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`Enormous thanks to:Allan MacDonald Joaquin Fernàndez-Rossier Maxim Tsoi Alvaro Núñez Jairo SinovaOutlineSlonczewksi torque theory and comparison with experiments. Spin accumulation theory and comparison with experiments. Calculation of spin wave properties with Slonczewski torque. Spin transfer without spin conservation.Experimental Evidence of Spin TransferPredicted theoretically by Slonczewksi and Berger in 1996IMyers et al, Science, 285, 867 (1999)Spin Transfer Torque Picture&lt;S&gt; v vM1M2The spin of the conduction electron is rotated by its interaction with the magnetization.This implies the magnetization exerts a torque on the spin. By Conservation of angular momentum, the spin exerts an equal and Opposite torque on the magnetization.A more careful way of formulating spin transfer is To write down an equation of motion for spin.Continuity requires density to change when there is a nonzero current divergenceSpin density (magnetization)Spin currentCurrent dependent equation of motion for magnetizationStiles, Phys. Rev. B, 66, 014407 (2002)Landau-Lifshitz Equationv v E v v &amp; M = -M × v -  M × M M·BMv v v v v2 ^ E[M] = -M  Bapp - KM  c + A MExternal field Anisotropy StiffnessSpin Currents in TrilayersV1, + V2,t  + t r  + r V1,V2,Landau Lifschitz with Spin Transfer TorqueWhen all the spin fluxes are evaluated for the scattering problem, one finds a simple result.t v ^ ^ ^ Q = N( ST )1,2 = gI 1,2 × 1 × 2()New term in Landau Lifshitz equationIn experiments, one layer is pinned.Slonczewski, J. Magn. Magn. Mat., 159, L1, (1996).The modified Landau Lifshitz equation^ 1 d dtv ^ × Beff -   ×  × Beff + gI  ×  ×  fixed ^ ^ v ^ ^ ^ = -B()()The damping and spin transfer term fight with each other to determine if M will become aligned or anti-aligned with B (in the simplest geometry).I c   Beff tUsing new L.L. equation to understand experimental dataThickness (nm)I c  BeffTsoi, Phys. Rev. Lett., 93, 036602, (2004)Ic  tAlbert et al., Phys. Rev. Lett., 89, 226802, (2002)Current BKiselev et al., Nature, 425, 380, (2003)&quot;W&quot;Slonczewski Torque describes experiment wellWhat about the `W' region?Stable point of L.L. Equation under currentBazaliy, Jones, Zhang, Phys. Rev. B, 69, 094421, (2004)ExperimentLandau Lifshitz with Slonczewski torqueSpin transfer in continuous caseexp [i ( w ( k ) t - kx ) ]Spin waves Spin current modifies spin wave spectrum in half metallic case '( k ) =  ( k ) + gJ S kStatic solutions to L.L. equations (like domain walls) move with velocity proportional to j.Bazaliy et al, Phys. Rev. B, 57, R3213 (1998). Rossier et al, cond-mat 0311522. Tatara et al, PRL, 92, 086601, (2004).Slonczewski torque term does a good job in describing experiments. There are some experiments which may not be well described by this term. There is another way of understanding spin transfer that may be useful.Diffusive spin transport in ferromagnetsOne can write down macroscopic transport equations For the spin currents. In steady state:e J ± = µm - µ±  ± x l sf 2 J± = -Steady state &quot;equation of continuity&quot; ± µ ±e xOhm's Law Solve the above for µ up and down, For a fixed J total, subject to continuity, etc.Valet, Fert, Phys. Rev. B, 48, 7099, (1993).There's more up current Coming in than going out ­ A net flux of up spin currentThis is counterbalanced In steady state By a nonzero µ, Which makes spins flip from Up to down.F µ = J  e (  F lSF )Explain how this can generate spin waves ­ write down expression for delta muSpin flips of conduction electronsµµh = µk = B + Ak 2Berger, Phys. Rev. B, 54, 9354, (1996) Tsoi, Phys. Rev. Lett., 80, 4281, (1998)µ must have right sign (&gt;0) to create spin waveSpin accumulation featuresThe critical current is inversely proportional to R Low k modes are generally excitedJ= µ  e ( lsf )1 Jc  Rµ &gt; B + Ak 2kPoint contact gives low k modesIndeed, low k modes are excited...Rippard et al., Phys. Rev. Lett., 92, 027201, (2004).Dependence of Ic on temperatureUsually damping increases with temperature*, so Ic would also increase with temperature.Tsoi, Phys. Rev. B, 69, 100406(R), (2004). *Stutzke et al., Appl. Phys. Lett., 82, 91, (2003)I c   (T ) ?Spin Accumulation picture seems to do better in this caseIc and 1/RTemperatureCritical current here is inversely proportional to the resistanceThe resistance goes up with increasing temperature, so the critical current will go down.Problems with spin accumulation picture.How can spin waves explain magnetization reversal? How can spin waves become coherent?Slide with chart of results and theoriesSpin Torque(I,B) phase diagram ­ switching and precession for pillar Ic proportional to damping? High k spin wave modes excitedSpin accumulationSpin wave =&gt; switching? Ic inversely proportional to R Low k spin wave modes excitedOther formulations of spin transfer:Shpiro, Levy, Zhang, Phys. Rev. B, 67, 104430 (2003). Heide, Phys. Rev. Lett., 87, 197201, (2001)...The Slonczewski torque predicts high frequency excitationSlonczewski considered the point contact geometry to linear order.Valid for linear order in the transverse magnetizationSlonczewski, J. Magn. Magn. Mat., 195, L261, (1999).Details of point contact calculation^ ^ d ^ × ( Beff ) -    × d   + gI ( x ) ×  ×  ^ ^ ^ ^ = -    dt dt   ()I ( x ) = I for x &lt; a I ( x ) = 0 for x &gt; aHDetails of point contact calculation u + (  - b - i  + ij )u = 02Dimensionless, linear L.L. equation with scaled variables x = Re[u ]  y = Im[u ]Consider harmonic time dependence Linearize the above about Mz=1, write transverse Magnetization as a complex number( + p )uin = 02 2p 2 = b -  - i + ij k 2 = b -  - i( + k )uout = 02 2Do 1-d caseuin ( r ) = sin( pr ) uout ( r ) = eikr p 2 = b -  - i + ij k = b -  - i2Boundary conditions on u Give the eigenvalue equation:ptan( p ) = ikAt the critical current,  is purely real (recall u has exp(-it) dependence). So find (,j) real so that eq. (1) is satisfied.For =0 case, a solution is: = 3.82305j = 3.35239  ka = 1.95604Steady state spin waves at the critical current=3a, as expected. Now investigate with greater current...For I&gt;Ic, the behavior can be much different.Time and length Scales increase by an order of Magnitude for I=1.2 Ic. This is sensitive to initial Conditions.B=.001 Alpha=.01  Ak + B2We may conclude that the Slonczewski torque in point contact geometries need not excited high k spin waves in general.Outstanding questions:What determines wavelength and frequency generally? (Current, wall width, etc). Do 2-d case.Slide with chart of results and theoriesSpin Torque(I,B) phase diagram ­ switching and precession for pillar Ic proportional to damping? High k spin wave modes excitedSpin accumulationSpin wave =&gt; switching? Ic inversely proportional to R Low k spin wave modes excitedThe real situation is more complicated than either one!Future workCurrent spreading Demagnetization effects98765432100123456789Another way to look at spin transferSpin transfer without spin conservationds/dt=0 for steady state. So the divergence of J equals the precession around the local moment.Nunez, MacDonald, cond-mat/0403710Spin transfer without spin conservationThe proposal is to say that the collective coordinate will precess around the nonequilibrium quasiparticle spins, in the same way that the nonequilibrium quasiparticle spins precess around the collective coordinatev vC v  tot =  ×  trCan evalute nonequilbrium magnetization directly: with Landauer formalism, or Keldysh formalism, etc. Núñez et al. showed spin torque strong in systems with spin-orbit couplingToy model descriptionConsider ballistic transport, and determine Spin density of transport states. Use Landauer formalism and take incoming Electrons to be eigenstates of 1-d tight-binding Hamiltonian.H FM^ p2 1 v v = - ( x ) 2m 2Spin Current0.02 0.015 0.01 0.005 Qx Qy Qz0.14 0.12 0.1 0.08 0.06S x S y SzSpin Density&lt;S&gt;Q0.04 0.02 00 -0.005 -0.01 -0.015 0-0.02 -0.0450100150200-0.06 050100150200positionPositionThe wave function above satisfies the following, As it should for steady state (energy Eigenfunction)unpolarizedt v Q + S ×  = 0Noncollinearity is Essential to spin transfer! In the low k limit, Ein = .01  spin filter: delta = .02Spin spiral for continuous magnetization1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 20 40 60 80 100 120S x S y SzEin = .19, =.22Spin Torque vs. Theta1 0.9 0.8 0.7LSpin torque for incoming electrons Polarized upTorque0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 140 160 180LThetaThe above is still in the low k limit, Ein=.19, delta=.21Future work with this model: Can investigate dynamics for case of no spin conservation...Movie of M(t)We can solve for the spin density assuming order parameter dynamics Are adiabatic compared to quasiparticle dynamics. Can then solve for the order parameter dynamics with AA's prescription...No units labelled! The length scales for L.L. dynamics and Sch. Equation are quite different... How to resolve??ConclusionThe (I,B) phase space behavior is well described by Slonczewski torque. Other experimental features aren't explained with simple torque model, but an alternative explanation is supplied by spin accumulation Local torque model need not imply high k spin waves in point contact geometries We have a theory and a calculation for systems in which spin is not conserved Can we find other qualitative features of data which simple models can help elucidate?`

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