Micromagnetics 101

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Micromagnetics 101
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Spin model Each site has a spin Si

• There is one spin at each site.
• The magnetization is proportional to the sum of
  all the spins.
• The total energy is the sum of the exchange
  energy Eexch, the anisotropy energy Eaniso, the
  dipolar energy Edipo and the interaction with the
  external field Eext.

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Exchange energy

• Eexch-J?I,d Si Si?
• The exchange constant J aligns the spins on
  neighboring sites ?.
• If Jgt0 (lt0), the energy of neighboring spins will
  be lowered if they are parallel (antiparallel).
  One has a ferromagnet (antiferromagnet

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Magnitude of J

• kBTc/zJ¼ 0.3
• Sometimes the exchange term is written as A s d3
  r r M(r)2.
• A is in units of erg/cm. For example, for
  permalloy, A 1.3 10-6 erg/cm

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Interaction with the external field

• Eext-g?B H S-HM
• We have set M?B S.
• H is the external field, ?B e/2mc is the Bohr
  magneton (9.27 10-21 erg/Gauss).
• g is the g factor, it depends on the material.
• 1 A/m4? times 10-3Oe (B is in units of G) units
  of H
• 1 Wb/m(1/4?) 1010 G cm3 units of M (emu)

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Dipolar interaction

• The dipolar interaction is the long range
  magnetostatic interaction between the magnetic
  moments (spins).
• Edipo?i,j MiaMjb?a,b/R3-3Rij,aRij,b/Rij5
• Edipo?i,j MiaMjb?ia?jb(1/Ri-Rj).
• Edipos r M( R) r M(R)/R-R
• If the magnetic charge qM-r M is small Edipo
  is small

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Anisotropy energy

• The anisotropy energy favors the spins pointing
  in some particular crystallographic direction.
  The magnitude is usually determined by some
  anisotropy constant K.
• Simplest example uniaxial anisotropy
• Eaniso-K?i Siz2

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Modifies Landau-Gilbert equation

• ? M /? t - ? MH rJm -? M/? h
• ? is the thermal noise.
• Ordinarily the magnetization current Jm is zero.
• H is a sum of contributions from the exchange,
  Hex the dipolar Hdipo, the anisotropy and the
  external field HHe Hex Hdipo Han HexJr2M
  HanK M0.

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Some mathmatical challenges

• The dipolar field is long range
• different scheme has been developed to take care
  of this. These include using fast Fourier
  transforms or using the magnetostatic potential.
  For large systems, the implicit scheme takes a
  lot of memory.
• Preconditioner Just the exchange. (it is
  sparse.) Physically the exchange energy is
  usually the largest term.

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Alternative approach

• Monte Carlo simulation with the Metroplois
  algoraithm.
• This is the same as solving the master equation
  dP/dtTP where T is the transition matrix.

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Physical understanding
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Three key ideas at finite temperatures

• Nucleation
• Depinning
• Spins try to line up parallel to the edge because
  of the dipolar interaction. The magnetic charge
  is proportional to , and this is reduced.

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Approximation

• Minimize only the exchange and the anisotropy
  energy with the boundary condition that the spins
  are parallel to the edge.

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Two dimension

• A spin is characterized by two angles ? and ?. In
  2D, they usually lie in the plane in order to
  minimize the dipolar interaction. Thus it can be
  characterized by a single variable ?.
• The configurations are then obtained as solutions
  of the imaginary time Sine-Gordon equation
  r2?(K/J) sin?0 with the parallel edge b.c.

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Edge domain Simulation vs Analytic approximation.

• ?tan-1 sinh(?v(y-y0))/(- v
  sinh(?(x-x0))),
• yy/l, xx/l the magnetic length lJ/2K0.5
  ?1/1v20.5 v is a parameter.

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Closure domain Simulation vs analytic
approximation

• ?tan-1A tn(? x', ?f) cn(v 1kg20.5y', k1g)/
  dn(v 1kg20.5 y', k1g),
• kg2A2?2(1-A2)/?2(1-A2)2-1,
• k1g2A2?2(1-A2)/(?2(1-A2)-1),
• ?f2A2?2(1-A2)2/?2(1-A2)
• v2?2(1-A2)2-1/1-A2.
• The parameters A and ? can be determined by
  requiring that the component of S normal to the
  surface boundary be zero

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For Permalloy

• For an important class of magnetic material, the
  intrinsic anisotropy constant is very small.
• r2?0. For this case, conformal mapping ideas
  are applicable.

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An example

• Constraint M should be parallel to the boundary!
• For the circle, a simple solution is ?tan-1y/x.
• Conformal mapping allows us to get the
  corresponding solution for the rectangle.

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Current directions

• Current induced torque
• Magnetic random access memory

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Nanopillar Technique (Katine, Albert, Emley)
Au (10 nm)
-Multilayer film deposited (thermal evaporation,
sputtering) on insulating substrate
Co (3 nm)
Cu (6 nm)
Co (40 nm)
Cu (80 nm)
-Electron-beam lithography, ion milling form
pillar structure (thicker Co layer left as
extended film)
-Polyimide insulator deposited and Cu top lead
connected to pillar
Cu
Polyimide insulator
-Current densities of 108 A/cm2 can be sent
vertically through pillar
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Magnetic Reversal Induced by a Spin-Polarized
Current
Large (107-109 A/cm2) spin-polarized currents
can controllably reverse the magnetization in
small (lt 200 nm) magnetic devices
Antiparallel (AP)
Positive Current
Parallel (P)
Cornell THALES/Orsay NIST
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Modifies Landau-Gilbert equation

• ? M /? t - ? MH rJm -? M/? h.
• The magnetization current Jm is nonzero.

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Charge and magnetization current

• Je-?r V -e Dr ? n -DM r (?M p0)
• J- ?M r ( V p0) - DM' r ?M - D' r ( ? n p0)
• p0M0/M0 M0 is the local equilibrium
  magnetization,
• V-ErW W(r)s d3r' ? n(r')/r-r

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Two perpendicular wires generate magnetic felds
Hx and Hy

• Bit is set only if both Hx and Hy are present.
• For other bits addressed by only one line, either
  Hx or Hy is zero. These bits will not be turned
  on.

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Coherent rotation Picture

• The switching boundaries are given by the line
  AC, for example, a field at X within the triangle
  ABC can write the bit.
• If Hx0 or Hy0, the bit will not be turned on.

B
A
X
Hy
C
Hx
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Bit selectivity problem Very small (green)
writable area

• Different curves are for different bits with
  different randomness.
• Cannot write a bit with 100 per cent confidence.

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Another way recently proposed by the Motorola
group Spin flop switching

• Electrical current required is too large at the
  moment

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Simple picture from the coherent rotation model

• M1, M2 are the magnetizations of the two
  bilayers.
• The external magnetic fields are applied at -135
  degree, then 180 degree then 135 degree.

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Magnetization is not uniform coherent rotation
model is not enough