Influence of shape anisotropy on magnetization reversal induced by nonlinear down-chirp microwave pulse
M. T. Islam, M. A. S. Akanda, M. A. J. Pikul, X. S. Wang, X. R. Wang
IInfluence of shape anisotropy on magnetization reversal induced bynonlinear down-chirp microwave pulse
M. T. Islam, a) M. A. S. Akanda, M. A. J. Pikul, X. S. Wang, and X. R. Wang
3, 4 Physics Discipline, Khulna University, Khulna 9208, Bangladesh School of Physics and Electronics, Hunan University, Changsha 410082, China Physics Department, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon,Hong Kong HKUST Shenzhen Research Institute, Shenzhen 518057, People’s Republic of China
It has been demonstrated that a circularly polarized linear down-chirp microwave field pulse (LDCMWP) reverses asingle-domain of high anisotropy magnetic nanoparticle but not efficient as expected. Therefore, based on the Landau-Lifshitz-Gilbert equation, we study the nonlinear down-chirp microwave field pulse (NLDCMWP) driven magneti-zation reversal, which can induce fast and energy-efficient reversal since the reversal time is close to the theoreticallimit and the required field amplitude and initial frequency band are smaller than that of the LDCMWP. The fast andenergy-efficient magnetization reversal is obtained because the frequency change of the NLDCMWP closely matchesthe frequency change of the magnetization precession which leads to efficient stimulated microwave energy absorption(emission) by (from) the magnetic particle before (after) it crosses over the energy barrier. Moreover, we found that theenhancement of hard anisotropy reduces the initial frequency band and the microwave amplitude of NLDCMWP andthe material with the larger damping is better for fast magnetization reversal. These investigations may give a pathwayto realize the fast and energy-efficient magnetization reversal.
I. INTRODUCTION
Achieving fast and energy-efficient magnetization reversalof high anisotropy materials is a central issue because of itspotential application in non-volatile data storage devices and fast data processing . To obtain high thermal stabilityand minimum errors in device application, high anisotropymaterials are required . But, the challenge is to find outthe way which can induce the fastest magnetization rever-sal with minimal energy consumption. Over the last twodecades, magnetization reversal has been investigated by aconstant magnetic field , by the microwave field of con-stant frequency, either with or without a polarized electriccurrent and by spin-transfer torque (STT) or spin-orbittorque (SOT) . However, in the case of external magneticfield, reversal time is longer and suffers from scalability andfield localization issues . For the microwave field of constantfrequency-driven magnetization reversal, the large field am-plitude and the long reversal time emerge as limitations .In the case of the STT-MRAM or SOT-MRAM based de-vices, the requirement of a large current density is a bottle-neck as it generates Joule heat which causes device durabilityand reliability issues . Moreover, there are several stud-ies showing magnetization reversal induced by microwavesof time-dependent frequency . In studies , magnetiza-tion reversal is obtained by a static field-assisted with a radio-frequency microwave field pulse. Here, a dc external field actsas the main reversal force. In study , the frequency of themicrowave is required to be the resonance frequency, and instudies, optimal microwaves are constructed. These kindsof microwave forms are difficult to realize in practice. Thestudy reports that magnetization reversal is induced by themicrowave pulse, but the pulse is applied such that magnetiza- a) Electronic mail: [email protected] tion just crosses over the energy barrier by resonantly energyabsorption, i.e., only positive frequency is employed.However, our recent study shows that the circularly andlinearly polarized linear down-chirp microwave pulse (LD-CMWP) is capable to induce fast magnetization reversal ofcubic / uniaxial nanoparticles. The physical strategy of thismodel is that the frequency of LDCMWP linearly decreases(from f to − f , with time) such that it roughly matchesthe resonant frequency change of magnetization precession.Thus, it triggers stimulated microwave absorption (emission)by (from) the magnetization before (after) it crosses over theenergy barrier. During the magnetization reversal process, themagnetization precession frequency decreases while the mag-netization climbs up and becomes zero momentarily whilecrossing the energy barrier and then increases with the op-posite precession direction while it goes down from the bar-rier. Indeed, in the hole process, the change of the frequencyof magnetization precession should not be linear. Therefore,it is meaningful to find an efficient down-chirp microwavepulse (DCMWP) so that its frequency change closely matchesthe frequency change of magnetization precession. Interest-ingly, in this study, we demonstrate a nonlinear down-chirpmicrowave pulse (NLDCMWP) to obtain faster magnetiza-tion reversal with the efficient stimulated microwave energyabsorption (emission) by (from) the magnetic particle. More-over, this study focuses significantly on the effect of shapeanisotropy and Gilbert damping on the magnetization rever-sal and the required parameter of NLDCMWP. We also findthat the initial frequency and microwave amplitude of NLD-CMWP decrease with the increase of hard anisotropy. Thematerials with the larger damping are better for fast magneti-zation reversal. These investigations might be useful in deviceapplications. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b nfluence of shape anisotropy on magnetization reversal induced by nonlinear down-chirp microwave pulse 2 m f -f (b)(a) τ t FIG. 1. (a) Schematic diagram of the system in which m represents aunit vector of the magnetization. A nonlinear down-chirp microwavefield is applied onto the single domain nanoparticle. (b) The fre-quency profile (sweeping from + f to − f ) of a nonlinear down-chirp microwave. II. ANALYTICAL MODEL AND METHOD
We assume a single-domain magnetic nanoparticle withuniaxial easy-axis anisotropy along the z -axis as shown inFIG. 1(a). The size of the nanoparticle is chosen so that themagnetization is considered as a macrospin represented by theunit vector m with the saturation magnetization M s . In the ab-sence of a microwave field, the ground state magnetization ofthe nanoparticle has two stable states, i.e., m parallel to ˆ z and − ˆ z .The magnetization dynamics m in the presence of circularlypolarized NLDCMWP is governed by the Landau-Lifshitz-Gilbert (LLG) equation d m dt = − γ m × h eff + α m × d m dt (1)where α and γ are the dimensionless Gilbert damping constantand gyromagnetic ratio respectively, and h eff is the total effec-tive field which includes the microwave magnetic field h mw ,the exchange field AM s ∇ m where A is the exchange stiffnessconstant, and the effective anisotropy field h k along z direc-tion, i.e., h k = h k m z ˆ z .The effective anisotropy field can be expressed in termsof uniaxial anisotropy h ani and shape anisotropy h shape as h k = h ani + h shape = [ h ani − µ ( N z − N x ) M s ] m z ˆ z where N z and N x are demagnetization factors and µ = π × − N / A is the vacuum magnetic permeability. Thus, the resonant fre-quency of the nanoparticle is obtained from the well-knownKittel formula f = γ π [ h ani − µ ( N z − N x ) M s ] . For microwave field-driven magnetization reversal from theLLG equation, the rate of energy change is expressed as˙ E = − αγ | m × h eff | − m · ˙ h mw (2)For positive damping, the first term is always negative.The second term can be either positive or negative for time-dependent external microwave field. Therefore, the mi-crowave field pulse can trigger the stimulated energy absorp- tion or emission, depending on the angle between the instan-taneous magnetization m and ˙ h mw .It has been shown that the fast magnetization reversal isachieved by LDCMWP with the physical picture: the fre-quency of LDCMWP roughly matches the frequency changeof magnetization precession and hence it triggers inefficientstimulated microwave absorption (emission) by (from) themagnetization before (after) it crosses over the energy bar-rier. To obtain close / better matching of the microwavefrequency with the magnetization precession frequency, theNLDCMWP is applied in the xy plane which takes theform h mw = h mw [ cos φ ( t ) ˆ x + sin φ ( t ) ˆ y ] , where h mw is theamplitude of the microwave field and φ ( t ) is the phase.In this study, we employ a NLDCMWP whose instanta-neous frequency f ( t ) sweeps nonlinearly with time at atime-dependent chirp rate η ( t ) (in units of ns − ) as shownin FIG. 1(b). The phase φ ( t ) is φ ( t ) = π f cos ( π Rt ) t ,where R (in units of ns − ) is the fitting parameter. Theinstantaneous frequency is obtained as f ( t ) = π d φ dt = f [ cos ( π Rt ) − ( π Rt ) sin ( π Rt )] which sweeps nonlinearlywith time at a time-dependent chirp rate η ( t ) (in units ofns − ) as shown in FIG. 1(b). The instantaneous chirp rateis η ( t ) = − f (cid:104) ( π R ) sin ( π Rt ) + ( π R ) t cos ( π Rt ) (cid:105) . For simplicity, in this study, we take into account the aver-age chirp rate as η avg = f τ , where τ is the pulse duration sothat the frequency sweeps from initial f to final − f as shownin FIG. 1(b).According to the applied NLDCMWP, the second term ofEq. (2) can be expressed as˙ ε = − H mw sin θ ( t ) sin Φ ( t ) (cid:20) φ ( t ) t − ddt (cid:18) φ ( t ) t (cid:19) t (cid:21) (3)where Φ ( t ) is the angle between m t (the in-plane componentof m and h mw . Therefore, the microwave field pulse can trig-ger the stimulated energy absorption (for − Φ ( t ) ) which is oc-curred before crossing the energy barrier and emission (for Φ ( t ) ) after crossing the energy barrier.The parameters for this study are chosen from typicalexperiments on microwave-driven magnetization reversal as M s = A / m, h k = .
75 T, γ = . × rad / ( T · s ) , A = × − J / m and α = .
01. The cell size used through-out this study is ( × × ) nm . We solved the LLG equa-tion numerically using the MUMAX3 package for time-dependent circularly polarized microwave field. We considerthe switching time window of 1 ns and least magnetization re-versal m z = − . III. NUMERICAL RESULTSA. Magnetization reversal by NLDCMWP and LDCMWPand their comparison
We first study the possibility of magnetization reversaldriven by NLDCMWP of the nanoparticle size ( × × ) nm . Purposely, with the microwave amplitude h mw = .
045 T which is same as the parameter estimated in thenfluence of shape anisotropy on magnetization reversal induced by nonlinear down-chirp microwave pulse 3 (a) (b)(c) (e)(d) (f)
FIG. 2. Model parameters of nanoparticle of M s = 10 A/m, H k = 0 .
75 T, γ = 1 . × rad/(T.s), and α = .
01. (a) Temporal evolutions of m z of V = ( × × ) nm driven by the LDCMWP (with optimal f = . , η = . − , h mw = .
045 T) (black line) and NLDCMWP(with the optimal f = . , η avg = . − , h mw = .
045 T ) (red line). The corresponding magnetization reversal trajectories for (b)LDCMWP and (c) NLDCMWP. The energy changing rate ˙ ε of the magnetization against time (blue line) and the time dependence of m z (redline) driven by (d) LDCMWP and (e) NLDCMWP. (f) Comparison of the switching time t s as a function of h mw dependence for LDCMWPand NLDCMWP. The black solid line is the theoretical limit. study , we try to find the optimal initial frequency f and theoptimal average chirp rate η avg of NLDCMWP and found thatfor f = . η avg = . − the magnetization re-versal is fastest. Then we compare the time evolutions of m z obtained by the optimized LDCMWP ( f = . , η = . − , h mw = .
045 T) (black line) and NLDCMWP ( f = . , η avg = . − , h mw = .
045 T) (red line) asshown in FIG. 2(a). Note that the NLDCMWP induces fastermagnetization reversal with f = . f = . , is useful indevice applications. To understand why the faster reversal isachieved, one may look at the trajectories of magnetizationreversal induced by LDCMWP and NLDCMWP are shown inFIG. 2(b) and FIG. 2(c) respectively. For LDCMWP, some ad-ditional precessions are observed in the trajectory while cross-ing the barrier as shown in FIG. 2(b). However, for NLD-CMWP such additional precessions are absent as shown inFIG. 2(c). This is attributed as the frequency change of theNLDCMWP closely matches the frequency change of mag-netization precession which leads the maximal stimulated en-ergy absorption and emission by (from) the magnetic particleas shown by the dotted circle in FIG. 2(e). However, in thecase of LDCMWP, the frequency change of the LDCMWProughly matches or mismatches momentarily to the frequencyof magnetization precession, and this is why it leads the inef-ficient energy absorption or emission as shown by the dottedcircle in FIG. 2(d). It is mentioned that the frequency changeof magnetization precession is nonlinear during the magneti-zation reversal.Then, keeping the optimal f and η fixed, we study theNLDCMWP and LDCMWP-driven magnetization reversalof nanoparticle of uniaxial shape, i.e., ( × × ) nm asa function microwave amplitude ( h mw ). FIG. 2(f) showsthe comparison of the optimal switching times obtained asa function of microwave amplitude, for NLDCMWP (redsquare) and LDCMWP (blue triangle) to the theoretical limit(black solid line) and found that for smaller amplitude (i.e.,0 .
035 T , .
040 T , .
045 T), NLDCMWP induces faster and efficient reversal whereas, for larger amplitudes, switchingtimes for NLDCMWP and LDCMWP become similar.
B. Hard anisotropy K ⊥ dependence of magnetization reversal Subsequently, we investigate the influence of the shapeanisotropy field h shape or hard anisotropy K ⊥ , which is origi-nated from the dipole-dipole interaction, on the magnetizationswitching time, microwave amplitude and initial frequencyof NLDCMWP. The K ⊥ increases by increasing the cross-sectional area S with the fixed thickness d = V = S × d is the volume of nanoparticle. Specifically, westudy the magnetization reversal of the nanoparticles of thecross-sectional areas S = ×
10 nm , 12 ×
12 nm , 14 × ×
16 nm , 18 ×
18 nm , 20 ×
20 nm and 22 × driven by the NLDCMWP with the optimal f , η avg andfixed h mw = .
045 T. The temporal evolutions of m z for dif-ferent S are shown in FIG. 3(a). It is noted from FIG. 3(a) andFIG. 3(c) that with the increase of S i.e., K ⊥ , the magnetiza-tion reversal becomes faster for the cross-section S = × and when S increases further, the switching time showsslightly increasing trend but for S = ×
22 nm or larger S ,magnetization switching time suddenly drops to t s = .
34 ns.This indicates that there is an optimal S for which the switch-ing time significantly reduces and becomes close to the theo-retical limit. To understand the reason for achieving the fastestreversal, FIG. 3(b) is referred and noted that the increase ofthe shape anisotropy field h shape with S , which reduces theheight of the energy barrier (energy difference between theinitial state and saddle point), which leads to faster magneti-zation reversal as expected. C. Hard anisotropy K ⊥ dependence of f and h mw Here we present the effect of the hard anisotropy K ⊥ (i.e.,shape anisotropy field) on the initial frequency f and the mi-nfluence of shape anisotropy on magnetization reversal induced by nonlinear down-chirp microwave pulse 4 (a) (b)(c) (d) FIG. 3. (a) Temporal evolution of m z induced by NLDCMWP fordifferent cross-sectional area, S . (b) The energy landscape E alongthe line φ =
0. The symbols i and s represent the initial state andsaddle point. (c) The minimal switching time as a function of hardanisotropy K ⊥ . (d) The minimal f and h mw as a function of K ⊥ while switching time window 1 ns. (a) (b) FIG. 4. (a) Minimal switching time t s as a function of initial fre-quency of NLDCMWP with fixed chirp rate for of different S . (b)Minimal t s as a function of Gilbert damping α for different S . crowave amplitude h mw of NLDCMWP. We observe that theminimal f of NLDCMWP for fast magnetization reversal de-creases with increasing the shape anisotropy field h shape asshown in FIG. 3(d) (red circle). It is understandable sincethe h shape reduces H k and thus reduces the resonant frequency.For further justification, we theoretically calculate the corre-sponding resonant frequency of the nanoparticles and plottedin FIG. 3(d) (black line). It is noted that the minimal fre-quency f always smaller than the theoretical / resonant fre-quency which is useful for device application. It is empha-sized that the fast and efficient reversal is valid for a range of f . FIG. 4(a) shows the estimated frequency bands of f byvertical dashed lines for different S (e.g., f = . ∼ . h shape = .
096 T) provided that the switching time windowis 1 ns. Moreover, the optimal microwave amplitude h mw alsoshows a decreasing trend with the increase of K ⊥ despite somefluctuation as shown in FIG. 3(d) (blue square). This hap-pens as the height of the energy barrier decreases with h shape (Refers to FIG. 3(b)) and thus the smaller h mw induces fastermagnetization reversal. (a) (b) FIG. 5. Magnetization reversal trajectories of biaxial shape ( × × ) nm driven by NLDCMWP for (a) α = . α = . D. Gilbert damping α dependence of magnetization reversal In the magnetization reversal process, the magnetization re-quires crossing over an energy barrier that originated from theanisotropy field. Therefore, to obtain the fast and energy-efficient NLDCMWP-driven magnetization reversal, lower(higher) Gilbert damping α before (after) crossing the energybarrier is required. Therefore, it is meaningful to find the op-timal α for different S at which the reversal is fastest. Westudy the NLDCMWP-driven magnetization as a function of α for different S . FIG. 4(b) shows the dependence of switch-ing time on the Gilbert damping for different S . It is foundthat, for S = ×
10 nm , the switching time is lowest at α = . α = .
01 and α = .
045 respectively and observed thatfor α = . α shows faster magnetization reversal. IV. DISCUSSIONS AND CONCLUSIONS
We investigated the NLDCMWP-driven magnetization re-versal of a single domain of nanoparticle and then studied theinfluence of hard anisotropy K ⊥ on the parameters of NLD-CMWP at zero temperature limit. Although the required pa-rameters of NLDCMWP, except the chirp rate, are smallerthan that of LDCMWP, still there is a challenge to gener-ate such the circularly polarized NLDCMWP with a time-dependent chirp rate. In addition, the finite temperature isomnipresent and it may have a negative influence on the re-quired parameters of NLDCMWP. Fortunately, a recent studyusing LDCMWP shows that the LDCMWP-driven magne-tization reversal is valid even above the room temperatureand the finite temperature assists the magnetization rever-sal as the three controlling parameters of LDCMWP are re-duced significantly. Therefore, it is expected that the param-eters of NLDCMWP also would reduce at room temperature.Thus, it might be possible to generate such nonlinear down-chirp microwave pulse using the techniques referring to thestudies . The circularly polarized microwaves with time-nfluence of shape anisotropy on magnetization reversal induced by nonlinear down-chirp microwave pulse 5dependent frequency might be generated by coupling a mag-netic nanoparticle to a pair of weak superconducting links where the time dependence of the microwave frequency istuned by voltage. Spin torque oscillator (STO) integratingwith a field-generating layer is another way to generate thechirp microwave pulse . In the study , the generationof microwaves of time-dependent frequency is shown exper-imentally. The mostly used co-planar waveguide technologyis also a potential candidate to generate NLDCMWP. Usingtwo co-planar waveguides, one can generate a circularly po-larized NLDCMWP . In conclusion, we find that a nonlinealdown-chirp microwave pulse can efficiently reverse a mag-netic nanoparticle. We also find that with the increase of hardanisotropy, the initial frequency and microwave amplitude ofNLDCMWP decreases and thus it assists the magnetizationreversal. In addition, the materials with the larger dampingare better for fast magnetization reversal. These investigationsmight be useful in device applications. ACKNOWLEDGMENTS
This work was supported by the Ministry of Education(BANBEIS, Grant No. SD201997) as well as M. A. J. P. ac-knowledges Ministry of Science and Technology (Grant No.440 EAS).
Appendix A: Calculation of ˙ ε In this Appendix, we show the details of the derivation of ˙ ε in Eq. 3. The rate of change of h mw is˙ h mw = d h mw dt = ddt ( h mw [ cos φ ( t ) ˆ x + sin φ ( t ) ˆ y ])= h mw [ − sin φ ( t ) ˆ x + cos φ ( t ) ˆ y ] d φ dt = h mw [ − sin φ ( t ) ˆ x + cos φ ( t ) ˆ y ] (cid:20) φ ( t ) t − ddt (cid:18) φ ( t ) t (cid:19) t (cid:21) The magnetization is given by m = m x ˆ x + m y ˆ y = sin θ ( t ) cos φ m ( t ) ˆ x + sin θ ( t ) sin φ m ( t ) ˆ y where θ ( t ) is the polar angle and φ m ( t ) is the azimuthal angleof the magnetization m .Substituting m x and ˙ h mw in Eq. 2, we get,˙ ε = − m · ˙ h mw = h mw sin θ ( t ) [ − sin φ ( t ) cos φ m ( t ) + cos φ ( t ) sin φ m ( t )] · (cid:20) φ ( t ) t − ddt (cid:18) φ ( t ) t (cid:19) t (cid:21) = h mw sin θ ( t ) sin ( φ ( t ) − φ m ( t )) (cid:20) φ ( t ) t − ddt (cid:18) φ ( t ) t (cid:19) t (cid:21) Defining Φ ( t ) = φ m ( t ) − φ ( t ) , we have finally˙ ε = h mw sin θ ( t ) sin Φ ( t ) (cid:20) φ ( t ) t − ddt (cid:18) φ ( t ) t (cid:19) t (cid:21) where (cid:104) φ ( t ) t − ddt (cid:16) φ ( t ) t (cid:17) t (cid:105) represents ω ( t ) . REFERENCES S. Sun, C. B. Murray, D. Weller, L. Folks, and A. Moser, “Monodispersefept nanoparticles and ferromagnetic fept nanocrystal superlattices,” sci-ence , 1989–1992 (2000). S. Woods, J. Kirtley, S. Sun, and R. Koch, “Direct investigation of super-paramagnetism in co nanoparticle films,” Physical review letters , 137205(2001). D. Zitoun, M. Respaud, M.-C. Fromen, M. J. Casanove, P. Lecante,C. Amiens, and B. Chaudret, “Magnetic enhancement in nanoscale corhparticles,” Physical review letters , 037203 (2002). B. Hillebrands and K. Ounadjela,
Spin dynamics in confined magneticstructures I & II , Vol. 83 (Springer Science & Business Media, 2003). S. Mangin, D. Ravelosona, J. Katine, M. Carey, B. Terris, and E. E. Fuller-ton, “Current-induced magnetization reversal in nanopillars with perpen-dicular anisotropy,” Nature materials , 210–215 (2006). A. Hubert, “R. scha fer, magnetic domains: the analysis of magnetic mi-crostructures,” (1998). Z. Sun and X. Wang, “Fast magnetization switching of stoner particles: Anonlinear dynamics picture,” Physical Review B , 174430 (2005). G. Bertotti, C. Serpico, and I. D. Mayergoyz, “Nonlinear magnetizationdynamics under circularly polarized field,” Physical Review Letters , 724(2001). Z. Sun and X. Wang, “Strategy to reduce minimal magnetization switchingfield for stoner particles,” Physical Review B , 092416 (2006). J.-G. Zhu and Y. Wang, “Microwave assisted magnetic recording utiliz-ing perpendicular spin torque oscillator with switchable perpendicular elec-trodes,” IEEE Transactions on Magnetics , 751–757 (2010). J. C. Slonczewski et al. , “Current-driven excitation of magnetic multilay-ers,” Journal of Magnetism and Magnetic Materials , L1 (1996). L. Berger, “Emission of spin waves by a magnetic multilayer traversed bya current,” Physical Review B , 9353 (1996). M. Tsoi, A. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P. Wyder,“Excitation of a magnetic multilayer by an electric current,” Physical Re-view Letters , 4281 (1998). J. Sun, “Current-driven magnetic switching in manganite trilayer junc-tions,” Journal of Magnetism and Magnetic Materials , 157–162 (1999). Y. B. Bazaliy, B. Jones, and S.-C. Zhang, “Modification of the landau-lifshitz equation in the presence of a spin-polarized current in colossal-andgiant-magnetoresistive materials,” Physical Review B , R3213 (1998). J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph,“Current-driven magnetization reversal and spin-wave excitations in co / cu / co pillars,” Phys. Rev. Lett. , 3149–3152 (2000). X. Waintal, E. B. Myers, P. W. Brouwer, and D. C. Ralph, “Role ofspin-dependent interface scattering in generating current-induced torquesin magnetic multilayers,” Phys. Rev. B , 12317–12327 (2000). J. Z. Sun, “Spin-current interaction with a monodomain magnetic body: Amodel study,” Physical Review B , 570 (2000). J. Sun, “Spintronics gets a magnetic flute,” Nature , 359–360 (2003). M. D. Stiles and A. Zangwill, “Anatomy of spin-transfer torque,” PhysicalReview B , 014407 (2002). Y. B. Bazaliy, B. Jones, and S.-C. Zhang, “Current-induced magnetizationswitching in small domains of different anisotropies,” Physical Review B , 094421 (2004). R. Koch, J. Katine, and J. Sun, “Time-resolved reversal of spin-transferswitching in a nanomagnet,” Physical review letters , 088302 (2004). Z. Li and S. Zhang, “Thermally assisted magnetization reversal in the pres-ence of a spin-transfer torque,” Physical Review B , 134416 (2004). W. Wetzels, G. E. Bauer, and O. N. Jouravlev, “Efficient magnetizationreversal with noisy currents,” Physical review letters , 127203 (2006). nfluence of shape anisotropy on magnetization reversal induced by nonlinear down-chirp microwave pulse 6 A. Manchon and S. Zhang, “Theory of nonequilibrium intrinsic spin torquein a single nanomagnet,” Physical Review B , 212405 (2008). I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini,J. Vogel, and P. Gambardella, “Current-driven spin torque induced by therashba effect in a ferromagnetic metal layer,” Nature materials , 230–234(2010). I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache,S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, “Per-pendicular switching of a single ferromagnetic layer induced by in-planecurrent injection,” Nature , 189–193 (2011). L. Liu, C.-F. Pai, Y. Li, H. Tseng, D. Ralph, and R. Buhrman, “Spin-torqueswitching with the giant spin hall effect of tantalum,” Science , 555–558(2012). S. I. Denisov, T. V. Lyutyy, P. Hänggi, and K. N. Trohidou, “Dynamicaland thermal effects in nanoparticle systems driven by a rotating magneticfield,” Physical Review B , 104406 (2006). S. Okamoto, N. Kikuchi, and O. Kitakami, “Magnetization switching be-havior with microwave assistance,” Applied Physics Letters , 102506(2008). T. Tanaka, Y. Otsuka, Y. Furomoto, K. Matsuyama, and Y. Nozaki,“Selective magnetization switching with microwave assistance for three-dimensional magnetic recording,” Journal of Applied Physics , 143908(2013). J. Grollier, V. Cros, H. Jaffres, A. Hamzic, J.-M. George, G. Faini, J. B.Youssef, H. Le Gall, and A. Fert, “Field dependence of magnetization re-versal by spin transfer,” Physical Review B , 174402 (2003). H. Morise and S. Nakamura, “Stable magnetization states under a spin-polarized current and a magnetic field,” Physical Review B , 014439(2005). T. Taniguchi and H. Imamura, “Critical current of spin-transfer-torque-driven magnetization dynamics in magnetic multilayers,” Physical ReviewB , 224421 (2008). Y. Suzuki, A. A. Tulapurkar, and C. Chappert, “Spin-injection phenomenaand applications,” in
Nanomagnetism and Spintronics (Elsevier, 2009) pp.93–153. Z. Sun and X. Wang, “Theoretical limit of the minimal magnetizationswitching field and the optimal field pulse for stoner particles,” Physicalreview letters , 077205 (2006). X. Wang and Z. Sun, “Theoretical limit in the magnetization reversal ofstoner particles,” Physical review letters , 077201 (2007). X. Wang, P. Yan, J. Lu, and C. He, “Euler equation of the optimal trajectoryfor the fastest magnetization reversal of nano-magnetic structures,” EPL(Europhysics Letters) , 27008 (2008). C. Thirion, W. Wernsdorfer, and D. Mailly, “Switching of magnetizationby nonlinear resonance studied in single nanoparticles,” Nature materials ,524–527 (2003). K. Rivkin and J. B. Ketterson, “Magnetization reversal in the anisotropy-dominated regime using time-dependent magnetic fields,” Applied physicsletters , 252507 (2006). Z. Wang and M. Wu, “Chirped-microwave assisted magnetization reversal,” Journal of Applied Physics , 093903 (2009). N. Barros, M. Rassam, H. Jirari, and H. Kachkachi, “Optimal switchingof a nanomagnet assisted by microwaves,” Physical Review B , 144418(2011). N. Barros, H. Rassam, and H. Kachkachi, “Microwave-assisted switchingof a nanomagnet: Analytical determination of the optimal microwave field,”Physical Review B , 014421 (2013). G. Klughertz, P.-A. Hervieux, and G. Manfredi, “Autoresonant controlof the magnetization switching in single-domain nanoparticles,” Journal ofPhysics D: Applied Physics , 345004 (2014). M. T. Islam, X. Wang, Y. Zhang, and X. Wang, “Subnanosecond magneti-zation reversal of a magnetic nanoparticle driven by a chirp microwave fieldpulse,” Physical Review B , 224412 (2018). T. L. Gilbert, “A phenomenological theory of damping in ferromagneticmaterials,” IEEE transactions on magnetics , 3443–3449 (2004). J. Dubowik, “Shape anisotropy of magnetic heterostructures,” Physical Re-view B , 1088 (1996). A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez,and B. Van Waeyenberge, “The design and verification of mumax3,” AIPadvances , 107133 (2014). M. Islam, M. Pikul, X. Wang, and X. Wang, “Thermally assisted mag-netization reversal of a magnetic nanoparticle driven by a down-chirp mi-crowave field pulse,” arXiv preprint arXiv:2011.08610 (2020). W. Li and J. Yao, “Generation of linearly chirped microwave waveform withan increased time-bandwidth product based on a tunable optoelectronic os-cillator and a recirculating phase modulation loop,” Journal of LightwaveTechnology , 3573–3579 (2014). S. K. Raghuwanshi, N. K. Srivastava, and A. Srivastava, “A novel approachto generate a chirp microwave waveform using temporal pulse shaping tech-nique applicable in remote sensing application,” International Journal ofElectronics , 1689–1699 (2017). L. Cai, D. A. Garanin, and E. M. Chudnovsky, “Reversal of magnetiza-tion of a single-domain magnetic particle by the ac field of time-dependentfrequency,” Physical Review B , 024418 (2013). L. Cai and E. M. Chudnovsky, “Interaction of a nanomagnet with a weaksuperconducting link,” Physical Review B , 104429 (2010). J.-G. Zhu and Y. Wang, “Microwave assisted magnetic recording utiliz-ing perpendicular spin torque oscillator with switchable perpendicular elec-trodes,” IEEE Transactions on Magnetics , 751–757 (2010). J.-G. Zhu, X. Zhu, and Y. Tang, “Microwave assisted magnetic recording,”IEEE Transactions on Magnetics , 125–131 (2007). H. Suto, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato, “Nanoscalelayer-selective readout of magnetization direction from a magnetic multi-layer using a spin-torque oscillator,” Nanotechnology , 245501 (2014). H. Suto, T. Kanao, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato,“Subnanosecond microwave-assisted magnetization switching in a circu-larly polarized microwave magnetic field,” Applied Physics Letters110