Micromagnetic instabilities in spin-transfer switching of perpendicular magnetic tunnel junctions
MMicromagnetic instabilities in spin-transfer switching of perpendicular magnetictunnel junctions
Nahuel Statuto, Jamileh Beik Mohammadi, and Andrew D. Kent Center for Quantum Phenomena, Department of Physics,New York University, New York, New York 10003 USA Department of Physics, Loyola University New Orleans, New Orleans, LA 70118 USA (Dated: September 10, 2020)Micromagnetic instabilities and non-uniform magnetization states play a significant role in spintransfer induced switching of nanometer scale magnetic elements. Here we model domain wall me-diated switching dynamics in perpendicularly magnetized magnetic tunnel junction nanopillars. Weshow that domain wall surface tension always leads to magnetization oscillations and instabilitiesassociated with the disk shape of the junction. A collective coordinate model is developed thatcaptures aspects of these instabilities and illustrates their physical origin. Model results are com-pared to those of micromagnetic simulations. The switching dynamics is found to be very sensitiveto the domain wall position and phase, which characterizes the angle of the magnetization in thedisk plane. This sensitivity is reduced in the presence of spin torques and the spin current neededto displace a domain wall can be far less than the threshold current for switching from a uniformlymagnetized state. A prediction of this model is conductance oscillations of increasing frequencyduring the switching process.
I. INTRODUCTION
Spin transfer torque magnetization switching has beenextensively studied since it was first predicted theoret-ically and demonstrated experimentally in spin-valvenanopillars . The magnetic anisotropy of the free layerplays an important role in setting the switching current;materials with a uniaxial anisotropy exhibit far more ef-ficient spin-transfer switching . This has led to researchon easy axis perpendicular to the plane free layers. Ad-vances include the demonstration of spin-transfer switch-ing in perpendicularly magnetized spin-valve nanopil-lars and perpendicularly magnetized magnetic tun-nel junctions . When the free layer is in the shapeof a disk there is axial symmetry that simplifies theanalysis of the magnetization dynamics. Thus recentresearch has focused on understanding the magnetiza-tion switching mechanisms in this high symmetry sit-uation . Perpendicular magnetic tunnel junctionsnanopillars (pMTJs) are also under intense develop-ment for applications as magnetic random access memory(MRAM) .Such junctions consist of thin ferromagnetic metalliclayers, one with a magnetization free to reorient and theother with a fixed magnetization direction separated bya thin insulating barrier. The junction stable magneticstates are layers magnetized parallel (P state) or antipar-allel (AP state) and have conductances that differ by afactor of 2 (or more) with CoFeB electrodes and an MgOinsulating barrier . Current flow through the junc-tion leads to spin-transfer torques on the free layer mag-netization that can switch it between magnetic states. Inthe macrospin limit—where the switching between thesestates is by coherent spin rotation—there are analyticmodels that characterize the thermally activated switch-ing and spin-transfer driven switching . Experiments, however, suggest that the magnetizationreverses nonuniformly and the reversal process appears tobe reversed domain nucleation and expansion by domainwall motion . This is confirmed by micromagneticmodeling, which shows that the assumption of a coher-ent magnetization reversal does not capture the switchingdynamics above a critical diameter set by the exchangeinteraction strength, magnetic anisotropy and magneti-zation d c = (16 /π ) (cid:112) A/K eff , where A is the ex-change constant and K eff ( d ) = K p − µ M (cid:2) N zz ( d ) − (cid:3) / K p is the perpendicular magnetic anisotropy, µ the per-mability of free space, M is the magnetization and N zz ( d )is the demagnetization coefficient perpendicular to theplane of the free layer that depends on the element’s di-ameter d and its thickness. For state-of-the-art pMTJsthe critical diameter d c can be just 10 to 30 nm, as theexchange constant of the thin CoFeB free layer can bemuch less than the bulk value . As the diameter in-creases beyond d c for fixed current overdrive j/j c —thecurrent j divided by the threshold current for magne-tization switching j c —the switching time increases andthe average magnetization is a nonmonotonic function oftime .The micromagnetic simulation in Fig. 1 illustrates thisreversal process. The free layer’s spatially averaged per-pendicular ( z -axis) component of magnetization is plot-ted versus time along with magnetization images at var-ious times in the reversal process. The reversal startswith the formation of a reversed region in the center ofthe free layer which then experiences a drift instability,leading to a domain wall that traverses the element .There are then magnetization oscillations of increasingfrequency as the reversed domain expands to completethe reversal. This behavior appears generic to reverseddomain expansion by domain wall motion in a disk. a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Time ( ns ) m z j/j c = 1.23 FIG. 1. Time evolution of the spatially averaged normalizedz-axis disk magnetization during a nucleation process with j/j c = 1 .
23 and d/d c = 1 .
95 ( r = 15 nm). The other mate-rial parameters can be found in Appendix VII E. Snapshotsshow the disk magnetization at different times indicated inthe curve by red dots. The inset is a schematic of the tunneljunction nanopillar. This paper addresses the origin of the magnetizationoscillations in the switching process both with an ana-lytic model and micromagnetic simulations. A model forthe collective coordinates of a domain wall in a disk isdeveloped and its solutions illustrated. The origin of themagnetization oscillations is shown to be Walker break-down associated with domain wall surface tension whichproduces an effective force on the domain wall that varieswith its position. Micromagnetic simulations are used totest the model and determine features of the collective co-ordinate model that are preserved as more spin degreesof freedom are considered. We also discuss the impli-cations of these micromagnetic instabilities on magnetictunnel junction characteristics and propose experimentsto observe this phenomena.The structure of the paper is as follows. Section IIintroduces the analytic model and its solutions for mag-netization relaxation from different initial states. Sec-tion III presents micromagnetic simulations that are com-pared to the analytic model. In Sec. IV the effect ofa spin current is studied and in Sec. V some of theexperimental consequences of the model are described.The main results are then summarized. The Appen-dices include more details on the derivation of the an-alytic model (Secs. VII A-VII C) and micromagnetic sim-ulations (Secs. VII D-VII E).
II. ANALYTIC MODEL
The starting point for our analysis is the Landau-Lifshitz-Gilbert (LLG) equation with a spin-torque term: ˙m = − γµ m × H eff + α m × ˙m + a I m × ( m × p ) , (1)where m is a unit vector in the direction of magneti-zation of the free layer, γ is the gyromagnetic ratio, µ the permeability of free space, and H eff is the effective r r a) b) FIG. 2. a) Schematic of a reversed domain in a thin disk ofradius r . The domain boundary—the red line—is charac-terized by its position from the center of the disk q and theangle of the magnetization from the boundary normal φ . Thecurvature of the DW is represented by radius r . b) Energyof the DW as a function of position in the disk normalized toits energy U at q = 0, its energy at the center of the disk. field. The effective field is the variational derivative ofthe energy density, U , with respect to the magnetization: µ H eff = − (1 /M ) δU/ ( δ m ), where M is the magnetiza-tion of the free layer. The second term on the right isthe damping term, where α is the Gilbert damping con-stant ( α (cid:28) p is the direction of the spinpolarization and a I is proportional to the spin current a I = (cid:126) P j/ (2 eM t ), where j is the current density, (cid:126) isthe reduced Planck’s constant, P is the spin polarizationof the current and t is the thickness of the free layer.The magnetization texture can be written in cylindri-cal coordinates with the origin at the center of the arccharacterizing the DW (i.e. radius r ): m z ( r ) = − tanh(( r − r ) / ∆) ,m r ( r ) = cos φ/ (cosh(( r − r ) / ∆) ,m θ ( r ) = sin φ/ (cosh(( r − r ) / ∆) , (2)where r is measured from the center of this same circle.∆ characterizes the width of the domain wall; the fullwidth of the domain wall is typically taken to be π ∆ .The magnetic energy density is given by: U = A (cid:2) ( ∇ m x ) + ( ∇ m y ) + ( ∇ m z ) (cid:3) − K eff m z , (3)where A is the exchange constant of the material and K eff is the effective perpendicular magnetic anisotropy, thedifference between the perpendicular uniaxial anisotropyand demagnetizing energy (i.e dipole-dipole interactionsare described by an average demagnetization field) .The first term on the right hand side of Eq. 3 is thusthe exchange energy and the second term is the mag-netic anisotropy energy. Energy minimization gives ∆ =2 (cid:112) A/K eff . The total energy U is obtained by sub-stituting the expressions for m in Eqs. 2 into Eq. 3and integrating it over the volume of the disk (see Ap-pendix VII A): U = (cid:20) σ DW + ∆ µ H N M φ (cid:21) t(cid:96), (4) FIG. 3. Domain wall dynamics in the ( q, φ ) model startingat t = 0 in a Bloch configuration ( φ = π/
2) with q/r = 0 . q and the lower panel shows the phase,restricted to [0 , π ], as a function of time. There is no spin-polarized current or applied magnetic field. where (cid:96) is the length of the domain wall. The firstterm on the right in Eq. 3, σ DW , is the energy associ-ated with a Bloch wall ( φ = π/
2) per unit wall area, σ DW = 4 √ AK eff . The next term is the added energydensity associated with a N´eel wall ( φ = 0) and H N isthe applied field normal to the DW that would trans-form a Bloch wall into a N´eel wall, H N ≈ tM/ ( t + π ∆)(for (cid:96) (cid:29) ∆). We do not include a term in the energyassociated with the curvature of the DW (characterizedby radius r ) as this term only becomes significant whenthe DW curvature is of order of its width. Under theseconditions the DW exits the sample and the model as-sumptions for the domain wall energy no longer hold (seeAppendix VII B). The energy of the domain wall is plot-ted versus q in Fig. 2(b). It is thus clear that the shapeof element leads to a conservative force on the domainwall F q = − ∂ U /∂q that tends to expel the DW from thedisk.We derive an equation for the collective coordinatesand the generalized velocities of the DW in the disk fromEq. 1 following the approach outlined in Ref. (see Ap-pendix VII A for further details). The DW coordinates( q , φ ) satisfy a Thiele equation, similar to that of a DWmoving under the influence of spin-transfer torque in ananowire :˙ φ − α ∆ ˙ q + γ t(cid:96)M ∂ U ∂q = 0 (5) − ˙ q/ ∆ − α ˙ φ + γµ H N φ − πa I , (6)where an overdot indicates a derivative with respect totime. The last term in Eq. 5 is proportional to the con-servative force F q associated with the free layer’s shape.It has been denoted a Laplace pressure or a domainwall tension. Again, a I is proportional to the currentand the last term in Eq. 6 is the spin transfer torque FIG. 4. Domain wall dynamics in the ( q, φ ) model startingfrom q/r = 0 .
06 and different initial DW phases: N´eel ( φ =0) and Bloch ( φ = π/
2) and an intermediate phase, φ = π/ on the domain wall due to the spin-polarized currentfrom the reference layer. Eq. 5 indicates that the Laplacepressure tends to cause spins in the domain wall to pre-cess, while a spin torque couples directly to the domainwall displacement (Eq. 6). The wall velocity is related to H N and the spin-transfer torque with maximum domainwall velocity (for φ = π/ v max = ∆ γµ H N / φ = π/
2) is started at a position justto the right of the center of the disk, i.e. q (cid:39) q increasing) with its positionoscillating as a function of time. The domain wall phase φ runs, indicating precession of the spins in the DW.This behavior corresponds to DW motion motion in theWalker breakdown limit, i.e. when ˙ φ (cid:54) = 0 the domainwall position oscillates and moves at an average velocityless than v max . In the absence of a spin current and inzero magnetic field the condition for Walker breakdownis (see Appendix VII A):12 d(cid:96)M ∂U∂q (cid:39) σ DW M t(cid:96) d(cid:96)dq > α µ H N . (7)( d(cid:96)/dq ) /(cid:96) diverges as the domain wall reaches the ele-ment boundary q (cid:39) ± r . Thus the domain wall position always oscillates in a magnetization reversal process thatoccurs by reversed domain expansion in a disk, with anoscillation frequency that increases as the domain wallapproaches the element boundary. This is the charac-teristic seen in the micromagnetic modeling in Fig. 1 for m z < φ . Figure 4 shows DW FIG. 5. Relaxation phase diagram showing the final magne-tization state as a function of the initial conditions for theanalytic model. Red represents a final state with magnetiza-tion up and blue magnetization down. dynamics for the same initial position and different ini-tial phases and plots the average z-magnetization m z andthe total magnetic energy versus time. The oscillationfrequency and amplitude vary significantly with time.This behavior is in contrast to that of DW motion ina nanowire . Surprisingly, the final state changes withinitial φ , for φ = 0 and φ = π/ m z = −
1, a down magnetized domain. Whereas for φ = π/ m z = +1, an upmagnetized domain, even from an initial state m z < q and φ . The results are shown in Fig. 5.The blue color represents magnetization relaxation to adown state ( m z = −
1) and the red color represents mag-netization relaxation to an up state ( m z = +1). The in-tricate pattern highlights the sensitivity to the the initialDW position and phase. It is straightforward to includethe effect of an applied field which modifies the patternas discussed in Appendix VII C. III. MICROMAGNETIC MODEL
A basic question is to what extent this simple collec-tive coordinate model with two degrees of freedom ( q, φ )captures the DW dynamics in a ferromagnetic disk. Thefull problem is much more complex and can include vari-ations in the DW curvature (i.e. that the DW is notrigid), its width as well as non-local magnetic dipole in-teractions that are not considered in this simple model.For this reason, we performed micromagnetic simulationsand compared them to our collective coordinate model.
Switch:Relaxation: t=0.0ns 0.1ns 1.6ns 5.2ns 8.0nst=0.0ns 0.1ns 0.8ns 1.4ns 2.5ns =0= /4 FIG. 6. Time evolution of m z for the same initial DW positionbut different initial phases φ . We simulate a CoFeB disk withradius 15 nm and thickness 2 . q/r = 0 .
06 nm withno spin current or applied magnetic field. Starting from bothN´eel ( φ = 0) and a Bloch DW ( φ = π/
2) the disk relaxesto down state. Whereas for an initial phase of φ = π/ φ = 0) the disk relaxes todown state. Whereas in the panels labeled switch the initialphase is φ = π/ Micromagnetic simulations were performed using theopen-source MuMaX code . We model the magneticdisk using the same parameters as in the analytic modeland the initial states were two magnetically-opposed do-mains with a DW at the same position as in Fig. 2(a).Figure 6 shows the micromagnetic simulations for thesame initial conditions as those in Fig. 4. The centralpanel shows the time evolution of the normalized diskmagnetization, m z , for reversed domains starting at thesame position, q/r = 0 .
06 with m z <
0, and at threedifferent initial domain wall phases. For N´eel and Blochconfigurations, φ = 0 and π/ φ = π/
4, orange curve, the up mag-netic domain expands resulting in switching of the diskmagnetization. Top panels show snapshots of the magne-tization at different times for the initial N´eel configura-tion and the initial intermediate, φ = π/ a)b) FIG. 7. a) Relaxation phase diagram as a function of q and φ with no spin current or field applied. We simulate a disk ofradius 15 nm for different initial domain sizes and different ini-tial DW phases. The colorscale indicates the normalized mag-netization of the final state where blue represents relaxationto a down domain and red that the magnetization switches.The blue, orange and green dots ( q/r = 0 .
06 and φ = 0 , π/ π/
2) show the parameters used in Fig. 6. b) Relaxationphase diagram for the ( q, φ ) model. An asymmetry can beobserved between positive and negative angles.
Moreover, the bottom panel shows the time evolution ofthe phase in the initial Bloch configuration, which wascalculated from the magnetization vector images. Thephase, as in Fig. 4, oscillates from 0 to 2 π indicatingthat the DW spins are oscillating as a function of thetime and their frequency is increasing as the domain isexpelled from the edge of the disk. The same behavior isobtained no matter the initial phase: the magnetizationoscillates, the magnetic domain breathes and the domainwall moves back and forward indicating that the systemis in the Walker breakdown regime as seen in our analyticmodel.While there are differences in the timescales, the over-all domain dynamics is captured by our analytic model.Importantly, they confirm that there are oscillations inthe DW position and the phase runs, as in our analyticmodel. The micromagnetic simulations also corroboratethe very sensitive dependence of domain dynamics on theinitial conditions as the DW approaches the center of thedisk. To examine this in more detail the relaxation dynamicswere computed for a wide range of initial conditions andthe results again used to construct a relaxation phasediagram. Figure 7(a) shows this micromagnetic ( q, φ )phase diagram where we use the same colorscale as thatin Fig. 5 and blue, orange and green dots represent theinitial conditions for the curves in Fig. 6.This pattern is compared to that of the collectivecoordinate model in Fig. 7(b). The collective coordi-nate model is more intricate with many red/blue bound-aries, showing a stronger dependence on initial condi-tions. Nonetheless, there are similarities. There is a re-gion for small values of q where the domain relaxes (blueregion) and a red region in which switching is observed.In addition, the switching pattern is not mirror symmet-ric with respect to the initial phase. We attribute theasymmetry toward negative angles to the sense of gyro-scopic motion. Depending on the initial phase, the DWwill oppose or favor the rotation of the magnetic domainresulting in switching if the initial state favors the rota-tion. IV. DYNAMICS DRIVEN BYSPIN-POLARIZED CURRENT
We now consider the influence of a spin current from aperpendicularly magnetized polarizing layer p = ˆ z on theDW dynamics. The spin-current threshold, a c , needed toobtain a complete switch starting from a uniform stateis related to the material parameters and can be derivedfrom Eq. 1 to be a c = αγµ H k , where H k is the per-pendicular anisotropy field of the disk, H k = 2 K eff /M .Our micromagnetic modeling shows that for parameterstypical of state-of-the art magnetic tunnel junctions (seeAppendix VII E) reversal by formation of a single domainwall for 1 < d/d c < .In our analysis of the magnetic dynamics we considerthat a single DW has already formed in the disk andmodel its relaxation, as in previous sections, but nowin the presence of a spin-polarized current. Figure 8(a)shows a ( q, φ )-phase diagram calculated with micromag-netic simulations for the same parameters used in Fig. 7with a spin current, a I /a c = 0 .
1. As in the relaxationphase diagram, the final state seems to be sensitive to ini-tial parameters. However, even in the presence of a smallcurrent (relative to the threshold current a c ), the param-eters region resulting in a switched final state (red region)becomes bigger. This behavior is also captured by the an-alytic model in Fig. 8(b) where the effect of the currentis more evident. A small spin current, a I /a c = 0 .
07, in-creases the region with a switched final state (red region)and a current of a I /a c = 0 .
15 eliminates the sensitivity toinitial conditions and for q/r ≤ . a)b) FIG. 8. a) Relaxation phase diagram as a function of q and φ for a spin current a I /a c = 0 .
05, i.e. 5% of the thresholdcurrent for switching starting from a uniform magnetizationstate. We simulate a 15 nm radius disk for different initialDW positions and phases. The colorscale is the same as inFig. 7. b) Relaxation phase diagram for the parameters q and φ calculated with the ( q, φ )-model for a I /a c = 0 . to switch the disk’s magnetization as a function of thesize of the initial domain for three fixed initial phases φ = 0 , π/ , − π/ q/r , the switching current in-creases monotonically when the size of the domain is re-duced, whereas for bigger domains—small q/r —regionscan be found that switch without current, which corre-spond with the red regions in Fig. 7(b). Similar behaviorfor the minimum current needed to switch the magneti-zation was observed in micromagnetic simulations thatare shown in Appendix Sec. VII E. V. EXPERIMENTAL CONSEQUENCES
The reversal mode and the micromagnetic instabili-ties will influence a pMTJ’s electrical response to voltage . . . . . q/r . . . . . . a I / a c Initial phase φ = 0 φ = π/ φ = − π/ FIG. 9. Switching current as a function of the initial positionof the DW for three different initial phases. Each point indi-cates the minimum current required for magnetization switch-ing. pulses. This is because the junction conductance is re-lated to the position of the DW. To a good approxima-tion (neglecting the finite area of the DW) the junctionconductance depends on the area of the reversed domain: G = g P A ↑ + g AP A ↓ , (8)where g P and g AP are the specific junction conductanceswhen it is magnetized in the parallel and antiparallel tothe reference layer, respectively (units of 1 / (Ωm )). Asthe area of the disk (again, neglecting the finite area ofthe wall) is A = A ↑ + A ↓ : G = ( g AP − g P ) A ↓ + g P A. (9)The last term g P A is independent of time and since weare interested in the time dependence of the conductancewe do not need to consider it further.This shows that the conductance is proportional toarea A ↓ . This area is in turn directly related to the nor-malized perpendicular magnetization of the disk, A ↓ = A ( m z / / m z versus time, e.g. that shown in Figs. 3and 6.Direct imaging experiments may also be possible butare challenging at the length and time scales involvedin these processes . Single shot time resolved electricalstudies would seem to be a promising means of observingthese instabilities. The key model prediction is conduc-tance oscillations that vary in frequency, increasing as thereversal proceeds to completion. In addition the modelpredicts that once a DW is nucleated lower spin currentscan be used to displace it and reverse the magnetization. VI. SUMMARY
In summary, we have considered the DW mediatedmagnetization switching of a disk in the presence of spintransfer torques; a geometry highly relevant to state-of-the art magnetic random access memory based perpen-dicular magnetized magnetic tunnel junctions. The re-sults show a great sensitivity to initial conditions and,in particular, to the DW phase. An analytical modelshows that DW surface tension leads to DW motion inthe Walker breakdown limit. Key features of a simplecollective coordinate model are found in micromagneticsimulations, including sensitivity to initial conditions andDW oscillations in the reversal process. These effectsshould be observable in experiments through measure-ments of tunnel junction conductance versus time. How-ever, noise or other factors may modify the dynamics inreal tunnel junctions. For example, noise can reduce thesensitivity to initial conditions and DW pinning associ-ated with spatial variations in material parameters (e.g.anisotropy, magnetization, etc.) will also effect the dy-namics. These effects can be included in more realisticmodels that build on this research.
ACKNOWLEDGMENTS
We acknowledge useful discussions with ChristianHahn and Georg Wolf. This research was supported inpart by National Science Foundation under Grant No.DMR-1610416. JMB was supported by Spin MemoryInc.
VII. APPENDIXA. Collective coordinate model
An equation for the collective coordinates, ξ j and thegeneralized velocities ˙ ξ j , of a domain wall in disk can bederived from Eq. 1 following the approach outlined inRef. G ij ˙ ξ j + F i − Γ ij ˙ ξ j + S φ = 0 , (10)where: G ij ( ξ ) = J (cid:90) Ω m · (cid:18) ∂ m ∂ξ i × ∂ m ∂ξ j (cid:19) dV , (11) F i ( ξ ) = − (cid:90) Ω ∂U∂ξ i dV = − ∂ U ∂ξ i (12)Γ ij ( ξ ) = αJ (cid:90) Ω (cid:18) ∂ m ∂ξ i (cid:19) · (cid:18) ∂ m ∂ξ j (cid:19) dV (13) S i ( ξ ) = a I J (cid:90) Ω (cid:18) ∂ m ∂ξ i (cid:19) · ( m × p ) dV, (14)where Ω is the region occupied by the ferromagnet, J is the angular momentum density, J = M/γ and U isthe total energy of the system. The spin polarizationis chosen to be perpendicular to the plane of the disk, p = ˆ z , in what follows. Substituting the magnetizationtexture of Eqs. 2 into Eqs. 11-14 and integrating gives the coefficients in Eq. 10 and resulting Eqs. 5 and 6 inthe main text.To solve Eqs. 5 and 6 numerically we rewrite them inthe form: d y dt = F ( y , t ) , (15)that is, we want the differential equations in a form inwhich there are no explicit time derivatives on the righthand side.We thus rewrite Eqs. 5 and 6 as follows:(1 + α ) ˙ φ = − γ t(cid:96)M ∂ U ∂q + α (cid:20) γµ H N φ − πa I (cid:21) (16)(1 + α ) ˙ q = γµ H N φ − πa I α γ t(cid:96)M ∂ U ∂q (17)To make the equations simpler we redefine the time asfollows ˜ t = t/ (1 + α ) and dφd ˜ t = (1 + α ) dφdt (18)If we now write derivatives with respect to ˜ t as φ (cid:48) and q (cid:48) and the equations of motion become: φ (cid:48) = − γ t(cid:96)M ∂ U ∂q + α (cid:20) γµ H N φ − πa I (cid:21) (19) q (cid:48) = γµ H N φ − πa I α γ t(cid:96)M ∂ U ∂q . (20)Walker breakdown corresponds to φ (cid:48) (cid:54) = 0. The first termon the right hand side of Eq. 20 indicates that the domainwall position q oscillates when φ (cid:48) (cid:54) = 0. The condition forWalker breakdown, Eq. 7 in the main text, follows fromEq. 19 with a I = 0.Eq. 20 shows that in the absence of a spin torque( a I = 0) and for magnetic fields less than the Walkerbreakdown field, the wall velocity is set by the phase φ . The maximum velocity occurs when φ = π/ v max = ∆ γµ H N /
4. This is the maximum velocitybefore Walker breakdown and we can write Eq. 20: q (cid:48) ∆ = v max sin 2 φ ∆ − a I π/ α γ t(cid:96)M ∂ U ∂q . (21) B. Domain wall curvature
As noted in Sec. II, there is a term in the energy den-sity associated with the curvature of the DW. This termcomes from the exchange energy, the first term on theright hand side of Eq. 3. Including this term Eq. 22 be-comes: U = (cid:20) σ DW + ∆ µ H N M φ + 2∆ Ar (cid:21) t(cid:96). (22)This term leads to an infinite energy when q = ± r , i.e.when the domain wall is at the boundary of the element.However, this energy only becomes important when thethe DW is within its width ∆ of the edge of this disk. Inthis case, the DW energy can decrease as the wall exitsthe disk. As result the DW energy is always finite andwe can neglect this energy term in our analysis. C. Applied Field
It is straightforward to include applied magnetic fieldsin the model. An applied field H in the z direction leadsto the following set of equations of motion: φ (cid:48) = − γµ H − γ t(cid:96)M ∂ U ∂q + α (cid:20) γµ H N φ − πa I (cid:21) (23) q (cid:48) = γµ H N φ − πa I α (cid:20) γ t(cid:96)M ∂ U ∂q + γµ H (cid:21) . (24)Figure 10 shows the effect of the magnetic field on thefinal magnetization state as a function of the DW’s initialconditions ( q, φ ). Figure 10(a) shows the results for afield applied in the negative z direction and Fig. 10(b)shows the same diagram for a field applied in the oppositedirection. In both cases the magnitude of the field is halfthe coercive field, H = H c /
2, where H c is defined as fieldthat just renders the metastable magnetic state ( q = r or q = − r ) unstable. D. Spin-polarized current with micromagnetics
We also determined the current required for magneti-zation switching from different initial states with micro-magnetics. Figure 11 shows the current needed to switchthe disk’s magnetization as a function of the initial DWposition for φ = 0. The behavior is qualitatively similarto that shown for the ( q, φ ) model in Fig. 9, the currentis small for small q and increases as q increases. Theswitching current is also a non-monotonic function of theDW’s position q . E. Material parameters
The parameters used in both the analytical model andthe micromagnetic simulations were: saturation mag-netization, M = 1 . × A/m, damping constant α = 0 .
03, uniaxial anisotropy constant K p = 1 . × J/m , exchange constant A = 4 × − J/m , a diskdiameter of 30 nm with thickness t = 2 . N zz − / ≈ . . We per-formed micromagnetic simulations using the open-sourceMuMax code with a graphics card with 2048 process-ing cores. We considered the effects of Oersted fields a)b) FIG. 10. Relaxation phase diagram showing the final mag-netization state as a function of the initial conditions for theanalytic model in the presence of an field applied. (a) Fieldin the − z direction and (b) field in the + z direction. In bothcases the applied field magnitude is half the coercive field H = H c / but not finite temperature. For a perpendicular mag-netic tunnel junction, the zero-temperature critical cur-rent density is related to materials parameters as fol-lows: j c = 2 eαµ M H k t/ ( (cid:126) P ), where α is the dampingparameter, e the charge of the electron, H k is the effec-tive perpendicular anisotropy field that depends on thesize of the junction , t is the thickness of the disk and P is the current polarization coefficient. For parameters weused in the calculations j c = 9 . × A/m . We notethat the characteristic field µ H N = 0 . µ H W = αµ H N / F. DW length and reversed domain area
The following are some useful mathematical relationsto compute the length of the DW and reversed domainarea given the DW position. The angle betwen the disk’sradius vector r and DW position q in Fig. 2(a) ϑ (0 ≤ .
05 0 .
10 0 .
15 0 .
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