Chiral magnetism of magnetic adatoms generated by Rashba electrons

We investigate long-range chiral magnetic interactions among adatoms mediated by surface states spin-splitted by spin-orbit coupling. Using the Rashba model, the tensor of exchange interactions is extracted wherein a pseudo-dipolar interaction is found besides the usual isotropic exchange interaction and the Dzyaloshinskii-Moriya interaction. We find that, despite the latter interaction, collinear magnetic states can still be stabilized by the pseudo-dipolar interaction. The inter-adatom distance controls the strength of these terms, which we exploit to design chiral magnetism in Fe nanostructures deposited on Au(111) surface. We demonstrate that these magnetic interactions are related to superpositions of the out-of-plane and in-plane components of the skyrmionic magnetic waves induced by the adatoms in the surrounding electron gas. We show that, even if the inter-atomic distance is large, the size and shape of the nanostructures dramatically impacts on the strength of the magnetic interactions, thereby affecting the magnetic ground state. We also derive an appealing connection between the isotropic exchange interaction and the Dzyaloshinskii-Moriya interaction, which relates the latter to the first order change of the former with respect to the spin-orbit coupling. This implies that the chirality defined by the direction of the Dzyaloshinskii-Moriya vector is driven by the variation of the isotropic exchange interaction due to the spin-orbit interaction.


Introduction
A lack of inversion symmetry, paired with strong spin-orbit (SO) coupling, generate the Dzyaloshinskii-Moriya (DM) interaction [1,2], a key ingredient for non-collinear magnetism, which is at the heart of chiral magnetism. The DM interaction defines the rotation sense of the magnetization, rotating clockwise or counterclockwise along a given axis of a magnetic material. This is the case of spin-spirals in two-dimensional [3][4][5] or onedimensional systems [6,7] down to zero-dimensional non-collinear metallic magnets [8][9][10]. This type of interactionis decisive in the formation of the recently discovered magnetic skyrmions (see, e.g. [11][12][13][14]), a particular class of chiral spin texture, theexistence of whichwas predicted three decades ago [15,16]. These structures are believed to be interesting candidates for future information technology [17][18][19][20] since lower currents are required for their manipulation, in comparison to conventional domain walls [21,22].
The ever-increasing interest in understanding the properties of the DM interaction and the corresponding vector is, thus, not surprising. Although the symmetry aspects of these interactions were discussed in the seminal work of Moriya [2], the ingredients affecting the magnitude and the particular orientation of a DM vector have been very little explored but are certainly related to the details of the electronic structure. In the context of longrange interactions mediated by conduction electrons, the DM interaction was addressed by Smith [23], and Fert and Levy [24]. They found a strong analogy withRuderman-Kittel-Kasuya-Yosida (RKKY) interactions [25][26][27]. Indeed, the long-range DM vector oscillates in magnitude and changes its orientation as function of

Description of the model
The investigation of the magnetic behavior of the nanostructures is based on an embedding technique, where magnetic impurities are embedded on a surface characterized by the Rashba spin-splitted surface states. Once the electronic structure is obtained, we extract the tensor of the magnetic exchange interactions as given in an extended Heisenberg model utilizing the mapping procedure described below.

Rashba model and embedding technique
The two-fold degenerate eigenstates of a two-dimensional electron gas confined in a surface or an interface, i.e.a structure-asymmetric environment, experiencesspin splitting induced by the SOinteraction. Within the model of Bychkov and Rashba [32,33], this splitting effect is describedby the so-called Rashba Hamiltonian x y x y y x R 2 2 2 so where g p ,g Î { } x y , , isthe componentof the momentum operator  p in a Cartesian coordinate system with x y , coordinates in the surface plane whose surface normal points alongê z ,m * is the effective mass of the electron,s g are the Pauli matrices, and  2 is the unit matrix in spin-space, with the z-axis of the global spin frame of referenceparallel toê z . a so is the Rashba parameter, a measure of the strength of the SO interaction and the parameter that controls the degree of Rashba spin splitting.
The energy dispersion of the Rashba electrons is characterized by the k-linear splitting of the free-electron parabolic band dispersion: 2 so 2 4 We note that in addition to theAu(111) surface, several systems carry Rashba spin-splitted states (see, e.g. [47][48][49][50][51][52]).  [37]. We want to calculate the magnetic interactions between magnetic adatoms immersed in a Rashba electron gas. Therefore, we use an embedding technique, where we connect the Rashba Green function G 0 to the Green function G of the system Rashba electron gas and magnetic adatoms via a Dyson equation. G 0 , connecting two points separated by  R, is given by: where G D and G ND , as defined inappendix A, depend on the position  R and energy E, while β is the angle between  R and the x-axis. When magnetic adatoms are present, the Green function connecting the adatoms sites i and j can be obtained from the Dyson equation: In the right-hand side ofequation (8), the first term is the isotropic exchange, while J ij A is the anti-symmetric part: which is connected to the DM vector components via: The last term of equation (8), J ij S , is the symmetric part that describes pseudo-dipolar interactions: For the Rashba model, we will see that there is a more natural way of decomposing the tensor, which is given in equation (18).
To find the magnetic ground state, we employ the magnetic exchange interaction tensor. We start from different initial configurations, evaluate the torque on each magnetic impurity and use it to iterate to new spin directions. Once the torque is below a numerical threshold, we take the magnetic configuration to define a candidate ground state. Then we compare the energies of the candidates and keep the one with thelowest energy.

Mapping procedure
The strategy is to consider the Hamiltonian describing the electronic structure of the nanostructures and perform the same type of differentiation as in equation (7) in order to identify the tensor of magnetic exchange interactions. We use Lloyd's formula [40], which permits the evaluation of the energy variation due to an infinitesimal rotation of the magnetic moments, starting from a collinear configuration [39,41,42]. In general, the contribution to the single-particle energy (band energy) after embedding the nanostructure is given by: the trace is taken over the spin-index, and a t i is simply the derivative of t with respect to a e i . Since the t-matrix can be written as: The final form of the tensor of magnetic exchange interactions is then finally given by: We see that equation (16) depends on the magnetic structure of the impurity cluster. In practice, we consider three different ferromagnetic configurations, aligned along the x-, y-and z-axes, compute the respective exchange tensors and keep the transverse blocks (e.g. for the ferromagnetic configuration along z we keep the xy block); elements that occur repeatedly are averaged.

Magnetic properties of dimers
3.1. RKKY approximation Before the numerical evaluation of the exchange tensor in nanostructures from equation (16), it would be interesting to have an approximate analytic form. This is achievable by considering in equation (16) the unrenormalized Green functions, G 0 , instead of G. Here we recover the RKKY approximation, expected from second-order perturbation theory and used, for example,in [31]. In the particular case of a two-dimensional Rashba electron gas, the Rashba Green function can be expressed using Pauli matrices: Surprisingly, we found anisotropies in the diagonal part of the exchange tensor that are generally neglected in the literature. The physical meaning of these anisotropies can be traced back to the extended Heisenberg model defined by the tensor of magnetic exchange interactions. In fact, by defining the x-axis as the line connecting the two sites i and j, we show in appendix B that the extended Heisenberg Hamiltonian describing the corresponding magnetic coupling can be written as: along the x-axis, I is given by -J J ij yy ij zz . This anisotropy is finite because of the two-dimensional nature of the Rashba electrons, so the x-and y-directions are non-equivalent to the z-direction. Here, I favors a collinear magnetic structure along the y-axis and counteracts the DM interaction. The analytical forms of the magnetic exchange interactions allow us to understand their origin in terms of the magnetic Friedel oscillations generated by single atoms [36]. These oscillations carry a complex magnetic texture that can be interpreted in terms of skyrmionic-like waves. Within the RKKY approximation and neglecting the energy dependence of D i , the isotropic interaction, J, connecting two impurities at sites i and j, is proportional to the z-component of magnetization generated at site j by a single impurity at site i. In other words, the impurity at site j feels the effective magnetic field generated by the magnetization at that site but induced by the adatom at site i. D, however, is defined by the in-plane component of the induced magnetization. This is a central result of our work. Here, the corresponding magnetic field felt by the second impurity has an in-plane component and naturally leads to a non-collinear magnetic behavior, i.e. the natural impact of the DM vector. I does not have a simple interpretation, but it can be related to the anisotropy (difference) of the induced magnetization parallel to the impurity moment upon its rotation fromout-of-plane to in-plane. In the following we proceed to the analytical evaluation of (J D , ,and I) from the equations above. The details of the integration are given in appendix C.
Evaluation of J. In order to derive analytically the exchange interactions, we use an approximation for the t-matrices. We assume that they are energy independent (resonant scattering for the minority-spin channel, i.e. . This approximation, used in [36], is reasonable for an adatom like Fe deposited on aAu(111) surface. Then, we find the asymptotic behavior of G D and G ND for large distances R (see appendix C). The isotropic exchange constant can be expressed as: where ( ) x SI is the sine-integrated function of x. J is found to be the sum of two functions. The first one evolves as a function of R 1 2 , as expected for regular two-dimensional systems, but the second function decays like R 1 , which has been neglected in the work of [31]. The R 1 decay leads to a slower decay of J than what is known for a regular two-dimensional electron gas. The origin of this term is the Van Hove singularity at the bottom of the two bands; the density of states of the Rashba electron gas resembles that of a one-dimensional electron gas between E R and E=0, where the two bands cross. At very large distances, ( ) x SI converges to a constant p ( ) . Naturally, when k so is set to zero we recover the classical form of the RKKY interaction without SO coupling for a free-electron gas in two-dimensions, i.e.J evolves like Evaluation of D. We consider the same approximations used above to calculate the y-component of the DM vector (D) and find: Similar to the isotropic exchange constant, D is a sum of two terms. The first term decays as . Evaluation of I. In appendix C, we show that I is a sum of two integrals over the energy because of a branch cut in the Hankel functions. The first integral, denoted I 1 , goes from E R to zero and the second, and if we sum up the two terms: The integral involving ( ) qR cos 2 is important at short distances since it competes with one of the terms defining -J . In fact, it has the opposite sign of - (22)). This reduces considerably the value of I compared to J. The second integral involves ( ) qR sin 2 and therefore it leads to a small contribution for low values of k so . A perturbative development of I in terms of k so shows that I is second order in SO coupling µ ( ) k so 2 . In figure 1(a), we plot the magnetic exchange interactions J D , and I as function of the distance between two magnetic adatoms. The black curve depicts J, which at short distances is characterized by a wavelength l = » p 18.5 k F Å. In figure 1(a) we see a beating of the oscillations, which can be understood by looking at the first term in equation (22). Writing it as so , the superposition of these two wave vectors causes a beating effect at  (1)). (a) We use the RKKY approximation (see equations (19), (20), and (21)) and assume a maximal scattering cross section for the minority-spin channel d =  (4) and (16)). The vertical lines define a magnetic phase diagram indicating the nature of the orientation of the two magnetic moments as function of their separation. C indicates the collinear phase of the magnetic moments and NC the non-collinear phase.
due to the second term in equation (22), since One notices that for a large range of distances ( > R 40 Å) the magnetic interactions do not oscillate around the y=0 axis, which is due to the ( ) x SI term present in equations (22) and (23) for J and D, while for I, the shift comes from the last term in equation (25). All these terms come from the Van Hove singularity at the bottom of the bands. Similar to J, D is negative for distances larger than 25 Å, which means thatwithin the RKKY approximationthe chirality defined by the sign of the DM interaction changes only for dimers separated by rather small distances. We notice also that D and I are oscillating functions that can be of the same magnitude as J. Thus, we believe that such systems provide the perfect playground to investigate large regions of the magnetic phase diagram inaccessible with theusual magnetic materials.

Beyond the RKKY approximation
The deposited magnetic impurities naturally renormalize the electronic properties of the Rashba electrons. We can now prove that the contributions of the sine integral to the magnetic interactions are artifacts of the RKKY approximation. When the energy approaches the Van Hove singularity, , the multiple scattering series cannot be truncated, and theRKKY approximation cannot be made. The Green function connecting two impurities is given by: where the second equality corresponds to the multiple scattering, or Born series.
. However, from the first equality in equation (26) for  E E R , therefore, the Van Hove singularity will not contribute to the exchange interactions computed from equation (16) and the contribution from ( ) x SI vanishes. To quantify the impact of the renormalization on the electronic states mediating the magnetic exchange interaction, we numerically compute G, by considering consistently the multiple scattering effects. This is done first via considering an energy dependence in the t-matrix, assuming that they correspond to a Lorentzian in the electronic structure of the impurities, and thus the phase shift is given by  (4)) giving G. The evolution of the three exchange interactions after renormalizing the Green function is given in figure 1(b). As expected, we note the disappearance of the RKKY approximation artifact leading to the apparent offset of the oscillations beyond R=40 Å(see figure 1(a)). The beating effect in J occurs at the same distance as in the RKKY approximation because it is an intrinsic property of the Rashba electron gas. At large distances the intensities of J and Ddecrease quickly, but I keeps oscillating up to a distance of »200 Å where it decreases quickly to zero.

Magnetic configurations of dimers
Having established the behavior of the tensor of magnetic exchange interactions as a function of distance, we investigate now the magnetic ground state of different nanostructures characterized by different geometries and different sizes. After obtainingthe magnetic interactions usingthe mapping procedure described in section 2.3, we minimize the extended Heisenberg Hamiltonian with respect to the spherical angles, q f ( ) , i i , defining the orientation of every magnetic moment cos sin , sin sin , cos In order to check the stability of the magnetic ground state, we often add to the extended Heisenberg Hamiltonian the term å ( ) K e i i z 2 , where K is a single-ion magnetic anisotropy energy favoring an out-of-plane orientation of the magnetic moment,asis the case for an Fe adatom on Au(111). We choose as a typical value = -K 6 meV for all the investigated nanostructures [44].
For the particular case of the dimer, an analytical solution is achievable by noticing that two magnetic states are possible: collinear (C) and non-collinear (NC). This is counter-intuitive, since the presence of the DM interaction leads usually to a non-collinear ground state. The presence of the pseudo-dipolar term I makes the physics richer and stabilizes collinear magnetic states. Once more, because of the particular symmetry provided by the Rashba electron gas, within the non-collinear phase, the only finite component of the DM vector, D y , enforces the two magnetic moments to lie in thexz plane perpendicular to the DM vector. Within the collinear phase, I enforces the moments to point along the y-axis.
Non-collinear phase. Here the magnetic moments lie in the xz plane and the pseudo-dipolar term does not contribute to the ground state energy. The ground state is then defined by the angle, q = atan xz -plane withenergy J. However, these last two solutions will not occursince the NC phase is lower in energy.
There is competition between the collinear phase C and the non-collinear phase NC, which depends on the involved magnetic interactions. Without I, figure 1(b) will consist of one single phase, the NC phase. Thanks to I, there is an alternation of the two phases depending on the interadatom distance. The magnetic anisotropy K favors an out-of-plane orientation of the moments and tends to decrease the spatial range of the collinear phase where the moments point along the y-axis.
Phase diagram. In figure 2, we plot the phase diagram of the dimers = ( ) K 0 meV . The color scale shows the energy difference DE between the ground states found in the NC phase and C phase normalized by | | J . A negative (positive) energy difference corresponds to a NC (C) ground state. Thus the blue region corresponds to a C phase and the red region to a NC phase: , which define the magnetic phases plotted in figure 2. We notice that when I and J are of the same sign, the dimers are mostly characterized by a C ground state. The corresponding C phase is separated from the NC phase by a parabola, as expected from the term -D J 2 2 2 . Moreover we note that even within the NC phase, a transition occurs when the sign of I J changes. This is related to the nature of the NC phase that changes by switching the sign of The sum over sites j is limited by the size of the nanostructure but it can be infinite, e.g. if dealing with a monolayer or an infinite wire. We checked the validity of the previous relation utilizing the analytical forms of J and D obtained in the RKKY approximation, i.e. equations (22) and (23), and found that equation (28) can be recovered for  k R 1 so but the error is proportional to the term involving the sine integral ( ) k R SI 2 F . Therefore,if one neglects the contribution of the Van Hove singularity of the Rashba electron gas, one arrives at the formula of Kim et al [45] However, we proved that the multiple scattering precisely cancels this extra contribution, so we propose the following relation to hold: First we compared the RKKY expressions in figure 3(a), inserting the result of equation (22) intoequation (29), and then the RKKY expression for D given in equation (23). The agreement wasvery poor, as expected. Second, in figure 3(b) we extracted J from equation (16) and numerically evaluated equation (29), and then compared with D,also givenby equation (16). So for the more realistic case (using the renormalized electronic structure) we foundthat equation (29) is a very good approximation. The intriguing implication of equation (29) is that it gives an interpretation for the origin of the chirality being leftor righthanded according to the sign of D. For a given distance R, D can be of the same (opposite) sign of J if the latter's magnitude increases (decreases) with the SO interaction.

Magnetic properties of other structures
In this section we build magnetic nanostructures of different sizes and shapes made of Fe adatoms deposited on Au(111) according to the parameters given in section 3.2. The distance between the first nearest neighbors is  (16), i.e. beyond the RKKYapproximation, and from equation (29). As explained in the main text, the contribution from the Van Hove singularity that leads to the discrepancy seen in panel (a) is spurious.
chosen to be d=10.42 Åfor all structures, corresponding to the seventh nearest-neighbordistance on the Au(111) surface (lattice parameter a=2.87 Å). This is very close to what is accessible experimentally [29].
We compute the magnetic interactions for the considered nanostructures. For the chosen interadatom distance for building the magnetic nanostructures, interactions beyond nearest neighbors play no significant role. For that reason, we report in table 1 only the average nearest-neighbor interactions, although all interactions are taken into account when determining the magnetic ground states. The z-component of the DM vector is two orders of magnitude smaller than the in-plane components for all the considered nanostructures, thereforeit will be omitted when discussing the magnetic ground states. A summary of the obtained average magnetic interactions between nearest neighbors is provided in table 1.

Magnetism of linear chains
In addition to dimers, we investigated several linear chains of different sizes;all of them presented the same characteristics. Here we discuss the example of a wire made of 14 adatoms. In this case, the isotropic exchange interaction between the nearest neighbors is antiferromagnetic. On average it is equal to 6.90 meV, i.e.doublethe isotropic interaction obtained for the dimer, which highlights the impact of the nanostructure in renormalizing the electronic structure of the system. Within the RKKYapproximation, the magnetic interactions would be independent ofthe nature, shape, size of the deposited nanostructures. Due to the Moriya rules, the DM vector lies along the y-direction within the surface plane, similar to the dimer case. It is thus perpendicular to the x-axis defined by the chain axis. The DM interaction is around 2 meV between nearest neighbors, i.e. once moredoublethe value obtained for the dimer.
The magnetic ground state is a spiral contained in the ( ) xz plane with an average rotation angle of 110°b etween two nearest-neighbor magnetic moments (see figure 4). Interestingly, this angle is much smaller than the one found for the dimer (164°),but similar to that found for intermediate chain sizes. The pseudo-dipolar term is around I=0.26 meV, and it has no impact on the ground state. This situation is equivalent to the NC phase of the dimer. Of course, choosing an interatomic distance with a large pseudo-dipolar term for the dimersleads generally to stable collinear magnetic wires (not shown here). We noticed that the effect of the magnetic anisotropy energy ( = -K 6 meV ) is mainly on the edge atoms. Indeed, the rotation angles between adjacent innermoments remain ataround 110°,while at the edgesthe magnetic momentspoint more along the z-direction. The rotation angle between the magnetic moment at the edge and the z-axis is reduced to 25°.

Magnetism of compact structures
After the one-dimensional case, we address in this section compact structures with the same interatomic distance as the one considered for the wire.
Trimer. We studied a trimer forming an equilateral triangle. The isotropic exchange constant J is equal to 3.51 meV, favoring antiferromagnetic coupling, a value close to the one found for the dimer. The frustration is large in this case, leading to a non-collinear ground state even without SO coupling [10,46]. The magnetic moments lie in the same plane, e.g. the surface plane, with an angle of 120°between two magnetic moments. This state has continuous degeneracy, since rotating each magnetic moment in the same way leaves the energy  invariant. If we now consider the DM interaction, we find that  D, with a magnitude of 1.0 meV (similar to the dimer value), lies in the xy-plane and perpendicular to the axis connecting two adatoms (see figure 5(c)). This interaction lifts the degeneracy present without D, stabilizing the magnetic structure shown in figures 5(a) and (b). The pseudo-dipolar term I is equal to 0.13 meV and is small compared to J and D, therefore the noncollinear phase is more stable. The isotropic interaction keeps the angle between the in-plane projections of the moment at 120°, while the DM interaction generates a slight upward tilting (81°instead of 90°). In fact, every DM vector connecting two sites favors the non-collinearity of the related magnetic moments by keeping them in the plane perpendicular to the surface and containing the two sites. This is,however,impossible to satisfy at the same time for the three pairs of atoms forming the trimer, which leads to the compromise shown in figures 5(a) and (b). The magnetic anisotropy reduces ( = -K 6 meV) considerably the non-collinearity and the three moments are forced to point almostparallel to the z-axis. Two of the magnetic moments are characterized by an angle of 10°instead of 81°with respect to the z-axis, while the angle of the third moment is 173°, as shown in figure 5(d). This is an interesting outcome compared to the behavior of the wire, which is characterized by a large averaged DM interaction in comparison to the trimer. Obviously the shape of the nanostructure is important in stabilizing non-collinear magnetism. The interadatom distance is = d 10.42 Å, while the average nearest-neighborisotropic exchange interaction is J=6.90 meV and the nearest-neighborDM vector points along the y-axis with an average intensity D=2 meV. The magnetic magnetic anisotropy K=0 meV.
Hexagon. We consider now a system of six atoms forming a hexagonal shape with the same interatomic distance as the one considered earlier. The magnetic ground state configuration is non-collinear as shown in figures 6(a) and (b). The isotropic magnetic exchange interaction, J, between nearest neighbors is of antiferromagnetic type, similar to the value obtained for the other nanostructures studied so far. J reaches a value of 5.64 meV, which is rather close to the interaction found for the wire. In fact one could consider this hexagonal structure as a closed wire. The magnitude of the DM vector connecting two nearest neighbors is large, 1.67 meV, but not as large as the one of the wire. The non-collinear state is better appreciated when plotting the projection of the momentunit vectors on the surface plane in figure 6(c) and along the z-axis in figure 6(d). The polar angle is either 16°or 164°, according to the antiferromagnetic nature of the interactions. The azimuthal angle follows the symmetry of the hexagon, leading to an azimuthal angle difference of 120°between adjacent moments. The magnetic texture is a compromise involving the antiferromagnetic J and the DM vectors (plotted in figure 6(e)). While J tries to make the moments anti-parallel to each other, the DM vector tends to make them lie in the plane perpendicular to the surface and containing at the same time the two pairs of atoms (similar to the dimer configuration). However, the magnetic moment has to satisfy the DM vectors arising from its nearest neighbors and, therefore, the moment compromises and lies in the plane perpendicular to the surface, and containing the atom of interest and the center of the hexagon. This is similar to what was found for the compact trimer. To test the stability of the non-collinear structure, we add the magnetic anisotropy energy and the polar angles become either 9°or 171°), i.e.a change of≈5°, which shows that K has a smaller impact on the hexagon than on the trimer. Heptamer. We add to the previous structure an atom in the center of the hexagon. Contrary to the other atoms this central atom has six neighbors and the magnetic ground state is profoundly affected by this addition as shown in figures 7(a)-(b). The nearest-neighbor isotropic exchange constant J, 4.69 meV, decreases slightly in comparison to the value found for the open structure. The obtained magnetic texture can be explained from the nearest-neighbor DM interaction (1.37 meV) with the corresponding vectors plotted in figure 7(e). The addition of the central atom creates frustration similar to the trimer case. Ideally, every pair of nearest-neighbor moments have to lie in the same plane. Thus, the central magnetic moment has to lie within one of the three planes orthogonal to the surface,and passing by two of the outer atoms and the central one. In this configuration, the three atoms are satisfied and the four atoms left have the direction of their moments adjusted, which leads to the final spin texture. Figures 7(c) and (d) show, respectively, the projection of the magnetic moment along the zaxis and in the surface plane. Interestingly, when the single-ion magnetic anisotropy is added only the central moment is affected. It experiences a switch from the in-plane configuration to a quasi out-of-plane orientation. A side view is shown in figure 7(f). This is another nice example showing how the stability of the non-collinear behavior is intimately related to the nature, shape, and size of the nanostructure.

Conclusions
We investigated the complex chiral magnetic behavior of nanostructures of different shapes and sizes wherein the atoms interact via long-range interactions mediated by Rashba electrons. We used an embedding technique based on the Rashba Hamiltonian and the s-wave approximation followed by a mapping procedure to an extended Heisenberg model. The analytical forms of the elements of the tensor of the magnetic exchange interactions werepresented within the RKKYapproximation, i.e.without renormalizing the electronic structure due to the presence of the nanostructure. We demonstrated the deep link between the magnetic interaction and the components of the magnetic Friedel oscillations generated by the single adatoms. The isotropic interaction and the DM interactions corresponded, respectively, to the induced out-of-plane and inplane magnetization. In addition tothese two interactions, the pseudo-dipolar term, already found in [31], wasshown to be large, generating a collinear phase competing with non-collinear structures induced by the DM interaction. We wentbeyond the RKKYapproximation by considering energy dependent scattering matrices and multiple scattering effects to demonstrate that the size and shape of the nanostructures have a strong impact on the magnitude and sign of the magnetic interactions. We proposed an interesting connection between the DM interaction and the isotropic magnetic exchange interaction, J. The DM interaction can be related to the first-order change inJ with respect to the SO interaction and,even more importantly, the origin of the sign of the DM interaction, i.e. defining the chirality, can be interpreted by the increase or decrease inJ upon application of the SO interaction. We considered nano-objects that can be built experimentally (see e.g. [8,29,34]), and show that each of the objects behave differently and the stability of their non-collinear chiral spin texture is closely connected with the type of structure built on the substrate.
The Green function for the Rashba electron gas can be calculated using the spectral representation: , with = -¢    R r r . After performing the sums over  k and n, the diagonal and off diagonal spin elements of the Green function G 0 of the Rashba electrons are given as: As mentioned in the main text, the vectors k 1 and k 2 are given by

Appendix B
In this appendix we derive the generalized Heisenberg Hamiltonian =  H e m i  e J ij j , which was simplified to the form given by equation (18). For this purpose, we need to calculate the elements of the tensor of exchange interactions showing up in equation (16), i.e. s a { G Tr ij s b } G ji , considering that G can be expressed in terms of G D and G ND (see equation (17)). This can be evaluated via the following trace Using the properties of the Pauli matrices, we know that for two vectors  A and  B , the following relation holds: The terms proportional to e e i x j x and e e i y j y will lead topseudo-dipolar-like terms after performing the energy integration given in equation (16). which leads to the final form of the Hamiltonian given in equation (18), and to the identification of the different magnetic interaction terms as presented in equations (22), (23), and (25).
While a positive k 1 leads to: