Transcript LEFT-HANDED NANOCRYSTALLINE MAGNETIC COMPOSITES

MRAM (Magnetic random
access memory)
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.
An array of magnetic elements
Schematic MRAM
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.
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 Hx=0 or Hy=0, the
bit will not be turned
on.
B
A
X
Hy
C
Hx
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)
cm^2, may save
power
Switching of magnetization of
small structures
Understanding the basic
physics: different approaches
Semi-analytic approaches
Solition solutions
Conformal Mapping
Edge domain: Simulation vs
Analytic approximation.
• =tan-1 [sinh(v(y’y’0))/(- v sinh((x’x’0)))],
• y’=y/l, x’=x/l; the
magnetic length
0
l=[J/2K] .5;
=1/[1+v2]0.5; v is a
parameter.
Closure domain: Simulation vs
analytic approximation
•
•
•

•
•
=tan-1[A tn( x', f) cn(v
[1+kg2]0.5y', k1g)/ dn(v [1+kg2]0.5
y', k1g)],
kg2=[A22(1-A2)]/[2(1-A2)2-1],
k1g2=A22(1-A2)/(2(1-A2)-1),
f2=[A2+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
Conformal mapping
From circle to square: Spins
parallel to boundaries
Navier Stokes equation (Yau)
Numerical methods
• Numerical studies can be carried out by
either solving the Landau-Gilbert equation
numerically or by Monte Carlo simulation.
Landau-Gilbert equation
• (1+2)dmi/d=hieffmi–(mi(mihieff))
• i is a spin label,
• hieff=Hieff/Ms is the total reduced effective field
from all source;
• mi=Mi/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: Hieff=hiex+hidemg+hiani .
Approximate results
• E=Eexch+Edip+Eanis.
• Between neighboring spins Edip<<Eexch.
• 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.
Reliability problem of switching of
magnetic random access
memory (MRAM)
Fluctuation of the switching field
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.
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 Hx=0 or Hy=0, the
bit will not be turned
on.
B
A
X
Hy
C
Hx
Experimental hysteresis curve
• J. Shi and S. Tehrani,
APL 77, 1692 (2000).
• For large Hy, the
hysteresis curve still
exhibits nonzero
magnetization at Hcx
(Hy=0).
Edge pinned domain proposed
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
• For finite Hy, curves with large Hsx are
usually associated with an intermediate
state.
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.
Another way recently proposed
by the Motorola group: Spin flop
switching
Two layers antiferromagnetically
coupled.
• 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.
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.
Switching boundaries
• Paper presented at
the MMM meeting,
2003 by the Motorola
group.
This solves the bit selectivity
but the field required is too big
Stronger field, -135: Note the edgepinned domain for the top layer
H
Very similar to the edge pinned
domain for the monlayer case.
• Switching scenario involves edge pinned
domain, similar to the monolayer case and
very different from the coherent rotation
picture.
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.
Bit selectivity vs interlayer
coupling: Magnitude of the
switching field
Temperature dependence
• Hc (bilayer) >>Hc (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.
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.
Energy between spins
H=0.5 ij=xyz,RR’ Vij(R-R’)Si(R)Sj(R’) ,
V=Vd+Ve+Va
The dipolar energy Vdij(R)=gij (1/|R|);
The exchange energy Ve=-J (R=R’+d)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.
 (Eexch+Eani)/ = AS22 -K sin =0.
2=x2-iy2.
This is the imaginary time sine Gordon
equation and can be exactly solved.
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|).
• Edipo=s 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.
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 SineGordon equation r2+(K/J) sin=0 with the
“parallel edge” boundary condition.