Direct imaging of long-range ferromagnetic and antiferromagnetic order in a dipolar metamaterial

This paper demonstrates stabilization of long-range ferro- and antiferromagnetic order in a magnetic metamaterial. In arrays of dipolar-coupled ferromagnetic nanodisks, ferromagnetic order is observed when the disks are arranged in a hexagonal lattice, whereas antiferromagnetic order prevails for the square lattice.


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In 1946, Luttinger and Tisza predicted that the magnetic order in a lattice of point dipoles is governed by the symmetry of the dipole lattice [1], suggesting a novel mechanism for ferromagnetic (FM) order not based on exchange interactions. However, in an atomic crystal lattice the magnetostatic interaction between individual atoms is relatively weak and results in Curie temperatures in the sub-100 mK regime [2].
Monodomain nanomagnets can serve as mesoscale analogues to atomic magnetic moments and are used extensively in the study of frustration in artificial spin ice [3], emergent magnetic monopoles [4,5], and dipolar magnetic order [6][7][8]. Magnetic elements below a critical size will be in a monodomain state, and the magnetization of each element can be described in terms of a single macrospin [9,10]. The ground state ordering of these macrospins is determined by the geometric arrangement of the elements [11] as well as their shape.
To first order, the total magnetization of a monodomain disk can be approximated as a point dipole. The ground state configuration in a lattice of such dipoles is well established and is predicted to be FM for a hexagonal lattice [12,13]. Collective ferromagnetic ordering has been shown in assemblies of close-packed monodisperse nanoparticles [14][15][16]. For a square lattice, the predicted ground state is two-fold degenerate, with stripe-ordered However, models including higher order moments [17] or spin-wave stiffness [18] show that this degeneracy is lifted and favor AF order. Recent experiments for a square lattice were found to support the presence of long-range order, compatible with this theoretically predicted behavior [7,19]. 3 Here, we directly image emergent long-range magnetic order in arrays of magnetostatically coupled nanoscale permalloy (Py; Ni 81 Fe 19 ) disks arranged in square and hexagonal lattices. Depending on the lattice symmetry, FM or AF order is stabilized. We also investigate magnetization reversal of these lattices in an applied field, as well as thermal relaxation of the magnetization in the square lattice. To this end, we use soft x-ray magnetic circular dichroism photoemission electron microscopy (XMCD-PEEM). This synchrotronbased technique with sub-100 nm spatial resolution relies on magnetic dichroism in the x-ray absorption to provide magnetic contrast.  Fig. 1(d)]. The results demonstrate that magnetostatic coupling supports long-range order in these magnetic metamaterials. Furthermore, this magnetic order depends directly on the lattice geometry.
The switching behavior of these arrays was investigated by applying small in-plane magnetic field pulses in situ, followed by XMCD-PEEM imaging in remanence. to a saturated state in the opposite direction. To maximize the magnetostatic interaction the disks were made as large as possible while still preserving a monodomain ground state. Due to variation in size, some magnets may have entered a flux-closure configuration [9]. The speckles observed in the magnetization maps in Fig. 2 may be attributed to such flux-closure configurations. We note a predominant orientation of the domain walls in Fig. 2(b) along the same direction as the average elliptic distortion of the disks (∼20° with the horizontal).
The corresponding magnetization reversal for the square lattice is shown in Fig. 3.
After initialization, this lattice is predominantly magnetized in one direction in remanence [ Fig. 3(a)]. However, we note the presence of short chains of disks with opposite 5 magnetization not seen in the hexagonal lattice. We attribute this observation to the fact that the square lattice is far from its dipolar-coupled ground state when saturated.
Consequently, some macrospins reverse their direction of magnetization to locally reduce the magnetostatic energy upon removal of the external field. The observation of a saturated state at remanence suggests that the anisotropy of the individual disks prevents relaxation of the array to its AF ground state, i.e., the system is in the blocked regime. When a small reverse field of 2 mT is applied, the array passes through a state of predominantly AF order The selection of AF over MV order has been previously explained by invoking higher order moments [17] to account for deviations from a purely dipolar field distribution due to the finite size of the disks. We have used micromagnetic modelling [21] to quantify the demagnetization energy for the FM, AF and MV order in the square lattice, see appendix B.
For perfectly circular disks ( = 1.00), we find that the AF and MV spin configurations are lowest in energy. The difference in demagnetization energy between these spin textures are within the numerical accuracy of this analysis (<< k B T) and are thus considered degenerate.
Thus, the selection of AF order in our system cannot be directly attributed to non-dipolar field distribution. However, if disks with elliptic distortions representative of our experiment ( = 1.05 along 20 degrees to the horizontal) are introduced, the degeneracy between AF and MV order is lifted, selecting the collinear AF ground state. This analysis shows that the measured elliptic distortion offers an independent mechanism for selection of the AF order.
The elliptic distortion of the disks will also affect the blocking temperature. This is briefly discussed in appendix B. We find that the average elliptic distortion in this experiment results in an energy barrier for magnetization reversal of 3.0 eV for individual disks at room temperature. However, for disks on a square lattice the activation barrier for switching from a saturated state to AF order is considerably reduced due to the dipolar coupling with the surrounding disks. For an elliptic distortion of = 1.05 this barrier is reduced to 1.4 eV. This finding is in keeping with the observed thermal relaxation observed for a saturated square lattice at 210°C [Fig. 5]. 7 In conclusion, we show by direct imaging that lattices of dipolar coupled nanomagnets can support long-range magnetic order. This ordering depends on the lattice geometry, with hexagonal and square lattices supporting FM and AF order, respectively. We find that the magnetic ground state of the arrays is affected by the shape of the nanomagnets and note that a small directional elliptic distortion of the disk-shaped elements on a square lattice favors collinear spin arrangements. The present work may prove useful to engineering of magnetic metamaterials and stimulate further investigations of dipolarcoupled systems. 8 In Fig. S1, ellipses are fitted to the nanodisks in the scanning electron micrographs of the square array, with a nominal disk diameter of 100 nm and a pitch of 130 nm. The scanning electron micrograph is shown as recorded in Fig. S1(a) and with the fitted ellipses in Fig. S1(b). The elliptic distortion = is plotted versus rotation of the major axis in Fig.   S1(c). The polar histogram in the inset shows the distribution of major axis orientations. The elements have an elliptic distortion of up to = 1.16, with an average of = 1.05. The polar histogram reveals a preferential orientation of the major axis at 20° with respect to the horizontal. We note that this preferential orientation is systematic and possibly due to a deviation from circularity of the electron beam. These results are also representative for the hexagonal lattice.

APPENDIX B: MICROMAGNETIC MODELLING
Here, the micromagnetic modelling is described in detail. We have used these models to make rough estimates of the effect of the elliptic distortion of the disk on the long-range order and blocking temperature. Typical material parameters for Py were used, with = 860 kA/m and a small cell size of 0.5x0.5x0.5 nm 3 to reduce effects of projecting circles onto a discrete simulation lattice.

I. Effect of elliptic distortion on the magnetic ground state
Here, the effect of the preferential disk ellipticity on the demagnetization energy for the FM, MV and AF order is calculated for a unit cell of 2x2 disks with 130 nm pitch, repeated for an overall array of 154x154 disks, corresponding to the fabricated sample. The demagnetization energy per disk was calculated for arrays initialized with FM, AF and MV order, respectively, and with uniform magnetization within each disk [ Fig. S2]. For perfectly 9 circular disks, the MV and AF configurations were degenerate at room temperature with For an elliptic distortion of = 1.05 along the horizontal direction, the degeneracy is lifted, and a gap of Δ = 1.8 eV opens with the AF order being lowest in energy. Thus, for the square lattice this simple micromagnetic analysis predicts a degenerate ground state for perfectly circular disks. However, this degeneracy is lifted when a preferential elliptic distortion is present.

II. Single disk blocking temperature
In the following section, the effect of elliptic distortion on the shape anisotropy and blocking temperature ( ) is estimated for single disks. Distortions up to 20% were simulated, corresponding to the range observed experimentally. The shape anisotropy was assessed from the difference in demagnetization energy between uniformly magnetized elements oriented along the major and minor axes, respectively. For an elliptic distortion = 1.05 the energy difference is ℎ = 3 eV. The Néel-Brown expression can be used to estimate the relaxation time for a magnet with an energy barrier Δ , where we use 0 = 10 −10 s as the inverse attempt frequency [22]. For Δ = ℎ = 3 eV, the relaxation time = 2.0 ⋅ 10 21 s for T = 210°C, which implies that the magnetization of the average disk is blocked even at the highest temperature accessed in our experiment.

III. Lowering of activation barrier due to magnetostatic coupling
The magnetostatic coupling of neighboring magnets may reduce the activation energy for switching. This is the case for the square lattice when going from a saturated to 10 an AF configuration. To keep the model simple, a system of 3x3 magnets is considered as shown in Fig. S3. The array is initially magnetized to the right, except for one disk to the right of the center magnet pointing to the left. When the center magnet is rotated 90 degrees, the demagnetization energy is reduced by 1.6 eV.
If we assume that the energy barrier for a magnet with = 1.05 is reduced from 3 eV to 1.4 eV by the dipolar coupling, the relaxation time at a temperature of 210°C is reduced