Towards control of the chirality sign at ultrathin metal films: Bi at 2Ni/Co

Proximity effects can be used to introduce spin–orbit interactions in magnetic metallic layers in contact to a heavy metal (HM). This well known phenomenon has been exploited to induce chiral spin textures at Co ultrathin films, where the left- or right- handedness can be tuned by the HM layer position, based on the broken inversion symmetry of the film and the existence of an additive interface effect. Here we show that structural and chemical features introducing further symmetry reductions can be added to this scenario ultimately enabling control over the definition of a unique winding sense. We focus on 2Ni/Co heterostructures and Bi, a scarcely explored HM metal of large size, to combine a chemically inhomogeneous ferromagnetic stack along the normal to the surface with in-plane asymmetries. Our results are contrasted to 2Co layers combined with Ir.


Introduction
Recent years have witnessed an increasing interest in the investigation of artificial structures designed to host stable spin textures with well-defined chirality [1][2][3][4][5]. These systems could enable the development of novel concepts in emergent technologies related to spinorbitronics and neuromorphic computing [6][7][8][9].
Among them, ultrathin magnetic metal films are attractive as their broken inversion symmetry makes possible the stabilization of non-collinear spin couplings derived from the Dzyaloshinskii-Moriya interaction (DMI) [10][11][12][13]. This way the degeneracy between chiral spin textures can be lifted, allowing for example to design magnetic domain walls with unique handedness that can be moved by torques based on the spin Hall effect [14][15][16][17]. Paradigmatic systems are those based on the combination of ferromagnetic films with heavy metal (HM) layers, in particular those including Co/Pt and Co/Ir interfaces [18][19][20][21]. The dominant DMI term is local, and for simple geometries intralayer DMI dominates over interlayer DMI and the relevant chiral energy balance restricts to the interface layers in contact [5,18]. This allows to design the chiral spin texture of a metallic film by adding contributions from different interfaces, exploiting an additive interface effect. A difficulty for this purpose is to predict the DMI strength and sign, that depends on local orbital hybridizations and charge density redistributions in a complex way [22], making necessary to analyze in detail the magnetic energy balance at each system of interest to actually determine the stability of non collinear spin alignments.
Adsorption of light elements involving extended s and p electronic states has proven useful for manipulating the DMI [23][24][25]. However, little attention has been paid to HMs that combine strong spin-orbit coupling (SOC) with a p valence band, such as Bi [26,27]. Nevertheless, these materials offer interesting features for achieving robust chiral objects: on the one hand, the hybridization of the HM p band and the d band of the ferromagnet fulfills one of the necessary conditions for a non-vanishing DMI [28]. On the other hand, the correlation between the DMI and the spin moment induced in the HM is unclear [28], while such an induced moment complicates the magnetic energy landscape by contributing to the net magnetization and introducing weak magnetic interactions that interfere with the control of the system's response. One advantage of Bi over 4d and 5d HM metals is that the p states are less prone to spin polarization through hybridization, making a clean introduction of SOC effects in the magnetic stack possible. Furthermore, this feature opens up the possibility of using Bi as a non-magnetic spacer for complex exchange interactions [29].
The purpose of our study is to explore the ability of Bi to induce chiral spin textures in the simplest scenario: spin spirals in an ultrathin magnetic 3d film. As we will show, the big size of Bi introduces structural symmetry reductions that may inhibit the sign reversal of the DMI just by inverting the HM position, opening the path to the creation of homochiral spin structures. We will further explore this idea choosing a 2Ni/Co heterostructure as magnetic film. The chemical asymmetry of the 2Ni/Co stack offers an additional degree of freedom to manipulate spin-orbit (SO)-derived spin textures. Opposite to Co, non-collinear spin textures at Ni have been much less studied. However, capping a Co layer with a Ni bilayer is known to stabilize a large perpendicular magnetic anisotropy (PMA), evidencing the ability of Ni to tune SO interactions in this system [30,31]. Here we will prove that: (i) chiral degeneracy can also be lifted at Ni films by contact to a HM layer, with a chiral sign opposite to Co; (ii) combination of the in-plane symmetry reductions induced by Bi and the broken internal inversion symmetry of the 2Ni/Co stacks serves to stabilize homochiral structures.

Theoretical method
We perform ab initio simulations using the Vienna ab initio Simulation Package (VASP) code [32] of 2Co and 2Ni/Co multilayers on Cu(111). Throughout the paper, the two-dimensional unit cell symmetries are referred to the (1 × 1) Cu unit cell. Both Cu and Ni adopt an fcc structure, and the Cu substrate has a lattice parameter close to those of Co and Ni, minimizing the emergence of strain effects. Our models are based on the in-plane Cu lattice parameter, while interatomic distances are allowed to vary by relaxing atomic positions until the forces on the atoms are below 0.01 eV Å −1 . The Co film thickness is limited to two layers to retain fcc stacking, as Co grows naturally in hcp form (see supplementary material S1). The surface region in our slabs is modeled inserting a vacuum region of 10-15 Å that guarantees both slabs surfaces do not interact. Similarly, three Cu planes are proved to be enough to include bulk-like substrate effects in the magnetic heterostructure (supplementary S1). As HMs we consider both Ir and Bi. Ir is an fcc element with a slightly larger atomic radius than Cu, that can be modeled assuming the Cu lattice parameter. Bi crystallizes in a rhombohedral structure that can be interpreted as an irregular stacking of fcc-like planes. The large size of Bi atoms imposes the use of surface (2 × 2) unit cells that accommodate only two Bi atoms per plane, arranged forming a honeycomb layer, as shown in figure 1. For all structures under study we consider the HM layer at two different positions: the interface with Cu (labeled HM interface) and on top the outermost surface plane (HM surface). Table 1 compiles the total energies of the different structures considered, including also 2Ni films on Cu(111). In general, HM layers prefer to be at the surface, except for Ir combined with 2Ni/Co stacks. However, the variation of the energy depending on the HM layer position is much larger in the case of Bi. It reflects the well-known surfactant property of Bi, and might also evidence a larger difficulty to stabilize homogeneous Bi interface layers without Bi segregation towards the surface.
Our simulations employ the perdew burke ernzerhof (PBE) parameterization of the exchange-correlation functional and projected augmented wave potentials [33,34]. For all systems, also the in-plane atomic positions have been relaxed until the forces on the atoms were below 0.01 eV Å −1 . The self-consistent ground state is obtained using an energy cutoff of 500 eV for the plane-wave basis set and a k-mesh of 16 × 16 × 1. Determination of local charges and magnetic moments is based on a Bader analysis, that correctly integrates the entire real space charge distribution avoiding space filling problems linked to atomic spheres [35].
Magnetic chirality is evaluated from the energy difference between clock-(CW) and anticlock-wise (ACW) real space spin spirals including the SOC [36]. This energy is proportional to the strength of the DMI, and thus to the tendency of the system to stabilize non collinear spin orders. The spiral orientation is controlled using the constrained magnetic moments method, following the prescription at [18]. Details are provided in the supplementary material S2. K-meshes of 3 × 6 × 1 (2 × 5 × 1) are used for 4 × 1 (4 × 2) cells, ensuring a precision in the energy difference between CW and ACW spirals below the meV range. We have checked that our results are robust upon increase of the k-mesh and the spiral length [37].
We have also estimated the PMA of the films under study, based on self-consistent total energy calculations including the SOC. The magnetic anisotropy energy (MAE) is defined as the energy difference between the structure with the magnetization along the normal to the surface and that with the magnetization at an in-plane high symmetry axis. Under this definition, negative MAEs correspond to PMA. Reciprocal space integrations with k-meshes of 10 × 10 × 1 have been used, to guarantee a precision in the MAE below the meV range. In figure 2 we provide the MAE of the system with largest anisotropy, the 2Ni/Co film with Bi, for different Bi positions and surface stacking sequences. As will be evidenced in section 3.2   (figure 5) the MAE is one order of magnitude lower than the magnetic chiral energy, ensuring the DMI as the dominant SO term in the systems under study. Our calculations do not include the magnetostatic dipole energy (E dd ), a necessary term to evaluate the actual magnetic anisotropy. E dd is proportional to the local magnetic moments, and opposes to the PMA adding a positive contribution that brings the magnetization in-plane [38]. Measurements of PMA in 2Ni/Co films serve to estimate the upper limit of E dd , evidencing that it is lower than the SO derived anisotropy contribution [30,31]. In principle we can assume a similar situation when Bi occupies HM interface positions. When Bi is placed at the surface, it lowers the surface magnetization thus reducing magnetostatic flux lines, so that the E dd contribution is expected to be even lower than for the bare 2Ni/Co film.

Ultrathin Co films
As mentioned in the introduction, Co layers have been proved to hold robust chirality when placed in contact to HMs such as Pt or Ir. Here we focus on two Co layers grown on Cu(111). At this reduced thickness, Co adopts the Cu lattice, both regarding the (similar) in-plane lattice parameter and the fcc stacking sequence.
We explore the magnetic chirality induced by a Bi layer in contact to the Co film, comparing it to the case of Ir. Consideration of Bi introduces a p valence band as carrier of SOC, opposite to the localized d states of traditionally studied HM layers. An additional feature is the big size of Bi atoms, that forces to use (2 × 2) unit cells with a honeycomb HM layer that does not fill all available fcc atomic positions. This causes   1) cell, determined as the difference between CW and ACW spin spirals, of 2Co and 2Ni films on Cu(111) in contact to a Bi or a Ir layer placed either at interface or surface positions. For Ir at 2Co/Cu(111), besides the full monolayer coverage of (1 × 1) symmetry, a 0.5 monolayer coverage of (2 × 2) symmetry arranged in a honeycomb structure similar to Bi is considered. magnetic asymmetries in the Co layer in contact to Bi, as we will explain now. The Bi-free Co film is characterized by magnetic moments of 1.73 and 1.62 µ B at the inner and surface planes, respectively. The increase of the surface magnetic moments is a well known effect arising from reduction in the number of bonds at the outermost surface [39]. Figure 3 shows that bonding to Bi slightly reduces the Co spin moment around 1.56 µ B , no matter the Bi layer position. But further the incomplete filling of fcc hexagonal sites at the Bi layer lowers the number of close neighbors of some Co atoms (highlighted in cyan at figure 3), mimicking a surface-like effect. Consequently, the corresponding Co atoms enhance their magnetic moment to values over 1.80 µ B . The result is that the in-plane distribution of spin moments becomes inhomogeneous when Bi is introduced. Additionally, it depends on the interface or surface position of the Bi layer, even though the integrated magnetization of the entire film differs only moderately between both situations, as also shown in figure 3. In general, Bi atoms do not acquire any significant spin polarization, thus not contributing to the system magnetization. This is at difference with Ir, where the d valence band has more tendency to polarize magnetically, leading to a HM moment around 0.2 µ B . Figure 4 shows the magnetic chiral energy of the 2Co/Cu(111) system in contact to a HM layer, defined as explained in the previous section and normalized to a (1 × 1) unit cell. With our prescription, negative (positive) chiral energies correspond to stability of CW (ACW) spin spirals. The two upper rows at the figure show the contribution corresponding to the maximum coverage provided by a Bi and an Ir layer, considering them either at interface or surface positions. In good agreement with previous calculations, Ir induces a large chiral energy at the Co film, that further reverses sign depending on the Ir layer position, reflecting the well known additive effect of interfacial chiral interactions [40]. The Bi layer also introduces magnetic chirality at the Co film, emerging as an HM alternative to tune the handedness of Co spirals.
An additional aspect worth to remark is that both the chiral energy and the interface additive effect are weakened in Bi as compared to Ir. Both changes can be partially assigned to the incomplete coverage induced by the Bi layer. To explore this hypothesis, we have performed calculations of an artificial honeycomb Ir layer Table 2. Spin moment (in µB) at the different Ni sites labeled following the 2Co structure of figure 3. As a reference, the corresponding moments at a Bi-free Ni film are 0.65 µB (outer layer) and 0.49 µB (inner layer).

System
Bi interface Bi surface  figure 4, proving that effectively the HM honeycomb arrangement largely reduces the chiral energy and can even suppress the additive interfacial effect. The first feature is expected as HM induced chirality arises from local proximity interactions, so that reducing the number of Co atoms in contact to HMs should lower the DMI weight. While the second is less intuitive. Incomplete coverages are scenarios for the emergence of anisotropic DMI, where the DM strength cannot be reduced to a single parameter D. This adds complexity to the correlation between the DM induced changes of the electronic properties and the resulting handedness. At interfaces of symmetry lower than C 3v , the combination of DM vectors has been explored to create complex multichiral objects [41]. In our case, the results for Ir indicate that the symmetry lowering effects can be used to invert the chirality sign of simple flat spin spirals. Another relevant aspect evidenced by the calculations of Ir honeycomb layers is that, when Co atoms have the same number of Ir and Bi neighbors, the DMI strength carried by any of them has similar magnitude, confirming Bi can be a robust HM candidate to introduce magnetic chirality in metallic films.

2Ni/Co heterostructures
The previous section has evidenced that Bi can be used to manipulate the DMI in order to tune the handedness of induced spin spirals. Here we will explore chemical inhomogeneities along the normal to the surface in the ultrathin metallic film, choosing as model system the 2Ni/Co heterostructure. As mentioned in the introduction, this system holds strong SOC-derived interactions even without proximity to a HM. For our present purpose, the broken chemical inversion symmetry at the trilayer offers a playground to overcome the interface additive effect of chiral interactions derived from varying the HM layer position.
Before addressing the combined 2Ni/Co film, we will estimate the strength of DMI interactions at Ni as compared to Co. In order to do this, we calculate the chiral energy of Ni spin spirals at the same conditions previously considered for Co: a two layers film placed in contact to a surface or interface HM layer, either Bi or Ir. The results are shown at the two bottom rows of figure 4. The magnetic chirality at Ni films reverses the sign found at Co in all situations, conferring an additional interest to the 2Ni/Co system. It is also clear that the magnetic chiral energy is considerably weaker at Ni than at Co in all configurations under study. This can be partially understood as the contribution to the micromagnetic energy carried by the DMI in ultrathin magnetic films scales with the magnetic moment of the 3d atom [28,42], Ni moments being roughly half those of Co, as can be seen at table 2. In spite of this, the table also shows that the penetration depth of the changes induced by the Bi surface layer is longer at Ni. This suggests that adding Co to the Ni bilayer, it might be sensitive to the local DMI interactions induced at the Ni surface.
We now calculate the magnetic chiral energy of the system formed by 2Ni/Co on a Cu(111) substrate, again considering a HM formed either by Ir or Bi placed both at interface and surface positions. In principle, if only the local interactions of the HM with the adjacent metal layer were relevant, one would expect to recover for each system the same values obtained from the corresponding situations at the 2Co or 2Ni films. The results shown in figure 5 indicate a completely different situation: while a simple additive interface effect based on the local HM interactions with close neighbors would provide a positive magnetic chiral energy under all Bi layer positions, we obtain negative values that favor robust CW spin spirals. Particularly striking is the increase of the magnetic chiral energy when Bi is placed at the surface. Also for Ir the chiral magnetic response of the 2Ni/Co film breaks any trend derived from simple additive effects extracted from the 2Ni and 2Co systems, and in particular again placing Ir at the surface significantly increases the stability of CW spin spirals at the 2Ni/Co heterostructure. These results point out to a penetration of the DMI beyond the HM interface layer, as already claimed in Gr/Co/Pt systems [25].
Related to this, a natural question is then the robustness of the magnetic chirality obtained at the 2Ni/Co heterostructure as the thickness is enlarged. In order to explore such aspect, we have performed calculations of thicker 2Ni/Co stacks of two periods, the results are shown in figure 6. In the case of surface Ir, convergence was not achieved under a reasonable constraint parameter, as some deviation of the initially proposed spiral orientation always emerged, making impossible to extract a chiral energy as defined here. For the rest, the trends obtained for magnetic chiral energies at 1 period are reproduced at 2 periods within the precision of our magnetic constraints procedure (see supplemental material S2). This implies that the   homochiral features will be maintained at thicker films, which can be useful in the case of Bi to minimize effects derived from its tendency to float. Previous studies in the search for simple descriptors enabling estimation of the DMI have identified a correlation between the sign and magnitude of the interface electric dipole moment at the Co layer in heterostructures where it is sandwiched between Pt(111) and different non-magnetic transition metal cappings. The situation here is slightly different and more complex. On one hand, the magnetic layer in contact to the HM is also adjacent to another magnetic layer, and we have seen that there is some penetration of the SO-derived features. On the other, the charge distribution at interface layers is not homogeneous, as not all atoms are bonded to Bi, enabling non-uniform lateral dipole components. The relevance of the first aspect can be inferred from 2Ni/Co stacks combined with Ir, where uniform in-plane charge distributions can be assumed. Table 3 provides the electric dipole moments at the HM interface with Ir either at the surface or interface positions, obtained from the variation of the valence charge with respect to the atomic value, multiplied by the interatomic distance. Comparing to the energies at figure 5, there is a correlation between the chirality sign and the Co electric dipole when Ir is at the interface, a situation closer to the buried Co/Pt(111) heterostructures. But no correlation is observed with the electric dipole of Ni and the DMI sign when Ir moves to the surface.
In the case of Bi only those atoms bonded to it would contribute to the DMI. Table 3 also contains the corresponding electric dipole moments. It evidences a significant difference between Bi and Ir, regarding their relative electronegativity with respect to Co and Ni. Again, there is a correlation between the sign of the DMI and the electric dipole of Co for Bi at the interface, and also the relative reduction of the DMI as compared to the Ir interface follows the reduction of the Co electric dipole moment. At the surface, the sign of the DMI coincides with the sign of the electric dipole of Ni, but there is no correlation between the magnitude of the DMI or the Ni electric dipole as compared to the Ir surface. Even though it is tempting to ascribe the differences between Bi and Ir at interface positions to their opposite electronegativity, it should be taken into account that Co atoms not bonded to Bi contribute with an electric dipole of +0.24 eÅ, leading to a complex distribution of interacting in-plane electric dipoles.
Another relevant aspect not mentioned so far is that there is some arbitrariness in the choice of the Bi sites, arising from the availability of multiple stacking positions. Alterations of the stacking sequence are known to strongly modify SOC derived properties such as the magnetic anisotropy, but less is known regarding antisymmetric exchange. In our slabs we have adopted for Bi hollow bonding sites that follow the fcc stacking sequence, after evaluating that this corresponds to the most stable situation. As shown in figure 1, the honeycomb layer occupies two different stacking sites. Alteration of one of these sites introducing some on top bonds slightly increases the system energy by around 90 meV per (1 × 1) cell, and thus may occur in real samples. The rightmost column of figure 5 shows that this alteration of the stacking sequence does not influence the stability of CW spirals. Similar results have been obtained for Gr/Co/Pt(111) when modifying the stacking of Co [25]. This suggests that stacking faults are relevant to determine the MAE, but have a minor effect on the DMI.
As a final remark, all previous results are based on structures with the Cu(111) lattice parameter. Even under the honeycomb (2 × 2) arrangement, this leads to interatomic Bi-Bi distances of 2.95 Å, while values up to 3.25 Å have been reported at Ni-Bi intermetalloids [43]. Our slab models are restricted to use a common two-dimensional lattice at all layers, but we can compare structures obtained with different in-plane lattice parameters that are relaxed along the normal to the surface to relief strains. Based on this approach, we have considered an expanded in-plane lattice providing Bi-Bi interatomic distances of 3.05 Å. The resulting magnetic chiral energies are included in the last row of figure 5. They reflect that the expansion does not alter the handedness trends previously shown for Bi using the Cu lattice parameter: the sign is preserved, and the strength enhances more than 25%. This supports that realistic models taking into account non-uniform interatomic distances may still hold the large DMI and homochirality reported here for Bi layers at 2Ni/Co heterostructures.

Conclusions
We have performed ab initio simulations of a 2Ni/Co stack in contact to a Bi layer, exploring the emergence of chiral spin textures. The lack of spin polarization at the p valence band of Bi enables to induce SO effects without introducing any additional magnetic component in the system. Our results indicate that CW spin spirals are stable independently of the Bi layer position (on top or below the 2Ni/Co stack), overcoming the sign reversal derived from a simple additive interface effect. This feature seems to be related to the inhomogeneous in-plane distribution of atoms arising from the large Bi size and to the combination of interfaces involving transition metals with opposite sign of the DMI. The result is robust under moderate strain modifications, the addition of more 2Ni/Co stacks, and the presence of stacking faults that could occur in realistic samples. This opens exciting possibilities to build thicker heterostructures with a unique winding sense of chiral rotations, and to explore chiral effects using Bi as a non-magnetic spacer. Though further work is deserved to identify the ultimate origin of the homochiral property, our work suggests Bi as an interesting HM candidate with special features to stabilize chiral objects.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).