Analysis of switching in uniformly magnetized bodies
ABSTRACT A full analysis of magnetization reversal of a uniformly magnetized body by coherent rotation is presented. The magnetic energy of the body in the presence of an applied field H is modeled as E=(μ0/2)MT DM-μ0HTM, where T denotes a matrix transpose. This model includes shape anisotropy, any number of uniaxial anisotropies, and any energy that can be represented by an effective field that is a linear function of the uniform magnetization M. The model is a generalization to three dimensions of the Stoner-Wohlfarth model. Lagrange multiplier analysis leads to quadratically convergent iterative algorithms for computation of switching field, coercive field, and the stable magnetization(s) of the body in the presence of any applied field. Magnetization dynamics are examined as the applied field magnitude |H| approaches the switching field Hs, and it is found that the precession frequency f∝(Hs-|H|)(14)/ and the susceptibility χ∝(Hs-|H|)-(12)/.
Analysis of Switching in Uniformly Magnetized
Michael J. DONAHUE and Donald G. PORTER, Member, IEEE
Abstract— A full analysis of magnetization reversal of a
uniformly magnetized body by coherent rotation is pre-
sented. The magnetic energy of the body in the presence of
an applied field H is modeled as E = (µ0/2)MTDM−µ0HTM,
where T denotes a matrix transpose. This model includes
shape anisotropy, any number of uniaxial anisotropies, and
any energy that can be represented by an effective field
that is a linear function of the uniform magnetization M.
The model is a generalization to three dimensions of the
Stoner-Wohlfarth model. Lagrange multiplier analysis leads
to quadratically convergent iterative algorithms for compu-
tation of switching field, coercive field, and the stable mag-
netization(s) of the body in the presence of any applied field.
Magnetization dynamics are examined as the applied field
magnitude |H| approaches the switching field Hs, and it is
found that the precession frequency f ∝ (Hs− |H|)(1/4)and
the susceptibility χ ∝ (Hs− |H|)−(1/2).
Keywords—Stoner-Wohlfarth model, uniform rotation, co-
herent rotation, single-domain particles, standard problems,
Particularly useful are simple problems with solutions that
can be determined analytically. For example, it was de-
termined in  that an average field method is superior to
a sampled field method for calculating self-demagnetizing
fields due to its agreement with analysis of an ideal uni-
formly magnetized body. An assumption of uniform mag-
netization is one way to bring examination of magnetiza-
tion switching within the reach of analysis.
The study of magnetization switching in a uniformly-
magnetized body by uniform rotation has been a topic of
interest in its own right as well. Most widely known is
the Stoner-Wohlfarth model  that predicts the switching
field for a body with anisotropy completely characterized
by a single axis. This simple model has been extended in
many ways. Most relevant to our work is the extension to
an arbitrary shape anisotropy  and an additional uni-
axial anisotropy . Solutions of these extended models
have been determined by tabulation of energies sampled
over a two-dimensional space of magnetization directions,
and interpolation between tabulated values to find energy
In this paper, we consider and analyze a more general
class of uniformly magnetized bodies, any body for which
the magnetic energy in the presence of an applied magnetic
TANDARD problems have proven useful for verifying
the calculations of micromagnetic simulations , .
Manuscript received February 14, 2002. Revised May 15, 2002.
Authors are with the National Institute of Standards and
field H may be expressed by
E = (µ0/2)MTDM − µ0HTM,
where T denotes matrix transpose. Here D is any (3 × 3)
matrix, so this model may include shape anisotropy, any
number of uniaxial anisotropies, and indeed any form of
magnetic energy that can be represented by an effective
field that is a linear function of magnetization. Without
loss of generality, D is symmetric, because any asymmetric
part contributes nothing to the energy. Exchange energy
does not appear due to the assumption of uniform mag-
netization. This formulation is a three-dimensional form
of the Stoner-Wohlfarth model. In this paper we present
an analysis that classifies all stationary points of (1) and
computes them with a quadratically convergent iterative
algorithm over a single parameter.
An even more general three-dimensional model is con-
sidered in .Techniques for finding stationary points
by geometric construction are described. However, for ac-
tual calculation of numerical values, searches over a two-
dimensional space are prescribed. For the special case cov-
ered by (1) – described in  as “biaxial anisotropy of sec-
ond degree” – we offer a more direct calculation algorithm.
When uniform magnetization is assumed, and the geo-
metric boundaries of the body include sharp corners, the
demagnetizing field at the corners diverges. It has been
shown that micromagnetic calculations remain valid in that
situation , .
II. Lagrange Analysis
To analyze the switching properties of the model de-
scribed by (1), we seek the values of M that minimize E for
fixed |M| = M. This is a constrained optimization prob-
lem. Let M = Mm and H = Hh, where |m| = |h| = 1, so
we may consider the magnitudes and directions of M and
H separately. M and h are fixed quantities. The applied
field sweeps over a fixed axis by variation of H. The mag-
netization is free to rotate by variation of m. By choice of
the coordinate system, D = diag(Dx,Dy,Dz) is a diagonal
matrix with Dx< Dy< Dz. Introduce the Lagrange mul-
tiplier (µ0λ) on the constraint, and solve for the condition
∇ME = µ0DM − µ0H = (µ0λ)M
to be met by stationary values of M. Solving for M yields
M(λ) = Mm(λ) = (D − λI)−1H,
saddle local min
Fig. 1. Plot of g(λ). Dx< Dy < Dz. hν ?= 0,ν = x,y,z
and substitution back into the constraint yields
Fig. 1 contains a plot of g(λ) as a function of λ. Each value
of H determines a horizontal line in Fig. 1. Solutions of (4)
for a given H are the intersection of that line with g(λ),
indicated by the arrows. The number of solutions varies
between 2 and 6, depending on the value of H.
III. Classification of stationary points
Given H, let λ∗be a particular solution to (4), and m∗=
m(λ∗) be the corresponding stationary point of (1).
m, |m| = 1 is a small perturbation from m∗, then the
difference in energy is
E(m) − E(m∗) =1
2µ0M2Φ(m − m∗)(5)
Φ(m − m∗)
For λ∗< Dx, all diagonal values of D − λ∗I are positive,
so Φ(m −m∗) ≥ 0 for all m and m∗is a global minimum.
Likewise, for λ∗> Dz, m∗is a global maximum. To classify
m∗as a local minimum, we need to show that Φ(m−m∗) ≥
0 for all m in a sufficiently small neighborhood of m∗.
Let u be the projection of m − m∗onto the orthogonal
complement of m∗. Then if |m−m∗| is small, the difference
between Φ(m − m∗) and Φ(u) is O(|m − m∗|3). For our
purposes we only need expansions of E to second order,
so it suffices to consider Φ restricted to the 2-dimensional
Moreover, since Φ(ru) = r2Φ(u), it suffices to direct our
attention to finding extremal values of Φ on the closed,
bounded set |u| = 1, uTm∗= 0. Let a1and a2be respec-
tively the minimum and maximum values of Φ on this set.
Solving the constrained optimization problem leads to the
= (m − m∗)T(D − λ∗I)(m − m∗).
2M2(Dx− λ∗)(Dy− λ∗)(Dz− λ∗)g?(λ∗)(6)
a1+ a2= Dx+ Dy+ Dz− 3λ∗−HTm∗
Combining these two relations leads to a quadratic equa-
tion in either a1or a2. There are two real roots; the smaller
is a1, and the larger is a2.
However, explicit formulae for the ai’s are not needed to
classify the stationary points of (1). These values are also
the eigenvalues associated with restriction of the bilinear
form Φ to m∗⊥. Using the “interlacing property” , 
(also known as the Sturmian separation theorem), it follows
Dx− λ∗≤ a1≤ Dy− λ∗≤ a2≤ Dz− λ∗.
Now consider the case Dx < λ∗< Dy. Here a2 > 0, so
it follows from (6) that a1 has the sign opposite that of
g?(λ∗). Therefore, if g?(λ∗) < 0, then a1> 0 and thus m∗
is a local minimum. If g?(λ∗) > 0, then a1 < 0 and m∗
is a saddle point. A similar analysis for Dy < λ∗< Dz
confirms the remaining classifications indicated in Fig. 1.
IV. Calculation of Switching Field and Minima
A sketch of a hysteresis loop is depicted in the inset
of Fig. 1. The points on the hysteresis loop are minima
of (1). Saturation corresponds to λ = −∞. Remanence
corresponds to λ = Dx where magnetization is aligned
with the easy axis. The switching event is the irreversible
transition from a to b that occurs when the applied field
magnitude |H| reaches the switching field Hs, where the
local minimum ceases to exist. The switch from a to b
is also illustrated on the plot of g(λ) where the point a
is located at the coordinates (λs,(M/Hs)2), defined by
Dx < λs < Dy and g?(λs) = 0. Numeric determination
of λs is straightforward because λs is known to be in an
interval on which g?is strictly increasing. From an initial
estimate of λ1= (Dx+Dy)/2, the quadratically convergent
iteration derived from Newton’s method is
hT(D − λnI)−3h
hT(D − λnI)−4h.
Any iteration that produces a λn+1outside the known in-
terval containing the solution can be immediately replaced
by its average with λnuntil λn+1returns to the valid in-
terval. Once λnis sufficiently close to λs, one can calculate
g(λn) with (4) to find Hs. The error in Hsis proportional
to (λn−λs)2. Two iterations of this algorithm are sufficient
to determine the switching field in the small particle limit
for µMAG Standard Problem No. 2 to 16 digit precision.
Once we know λsand Hs, we can use a similar iterative
technique to find the local minimum for any given applied
field H, |H| < Hs. In this case we want to find λ satisfying
g(λ) − (M/H)2= 0 on the interval Dx < λ < λs. The
Newton iterate is
λn−?g(λn) − (M/H)2?/g?(λn)
hT(D − λnI)−2h − (M/H)2
hT(D − λnI)−3h
Again, the known interval containing the solution can be
used to assure proper convergence. Once λnhas converged,
use (3) to determine the corresponding M. This method
can also be used to find the global minimum, where the
relevant interval is λ < Dx.
V. Coercive field
Typically during the switching event, the magnetization
in the direction of the applied field, M·H, passes through
zero, i.e., the coercive field Hcis equal to the switching field
Hs. However, if the field is applied at an angle sufficiently
removed from the easy axis, then there can exist a local
minimum such that M·Hc= 0 with 0 < Hc< Hs. In this
circumstance it is possible to solve for Hcin closed form.
M · Hc
Dz− λ= 0.
One can obtain a quadratic equation in λ by multiplying
(11) by det(D − λI) = (Dx− λ)(Dy− λ)(Dz− λ). There
are two roots, λ1, λ2, satisfying Dx< λ1< Dy< λ2< Dz.
The coercive field corresponds to the smaller root, λc= λ1.
Substitution into (4) and (3) yields computed values for Hc
and M respectively.
VI. Precession Frequency
Away from equilibrium, the magnetization evolves under
where γ and α are the Landau-Lifshitz gyromagnetic ra-
tio and damping coefficients, respectively. If m is a small
perturbation from equilibrium position m∗, then the mag-
netization precesses around and gradually decays towards
m∗. The precession frequency f depends upon the cur-
vature of the energy surface in the neighborhood of the
f =γ√1 + α2M√a1a2
= −|γ|(M × Heff) −|αγ|
(M × (M × Heff)), (12)
The product a1a2is a function (6) of the applied field mag-
nitude |H| and λ∗. If λ∗corresponds to a local minimum,
it is natural to ask how f varies as |H| increases towards
the switching field, Hs.
Expanding (4) in λ about λs, one finds
g?(λ) = 2M
+ O(Hs− |H|),
which combines with (6) and (13) to yield
(1 + α2)M det(D − λsI)
as |H| ↑ Hs. This expression reveals the dependence of the
ring-down precession frequency f on the applied field mag-
nitude |H| is f ∝ (Hs− |H|)(1/4)immediately preceding
the switching event. Similar analysis establishes that the
susceptibility χ ∝ (Hs− |H|)−(1/2)just prior to switching
as well , .
VII. Special Cases
In all the preceding analysis it has been assumed that
Dx < Dy < Dz and hx ?= 0, hy ?= 0, hz ?= 0. If any
of these assumptions fail, then one or more terms of (4)
is removed and analysis of a simpler problem is possible.
If either hy = 0 or hz = 0, or if any two of Dx,Dy,Dz
are equal, then the model defined by (1) simplifies to the
Stoner-Wohlfarth model with its known solutions. In terms
of the plot in Fig. 1, one of the poles is removed, and the
saddle points disappear from the analysis.
In the case that Dx = Dy = Dz, the problem further
simplifies to be equivalent to magnetization reversal in a
sphere, with no anisotropy at all.
A three-dimensional generalization of
Wohlfarth model has been defined for the purpose of an-
alyzing magnetization reversal of a uniformly magnetized
body by coherent rotation. Analysis takes the form of solv-
ing a constrained optimization problem by use of Lagrange
multiplier techniques. For any body represented by the
model and any applied field axis, algorithms for computing
the switching and coercive fields have been derived. The
stable magnetization direction(s) for any applied field may
also be calculated using the techniques presented here. The
iterative algorithms are simple and converge quickly and
reliably. The behavior of precession frequency and suscep-
tibility as the switching field is approached have also been
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