Beyond CMOS computing

7. Basics of Magnetization Switching

Dmitri Nikonov

[email protected]

1 Nikonov 7. M Dynamics Outline

 Energies in a nanomagnet

 Precession in a magnetic field

 Anisotropies in a nanomagnet

 Hysteresis and astroid

 Spin torque and oscillations

2 Nikonov 7. M Dynamics Spin Dynamics and Its Uses

Nuclear Magnetic Resonance Imaging Nobel Prize, physics (NMR  MRI) 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 bio- Pre-requisite to molecules, 2002 understanding Nobel Prize, medicine spin transfer P. Lauterbur, P. Mansfield, torque MRI, 2003

3 Nikonov 7. M Dynamics Levels of Description Macrospin Micromagnetics (nanomagnetics)

mz



 mx

my

A nanomagnet described by one Looking into distribution of vector. magnetization inside. This lecture Later lecture

4 Nikonov 7. M Dynamics Spin and Magnetic Moment 1 S  spin magnetic 2 moment 1 magnetic moment, (-1) for negative charge m  gB (  1) e- 2 of an electron g  2 g-factor, depends spin on the material

e 24 2 Bohr magneton, B  9.27  10 Am  elementary 2me current * loop area

From here we use “spin” in place of “magnetic moment”.

5 Nikonov 7. M Dynamics Energies in a Nanomagnet

6 Nikonov 7. M Dynamics Zeeman Energy in Magnetic Field Two quantum levels, 1 “up” and “down” spin. gBB B 2 1 1 UZee mB    g B B  gB 2 2 B

gB gyromagnetic ratio,   G=Lande factor B  0H ext External magnetic field from - coil; - wires on chip; - neighbor nanomagnets

EVMBZee/  s Energy per volume Ms=magnetization

7 Nikonov 7. M Dynamics Unpolarized Spins

e- + e- = 2e-

Statistical sum of two spin polarized states can be an unpolarized state.

8 Nikonov 7. M Dynamics Equilibrium Polarization

Spin polarization Probability to be in an energy level from Boltzmann statistics PP m  ud mB PP ud PPudexp 1/ kTB Equilibrium polarization

mB 2 2 2 general, for spin=1/2 m0  m tanh  m g B / 4 kTB m2 B m0  in the limit of weak field, prop. to magnetic field, kTB inversely to temperature

m0  m in the limit of strong field, saturates to the maximum value

9 Nikonov 7. M Dynamics Equations for Precession

mx m   my B z  m z B y    []mB  t  t eff m y   m B  m B  t z x x z

mz   mx B y  m y B x  t

U BH   eff0 eff dm effective magnetic field, valid for any energy

Classical equations, derived from Hamiltonian mechanics

10 Nikonov 7. M Dynamics Precession in Magnetic Field B

Precession = Rabi oscillations of B=[0 0 0.12] projections

11 Nikonov 7. M Dynamics Precession in Magnetic Field

12 Nikonov 7. M Dynamics Relaxation, Bloch Equations

mmxx   []mB eff x  Random interactions = stray magnetic tT2 fields, spin-orbit coupling, magnetic mm impurities. yy  []mB   Approximation of constant relaxation tTeff y 2 rate works very well.

mzz m m0   []mB eff z  tT1 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.

tT1 mz ( t ) m0  1 e 

t T2 mx (t)  m0 xe cos(t  0 )

tT2 myy( t ) m00 e sin( t )

13 Nikonov 7. M Dynamics Damped Precession B

Spin projections are damped to B=[0 0 0.12] equilibrium values 1/T1=0.01 1/T2=0.03

14 Nikonov 7. M Dynamics Damped Precession

15 Nikonov 7. M Dynamics 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.

EVKmcry /~

2 2 2 2 2 2 U33 K V() mx m y  m x m z  m y m z

2 Uu K u V(1 m u ) K~1e2 to 1e6 J/m3

projection on the axis of anisotropy mu

16 Nikonov 7. M Dynamics Precession with Anisotropy B

Ku

Anisotropy forces the spin to tilt, B=[0 0 0.12] compromise with the magnetic 1/T1=0.01 field 1/T2=0.03 Ku=3e5J/m^3

17 Nikonov 7. M Dynamics Precession with Anisotropy

18 Nikonov 7. M Dynamics Ferromagnets, Spin Polarization

spins align = magnetization M mN

lower energy for spins internal magnetic field along 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 !

kT BTBC 1500 B

19 Nikonov 7. M Dynamics Bloch vs. LLG equations

Bloch Equations Landau-Lifshitz-Gilbert Equation m 1 mm   []()m  B  m  m  []m  Beff  m  tTeff 0 tt

for magnetic field Bz

1 T2 0 0  0 1   0 1 T 0  m  0 T  2  0   0 0 1 T1  m0 1.Describes paramagnetic solids, 1.Describes ferromagnetic solids liquids, or gases. below Curie temperature. 2.Does not preserve magnetization. 2.Preserves magnetization. 3.Damping has transverse and 3.Damping only changes longitudinal terms re. magnetic direction, uniformly re. to field. magnetic field.

For a unified picture see: example Phys. Rev. B 74, 094436 (2006).

20 Nikonov 7. M Dynamics 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.

 Mt2 EV/~0 s demag 2D D=size, t=thickness

21 Nikonov 7. M Dynamics Demagnetization = Shape Anisotropy Ly N xx 00  UVN 0 MM00 dem yy 2  Lz 00N zz Demagnetization tensor N NNN   1 Lx xx yy zz

Smallest energy to point along the longest axis.

LLLx y z

NNNxx yy zz

22 Nikonov 7. M Dynamics Landau-Lifshitz-Gilbert equations

Ms=magnetization effective field damping ddMM   Damping =   M × Beff  M × dt M dt phenomenological s description

gB gyromagnetic   Effective Field factor Damping

 E B  eff  M Precession Spin-Transfer E = all energy parts. M (See prior slides)

23 Nikonov 7. M Dynamics Precession of a ferromagnetic particle B

Spin approaches the direction of B=[0 0 0.12] the magnetic field, keeps the alpha=0.03 magnitude (i.e. wraps around a sphere)

24 Nikonov 7. M Dynamics Precession of a ferromagnetic particle

25 Nikonov 7. M Dynamics Precession of a Nanomagnet B

Demagnetization forces the spin to B=[0 0 0.12] the plane and then aligns it to the alpha=0.01 longest axis 120x60x3nm Nxx=0.0279;Nyy=0.0731; Nzz=0.8990;

26 Nikonov 7. M Dynamics Precession of a Nanomagnet

27 Nikonov 7. M Dynamics Magnetic Field from a Wire Biot – Savar law   Ir a=radius 0 , ra  2 a 2 of the wire Br()   I  0 , ra  2 r

a=30nm, r=3a, B=0.01T, need I=45mA.

28 Nikonov 7. M Dynamics Switching by Magnetic Field B

Magnetic field forces the spin from B=[-0.06 0 0] one to the other stable state along alpha=0.01 the long axis; Ringing determined 120x60x3nm by slow damping. Nxx=0.0279;Nyy=0.0731; Nzz=0.8990;

29 Nikonov 7. M Dynamics Switching by Magnetic Field

30 Nikonov 7. M Dynamics Magnetization Switching in MRAM

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

31 Nikonov 7. M Dynamics Hysteresis and Terminology

32 Nikonov 7. M Dynamics Hard and Soft Materials

Also magnetic RAM

33 Nikonov 7. M Dynamics In-Plane Energy

Very high energy for out-of plane magnetization. Approximation: only in-plane direction.

2 E  V Keff sin   0M s H ex cos  H  Effective in-plane anisotropy. Depends on the in-plane shape, aspect ratio.

2 Keff  Ku  0M s N yy  N xx / 2

Anisotropy field – related to coercive field 2Keff H K  0M s

34 Nikonov 7. M Dynamics Stoner-Wohlfarth Astroid

35 Nikonov 7. M Dynamics Astroid vs. Hysteresis

Reversal (switching) field = abrupt jump of M

Coercive field = when M.H=0

36 Nikonov 7. M Dynamics Limitations of Macrospin

37 Nikonov 7. M Dynamics Experimental Astroids

38 Nikonov 7. M Dynamics Landau-Lifshitz-Gilbert with spin torque

effective field damping spin transfer ddMM    M × B  M ×   M × M × p eff   dt Ms  dt g() Effective Field polarization current Damping  PI  Precession Spin-Transfer 2eM V s M magnetization volume

39 Nikonov 7. M Dynamics Direction of Spin Torque  M  M  p current g

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

40 Nikonov 7. M Dynamics Both magnetic field and current

41 Nikonov 7. M Dynamics Spin Torque Oscillator

~50 nm ~100Cu nm Oscillation region Free layer Spacer i Fixed layer B No oscillation

magnetic field Permanent current. M oscillating around one equilibrium. Frequency of oscillations depends on current.

42 Nikonov 7. M Dynamics 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 Nikonov 7. M Dynamics