## Abstract

We investigate an effect of *configurational anisotropy* in highly symmetric soft ferromagnetic nanoparticles. Using the micromagnetic variational principle and methods of Γ-convergence, we show that in ferromagnetic *generalized right prisms with symmetry*,^{1} there is a finite number of preferred magnetization directions and that these directions are independent of the shape of the magnet. This result provides a rigorous justification of work by Cowburn and Welland.

## 1. Introduction

In the late 1960s, Brown (1968) conjectured that as the size of a ferromagnetic particle goes to zero, at some point, the competition between magnetostatic energy and exchange energy requires uniform magnetization inside the magnet. This turned out to be not rigorously true for non-ellipsoidal particles. For example, in the case of a cubic nanomagnet, it is not energetically preferable to have several domains inside the particle but still the magnetization is ‘near-uniform’ rather than uniform.

It has been noticed by Cowburn & Welland (1998) that in a soft highly symmetric nanoparticle, the geometry affects its magnetic properties through a phenomenon called *configurational anisotropy*. The basic idea is that even in very small particles magnetization inside the magnet is not uniform but varies depending on the shape of the nanomagnet. These small deviations give rise to additional anisotropy energy terms which, in the case of a symmetric particle, can dominate magnetic properties and determine the preferred magnetization directions. For instance, square-planar elements in this regime may have several preferred states depending on the length–thickness ratio.

In a series of papers, Cowburn and co-workers (Cowburn & Welland 1998, 1999; Cowburn *et al.* 1999*a*,*b*; Cowburn 2000) investigated the configurational anisotropy of symmetric particles of different shapes and sizes experimentally, numerically and by some simple perturbation analysis. In particular, they discovered the formation of ‘leaf’ and ‘flower’ states in the case of a square-planar magnet and found that the energy surface between these states can be described by a fourfold-symmetric configurational anisotropy field which depends on the length–thickness ratio.

In the present paper, we give the first mathematically rigorous treatment of configurational anisotropy, placing the perturbation arguments of Cowburn *et al*. on a sound mathematical foundation. In particular, we investigate the configurational anisotropy of highly symmetric nanoparticles in the shape of a generalized right prism with symmetry (referred to in what follows simply as a prism with symmetry) using the micromagnetic variational principle. The problem of finding minimizers of the micromagnetic functional for small magnets in the vicinity of the corresponding isolated local minimizers for ‘0-size’ particle has been solved by DeSimone (1995). However, these results cannot explain the phenomenon of configurational anisotropy, because in the absence of crystalline anisotropy, the ‘0-size’ energy of a symmetric prism can have a continuum of global minimizers. Therefore, nothing can be said about the limiting behaviour of minimizers of the original micromagnetic energy. Ball *et al*. (2002) studied the problem of finding local minimizers of the micromagnetic energy in the small-particle regime by using the implicit function theorem. However, their analysis does not provide any information about the direction of the local minimizer for highly symmetric particles. A similar problem has also been studied for thin films by Kohn & Slastikov (2005), who showed that the full micromagnetic energy can be reduced to a simpler local two-dimensional energy, minimizers of which correspond to equilibrium states of the system. In this paper, we treat the case of a ferromagnetic nanoparticle, meaning that all lateral dimensions of the ferromagnet are comparable and small.

We propose to use variational methods to treat this problem. In particular, we are going to use methods of asymptotic development by Γ-convergence (Anzellotti & Baldo 1993) in order to gain information on the preferred directions of magnetization in small highly symmetric ferromagnets.

The paper is organized as follows. In §2, we introduce the variational principle of micromagnetics and give some properties of the demagnetizing tensor. In §3, we discuss the strategy to approach the problem. Finally, §4 is devoted to a proof of the Γ-convergence of the micromagnetic energy to a simpler problem that contains information on the behaviour of the magnetization and to the study of minimizers of limiting energy. For prisms with symmetry, we show that our limiting problem identifies the preferred magnetization directions, which are determined rather explicitly by the shape of the magnet. For prisms with symmetry, where *n*≠4, the limiting problem is degenerate; the configurational anisotropy of such particles is evidently much weaker, since it lies at the higher order in the asymptotic development.

## 2. Micromagnetic variational principle

In our study, we will follow a continuum model of micromagnetics. After a suitable non-dimensionalization, the micromagnetic energy functional looks like
2.1
The domain is the region occupied by a ferromagnet, the volume of *Ω* is 1 (*μ*(*Ω*)=1); is the direction of magnetization, constrained by
2.2
The function *u* is defined on and satisfies the following equation:
2.3
and *χ*(*Ω*) is an indicator of the set *Ω*. This tells us that magnetostatic energy is non-local in *m*.

The applied field acting on the ferromagnet is defined by *h*_{ext}; and *ϕ* is the internal anisotropy function. Parameters *ϵ* and *Q* are related to exchange length and anisotropy constant, respectively.

By micromagnetic variational principle, solutions of the following minimization problem 2.4 correspond to stable magnetization distributions. This minimization problem is non-convex, non-local variational problem that is impossible to solve in full generality using the current techniques.

Let us describe each of the terms in the energy equation (2.1) separately. More information on micromagnetics can be found in Aharoni (1996).

— The exchange energy penalizes spatial variations of magnetization and as

*ϵ*is small, it makes magnetization almost constant.— The anisotropy energy favours special directions of magnetization. In this paper, we study soft magnets where

*Q*is very small and crystalline anisotropy is not taken into consideration.— The magnetostatic energy prefers to have div

*m*=0 in*Ω*and*m*⋅*n*=0 on ∂*Ω*, where*n*is a unit normal to ∂*Ω*.— Zeeman energy favours magnetization to become parallel to the external field

*h*_{ext}.

### (a) Demagnetizing tensor

Using equation (2.3), it is easy to show that
2.5
where *Pm*=−∇*u* and is a linear continuous operator, also called Helmholtz projection of *m* onto the gradients. In the case when *m* is a constant over *Ω*, we have the following useful expression
2.6
and *D* is the effective demagnetizing tensor. It is a well-known fact that *D* is a symmetric, positive definite matrix. In the case when *Ω* is ellipsoid and *m* is a constant, we have even stronger result
2.7
and in this case, *D* is the real demagnetizing tensor.

We would like to prove several useful properties of the operator *P* and tensor *D* in the case when *Ω* is a prism with symmetry. We prove the following simple lemma.

### Lemma 2.1.

*Let Ω be a prism of unit volume with* *symmetry. Assume that m is a constant vector in* *and* *, then (Pm)(Rx)=(RPR*^{T}*m)(x). Moreover, if* *, then D=RDR*^{T}*, and therefore*
2.8

### Proof.

Let , by definition *Pm*(*x*)=−∇*u*(*x*), where
It is easy to see that if *m* is a constant vector, then
and therefore
2.9
Now, it is a straightforward calculation that (*Pm*)(*Rx*)=(*RPR*^{T}*m*)(*x*) for any .

In order to see that *D* has the form (2.8), we notice that for any
Since *Rn*(*x*)=*n*(*Rx*) for any , we have *RDR*^{T}=*D*, and this implies that *D* has the form (2.8). Lemma is proved. ■

## 3. Preliminary discussion

We are studying the dependence of magnetization distribution *m* and micromagnetic energy *E*_{ϵ}(*m*) on the geometry of the domain *Ω*. In general, this task is unsolvable by any of the current analytical techniques. We are investigating this problem for ‘sufficiently small’ (*ϵ*≪1) soft (*Q*=0) ferromagnetic particles, which have the shape of a regular prism. We also assume that the applied field is absent (*h*_{ext}=0), otherwise it will determine the preferred direction of magnetization.

Since stable configurations of magnetization correspond to local minimizers of micromagnetic energy, we are interested in finding minimizers of functional *E*_{ϵ}(*m*) subject to constraint |*m*(*x*)|=1 for *x*∈*Ω*. The problem of finding local minimizers of micromagnetic functional for small ferromagnetic particles has been studied by DeSimone (1995). We summarize some of his results in the following theorem.

### Theorem 3.1.

*For every fixed* , *the following energy*
3.1
*is the Γ*(*L*^{2})-*limit of E _{ϵ} in H*

^{1}(

*Ω*).

*In particular, given a sequence {ϵ*

_{j}} converging to zero, and denoting by m_{j}a minimizer of E_{ϵj}, there exist subsequences {ϵ_{j}} and {m_{j}}, not relabelled, such that*and*

*Moreover, given m*

_{0}

*be an isolated constant L*

^{2}-

*local minimizer of E*

_{0},

*there exists ϵ*

_{0}>0

*and a family*

*with ϵ*<

*ϵ*

_{0},

*such that m*

_{ϵ}is L^{2}-

*local minimizer of E*

_{ϵ}, andThis result guarantees that every sequence of minimizers of *E*_{ϵ} converges to some minimizer of *E*_{0} and, on the other hand, any isolated local minimizer of *E*_{0} can be approximated by a suitable sequence of local minimizers of *E*_{ϵ} as *ϵ*→0.

Since we neglect crystalline anisotropy and do not have an applied field, in our case
3.2
and the limiting energy is
We are interested in the behaviour of minimizers *m*_{ϵ} for small *ϵ*. The usual way of recovering information about *m*_{ϵ} is to find a minimizer *m*_{0} of the limiting problem and then deduce that *m*_{ϵ} is close to *m*_{0} in some norm. Unfortunately, it does not work here directly as *E*_{0}(*m*) has a continuum of minimizers (any *m*=(*m*_{1},*m*_{2},0) is a minimizer with an energy *E*_{0}(*m*)=*α*). The proof of the following lemma is straightforward.

### Lemma 3.2.

*If α≤β in formula (2.8) of D, then there exist a continuum of local minimizers of the following problem*
*Moreover, if α<β the set of minimizers is* *and if α=β, then any* *is a minimizer.*

This feature may be an intrinsic property of the original energy, but in many cases it is because of the fact that we lose some information taking a Γ-limit of the original energy. That is precisely what happens here. We know that minimizers *m*_{ϵ} of the energy *E*_{ϵ}(*m*) are close to some constant vector, but they are not really constants. Small variations of *m*_{ϵ} play an important role in determining preferred directions of the minimizers. Therefore, in order to obtain the right behaviour of *m*_{ϵ}, we need to find contribution of these small variations to the micromagnetic energy. A similar problem was studied by Ball *et al*. (2002). They were interested in the existence of solutions for problem (2.1) in the neighbourhood of a given solution for a problem with *ϵ*=0. In our case, the degeneracy of the problem does not allow for the direct use of these results. The point is that all the information about leading order minimizers is stored in the next order and therefore, it is necessary to look at the leading and the next order simultaneously. One way to accomplish this is to use a method of asymptotic development by Γ-convergence (see Anzellotti & Baldo 1993). The idea is to look at the next order term in the energy expansion with respect to *ϵ*. Therefore, we need to study the following energy:
3.3

It is obvious that minimizers of *F*_{ϵ} coincide with minimizers of *E*_{ϵ}, but it is also clear that the Γ-limit of *F*_{ϵ} (if it exists) is quite different from *E*_{0}(*m*). Now, our goal is to prove the Γ-convergence result for *F*_{ϵ} and deduce the behaviour of minimizers *m*_{ϵ} from the information about minimizers of the limiting energy.

## 4. Main result

In this section, we state the mathematical problem that we will solve. We consider one parameter family of micromagnetic energy functionals
4.1
where *u* satisfies
We also assume that *Ω* is a fixed prism with symmetry of volume |*Ω*|=1. Constant *α* is defined as , and we know that if *α*<*β*, minimum of *E*_{0}(*m*) is achieved on any constant vector with *ϕ*∈[0,2*π*) (see lemma 3.2). This definition of *α* is motivated by the fact that *E*_{ϵ}*E*_{0}, and we would like to select true minimizers out of the continuum set of minimizers of *E*_{0}(*m*). Therefore, below we assume that *α*<*β*.

Using the definition of demagnetizing tensor and denoting , we can simplify this energy by defining : 4.2

We will use the following notation: , .

### Theorem 4.1.

*Assume that α*<*β, define* , , *and suppose ϵ*→0 *then*

—

*if F*≤_{ϵ}(m_{ϵ})*C, then**in*,*v*→_{ϵ}*v weakly in**H**(maybe up to subsequence, not relabelled) and m*_{0}∈,*S**m*_{0}·*v*(*x*)=0*for all x*∈*Ω*.—

*F*_{ϵ}*F*_{0}*in*,*where*4.3

### Proof.

Let us prove the first part of the theorem. If *F*_{ϵ}(*m*_{ϵ})≤*C*, we will deduce that and is bounded in **H**. Since *F*_{ϵ} is bounded only on |*m*_{ϵ}(*x*)|=1, it is easy to deduce that . Now, we would like to show that *v*_{ϵ} is bounded in **H**. We notice that and , and therefore
4.4
By Poincare inequality, we have , where *C*>0 depends only on *Ω*. Using these facts and equation (4.4), it is easy to show that
Therefore, we obtain *v*_{ϵ} is bounded in **H** and using , we have in and *v*_{ϵ}→*v* weakly in **H** (maybe up to a subsequence, not relabelled). In order to see that *m*_{0}·*v*(*x*)=0 for a.e. *x*∈*Ω*, it is enough to notice that and pass to the limit as *ϵ*→0 a.e. *x*∈*Ω*.

Now we would like to prove a Γ-convergence result. Using inequality (4.4), it is easy to see that for any sequence *m*_{ϵ} such that in and *v*_{ϵ}→*v* weakly in **H**
We also have to prove that for any *m*_{0}∈**S** and *v*∈**H** with *m*_{0}·*v*(*x*)=0 for *x*∈*Ω*, there exists a sequence *m*_{ϵ}∈(*H*^{1}(*Ω*))^{3}, such that |*m*_{ϵ}(*x*)|=1 and in and *v*_{ϵ}→*v* in **H**. A natural choice for *m*_{ϵ} is
It is easy to see that *m*_{ϵ}∈(*H*^{1}(*Ω*))^{3} and |*m*_{ϵ}(*x*)|=1. Let us show that in and *v*_{ϵ}→*v* in **H**. By simple rearrangement, we have
Taking average on both sides, we obtain . Passing to a limit as *ϵ*→0, we have in and . We also see that
and it is easy to conclude using |*m*_{ϵ}(*x*)|=1 and *v*∈(*L*^{4}(*Ω*))^{3} that *v*_{ϵ}→*v* in *L*^{2}(*Ω*). Now we want to prove convergence of the gradients. Let us calculate the gradient of *m*_{ϵ}
Recalling that ∇*v*_{ϵ}=(1/*ϵ*)∇*m*_{ϵ}, from this formula we see
and therefore
It is easy to see that the right-hand side of the inequality is in *L*^{2}(*Ω*), but we need to show that it goes to 0 as *ϵ*→0 in *L*^{2}. Consider the second term in the right-hand side
Since by Chebyshev inequality, , we conclude that as *ϵ*→0
By the same arguments, we can deal with the first term in the right-hand side and therefore
It is also clear that . Now we can pass to the limit as *ϵ*→0 in energy (4.4) and obtain
The theorem is proved. ■

### (b) Minimizers

In this subsection, we find the preferred directions of magnetization for prisms with symmetry. In order to do this, we are going to use a general result of theorem 4.1. The behaviour of minimizers of energy (4.3) provides us with the information about the preferred magnetization directions. Therefore, we are studying the following problem
4.5
We first fix *m*_{0} and minimize in *v*. It is not difficult to see that the minimizer satisfies the following Euler–Lagrange equations
4.6
4.7
and
4.8
Now, in order to find preferred directions of magnetization, we minimize the following energy with respect to *m*_{0}:
To solve this problem, we proceed as follows. We fix coordinate axes in such a way that *z*-axis is normal to the base of the prism and *Ω* is symmetric with respect to *y*-axis. We also choose the basis vectors *e*_{1}, *e*_{2} and *e*_{3} to be parallel to *x*-, *y*- and *z*-axes, respectively.

We may represent *m*_{0} in the form and solving equations (4.6)–(4.8) find the dependence of solution *v* of angle *ψ*. This will give us the dependence of the energy *F*_{0}(*m*_{0},*v*(*m*_{0})) of the angle *ψ*. After that, we can minimize in *ψ* and see what are the minimizing directions.

Let us proceed with this plan. Using the linearity of the operator *P*, we have . Below, we will use the following notation *P*_{ij}=*P***e**_{i}·**e**_{j}. Now it is clear that
4.9
Now we can define *v*(*m*_{0}) in terms of functions *m*_{11−22}, *m*_{12}, *m*_{21}, *m*_{13} and *m*_{31}, which are determined by the following equations:
and for *i*≠*j*

It is not difficult to see that these equations are solvable, indeed
and
Note that . It is a well-known fact that solutions *m*_{11−22} and *m*_{ij} are determined up to some constant, and we will later show that these constants do not contribute to the energy. Now by linearity we can determine *v*(*m*_{0}):
4.10
It is not difficult to see that *v*(*m*_{0})·*m*_{0}=0 identically.

We can easily find the energy *F*_{0}(*m*_{0},*v*(*m*_{0})) in terms of *ψ*
So far, we have not used any symmetry of domain *Ω* and that is what we are going to do in our next step. Using the fact that (*P***e**_{i})(*Rx*)=(*RPR*^{T}**e**_{i})(*x*) (see lemma 2.1) and taking to be reflection through *y*–*z* plane, we can deduce
Therefore, using the definition of *m*_{11−22} and *m*_{ij}, we obtain
Taking to be a rotation by angle 2*π*/*n* and using the formula (*P***e**_{i})(*Rx*)=(*RPR*^{T}**e**_{i})(*x*), we can find *P*_{ij}(*Rx*) and *m*_{ij}(*Rx*) in terms of *P*_{lk}(*x*) and *m*_{lk}(*x*). Since
after simple calculation, we obtain
and
and this implies that .

It is also straightforward to obtain
and this implies
Therefore, the energy simplifies to
We see that if domain *Ω* has a symmetry with *n*≠4, then the energy is independent of the angle *ψ* and therefore it is impossible to obtain a preferred direction of magnetization at this order. One has to go to the next order in the asymptotic expansion of energy to extract this information. For domains with symmetry, we can find the preferred direction of magnetization by minimizing *F*_{0}(*ψ*). It is obvious that there are two sets of critical points of the energy
Depending on the sign of expression , they deliver minimum (or maximum) of the energy. In general, the sign of this integral can be determined by numerical computations. For instance, as noticed in Cowburn & Welland (1998) in the case of a square base: for small prism heights, the preferred direction is along the diagonal (i.e. *ψ*_{1}) and as the height increases, it changes to *ψ*_{2}. Equilibrium states appear as shown in figure 1.

## 5. Conclusion

We have studied the full micromagnetic model for ferromagnetic nanoparticles with symmetry. Using methods of asymptotic development by Γ-convergence, we proved that there is a discrete set of magnetization directions *ψ*_{1}=(*π*/4)+(*π*/2)*k* and *ψ*_{2}=(*π*/2)*k* which, depending on the geometric parameters of the ferromagnet, correspond to equilibrium states of the system. We also showed that in order to treat ferromagnetic particles with symmetry , *n*≠4, it is necessary to go to the next order in the asymptotic expansion of the energy with respect to *ϵ*. In particular, this means that the effect of configurational anisotropy in this case is much weaker than for particles with symmetry.

## Acknowledgements

I am very grateful to Robert V. Kohn for suggesting the problem and plenty of useful discussions on the subject during my study at Courant Institute. I would also like to thank Jonathan Robbins and an anonymous referee for useful comments on the paper. This work was partially supported by the Nuffield Foundation (grant no. NAL32562.)

## Footnotes

↵1 By a generalized right prism with symmetry, we mean a right prism with a possibly curvilinear base that has the symmetry of a regular

*n*-sided polygon.- Received February 8, 2010.
- Accepted March 26, 2010.

- © 2010 The Royal Society