Read Back Signals in Magnetic Recording - Research Group Fidler

4 Numerical MethodsEquation
Numerical Methods
Section (Next)
Micromagnetic simulations require a discrete representation of the continuous magnetization
distribution on a computer. Due to the strong ferromagnetic coupling the magnetization does
not vary rapidly over short distances. So the discretization size can be much larger than the
atomic lattice constant. Usual lengths of significant magnetization changes (widths of domain
walls) are in the order of some nanometers for hard magnetic materials, for soft magnetic
materials even one or two orders of magnitude larger. To keep numerical errors small, the
discretization size should be smaller than the domain wall widths.
Dynamic micromagnetic calculations require two major tasks. The first part is to calculate the
stray field. For this purpose a partial differential equation (PDE) of the form
Lu [ ( r)] = s(
r )
(4.1)
has to be solved. Here L denotes the differential operator, s is called source term and u is the
unknown function, which fulfills this PDE. For stray field calculations u represents the
magnetic potential. We also have to solve an equation of type (4.1) for the electric potential of
an inhomogeneous conductor (see (2.31)). For this purpose the Finite Element Method is
used.
As second task we have solve an initial-value problem (IVP) of the form:
y� = f(, t y), y( t ) = y . (4.2)
0 0
In case of the LLG equation y represents the magnetization. Therefore y 0 would be the initial
magnetization. Here the time has to be discretized. We use a time integration routine, which
automatically adapts the time steps. So for fast physical processes a smaller time step is
chosen to increase accuracy.
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