## Abstract

We study the domain wall structure in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Using the reduced one-dimensional thin-film micromagnetic model, we analyse the critical points of the obtained non-local variational problem. We prove that the minimizer of the one-dimensional energy functional in the form of the Néel wall is the unique (up to translations) critical point of the energy among all monotone profiles with the same limiting behaviour at infinity. Thus, we establish uniqueness of the one-dimensional monotone Néel wall profile in the considered setting. We also obtain some uniform estimates for general one-dimensional domain wall profiles.

## 1. Introduction

Thin soft ferromagnetic films have been widely used as a data storage solution in modern computer technology [1–3]. It is well established that for sufficiently thin films, the magnetization vector of the material lies almost entirely in the film plane. Such ultra-thin ferromagnetic films often exhibit magnetization patterns consisting of domains in which the magnetization vector is nearly constant and is aligned along one of the directions of the easy axis of materials. Domains with different orientation of the magnetization are separated by thin transition layers called *domain walls* in which the magnetization vector rotates rapidly from one direction to another.

The study of the domain wall structure in ferromagnetic materials has attracted a lot of attention. One of the common domain wall types in ultrathin ferromagnetic films is the *Néel wall*. In this wall type, the magnetization vector exhibits an in-plane 180° rotation in the absence of an applied magnetic field. At present, the structure of the Néel wall is rather well understood. Within the framework of micromagnetic modelling, the overall physical picture has been summarized in books [2,4] (see also [5–8], etc.). Experimental observations of the one-dimensional Néel wall profiles can be found in [9–11]. Rigorous mathematical analysis of Néel wall is more recent, starting from the work of García–Cervera on the analysis of the associated one-dimensional variational problem [5,12]. Melcher studied one-dimensional energy minimizers in thin uniaxial films and obtained symmetry, monotonicity of the one-dimensional minimizing profile, as well as the logarithmic decay beyond the core region for very soft films [13]. Linearized stability of the one-dimensional Néel wall with respect to one-dimensional perturbations in a reduced thin film model was proved in [14]. Asymptotic stability of one-dimensional Néel walls with respect to large two-dimensional perturbations in a reduced two-dimensional thin film model was demonstrated in [15].

Recently, Chermisi & Muratov [16] studied the reduced one-dimensional energy in the presence of an applied in-plane magnetic field in the direction perpendicular to the easy axis. They expressed the magnetic energy in terms of the phase angle rather than the usual two-dimensional unit vector representation of the magnetization. They obtained uniqueness and strict monotonicity of the angle variable for the minimizing Néel wall structure. Moreover, they proved precise asymptotic behaviour of the minimizing Néel wall profiles at infinity.^{1} The associated Euler–Lagrange equation in their setting is expressed as an ordinary differential equation for the phase angle with a non-local term present.

We note that while from the physical point of view the Néel walls are believed to be the energy-minimizing configurations of the magnetization connecting the two oppositely oriented domains in uniaxial films, it is natural to ask whether other, metastable Néel wall-type configurations connecting the two domains, are also possible. For example, in the presence of a transverse in-plane magnetic field, one can distinguish normal and reverse domain walls, which differ by the rotation sense of the magnetization [17]. Clearly, the reverse domain wall is not an energy minimizer, because the magnetization vector opposes the applied field in such a wall. Still, in view of the highly nonlinear and non-local character of the problem it is not *a priori* clear whether there could exist other one-dimensional domain wall profiles connecting the domains of opposing magnetization which are only local, but not global minimizers of the micromagnetic energy.

In this paper, we follow the variational setting introduced in [16] and consider the *critical points* of the associated energy functional which are monotone in the angle variable. We prove that any monotone critical point of the reduced one-dimensional energy is unique (up to translations) and, therefore, is the minimizer. Thus, we establish that monotone one-dimensional magnetization profiles that are not global energy minimizers do not exist, corroborating the expected physical picture. This also provides a better understanding of the results of the numerical solution of the considered problem and allows to conclude that the obtained one-dimensional profiles [7] indeed correspond to the Néel walls. In addition, we address the question of uniform regularity of the critical points of the one-dimensional energy and establish uniform bounds and, hence, decay of all the derivatives of such solutions at infinity. This result can be applied to other types of domain wall profiles of interest, such as those of the 360° walls [18,19].

The rest of this paper is organized as follows: in §2, we recall some basic facts about the micromagnetic energy and the reduced one-dimensional energy in the presence of an applied in-plane field oriented normally to the easy axis. The main results are stated at the end of §2. The proof of the uniqueness theorem is presented in §3, and the proof of the uniform estimates for the derivatives of domain wall solutions is given in §4. Finally, we briefly revisit the question of the decay of Néel walls at infinity in appendix A and present the proof of a technical lemma in appendix B.

## 2. Variational setting and statement of the main result

In this paper, we are interested in the analysis of magnetization configurations in thin uniaxial ferromagnetic films of large extent with in plane easy axis and applied in-plane field normal to the easy axis. The energy functional related to such a system, introduced by Landau and Lifschitz, can be written in CGS units as a combination of five terms:
*Ω* and *Ω*, the positive constants *M*_{s}, *A* and *K* are the material parameters denoting the saturation magnetization, exchange constant and the anisotropy constant, respectively,

In the case of extended monocrystalline thin films with an in-plane easy axis, we have *i*th coordinate direction. For moderately soft thin films, a *reduced thin* *film energy* has been derived [7,20,21], providing a significant simplification to the considered variational problem. For a better understanding of the parameter regime, we introduce the following quantities
*ultra-thin* and *soft* film, we have *Ld*∼*l*^{2}. We can then introduce a dimensionless parameter
*h* is the dimensionless applied magnetic field.

To study one-dimensional Néel wall profiles, we assume further that *θ*=*θ*(*x*) that represents the angle between *θ* as
*d*^{2}/*dx*^{2})^{1/2} represents the linear operator whose Fourier symbol is |*k*| and can be understood as a bounded linear map from *h*|<1, we shall always assume that 0≤*h*<1 in most of the paper.

Let *η*_{h}. The following result was obtained in [16] addressing the uniqueness, strict monotonicity, symmetry properties and decay of one-dimensional Néel walls.

### Theorem 2.1 ([16])

*For every ν>0 and every h∈[0,1), there exists a minimizer of E(θ) in* *which is unique (up to translations), strictly decreasing with the range equal to (θ*_{h}*,π−θ*_{h}*) and is smooth. Moreover, if θ is a minimizer satisfying θ(0)=π/2, then θ(x)=π−θ(−x), and there exists a constant c>0 such that*

The Euler–Lagrange equation associated with the functional in (2.2) is given by

### Theorem 2.2

*For every ν>0 and every h∈[0,1), there exists a unique (up to translations) non-increasing smooth solution of (2.3) which satisfies the conditions at infinity in (2.4) and has bounded energy.*

Thus, the only possible monotone Néel wall profile is that of the minimizer of the energy in (2.2), whose existence and uniqueness was established in theorem 2.1. This confirms the long-standing physical intuition that the Néel wall profiles observed in ultrathin uniaxial ferromagnetic films minimize the one-dimensional micromagnetic energy among all such profiles.

We also obtain the following estimates for the general one-dimensional domain wall profiles. Here, by a one-dimensional domain wall profile, we mean a smooth solution of (2.3) connecting zeroes of *θ* of (2.3) with bounded energy is smooth, and it is easy to see that any solution of (2.3) with bounded energy should approach a zero of

### Theorem 2.3

*There exist C*_{i}*>0, i=1,2,…, such that for any solution θ of (2.3) with* *we have
**where C*_{i}*=C*_{i}*(ν,h,E(θ)). Moreover, all the derivatives of θ vanish at infinity.*

The main idea to prove the uniqueness result is as follows. Given any two monotone solutions *θ*_{1} and *θ*_{2} of (2.3) satisfying (2.4) and *θ*_{1}(0)=*θ*_{2}(0)=*π*/2, consider a suitable curve *γ* connecting *θ*_{1} and *θ*_{2}. The curve *γ* is chosen in such a way that any *θ*^{t}∈*γ* satisfies *t*∈[0,1]. We then show that if *f*(*t*):=*E*(*θ*^{t}), then *f*∈*C*^{2}([0,1]) and *f*^{′′}(*t*)>0 for any *t*∈[0,1], which implies strict convexity of *f*. At the same time, because *θ*_{i} are solutions of (2.3), we must have *f*^{′}(*t*)|_{t=0,1}=0, which is impossible. A similar argument, using a hidden convexity of the considered energy functional, was used recently in [22] to prove uniqueness of solutions for a very different variational problem.

The uniform-bound theorem relies on the uniform estimate on the non-local term in (2.3). To obtain the estimate on the non-local term, we used local smoothness of the solutions, together with the integral representation of the non-local term and energy-type estimates for the first derivatives. Decay property of derivatives of solution at infinity follows directly once we get those uniform derivative bounds.

## 3. Uniqueness of the critical point

Assume that *θ*_{1}≢*θ*_{2} are two non-increasing solutions of (2.3) satisfying (2.4) and *θ*_{i}(0)=*π*/2. Let now
*θ*_{i} are smooth and *dθ*_{i}/*dx*<0 on *θ*^{t}. We note that the latter is not obvious *a priori*, because the definition of *θ*^{t} involves the arcsine function, which is *not* differentiable when its argument equals *π*/2. This could potentially create problems near *x*=0. In fact, the conclusions of this section would clearly be incorrect, if there were multiple points at which either *θ*_{1} or *θ*_{2} equals *π*/2. Indeed, uniqueness of solutions of (2.3) and (2.4) with finite energy is false in view of the translational symmetry of the problem. Therefore, the somewhat delicate estimates near *x*=0 in the lemmas in the following are not merely technical, they are what enables the intuitive arguments of [14,16] to be used to establish uniqueness of the solutions that are translated so as to equal *π*/2 at *x*=0.

In the following, the subscripts *x* and *t* denote the partial derivatives with respect to the corresponding variables.

### Lemma 3.1

*For any* *t*∈[0,1], *the function* *θ*^{t}(*x*) *is continuously differentiable with respect to* *For any* *is twice continuously differentiable with respect to* *t* *on* [0,1], *with the understanding of one-sided derivatives at the boundary. All derivatives* *and* *are continuous functions of* *x* *and* *t* *separately on* *Moreover, there exists a constant* *K*>0 *depending only on* *θ*_{1} *and* *θ*_{2} *such that for all*

We present the proof of lemma 3.1, which is a rather tedious exercise in calculus, in appendix B. To proceed with the proof of our theorem 2.2, we first prove differentiability of *E*(*θ*^{t}).

Recall that
*f*(*t*)=*E*(*θ*^{t}) for shorthand. Lemma 3.2 is a direct corollary of lemma 3.1.

### Lemma 3.2

*We have* *for all* *t*∈[0,1]. *Moreover*, *f*(*t*) *is twice continuously differentiable, and* *f*^{′′}(*t*)>0 on [0,1].

### Proof.

By lemma 3.1 and (3.2), we have
*f*(*t*)=*E*(*θ*^{t}) is well defined. To ensure that *f*(*t*) is sufficiently regular, observe that from (3.2), we can write
*P*_{2}(*t*) is a quadratic polynomial in *t* with bounded coefficients depending on *θ*_{1,} *θ*_{2}. The question of differentiability of *f*(*t*) thus reduces to that of
*t* on [0,1] for each *t*∈[0,1]
*g*^{′}(*t*) and *g*^{′′}(*t*) are both continuous on [0,1]. A direct computation then yields

### Proof of theorem 2.2

Existence and smoothness of solutions follows from theorem 2.1 in [16]. We argue by contradiction and assume that *θ*_{1}≢*θ*_{2} are two monotone decreasing solutions of (2.3) satisfying (2.4), together with *θ*_{i}(0)=*π*/2. Let *θ*^{t} be defined by (3.1) and let *f*(*t*)=*E*(*θ*^{t}). Differentiating (3.2) at *t*=0, we get

Therefore, one cannot have (3.8) and (3.9) to hold at the same time, a contradiction. ▪

### Remark 3.3

Our proof of uniqueness works as long as *θ*(*x*) has range (*θ*_{h},*π*−*θ*_{h}), satisfies (2.4) and passes through *π*/2 only once.

## 4. Uniform bounds and decay of the derivatives

### (a) Uniform bound for solutions with bounded energy

Let
*θ* of (2.3) with bounded energy is smooth. We shall use this fact for the rest of the section.

### Lemma 4.1

*Let* *ν*>0, *let* *and let* *θ* *be a solution of* (2.3) *such that* *Then, there exists a constant* *C*=*C*(*ν*,*h*,*E*(*θ*))>0 *such that* |*v*(*x*)|≤*C* *for all*

### Proof.

Using the identity (see, for example, formula (3.1) in [23])
*δ*>0, we have
*δ* after direct integration. Because *θ* is smooth, it follows from Taylor expansion that the third term on the right-hand side of (4.2) can be bounded by

To estimate the first term on the right-hand side of (4.4), we use (2.3) to obtain
*θ*_{x}, we observe that because *θ*_{x}(*x*_{n}) *θ*_{x} and integrating from *x*_{n} to *x*, we get
*u*(*x*)+*h*|≤1, we get
*δ*=*π*/*ν*, we get

### Corollary 4.2

*There exists* *C*_{i}=*C*_{i}(*ν*,*h*,*E*(*θ*))>0 (*i*=1,2,…,) *such that, given any solution* *θ* *of* (2.3) *with* *we have*

### Proof.

The estimate for *θ*_{x}, *θ*_{xx} follows directly from (4.5), (4.8) and lemma 4.1. To estimate *θ*_{xxx}, differentiate (2.3). We have
*v*_{x}| and, thus, a bound on |*θ*_{xxx}|. Differentiating repeatedly, we obtain similar estimates for all derivatives. ▪

Because any solution of (2.3) with bounded energy is in *θ*, we conclude that any solution of (2.3) with bounded energy must have all its derivatives vanish at infinity.

## Ethics

This research does not contain human or animal subject.

## Data accessibility

This paper has no supplementary material.

## Authors' contributions

Both authors worked together on all sections of this paper. The research and writing of this article were carried out by both authors. The final version has been approved for publication by both authors.

## Competing interests

We have no competing interests.

## Funding

The work of C.B.M. was supported, in part, by NSF via grants nos DMS-0908279 and DMS-1313687.

## Acknowledgements

X.Y. thanks Christof Melcher for helpful discussions.

## Appendix A. Decay of Néel walls

Here, we revisit the question of the asymptotic decay of Néel wall solutions, whose existence and uniqueness is guaranteed by theorem 2.1. Let *θ* be the unique minimizer of *E* in *θ*(0)=*π*/2, and introduce
*ρ*(*x*) is a smooth even function of *x*, except at *x*=0, where *ρ*_{x} undergoes a jump discontinuity.

Proceeding as in step 4 of the proof of theorem 2.1 in [16], we observe that *ρ* satisfies distributionally
*a*=2|*θ*^{′}(0)|>0 and *δ*(*x*) is the Dirac delta function. The last term in the right-hand side of (A 2) was inadvertently omitted in [16]. Nevertheless, its presence does not affect the rest of the proof. Namely, we invert the operator *L* with the help of the fundamental solution *G* (see [16], lemma A1 for an explicit definition and properties of *G*). In particular, we have
*aG* term in the right-hand side of (A 4) leaves all the remaining estimates unchanged, in view of the fact that 0<*G*(*x*)≤*C*/|*x*|^{2}, for some *C*>0.

## Appendix B. Proof of lemma 3.1

Here, we present the proof of lemma 3.1.

### Proof.

By our assumption, when *x*≠0, we have
*t*∈[0,1]. Because arcsin(*u*) is differentiable for *u*<1, chain rule applies when taking derivative of *θ*^{t}(*x*) with respect to *x* at *x*≠0 for any *t*∈[0,1]. From the assumption on *θ*_{i} and the definition of *θ*^{t}, we have
*x*≠0, the function *t* for any *t*∈[0,1]. Differentiating (B 1) with respect to *t*, we get for *x*≠0

From (B 1) to (B 3), continuity of *x* for all *x*≠0 follows. For *x*=0, we calculate the derivatives of *θ*^{t} via the definition as follows. By assumption, we have 0<*θ*^{t}(*x*)<*π*/2 when *x*>0 and *π*/2< *θ*^{t}(*x*)<*π* when *x*<0. From this, we obtain

We calculate the derivative of *θ*^{t}(*x*) with respect to *x* at *x*=0 as follows
*θ*_{ix}(0)<0. It then follows from (B 6) that
*x*=0 for any *t*∈[0,1]. Continuity of *t* is obvious from (B 1) and (B 5).

Next, we evaluate *x*=0. Recall that *θ*_{ix}(0)<0 and differentiate (B 5) with respect to *t*. We get
*x*=0 as
*x*=0 and continuity of *t* follows from (B 2) and (B 8). Lastly, recall that *θ*_{ix}(0)<0 and differentiate (B 8) with respect to *t*. This yields

To derive continuity of *x*=0, we calculate the limit of *x*=0. By (B 6) and (B 9),

Continuity of *x*=0 follows from (B 11) and (B 12). Continuity of *t* variable follows from (B 3) and (B 11).

Finally, we derive the bounds on those derivatives. When *x*=0, it follows directly from (B 5), (B 8) and (B 11) that
*x*≠0, we write
*t*≤1 and *x*≠0, we have
*δ*_{0}>0 such that for *i*=1,2, we have
*x*|<*δ*_{0,} we obtain
*x*|≥*δ*_{0}, we get

Let
*K*=*M*^{5}+*M*. ▪

## Footnotes

- Received November 2, 2015.
- Accepted February 19, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.