7. Basics of Magnetization Switching

Transcription

1 Beyond CMOS computing 7. Basics of Magnetization Switching Dmitri Nikonov 1

2 Outline Energies in a nanomagnet Precession in a magnetic field Anisotropies in a nanomagnet Hysteresis and astroid Spin torque and oscillations 2

3 Spin Dynamics and Its Uses Nuclear Magnetic Resonance Imaging (NMR MRI) Pre-requisite to understanding spin transfer torque Nobel Prize, physics I. Rabi, NMR, 1944 F. Bloch, E. Purcell, NMR, 1952 P. Kusch, magnetic moment of electron, 1955 Ideas re-used in lasers. Nobel Prize, chemistry R. Ernst, NMR spectroscopy, 1991 K. Wüthrich, NMR for biomolecules, 2002 Nobel Prize, medicine P. Lauterbur, P. Mansfield, MRI,

4 Levels of Description Macrospin Micromagnetics (nanomagnetics) m z m x m y A nanomagnet described by one vector. This lecture Looking into distribution of magnetization inside. Later lecture 4

5 Spin and Magnetic Moment e- magnetic moment spin m S g g B ( 1) 2 2 spin magnetic moment, (-1) for negative charge of an electron g-factor, depends on the material B e A m 2m e 24 2 Bohr magneton, elementary current * loop area From here we use spin in place of magnetic moment. 5

6 Energies in a Nanomagnet 6

7 Zeeman Energy in Magnetic Field Two quantum levels, up and down spin. 1 U Zee m B gbb 2 g B gyromagnetic ratio, G=Lande factor B 1 2 g B 1 2 E / V M B Zee s B g B B H 0 ext External magnetic field from - coil; - wires on chip; - neighbor nanomagnets Energy per volume Ms=magnetization B 7

8 Unpolarized Spins + = e- e- 2e- Statistical sum of two spin polarized states can be an unpolarized state. 8

9 Equilibrium Polarization Spin polarization u d m P u P d m 0 m 0 P mb m tanh kt B 2 m B kt B P Equilibrium polarization Probability to be in an energy level from Boltzmann statistics P general, for spin=1/ m / 4 g B in the limit of weak field, prop. to magnetic field, inversely to temperature u mb exp 1/ P kt B d m0 m in the limit of strong field, saturates to the maximum value 9

10 Equations for Precession m t x m t y m B m B y z z y m B m B z x x z m t [ m B ] eff m t z m B m B x y y x B eff H 0 eff U dm effective magnetic field, valid for any energy Classical equations, derived from Hamiltonian mechanics 10

11 Precession in Magnetic Field B Precession = Rabi oscillations of projections B=[ ] 11

12 Precession in Magnetic Field 12

13 Relaxation, Bloch Equations m t m t x y m [ m Beff ] x T m [ m Beff ] y T x 2 y 2 Random interactions = stray magnetic fields, spin-orbit coupling, magnetic impurities. Approximation of constant relaxation rate works very well. m z z 0 [ eff ] m m B m z t T1 T1. Longitudinal relaxation. Spin relaxation. Spin-lattice interaction. T2.Transverse relaxation. Spin dephasing. Decoherence. Spin-spin interaction. Solution known, can extract relaxation rates from response. m ( t) m 1 e m z x y 0 tt t T2 ( t) m0 xe cos( t 0 ) m t m e t tt2 ( ) sin( ) 0y

14 Damped Precession B Spin projections are damped to equilibrium values B=[ ] 1/T1=0.01 1/T2=

15 Damped Precession 15

16 Magneto-Crystalline Anisotropy Spins in a solid have different energy depending on direction. Influenced by crystal structure (3-axial anisotropy) strain (uniaxial anisotropy). Can be induced by magnetic annealing. E / V ~ K mcry U K V m m m m m m ( x y x z y z ) U K V m 2 u u (1 u ) K~1e2 to 1e6 J/m 3 projection on the axis of anisotropy m u 16

17 Precession with Anisotropy B Ku Anisotropy forces the spin to tilt, compromise with the magnetic field B=[ ] 1/T1=0.01 1/T2=0.03 Ku=3e5J/m^3 17

18 Precession with Anisotropy 18

19 Ferromagnets, Spin Polarization spins align = magnetization M mn lower energy for spins along magnetic field internal magnetic field Spin polarization close to saturation, if well below Curie temperature Tc. m ~1 Estimate of internal magnetic field, for Tc=1000K, much more than any external field! B kt B B C 1500T 19

20 Bloch vs. LLG equations 1 T Bloch Equations m 1 [ m Beff ] ( m m0) t T for magnetic field Bz 1 T T 2 0 m T 1 m 0 1.Describes paramagnetic solids, liquids, or gases. 2.Does not preserve magnetization. 3.Damping has transverse and longitudinal terms re. magnetic field. Landau-Lifshitz-Gilbert Equation m t m [ m Beff ] m t 1.Describes ferromagnetic solids below Curie temperature. 2.Preserves magnetization. 3.Damping only changes direction, uniformly re. to magnetic field. For a unified picture see: example Phys. Rev. B 74, (2006). 20

21 Dipolar Energy Intuitive: magnetization poles pointing out of the surface create magnetic field. Between nanomagnets, esp. layers in the same pillar = dipolar field or Nanomagnet with itself = demagnetization Promotes magnetization along the surface. E demag s / V ~ 2 D 2 0M t D=size, t=thickness 21

22 Demagnetization = Shape Anisotropy Lz Lx Ly xx 0 U dem V M N yy 2 N Demagnetization tensor N N xx N yy N zz 1 N zz M Smallest energy to point along the longest axis. L L L x y z N N N xx yy zz 22

23 Landau-Lifshitz-Gilbert equations Ms=magnetization effective field damping dm dm eff dt M B M M Damping = dt s phenomenological description g B gyromagnetic factor Effective Field Damping B eff E M E = all energy parts. (See prior slides) Precession M Spin-Transfer 23

24 Precession of a ferromagnetic particle B Spin approaches the direction of the magnetic field, keeps the magnitude (i.e. wraps around a sphere) B=[ ] alpha=

25 Precession of a ferromagnetic particle 25

26 Precession of a Nanomagnet B Demagnetization forces the spin to the plane and then aligns it to the longest axis B=[ ] alpha= x60x3nm Nxx=0.0279;Nyy=0.0731; Nzz=0.8990; 26

27 Precession of a Nanomagnet 27

28 Magnetic Field from a Wire Biot Savar law 0Ir, r a 2 2 a Br ( ) 0I, r a 2 r a=radius of the wire a=30nm, r=3a, B=0.01T, need I=45mA. 28

29 Switching by Magnetic Field B Magnetic field forces the spin from one to the other stable state along the long axis; Ringing determined by slow damping. B=[ ] alpha= x60x3nm Nxx=0.0279;Nyy=0.0731; Nzz=0.8990; 29

30 Switching by Magnetic Field 30

31 Magnetization Switching in MRAM The element at the crossing of a word line and a bit line switches. One field should be not enough to switch. Range of fields for switching 31

32 Hysteresis and Terminology 32

33 Hard and Soft Materials Also magnetic RAM 33

34 In-Plane Energy Very high energy for out-of plane magnetization. Approximation: only in-plane direction. E V K eff 2 sin 0M s H Effective in-plane anisotropy. Depends on the in-plane shape, aspect ratio. ex cos H K eff K u M N N / s yy xx Anisotropy field related to coercive field H K 2K eff M 0 s 34

35 Stoner-Wohlfarth Astroid 35

36 Astroid vs. Hysteresis Reversal (switching) field = abrupt jump of M Coercive field = when M.H=0 36

37 Limitations of Macrospin 37

38 Experimental Astroids 38

39 Landau-Lifshitz-Gilbert with spin torque effective field damping spin transfer dm dm eff dt M B M M dt g( ) M M p s polarization current Effective Field Damping PI 2 s em V Precession M Spin-Transfer magnetization volume 39

40 Direction of Spin Torque current g M M p torque M p Results in a component of pinned layer magnetization (p) perpendicular to the free layer magnetization (M). I.e. torque is ZERO for collinear magnetizations Rule: current from free to pinned layer tries to align the free layer magnetization to the pinned layer magnetization 40

41 Both magnetic field and current 41

42 Spin Torque Oscillator ~50 nm Cu ~100 nm Oscillation region Free layer Spacer Fixed layer i B No oscillation magnetic field Permanent current. M oscillating around one equilibrium. Frequency of oscillations depends on current. 42

43 Summary Dynamics determined by energies in a nanomagnet Magnetic field causes precession of nanomagnets Anisotropies cause evolution to stable states As external magnetic field is varied, nanomagnets exhibit hysteresis Spin torque and magnetic field can cause steady-state oscillations in nanomagnets 43