# Stripe width and non-local domain walls in the two-dimensional Dipolar Frustrated Ising Ferromagnet

###### Abstract

We describe a novel type of magnetic domain wall which, in contrast to Bloch or Néel walls, is non-localized and, in a certain temperature range, non-monotonic. The wall appears as the mean-field solution of the two-dimensional ferromagnetic Ising model frustrated by the long-ranged dipolar interaction. We provide experimental evidence of this wall delocalization in the stripe-domain phase of perpendicularly magnetized ultrathin magnetic films. In agreement with experimental results, we find that the stripe width decreases with increasing temperature and approaches a finite value at the Curie-temperature following a power law. The same kind of wall and a similar temperature dependence of the stripe width is expected in the mean-field approximation of the two-dimensional Coulomb frustrated Ising ferromagnet.

###### pacs:

75.60.Ch, 05.70.Fh, 64.60.CnIntroduction.— Recent rigorous works Lieb ; Kiv1 state that the spontaneous magnetization of a two-dimensional (2d) Ising ferromagnet vanishes exactly if a dipolar coupling is present (Dipolar Frustrated Ising Ferromagnet, DFIF). Frustrating interactions on different spatial scales occur e.g. in ultrathin magnetic films where the spins point perpendicular to the film plane Lieb ; Kooy ; Seul ; Garel ; Van ; Sornette ; Villain1 ; Poki ; Cannas ; Rolf ; Qiu ; Muratov ; Singer . Several approaches indicate a striped ground state Lieb ; Kooy ; Van ; MacIsaac ; Castro with spin modulation along one in-plane direction and uniform spin alignment along the orthogonal in-plane direction. At finite temperatures, such spin “microemulsions” Kiv2 suffer from the Landau-Peierls instability Garel ; Poki ; Cannas ; Rolf ; Qiu ; Muratov ; Singer ; Lieb that delocalizes, in the thermodynamic limit, the stripe position, thus reducing the positional order to quasi-long range Garel ; Poki ; Kiv1 . Notice, however, that the stripe width remains well defined at finite temperatures even in the thermodynamic limit Janco , and that experiments on real systems do indeed observe the persistence of stripe order at finite temperatures Seul ; Sornette ; Rolf ; Qiu ; Oliver1 , possibly because of domain-wall pinning Poki ; Note1 ; Lemerle . It is thus worthwhile to study the Mean-Field (MF) behavior of a DFIF at finite temperature in the hope that some characteristics are “robust” enough to be valid beyond the MF approximation and to be observable in experiments. Competing interactions acting on different length scales are fundamental to many different chemical and physical systems And ; Seul ; Ball ; Keller ; Bates so that robust MF results on such a general model like the DFIF may have a wide significance.

A central question that motivated this work is the equilibrium stripe width (number of lattice parameters) at finite temperatures. One result appears to be well established in 2d: The stripe width in the ground state depends exponentially on the ratio between the exchange () and the dipolar () energy per atom Villain1 ; Kiv2 ; Poki ; MacIsaac ; Lieb . The temperature dependence, instead, is controversial. Within the MF approximation, is expected to decrease with temperature, because it should reach a finite value on the order of at the temperature of the MF transition to the paramagnetic state where the spin averages to zero at every site Villain1 ; Castro . Theoretical arguments based on sharp interfaces Gehring ; Connell ; Sto predict a (stretched) exponential decrease of down to an atomic length scale at the transition temperature . Within the spherical approximation, the modulation length of a related model (the Coulomb Frustrated Ferromagnet) “monotonically increases with temperature until it diverges at a disorder line temperature”, and this independently of the dimensionality of the system Nussinov . Experimental results on the temperature dependence of the stripe width are controversial as well Qiu ; Oliver1 ; Keller and in some cases experiments do not show any change of with temperature Seul .

In spite of its apparent simplicity, a detailed study of the stripe width of a DFIF as a function of temperature within the MF approximation has not been reported yet. Here, we fill this gap and solve the relevant MF equations for the DFIF model on a discrete lattice, finding a number of unexpected results. The sharp-interface assumption gives the correct stripe width at low temperatures, but fails to reproduce the results of the full MF calculation close to the transition temperature . Near , the temperature dependence of crosses over to a power law. This cross-over is accompanied by a delocalization of the wall between adjacent stripes: The profile changes from square-like at low temperatures to cosine-like at . At intermediate temperatures, the interface between two adjacent stripes develops a pronounced non-monotonic “shoulder” tailing down toward the center of the stripe according to a power law. The non-local walls are in striking contrast to the profile of the domain walls encountered in the Ising model without dipolar interaction Ising and also with the shape of conventional Bloch or Néel walls dividing domains in typical Heisenberg or planar ferromagnets Landau . The strength of the “shoulder” structure at intermediate temperatures depends on the relative strength of and , but it occurs over mesoscopic scales and in a sizeable range of temperatures so that it should be observable by spatially resolved experiments which have a high enough signal-to-noise ratio. The cosine-like profile, instead, is realized sufficiently close to independently of the ratio . Here we provide experimental evidence that the spin profile of the stripes changes indeed from square-like at low temperatures to cosine-like at .

The model.— The DFIF Hamiltonian on a discrete lattice reads

with respect to .

The stripe width.— Typically, in thin magnetic films . We are thus interested in large ratios , for which we recover the value of known from Ref. MacIsaac and the finite value known from Ref. Villain1 . and the limiting magnetization profile (see Fig. 2a) are found as the maximum eigenvalue and the corresponding eigenvector of explanation .

In Fig. 1, we plot for . The polygonal appearance of the graphs is due to changing only by in a discrete model. In order to understand the low-temperature behavior of and the low-T plateau, we introduce a manageable “sharp-interface” two-spin model, where and for (lower left inset of Fig. 1). In the limit of large , is just the MF value for a ferromagnetic Ising model, , and is the corresponding MF value for a spin adjacent to a domain wall, . Due to the reduced exchange energy in the argument of the , (see later Fig. 2a). Inserting and into the sharp-interface free energy and minimizing it with respect to , we obtain , where . The dashed curve in Fig. 1 representing for reproduces the low-temperature behavior of the numerical solution but fails at higher temperatures, where it gives . In the upper inset of Fig. 1, we plot vs for the full MF calculation close to in a log-log plot, showing that the domain width behaves asymptotically according to a power law , with as discussed in the figure caption. This numerical outcome seems to confirm the conjecture of Ref. Oliver1 , but is at odds with the sharp-interface limit of Refs. Gehring ; Connell ; Sto . It would be interesting to review the experimental results of Ref. Qiu ; Keller under the point of view of a power law. The argument of Ref. Oliver1 associates the cross-over to the power-law behavior with the higher harmonics (responsible for the sharp interface at low temperatures) vanishing with increasing temperature and thus with the broadening of the spin profile to a cosine-like profile close to .

The magnetization profile.— The spin profiles at different temperatures, obtained from the transcendental MF equations, are plotted in Fig. 2a for selected temperatures (marked with dots in Fig. 1). We identify three regimes: (i) A low- regime, corresponding to the plateau in the curves, with a square-like profile. (ii) An intermediate regime, corresponding to the steep descent of . Here, a novel feature consisting of a double-shoulder and a wall delocalization are observed. (iii) A high- regime, corresponding to the critical region, where the magnetization has indeed the cosine-like profile also expected from analytical considerations and leading to the power-law behavior of the equilibrium stripe width Oliver1 . Notice that only within the first regime is the interface sharp and does the bulk magnetization (circles) not change very much, so that the two-spin model (dashed curve in Fig. 1) is indeed justified. In order to understand the origin of the non-monotonic shoulder and the wall delocalization we have solved the MF equations for a single domain wall. In Fig. 2b we compare the profiles in the absence (dashed line) and in the presence (solid line) of the dipolar interaction. For (), the profile is of the Landau-type; it increases monotonically and attains the asymptotic value exponentially. For finite , the profile is not monotonic: it has a shoulder close to the wall center and then decays to as the inverse of the distance (lower inset). Both features derive from the dipolar (demagnetizing) field , which is plotted for a step-like profile in the upper inset. Close to the center of the wall, almost vanishes because of the compensation between the fields generated by up and down spins. There, the deviation from the Landau-type wall (dashed line) is small. The approach to the value occurs as and depresses the spin profile below the asymptotic value for . We have also checked within a continuum model that the demagnetizing field far from the wall vanishes in the infinite-thickness limit so that the three-dimensional, monotonic and localized “Landau” wall is recovered in 3d. Thus, the formation of the non-monotonic long-ranged wall is a purely two-dimensional effect.

The square-like profile at low temperatures delocalizes into a cosine-like profile close to , no matter how small the dipolar interaction is. We provide experimental evidence of this wall delocalization in Fig. 3. The spin profile was measured in SEMPA Oliver1 experiments on
ultrathin Fe films grown epitaxially on Cu(100). These systems are
magnetized perpendicularly to the film plane and show the sought-for
stripe structure Oliver2 .
The two different profiles at low
temperature (empty dots) and close to the stripe-paramagnetic transition temperature (full dots) point to the realization of the MF cross-over shown in Fig. 2a.

Conclusions.— In summary, we have shown that when dealing with “spin microemulsions” (and probably with analogous pattern-forming systems), the range of validity of the sharp-interface limit must be evaluated carefully and that important physical features, like the stripe width, crucially depend on whether the actual interface is sharp or not. In addition, we have discovered that the “Landau”-type walls (and probably the Bloch- and Néel-type walls as well) must be modified in low-dimensional systems because the dipolar interaction produces a non-monotonic, long-range tail which is absent in 3d systems such as those considered by Landau Landau . Our study has focused on the DFIF model on a discrete lattice. However, its results appear to be relevant future for the Coulomb Frustrated Ising Ferromagnet cfif as well. In the Coulomb system, antiferromagnetic interactions decay as rather than as the dipolar interaction, but preliminary results indicate that the phenomenology (domain shrinking, power-law approach to a finite value at , delocalization and presence of a shoulder in the domain wall profile) is the same. Although our work is based on the MF approximation, it produces results which appear to be realized in real pattern-forming systems, such as the power-law dependence of the stripe width Oliver1 and the spin profile (Fig. 3c), and it might provide a reasonable starting point for more sophisticated theoretical work.

We acknowledge financial support by ETHZ and the Swiss National Science Foundation and fruitful discussions with S.A. Cannas, E. Tosatti, G.E. Santoro and Z. Nussinov.

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