17D. Bhowmik, M. E. Nowakowski, L. You, O. Lee, D. Keating, M. Wong, J. Bokor, and S. Salahuddin, “Deterministic Domain Wall Motion Orthogonal To Current Flow Due To Spin Orbit Torque,” e-print arXiv:1407.6137 . 18X. Fong, S. K. Gupta, N. N. Mojumder, S. H. Choday, C. Augustine, and K. Roy, in Proceedings of the International Conference on Simulation of Semiconductor Processes and Development (SISPAD ) (IEEE, 2011), p. 51. 19J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989). 20E. Martinez, S. Emori, and G. S. D. Beach, Appl. Phys. Lett. 103, 072406 (2013). 21A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez,and B. V. Waeyenberge, AIP Adv. 4, 107133 (2014). 22S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D. Beach, Nat. Mater. 12, 611 (2013). 23C. Augustine, X. Fong, B. Behin-Aein, and K. Roy, IEEE Trans. Nanotechnol. 10, 778 (2011).093704-5 Sengupta et al. Appl. Phys. Lett. 106, 093704 (2015) This article is copyrighted as indicated in the article. 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1.1689171.pdf

Spin transfer excited regular and chaotic spin waves in current perpendicular to plane spin valves Xiaochun Zhu, Jian-Gang Zhu, and Robert M. White Citation: Journal of Applied Physics 95, 6630 (2004); doi: 10.1063/1.1689171 View online: http://dx.doi.org/10.1063/1.1689171 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/95/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The effect of film and interface structure on the transport properties of Heusler based current-perpendicular-to- plane spin valves Appl. Phys. Lett. 98, 242508 (2011); 10.1063/1.3600792 Current driven magnetization reversal in microstructured spin valve with current-in-plane configuration J. Appl. Phys. 105, 07D118 (2009); 10.1063/1.3068483 Spin transfer switching of closely arranged multiple pillars with current-perpendicular-to-plane spin valves J. Appl. Phys. 103, 07A713 (2008); 10.1063/1.2838473 Nanocontact spin-transfer oscillators based on perpendicular anisotropy in the free layer Appl. Phys. Lett. 91, 162506 (2007); 10.1063/1.2797967 Spin momentum transfer in current perpendicular to the plane spin valves Appl. Phys. Lett. 84, 3103 (2004); 10.1063/1.1707227 [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: 130.209.6.50 On: Mon, 22 Dec 2014 01:10:32Spin transfer excited regular and chaotic spin waves in current perpendicular to plane spin valves Xiaochun Zhu,a)Jian-Gang Zhu, and Robert M. White Department of Electrical and Computer Engineering, Data Storage Systems Center, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890 ~Presented on 6 January 2004 ! Spin polarized current induced spin wave excitations have been studied with utilizing the spin transfer modified Gilbert equation in micromagnetic modeling. An elliptical shaped spin valve atdeep submicron dimension is modeled. It is found that stable spin waves with extremely narrowlinewidth are excited in the free layer when the perpendicularly injected direct current is slightlyabove a threshold. The spin transfer effect yields self leveling of the generated spin waves andresults in spin waves with stable amplitude for each excited mode.At high current level, the excitedspin waves become chaotic in nature, causing a pronounced 1/ f-like spectral content in the magnetoresistive output. © 2004 American Institute of Physics. @DOI: 10.1063/1.1689171 # INTRODUCTION It has been predicted theoretically and observed experi- mentally that spin transfer effect in current perpendicular toplane ~CPP!configuration can produce magnetization switching at low fields and excite spin waves at relativelyhigh fields. 1–4It is important to understand both magnetiza- tion switching and spin wave generation which both couldundermine the application of CPP giant magnetoresistivesensors at useful current densities. In this article, we present a micromagnetic modeling in- vestigation of spin current induced stochastic spin waves,hence noise, in a single free layer CPP device at deep sub-micron dimension. MODEL A single layer Ni 80Fe19thin film element patterned into an ellipse is modeled as shown in Fig. 1. The ellipticalshaped element has a long-axis of 100 nm and a short axis of50 nm. The thickness of the layer is 30 Å. The element isdescritized into a two-dimensional array of mesh cells andthe lateral dimension of each mesh cell is 2 nm 32n m square. An external field His applied along the long-axis of the ellipse element, parallel to the initial magnetization di-rection. A dc perpendicularly injected current with a spinpolarization factor P 050.25 was used in the calculation. The perpendicular current density was assumed to be spatiallyuniform and Amperean field arising from the current flowwas also included. The spin transfer torque modified Gilbertequation is used: 2,3 dM dt52gM3H1gp0J\ eMdM3Mˆ03Mˆ1a MM3dM dt,~1! where the second term on the right hand side of Eq. ~1! represents the spin transfer torque with P0andMˆ0denoting the polarization factor and the polarization direction of thespin current, respectively, dthe layer thickness, Jthe current density, and Hthe effective magnetic field. The Gilbertdamping constant a50.02 was used in the calculations. The coupled equations were transformed into the modified Gil-bert equation into the explicit Landau–Lifshitz form andthen were solved using the Adams method. The time stepsize is varied during the integration with strict numericalerror tolerance and the maximum time step size is below 1ps. RESULTS AND DISCUSSIONS Figure 2 shows the long-axis magnetization component, averaged over the entire element, at three different currentmagnitudes. Below 0.65 mA, the magnetization in the ellipseelement is absolutely static.At 0.65 mA, a sustained magne-tization precession occurs in the element.The long-axis mag-netization component shows characteristic two-frequencyquasiperiodic oscillations near a frequency that is twice theferromagnetic resonance frequency ~around 10 GHz in this case!. The oscillation of the magnetization component arises from the spin current induced quasispatially uniform magne-tization precession around the external field direction, i.e.,the long axis of the element.At 0.8 mA, the amplitude of theaveraged magnetization precession becomes rather random.At 1.5 mA, not only the amplitude of the oscillation becomesrandom, the period of the oscillation also varies with time. Figure 3 shows the corresponding power spectral densi- ties for the above three cases.At 0.65 mAcurrent, two spec-tral peaks are present. The higher frequency peak, which hasrelatively higher amplitude, corresponds to the magnetostatic a!Author to whom correspondence should be addressed; electronic mail: xiaochuz@andrew.cmu.edu FIG. 1. Schematic drawing of modeled free layer. The external field isapplied in the element length direction.JOURNAL OF APPLIED PHYSICS VOLUME 95, NUMBER 11 1 JUNE 2004 6630 0021-8979/2004/95(11)/6630/3/$22.00 © 2004 American Institute of Physics [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: 130.209.6.50 On: Mon, 22 Dec 2014 01:10:32mode of the excited spin waves with the half wavelength equal to the length of the ellipse while the lower frequencypeak corresponds to the ferromagnetic resonance mode. Notethat the linewidths of both peaks are extremely narrow, indi-cating the net damping of the system is virtually zero.At 0.8mA, the resonance peak shifts to lower frequencies and the peak broadens. At 1.5 mA, the down shift of the resonancefrequency and broadening and lowering of the resonancepeak have continued and, more significantly, substantial1/f-like content has appeared in the spectrum. Figure 4 plots the variance, i.e., rms power, of the long-axis magnetizationcomponent as a function of the current. The variance showsan evident criticality with a critical current at 0.65 mA. In-creasing the current magnitude, the excited uniform magne-tization precession quickly becomes stochastic with its am-plitude increases rapidly. At high current magnitude, thetime-domain averaged magnetization component along thelong axis reverses its direction towards the spin current po-larized direction, the variance starts to decrease with increas-ing current. The above spin current induced magnetization behavior can be understood in terms of energy analysis of the spinsystem. Since the effective field in the Gilbert equation isdefined as: 5,6 H52]E ]M, ~2! whereEis the energy density of the system and Mis the local magnetization. The energy rate equation of the system,thus, can be derived as the following: 7,8 E vdE dtdv5E vS2aMH2g 11a2~hˆ3mˆ!21MHg 11a2 P0J\ eMd~~hˆmˆ!~mˆmˆ0!2hˆmˆ0!Ddv,~3! wherehˆ,mˆ, andmˆ0are the unit vectors of H,M,M0, and the integration is over the entire volume of the free layer.Thefirst term in the above equation is always negative in sign~except when it is zero !, corresponding to energy dissipation. For the given initial magnetization, field, and spin currentpolarization directions, a small magnetization precession willyield the second term positive, resulting in energy beingpumped into the system by the spin current. When the sec-ond term exceeds the first term, sustained magnetization pre-cession is excited. Since the net damping is effectively zeroat this point, the excited spin wave at the critical current FIG. 4. Calculated variance of the normalized longitudinal magnetization component as a function of driving current magnitude. Below 0.65 mA,magnetization is stationary.At 0.65 mA, spatial uniform magnetization pre-cession at the ferromagnetic resonance frequency is excited. When currentmagnitude exceeds this critical current, stochastic magnetization precessionis generated with its amplitude increases with current level initially. Thedecrease of the fluctuation amplitude is due to the mean magnetizationchange. FIG. 2. Current driven magnetization fluctuation in an elliptic magnetic filmelement. Normalized longitudinal magnetization component, spatially aver-aged over the element, is plotted at three different current amplitude. Theinsets are longer time scale plots. A 1000 Oe external magnetic field isapplied along the long-axis of the element. FIG. 3. Calculated power spectral densities of the longitudinal magnetiza-tion component for the three cases plotted in Fig. 1. At 0.64 mA, the onsetof the current driven fluctuations, a single dominant frequency peak alongwith a side peak is observed. Note the linewidth is extremely narrow. In-creasing current magnitude broadens and down-shifts the peak frequencies. At 1.5 mA, a pronounced 1/ fnoise becomes present.6631 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Zhu, Zhu, and White [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: 130.209.6.50 On: Mon, 22 Dec 2014 01:10:32value should have extremely narrow linewidth, just as the I 50.65mA case shown in Fig. 3. Figure 5 shows the rate of the total energy change, inte- grated over the free layer, for the I50.65mA case.When the current is turned on, the energy increases to initiate the spinwave excitation. As the spin wave starts to gain amplitude,the energy of the spin system starts to oscillate between gainand damping with a net increase. As the magnetization pre-cession angle with respect to the effective field increases, theoscillation amplitude in the dynamic energy change also in-creases. After a period of more than 20 ns, a dynamic equi-librium, consequently steady excited spin waves, is reachedwith the energy steadily oscillating between gain and damp-ing. This self-limiting characteristic of the system energy isthe reason for steady excited spin waves and is specificallydue to the nature of spin transfer. At a current significantly beyond the critical value, ex- cited spin waves are chaotic in nature. Figure 6 shows thedynamic oscillation of the long-axis magnetization compo-nent and corresponding time derivative of the total energy.The rapid rising and slow falling of the long-axis componentin time domain shown in the figure has been reported previ-ously in studies on field-driven chaotic spin waves. Asshown in the figure, the rapid rising of the magnetizationcomponent corresponds to a rapid-damping of the energy while the slow-falling corresponds to a slow-gain of the en-ergy. Increasing current yields the excited magnetization pre- cession trajectory to move out of the film plane. At suffi-ciently large current level, the precession trajectory can becompletely either above, or below, the film plane, and withchaotic transition between them, yielding a random telegraphnoise in the magnetoresistance output, as shown in Fig. 7.The onset of chaotic spin waves are marked by excitation ofa large number of multiple spin waves modes, accompaniedby the appearance of 1/ flike noise at low frequencies. Even though thermal excitation has been neglected in this study for the purpose of obtaining a better understandingspin transfer, it should not introduce any significant alter-ations to the results presented here.At room temperature, thethermal excitation energy is significantly smaller than all theenergy terms in this study, especially with an external field of1000 Oe or higher applied. 9,10 CONCLUSIONS AND REMARKS Spin polarized current excited spin waves in a deep sub- micron size elliptical spin valve element have been studiedwith the utilization of the spin transfer torque modified Gil-bert equation. Sustained spin waves are excited when thespin transfer resulted energy gain exceeds the intrinsic en-ergy damping. The excited spin wave causes the system en-ergy to oscillate between the energy gain and energy damp-ing to reach a dynamic energy balance. This self-levelingcharacteristic leads to extremely steady spin waves providedcurrent is only slightly above the critical value. For currentssignificantly above the critical value, the excited spin wavebecomes chaotic in nature. The chaotic spin waves yield asubstantial 1/ f-like noise at low frequencies. 11 1L. Berger, Phys. Rev. B 54, 9353 ~1996!. 2J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1~1996!. 3J. C. Slonczewski, J. Magn. Magn. Mater. 247,3 2 4 ~2002!. 4J. A. Katine, F. J. Albert, and R. A. Buhrman, Phys. Rev. Lett. 84, 3149 ~2000!. 5T. L. Gilbert, Phys. Rev. 100, 1243 ~1955!. 6J.-G. Zhu, Ph.D. thesis, University of California at San Diego, San Diego, CA, 1989. 7J.-G. Zhu and X. Zhu, IEEE Trans. Magn. 40~2004!~to be published !. 8Z.. Li and S. Zhang, Phys. Rev. B 68, 024404 ~2003!. 9P. H. Bryant, C. D. Jeffries, and K. Nakamura, Nonlinear Phenomena and Chaos in Magnetic Materials , edited by P. E. Wigen ~World Scientific, Singapore, 1994 !, pp. 83–111. 10X. Zhu and J. Zhu, J. Appl. Phys. 95, 7318 ~2004!, these proceedings. 11J. Zhu, N. Kim, Y. Zhou, Y. Zheng, J. Chang, K. Ju, X. Zhu, and R. M. White, IEEE Trans. Magn. ~to be published !. FIG. 5. The rate of energy change in the 0.65 mAcase shown in Figs. 2 and 3. The figure shown at the right shows the energy change rate after steadystate is reachedApoint above the middle dashed line represents energy gainand below represents energy damping. FIG. 6. Long-axis magnetization component ~top!and the corresponding rate of energy change as a function of time.The rapid rising and slow falling inMxis characteristic for chaotic spin waves with similar observations reported previously on field-driven chaotic spin wave excitation. FIG. 7. Chaotic magnetization precession trajectory at I56 mA. The two dynamic attractors are significantly above and below the film plane and thetransitions between the two attractors are chaotic in nature.6632 J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004 Zhu, Zhu, and White [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: 130.209.6.50 On: Mon, 22 Dec 2014 01:10:32