Non-Maxwell-Boltzmann statistics in spin-torque devices: calculating switching rates and oscillator linewidths

Transcription

1 Non-axwell-Boltzmann tatitic in pin-torque device: calculating witching rate and ocillator linewidth P. B.Vicher and D.. Apalkov Department of Phyic and Atronomy Thi project wa upported by NSF grant # ECS , DOE Computational aterial Science Network, and DR grant #

2 Introduction & Outline Relevance of pin torque: Application include witching reader noie GHz ocillator Approache to prediction of pin torque effect LL imulation ( brute force ) calculate energy ditribution (Fokker-Planck equation) Preent work Extend Brown FP equation to pin torque convert to equation for energy ditribution Solve in mall-ocillation limit ( effective-temperature theory) olve for non-equilibrium teady tate when amplitude i large; ueful for Equilibrium witching rate (telegraph noie) GR reader noie GHz pin-torque ocillator olve tranient problem: witching by nanoecond current pule

3 Landau-Lifhitz equation for pin-torque device LL equation for (magnetization of free layer) i modified to include Slonczewki pin torque: & = & + & + & + & con LL where LL damping torque i Slon ( m H) & = αγ mˆ ˆ & LL S rand H (α i the LL damping contant, i the aturation magnetization) = random thermal torque rand & LL Conervative torque -- γ i the gyromagnetic ratio & = γ H con ( mˆ mˆ ) & Slon = γj Smˆ p (Slonczewki form of the pin torque -- J ~ current denity) Thin F layer (Co,..) N Spacer Thick F layer (pinned) p electron current GR pillar device (Albert et al, 2000) Unit vector: mˆ ˆm p = / (free layer) = p / (pinned layer)

4 Landau-Lifhitz Trajectorie on the -phere The magnetization of the free layer i repreented by a point on a phere of radiu ; the Stoner- Wohlfarth orbit (no damping or pin torque) i a cloed curve on the phere. It i convenient to draw the orbit (contant-energy contour) on a planar projection of the phere: Unwitched energy well Switched energy well & Switching proce: pin torque increae energy in left (red) well until i on black orbit, which carrie it to right well where LL damping trap it. The LL damping torque move toward lower energie The conervative torque move along the orbit The Slonczewki pin torque move toward higher energie

5 Fokker-Planck Theory FP equation decribe evolution of probability ditribution for magnetization (repreent enemble by dot on the phere, denity i ρ) dot: ρ(,t=0) ρ t (, t) = J (, t) (continuity equation) where convection diffuion J(, t) = ρ(, t) & D ρ(, t) d dt preceion γ α = γ H Landau-Lifhitz damping ( H) + & pin torque SpinTorque

6 Fokker-Planck Equation in Energy Frequently ρ(,t) depend only on energy: we have for the firt time derived a FP equation in energy. Thi alo take the form of a continuity equation for a 1D probability ditribution ρ (E): γ ( ) ρ '( E, t ) S P E = j E ( E, t ) µ 0 t E where j E (E,t) i the number of ytem per unit time croing a contant-energy contour. Sytem change energy becaue LL damping give energy lo proportional to α: αγ H cond Spin torque increae energy, proportional to current J: + γj mˆ p d mˆ J 2 Thi can be repreented by an effective α: αeff = α η( E) 1.5 mˆ p [ ] where ( ) d mˆ η(e) η E = 1 i a dimenionle pintorque damping coefficient 0.5 H cond 0 (ratio of work done by pin torque to work done by LL damping) E/K Computing the energy-current j E from the 2D probability current j, and etting j E (E,t)=0 (teady tate) give ln ρ' ( E) γ η( E) E = D S α J [ ]

7 ln ρ' E ( E) γ η( E) D S Steady-tate energy ditribution Fokker-Planck equation in energy: = α J Vβ ( E) E/K If we approximate η(e) by a contant independent of E, the FP equation above ha the trivial (Boltzmann) olution ρ = exp(-βe) where β i an effective invere temperature. η(e) η ( E) = work done by pin torque work done by LL damping Thi mall-amplitude approximation i often adequate at low current (example: telegraph noie rate) However, the variation of η(e) with E lead to entirely new phenomena at higher current. Dwell time (n) Dwell time in well #1 S. Urazhdin et al [Phy. Rev. Lett. 91, (2003)] (quare and circle) Line: fit uing mall-amplitude approx. multilayer: Py(20 nm)/cu(10)/py(6) lateral dimenion 130x60 nm 2. Current I, ma Dwell time in well #2

8 T=0 P E Behavior at T=0 and T= 20K (table) φ Saddle E E P well bottom P table I D table E ln ρ' E P untable ( E) γ = D I w (T=0) S effective damping η α ( E) J T=20K 0 E ln ρ P current P metatable τ ~ e 1100 n; ln τ/n~1100 table D metatable ln ρ I w (T=20K) D ln τ/n~23 (τ ~ 1; witche)

9 Switching at T=300 K: No Dynamical State At higher temperature, the ytem witche before reaching the dynamical tate. 0 I w I D current E ln(ρ) P P mot probable in equilibrium equally probable in equilibrium, but P metatable P ln τ ~ 23 witche

10 Concluion Fokker-Planck theory ha been extended to include non-conervative current-induced pin torque Small-amplitude approximation work for low current, but doe not decribe witching quantitatively and doen t decribe dynamical preceion (GHz noie) at all Full nonlinear theory allow calculation of noie pectra and ocillator linewidth, a well a pintorque witching rate.