Robust ferromagnetism carried by antiferromagnetic domain walls

Ferroic materials, such as ferromagnetic or ferroelectric materials, have been utilized as recording media for memory devices. A recent trend for downsizing, however, requires an alternative, because ferroic orders tend to become unstable for miniaturization. The domain wall nanoelectronics is a new developing direction for next-generation devices, in which atomic domain walls, rather than conventional, large domains themselves, are the active elements. Here we show that atomically thin magnetic domain walls generated in the antiferromagnetic insulator Cd2Os2O7 carry unusual ferromagnetic moments perpendicular to the wall as well as electron conductivity: the ferromagnetic moments are easily polarized even by a tiny field of 1 mT at high temperature, while, once cooled down, they are surprisingly robust even in an inverse magnetic field of 7 T. Thus, the magnetic domain walls could serve as a new-type of microscopic, switchable and electrically readable magnetic medium which is potentially important for future applications in the domain wall nanoelectronics.


S1. Ferromagnetic component above TN: Mex
Cd2Os2O7 shows the ferromagnetic component Mex which is present above TN (Fig. 2b) and is almost temperatureindependent upon cooling across TN, in addition to the linear component Mlin from a paramagnetic or an antiferromagnetic state of bulk and the robust ferromagnetic component Mrob originated in the {001} MDWs. The Mex of crystal A saturates already at 1 T to a small value of 6.7 × 10 −5 µB/Os, which corresponds to ~0.05% of the magnetic moment of Os (µOs = 1-1.5 µB; Yamauchi, I. & Takigawa, M. in preparation), even above room temperature. Hence it must come from a ferromagnetic moment, though the magnetic hysteresis is almost absent. We have examined several crystals and always observed similar Mex. The origin of Mex is still unclear, but must be irrelevant to the AIAO order. We speculate that the Mex is due to a tiny amount of ferromagnetic inclusion or a ferromagnetic layer formed on the crystal surface 1 . Then, d is calculated as 2-3 µm. This value is considerably smaller than the actually observed value of ~20 µm by the circular polarized resonant X-ray diffraction imaging technique 2 . However, the surface density of MDWs can be smaller than the bulk density as a result of annihilation of small domains at the surface. Moreover, there is a significant sample dependence in the density. Thus, we think that our MDW model reasonably account for the magnitude of the observed Mrob.

S3. Stability of MDWs
The local structures of MDWs are considered in terms of the classical spin model. In the extreme limit of the strong Ising anisotropy 3 , four magnetic moments on the vertices of tetrahedron can point only in or out to the center of the tetrahedron. As a result, three types of spin configurations are possible: 4-in or 4-out (denoted as 4-0), 3-in/1-out or 1-in/3-out (3-1), and 2-in/2-out (2-2). The energies of these configurations are given as E40 = −6Jeff, E31 = 0, and E22 = 2Jeff, where Jeff is the nearest-neighbor effective antiferromagnetic interaction (Jeff > 0). The stability between these MDWs can be compared in terms of these energies.
The energy of a MDW EMDW defined as the energy cost per area is calculated as 2(E22 − E40)/a 2 = 16Jeff/a 2 for a (001) MDW consisting of only 2-2 tetrahedra (Fig. S1a) On the other hand, the EMDW of a (001) MDW consisting of only 3-1 tetrahedra (Fig. S1b) is 24Jeff/a 2 . More complex (001) MDWs containing both 2-2 and 3-1 tetrahedra apparently take larger energies than 16Jeff/a 2 . Therefore, the 2-2 structure of Fig. S1a Fig. S1c and d are most stable with EMDW ~17.0Jeff/a 2 and ~13.9Jeff/a 2 , respectively. Therefore, the three kinds of MDWs can coexist in a crystal, because their EMDW values are relatively close to each other. In addition, the potential barriers between them and also a pinning by defects may help the coexistence. Figure S4. Conceptual representation of a microscopic magnetic memory device utilizing the robust ferromagnetic moment of the (001) MDW of the AIAO order for information storage. In the writing action, a crystal containing a (001) MDW at a low temperature below TN is instantaneously heated above TN by a pulse current and immediately cooled in a small magnetic field so that the ferromagnetic moment of the MDW align up or down along the field. In the reading action, a Hall voltage, that is induced by conduction electrons (e − ) confined around the MDW turning right or left depending on the directions of the ferromagnetic moments due to the anomalous Hall effect, is measured to know the memory.

S4. MDWs observed on the (111) crystal surface
The experimentally visualized domain pattern on a (111) facet of a single crystal provides important information about the coexistence of MDWs: two kinds of rectilinear MDWs running along the <110> and <112> direction are observed 2 . Figure S2 shows how the three kinds of MDWs appear on the (111)

S5. Magnetic properties of the {110} and {111} MDWs
The  Fig. 1c. Moreover, as shown in Fig. S3, the magnetization Mfree left after the subtraction of Mlin, Mex, and Mrob gradually grows upon cooling below 100 K and almost saturates at 3 K in 7 T. The lowesttemperature M-H curve can be fitted well by the Brillouin function, indicating the presence of non-interacting free magnetic moments; the Curie-like upturn must also come from this Mfree. The magnitude of Mfree is as small as 7 × 10 −5 µB/Os and happens to be nearly equal to that of Mrob. Thus, it is likely that the Mfree also comes from uncompensated moments at the {110} and {111} MDWs. However, an interesting question is why they do not freeze at such low temperatures. There is an energy barrier as large as 80 K due to the strong magnetic anisotropy 3 for a magnetic moment at the bridging site in the 3-1/4-0 pair to flip. Thus, the spin should freeze in the classical picture. Possibly, a certain quantum effect play a role in the flipping process. In fact, a quantum tunneling effect is pointed out to understand the quasi-free spins of MDWs in the distorted kagome antiferromagnets Na2Ba3[Fe3(C2O4)6][A(C2O4)3] (A = Sn, Zr) 4 . This may be also the case for Ising spins in the spin-ice system on the pyrochlore lattice, where spin flipping occurs at much lower temperatures compared to the magnetic interaction energy 5 .

S6. Possible application in the domain wall nanoelectronics
The {001} MDW found in this study is atomically thin and possesses a robust ferromagnetic moment together with electrical conductivity. The direction of the ferromagnetic moment can be controlled by a small field upon cooling, and the magnitude is reproducible in repetition, as demonstrated in Fig. 3a. Taking all these unique features of the MDW together, it would be possible to design a novel electronic device used in the domain wall nanoelectronics. Here we propose an example of such a novel magnetic memory device as illustrated in Fig. S4.
Let us assume a microscopic crystal containing a single or a set of parallelly aligned (001) MDWs. The direction of the robust ferromagnetic moments is used to store information; up ('0') or down ('1'). In the writing process, the crystal is heated instantaneously up to above TN by Joule heating using a pulsed electrical current and is immediately cooled in a small magnetic field generated by a mini coil, so that the robust ferromagnetic moment polarizes along the field. Once cooled down to a low enough temperature, the memory is kept robust against external fields. In the reading process, the anomalous Hall effect is used to detect the up or down polarization of the MDW magnetization. Since the MDW is atomically thin, a large response in the Hall voltage is expected even for a small polarization. In addition, a higher-density integration could be achieved because domain walls in antiferroic orders are more stable for miniaturization than those in ferroic orders. Therefore, employing the MDWs of the AIAO order would provide us with a novel route to a new type of magnetic memory to be used in the domain wall nano electronics in the future. One practical problem to get over for this is to find a more appropriate material that has a higher TN above room temperature to the AIAO order and is easy to handle in the manufacturing process than Cd2Os2O7. We believe that there is a chance to discover such a compound or other type of Ising antiferromagnets that carry useful MDWs in nature.