MRAM (Magnetic random access memory)

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MRAM (Magnetic random access memory)
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Outline

• Motivation introduction to MRAM.
• Switching of small magnetic structures a highly
  nonlinear problem with large mesoscopic
  fluctuations.
• Current theoretical approaches.
• Problems write reliability issues.

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An array of magnetic elements
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Schematic MRAM
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Write 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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Read Tunnelling magneto resistance between
ferromagnets

• Miyazaki et al, Moodera et al.
• room temperature magneto resiatance is about 30
• Fixed the magnetization on one side, the
  resistance is different between the AP and P
  configurations
• large resistance 100 ohm for 10(-4) cm2, may
  save power

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Switching of magnetization of small structures
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Understanding the basic physics different
approaches
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Semi-analytic approaches

• Solition solutions
• Conformal Mapping

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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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Conformal mapping
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From circle to square Spins parallel to
boundaries
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Navier Stokes equation (Yau)
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Numerical methods

• Numerical studies can be carried out by either
  solving the Landau-Gilbert equation numerically
  or by Monte Carlo simulation.

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Landau-Gilbert equation

• (1?2)dmi/d?hieff?mi?(mi?(mi?hieff))
• i is a spin label,
• hieffHieff/Ms is the total reduced effective
  field from all source
• miMi/Ms, Ms is the saturation magnetization
• ? is a damping constant.
• ?t???Ms is the reduced time with ? the
  gyromagnetic ratio.
• The total reduced effective field for each spin
  is composed of the exchange, demagnetization and
  anisotropy field Hieffhiexhidemghiani .

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Approximate results

• EEexchEdipEanis.
• Between neighboring spins EdipltltEexch.
• The effect of Edip is to make the spins lie in
  the plane and parallel to the boundaries.
• Subject to these boundary conditions, we only
  need to optimize the sum of the exchange and the
  anisotropy energies.

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Reliability problem of switching of magnetic
random access memory (MRAM)
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Fluctuation of the switching field
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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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Experimental hysteresis curve

• J. Shi and S. Tehrani, APL 77, 1692 (2000).
• For large Hy, the hysteresis curve still exhibits
  nonzero magnetization at Hcx (Hy0).

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Edge pinned domain proposed
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Hysteresis curves from computer simulations can
also exhibit similar behaviour

• For nonzero Hy switching can be a two step
  process. The bit is completely switched only for
  a sufficiently large Hx.

O
E
S
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• For finite Hy, curves with large Hsx are usually
  associated with an intermediate state.

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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

• Two layers antiferromagnetically coupled.

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• Memory in the green area.
• Read is with TMR with the magnet in the grey
  area, the same as before.
• Write is with two perpendicular wires (bottom
  figure) but time dependent.

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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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Switching boundaries

• Paper presented at the MMM meeting, 2003 by the
  Motorola group.

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This solves the bit selectivity but the field
required is too big
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Stronger field, -135 Note the edge-pinned domain
for the top layer
H
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Very similar to the edge pinned domain for the
monlayer case.
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• Switching scenario involves edge pinned domain,
  similar to the monolayer case and very different
  from the coherent rotation picture.

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Coercive field dependence on interlayer exchange

• For the top curve, a whole line of bits is
  written.
• For real systems, there are fluctuations in the
  switching field, indicated by the colour lines.
  If these overlap, then bits can be accidentally
  written.

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Bit selectivity vs interlayer coupling Magnitude
of the switching field
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Temperature dependence

• Hc (bilayer) gtgtHc (single layer). Hc (bilayer)
  exhibits a stronger temperature dependence than
  the monolayer case, different from the prediction
  of the coherent rotation picture.
• Usually requires large current.

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Simple Physics in micromagnetics

• Alignment of neighboring spins is determined by
  the exchange, since it is much bigger than the
  other energies such as the dipolar interaction
  and the intrinsic anisotropy.

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Energy between spins

• H0.5 ?ijxyz,RR Vij(R-R)Si(R)Sj(R) ,
• VVdVeVa
• The dipolar energy Vdij(R)g?i?j (1/R)
• The exchange energy Ve-J? (RRd)?ij d denotes
  the nearest neighbors
• Va is the crystalline anisotropy energy. It can
  be uniaxial or four-fold symmetric, with the easy
  or hard axis aligned along specific directions.

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Optimizing the energy

• Eexch-A? dr (? S)2.
• Eani-K ? dr Sz2.
• Let S lie in the xz plane at an angle ?.
• Eexch-AS2? dr (? ?)2.
• ? (EexchEani)/? ? AS2?2 ?-K sin ?0.
• ?2?x2-?iy2.
• This is the imaginary time sine Gordon equation
  and can be exactly solved.

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Dipolar interaction

• The dipolar interaction 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. The spins are parallel to the edges so
  that qM is small.

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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
  boundary condition.