A Relation between micromagnetic and spin-lattice model

Role of Dzyaloshinskii-Moriya interaction for magnetism in transition-metal chains at Pt step-edges

Abstract

We explore the emergence of chiral magnetism in one-dimensional monatomic Mn, Fe, and Co chains deposited at the Pt(664) step-edge carrying out an ab initio study based on density functional theory (DFT). The results are analyzed employing several models: (i) a micromagnetic model, which takes into account the Dzyaloshinskii-Moriya interaction (DMI) besides the spin stiffness and the magnetic anisotropy energy, and (ii) the Fert-Levy model of the DMI for diluted magnetic impurities in metals. Due to the step-edge geometry, the direction of the Dzyaloshinskii vector (-vector) is not predetermined by symmetry and points in an off-symmetry direction. For the Mn chain we predict a long-period cycloidal spin-spiral ground state of unique rotational sense on top of an otherwise atomic-scale antiferromagnetic phase. The spins rotate in a plane that is tilted relative to the Pt surface by towards the upper step of the surface. The Fe and Co chains show a ferromagnetic ground state since the DMI is too weak to overcome their respective magnetic anisotropy barriers. An analysis of domain walls within the latter two systems reveals a preference for a Bloch wall for the Fe chain and a Néel wall of unique rotational sense for the Co chain in a plane tilted by towards the lower step. Although the atomic structure is the same for all three systems, not only the size but also the direction of their effective -vectors differ from system to system. The latter is in contradiction to the Fert-Levy model. Due to the considered step-edge structure, this work provides also insight into the effect of roughness on DMI at surfaces and interfaces of magnets. Beyond the discussion of the monatomic chains we provide general expressions relating ab initio results to realistic model parameters that occur in a spin-lattice or in a micromagnetic model. We prove that a planar homogeneous spiral of classical spins with a given wave vector rotating in a plane whose normal is parallel to the -vector is an exact stationary state solution of a spin-lattice model for a periodic solid that includes Heisenberg exchange and DMI. In the vicinity of a collinear magnetic state, assuming that the DMI is much smaller than the exchange interaction, the curvature and slope of the stationary energy curve of the spiral as function of the wave vector provide directly the values of the spin stiffness and the spiralization required in micromagnetic models. The validity of the Fert-Levy model for the evaluation of micromagnetic DMI parameters and for the analysis of ab initio calculations is explored for chains. The results suggest that some care has to be taken when applying the model to infinite periodic one-dimensional systems.

pacs:
75.70.Tj, 71.15.Mb, 71.70.Gm, 73.90.+f

I Introduction

In a seminal work, Gambardella et al.Gambardella et al. (2002); Gambardella (2003) showed for the first time the presence of a truly one-dimensional (1D) metallic magnet. They succeeded in growing high-density arrays of monatomic Co chains on vicinal Pt(997) surfaces,Dallmeyer et al. (2000); Gambardella et al. (2000a, b) denoted as Co/Pt(997), and investigated the magnetic properties by X-ray magnetic circular dichroism (XMCD). They found that, below a blocking temperature of about , a long-range ordered collinear spin state is observed with magnetic moments aligned in the easy axis direction. The authors explained this ferromagnetic order with a large magnetic anisotropy energy (MAE) of that counteracts the magnetic fluctuations due to the finite temperature. The success in growing and measuring 1D magnetic monatomic chain structures as well as a detection of an unusual easy axis direction pointing perpendicular to the chain direction and tilted by an angle of towards the upper terrace triggered theoretical investigations based on density functional theory (DFT),Komelj et al. (2002); Shick et al. (2004); Újfalussy et al. (2004); Komelj et al. (2006); Baud et al. (2006a, b) that affirmed the presence of an unusual direction of the easy axis. The strong MAE could be traced back to the large spin-orbit coupling (SOC) contribution of the Pt substrate. Succeeding these pioneering experiments alternative 1D systems had been investigated, among those FePt alloysHonolka et al. (2009a) and submonolayer Fe stripes,Honolka et al. (2009b); Carbone et al. (2011) both on Pt(997), as well as Fe stripesHammer et al. (2003) and Co zigzag chains,Dupé et al. (2015) both on an Ir(001) (51) surface.

In this paper we address the question in how far these results and their interpretation remain unchanged in the light of the recently discovered interface induced Dzyaloshinskii-Moriya interaction (DMI).Bode et al. (2007) The DMIDzyaloshinskii (1957); Moriya (1960) appears in magnetic systems that lack inversion symmetry and exhibit strong SOC. Only recently it was found to be an indispensable ingredient to understand non-collinear magnetic structures of unique rotational sense observed in thin films, for the first time demonstrated by Bode et al.Bode et al. (2007) who measured and analyzed a Mn monolayer on a W(110) substrate. Up to now a number of similar systems are known in which the DMI leads to magnetic ground states that are described as cycloidal spin spiralsSantos et al. (2008); Ferriani et al. (2008); Zimmermann et al. (2014) or to the formation of a two-dimensional generalization of spirals with one-dimensional propagation vectors, the topological magnetic skyrmions.Heinze et al. (2011); Romming et al. (2013) Also for biatomic Fe chains deposited on an Ir(001) (51) surface such a DMI-induced non-collinear magnetic ground state has been predictedMokrousov et al. (2009); Mazzarello and Tosatti (2009) and experimentally verified shortly after.Menzel et al. (2012)

In the light of these analyses we turn to the Pt step-edge structure and investigate the leading magnetic interactions for different monatomic TM chains deposited along the step-edges. Due to the reduced symmetry occurring at step-edges, a complex interplay of DMI, MAE, and exchange interaction is found to determine the magnetic ground state or the rotation type within a domain wall.

The magnetic structures are explored in the context of a micromagnetic model that is introduced in Sec. II. There, we discuss consequences of the symmetry of the investigated structure on the magnetic anisotropy and DMI and derive two micromagnetic criteria that determine the appearance of homogeneous and inhomogeneous spin spirals as magnetic ground states. In Sec. III we give details on the unit cell and the performed DFT calculations. We proceed in Sec. IV with presenting the results of the performed calculations for the three investigated systems, monatomic chains of Mn, Fe, and Co at Pt(664) step-edges, and extract parameters for the previously discussed micromagnetic model. Based on these parameters we predict the magnetic ground state for each system and characterize possible domain wall structures. We conclude this paper with four appendices: In Appendix A we relate the micromagnetic parameters to the parameters of a lattice-spin model. In Appendix B we show that the spin spiral as calculated from first principles is a stationary state of the lattice-periodic spin model containing Heisenberg interaction and DMI. In Appendix C we relate the micromagnetic parameters with the spin-spiral energetics as calculated from first principles. In Appendix D we analyze the relation between the microscopic DM vectors as obtained from the Fert-Levy model and the micromagnetic DM vectors. DM vectors are evaluated and compared to the ab initio results from the main text.

Ii Micromagnetic analysis of the step-edge structure

ii.1 Symmetry considerations

Many of the systems, in which the DMI is known to lead to a non-collinear magnetic ground state, consist of one or more layers of transition-metal (TM) elements placed on top of a heavy element substrateBode et al. (2007); Ferriani et al. (2008); Zimmermann et al. (2014) and exhibit two mirror planes. This restricts the direction of easy, medium, and hard axis as well as the direction of the effective Dzyaloshinskii-vectorHeide et al. (2011) (-vector) to high-symmetry directions. Thus, the -vector always points along either easy, medium, or hard axis. In the step-edge structure discussed in this paper (see Fig. 1), however, only one mirror plane perpendicular to the chain direction remains. A consequence of this reduction of symmetry with respect to film structures is the previously mentioned easy axis direction for the Co chains, tilted by towards the upper terrace. Similarly, the rules of MoriyaMoriya (1960) only allow to reduce the possible orientation of the -vector to the plane perpendicular to the chain axis, which is why ab initio calculations become necessary to determine not only the strength of the DMI but also the direction of the -vector.

Due to this particular symmetry at hand, the search for the magnetic ground state takes place in a higher-dimensional space. Besides the strength of the -vector and the differences among easy, medium, and hard axes, one has to include in the final analysis the relative angle between and the principal axes of the anisotropy tensor.

ii.2 The micromagnetic model

To systematically study the magnetic phases in a solid from first principles one usually employs a multiscale approach. DFT calculations are performed that allow to extract system-specific parameters, which characterize the behavior of the system in terms of a suitable model, e.g., a (generalized) Heisenberg or spin-lattice modelLežaić et al. (2013) with spins placed on a discrete lattice. When the magnetic structure varies slowly across the crystal, meaning that the magnetic moments rotate on a length scale that is much larger than the interatomic distance, a micromagnetic model becomes favorable. Instead of a classical spin vector on each atomic site, such a model uses a continuous magnetization vector field (with ) with effective parameters in which atom-specific contributions are implicitly contained. In case one deals with an antiferromagnetic spin-alignment, the classical spin vector is replaced by a staggered spin vector where the difference of up and down spins on neighboring atoms form a new order parameter, that is treated then as a continuous field. Regarding the atomic structure we deal with in this paper, a linear chain of magnetic atoms along the direction as depicted in Fig. 1, the magnetic energy for such spin textures can be expressed by the micromagnetic energy functional

(1)

with and . The first term in Eq. (1) contains the spin stiffness, , and favors collinear spins (). In contrast, the second term is linear in and thus shows a preference for a certain rotational sense of with a strength and direction determined by the Dzyaloshinskii-vector, . Finally, the magnetic anisotropy is accounted for by the last term, that features the anisotropy tensor, , whose principal axes point along hard, medium, and easy axes.1 Note that in this work, without loss of generality, the energy-zero is given with respect to a magnetic configuration in which all spins are aligned along the easy axis. Furthermore, we point out that the local character of the integrand in Eq. (1) is reasonable as long as the range of the magnetic interactions is shorter than the characteristic length scale of the magnetic structure that is described.

In the following it is assumed to have knowledge of the model parameters , , and . Considering the symmetry of the step-edge structure (cf. Fig. 1 and discussion in previous Sec. II.1) the latter two are of the form

(2)

The direction of the Dzyaloshinskii vector, , is described with respect to the -axis by the angle2

(3)

The eigenvalues of the anisotropy ellipsoid , , , and , are the magnetic anisotropies along the principal axes. The principal axis corresponding to is parallel to the -axis. The axes and associated with and are obtained by a clockwise rotation, , of the magnetization around the -axis by an angle

(4)

which results to

(5)
(6)
(7)
Fig. 1: (color online) Step-edge structure, unit vectors and parameters used in the text with respect to the Cartesian coordinate system (): The -vector, being orthogonal to the -axis, points along and encloses an angle with the -axis. The pairwise orthogonal principal axes of the anisotropy tensor , , , and , are associated with , , and , respectively, where encloses an angle with the -axis. The rotation axis is perpendicular to the -axis and encloses an angle with the -axis. denotes the projection of the -vector onto . and are parallel and perpendicular to the -axis and are associated with and , respectively, the anisotropy components within the rotation plane perpendicular to . The magnetization density varies as function of distance along the step-edge (-axis) within the rotation plane (see semitransparent orange area) and encloses the spin-spiral rotation angle with . Note, that the angles , , and are positive (negative) when pointing towards the lower (upper) terrace of the step-edge.

ii.3 Homogeneous versus inhomogeneous flat spin spirals

It was first shown by DzyaloshinskiiDzyaloshinskii (1965) that the magnetization that minimizes the energy in functional (1) may correspond to spins that are periodically modulated rather than collinearly aligned along the easy axis. According to the analysis of Heide et al.Heide et al. (2011) such a non-collinear spin structure can be either a three-dimensional (3D) spin spiral or a flat spin spiral with a propagation vector along the step-edge ( direction) with magnetic moments rotating around the rotation axis of the spiral with an angle that encloses and the -axis (the surface normal) and that is restricted to . In the following we restrict our analysis to flat spin spirals, i.e., the magnetization direction is always perpendicular to the rotation axis and is independent of . For one part, this allows an analytical treatment of the problem, and for the other, we will show in Sec. IV that for all investigated systems the regime of truly 3D spin spirals can be excluded. Thus, the magnetization direction along the chain is given by

(8)

and depends on a 1D parameter, the spin-spiral rotation angle . The matrix describes a rotation around the -axis. Inserting this into the energy functional from Eq. (1), normalized to one period length one arrives at an expression for the average energy density,

(9)

with . is the projection of the Dzyaloshinskii-vector onto the rotation axis and reads

(10)

and denote the anisotropy components in the rotation plane of the magnetization perpendicular and parallel to the chain axis and are given by

(11)
(12)

Note, that and depend explicitly on the angle of the rotation axis, , such that the functional of the average energy density in Eq. (9) depends on as well. For later purposes we additionally define , , and the average . Note, that can become negative since the present formalism also accounts for the rotational sense of the spiral. We distinguish a right-rotating spiral for and (energetically preferred when , see Eq. (9)) and a left-rotating spiral for and (energetically preferred when ) following the convention that a left-rotating spiral rotates clockwise when projecting the magnetic moments onto the plane and reading the spiral rotation along the positive direction (see Fig. 1).

For a homogeneous spin spiral changes linearly with distance within the chain and one finds , where is the spin-spiral wave vector. Thus, , and Eq. (9) simplifies to

(13)

i.e., the energy density shows a parabolic behavior with respect to the inverse of the spiral length. Only when the minimum of this expression,

(14)

is below zero (corresponding to the energy of collinear spins aligned along the easy axis direction), a spiraling magnetic ground state can be established. This leads to the criterion for the appearance of a homogeneous spin spiral,

(15)

In the case of an inhomogeneous spin spiral () the energy-density functional in Eq. (9) can be minimized by means of the Euler-Lagrange formalismDzyaloshinskii (1965) resulting in

(16)
(17)

with the Lagrange multiplier . and are the complete elliptic functions of first and second kind,3 respectively, with the ellipticity . It can be shown that the average energy density in Eq. (16) gets minimal when . Together with Eq. (17) this leads to a conditional equation for ,

(18)

An inhomogeneous spiral appears for , leading to the criterion

(19)

where

(20)

is a factor that depends on the ellipticity of the anisotropy energy within the plane of rotation of the magnetic moments spiral rotation axis: If the ellipticity within the rotation plane is zero (), then , which means that both criteria, Eqs. (15) and (19), become identical. One can show that elsewise , meaning that the criterion for the appearance of an inhomogeneous spin spiral is always easier to be fulfilled than the criterion for the appearance of a homogeneous spiral, Eq. (15). For the case that the easy axis lies along the rotation axis () we have and Eq. (19) simplifies to

(21)

which has been already discussed in literature.Dzyaloshinskii (1965); Izyumov (1984)

As a final remark we state that the anisotropy term in Eq. (9) can also be written as , which leads to the same expressions as derived above.

ii.4 Micromagnetic Parameters

The three micromagnetic parameters , , and are related to the site-dependent microscopic parameters of a spin-lattice model via

(22)

where defines the distance between two neighboring atoms within the chain, and is the distance between atoms at sites and . , , and are the exchange interaction, the Dzyaloshinskii vector between a pair of atoms at sites and , and the on-site anisotropy at the representative atom labeled , respectively (see Appendix A and Ref. Zimmermann et al., 2014 for details).

The integrand of Eq. (1) is an energy density. For the quasi one-dimensional magnets studied in this work, it has the unit energy per length. Accordingly, the parameters , , and take the units energy times length, energy, and energy per length, respectively. However, it is often convenient to use another normalization, and represent energy densities in units of energy per TM atom. The conversion from the first normalization to the second one is done by multiplication with . In analogy, the units for the micromagnetic parameters , , and change to energy times area per TM atom, energy times length per TM atom, and energy per TM atom, respectively. For the rest of this paper we use the same symbols for the two different normalizations, and the used normalization can be inferred from the unit.

Notice, it is customary that both communities, the micromagnetic and the spin-lattice model community, refer to or , respectively, as the Dzyaloshinskii-vector, although they are obviously different. We follow this tradition, but refer in addition to the -vector in the spin-lattice model as microscopic -vector, and in the micromagnetic model either as the micromagnetic or effective -vector or as the spiralization,Freimuth et al. (2014) whenever necessary.

The spin stiffness and spiralization can be obtained directly from first-principles calculations invoking the homogeneous spin-spiral state. In Appendix B we prove that for each wave vector there are two flat homogeneous spin spirals of opposite handedness with a rotation axis parallel and antiparallel to the -vector of that given mode. The lowest energy is found for a wave vector with a spin-chirality opposite to the -vector of that mode. In Appendix C we show that if is in the vicinity of a high-symmetry point in the Brillouin zone, e.g., in case of the ferromagnetic state, typically this implies that the DMI is small compared to the exchange interaction. For the step-edge structure becomes one-dimensional and we obtain the spin stiffness from the curvature . The spiralization projected onto the direction of the DMI-vector is obtained from the slope of the energy calculated for wave vectors in the vicinity of the high-symmetry point. In the following Section we calculate and from for spin-spiral waves with -vectors of different length from first principles in two separate steps: At first, is calculated without spin-orbit interaction employing the generalized Bloch theorem, from which the spin stiffness is determined and for which the spiralization is zero by definition, and then the spiralization is determined by calculating the change of the total energy adding the spin-orbit interaction in first order perturbation calculated from electronic states related to the spin-spiral solution.

Iii First-principles theory

iii.1 Structural Model and Computational Details

Fig. 2: (color online) Sketch of the unit cell, a slab of a (664) vicinal surface decorated with a monatomic chain along the edge, and its repetition within the plane. The dark blue spheres correspond to the transition metals (Co, Fe, Mn) and the bright gray spheres represent the substrate atoms (Pt). The chain-to-chain distance is 13.24 Å and the nearest-neighbor distance within the chain is 2.82 Å. The inset in the lower left illustrates the use of the coordinates within the text. Note that the structure is periodic with respect to the plane. Although in the actual calculation all quantities are referenced with respect to , throughout this paper they are given with respect to , in accordance with Fig. 1.

The ab initio calculations based on DFT are carried out in film geometry of the full-potential linearized augmented plane-wave (FLAPW) methodKrakauer et al. (1979); Wimmer et al. (1981) as implemented in the fleur code.fle () The chains at stepped surfaces are modeled like in earlier studies,Baud et al. (2006a, b) where the chosen unit cell, a (664) step-edge structure, turned out to be a suitable structural model. The setup of the unit cell is inversion symmetric and consists of a tilted 8-layer Pt slab with two monatomic TMs deposited on both sides of the slab onto the step-edge. Throughout this investigation no relaxation of the structure is considered. This is motivated by the finding that relaxations can lead to an unphysically strong quenching of the orbital moment and, thus, to less accurate results for the MAE.Baud et al. (2006a, b); Mosca Conte et al. (2008) In Fig. 2 we sketch the structural model and indicate the unit cell by the darker spheres in the foreground. The chain axis and, thus, the propagation direction of the investigated spiral structure is chosen as -axis. To ensure a periodic repetition of the structure along the -direction, as required by a solid-state code, the steps of the surfaces with normal [111] direction are tilted by an angle of about , so that the -direction of the unit cell is [664] and the -direction is (see inset in the lower left of Fig. 2), resulting in a (61) surface unit cell. The used lattice constant is as calculated by Baud et al.Baud et al. (2006a) Along the direction the unit cell has the length of the distance of two vicinal TM chains, . The width corresponds to the distance between two neighboring TM atoms within one chain, . For all types of atoms within the 2D unit cell the muffin-tin (MT) radius is chosen to be . If not stated otherwise, all energies obtained from first-principles calculations refer to energies per computational unit cell. Depending then on the micromagnetic quantity under consideration, this energy can be related to energy per magnetic TM atom or per chain atom, respectively.

For the exchange and correlation functional we chose the local density approximation (LDA) as proposed by Moruzzi, Janak, and Williams.Moruzzi et al. (1978) The computational cutoff values for the expansion of the Kohn-Sham potential are for the potential and for the exchange-correlation potential. The Hamiltonian matrix elements for all atoms in the unit cell due to the non-spherical part of the potential are expanded up to . The spherical harmonics expansion of the LAPW basis includes functions up to within each MT sphere and all basis functions satisfying are included. If not stated otherwise, is used. All self-consistent calculations have been carried out with 128 -points in the full 2D Brillouin zone, whereas for one-shot calculations employing the force theorem of AndersenMackintosh and Andersen (1980) 512 -points have been used.

iii.2 Spin stiffness

The parameters , , and are calculated as outlined in Refs. Zimmermann et al., 2014 and Heide et al., 2009. The spin stiffness, , is obtained by determining the total energy of the system as function of flat homogeneous spin spirals with wave vectors of different lengths, all in the vicinity of the ferromagnetic or antiferromagnetic state and all along the chain direction. Since the lengths of the -vectors are small we applied the force theorem of AndersenMackintosh and Andersen (1980) to obtain these energies as deviations from the collinear state whose densities are calculated self-consistently employing the scalar relativistic approximation and which served as the initial state from which the force theorem is applied. To avoid numerical errors the magnetization in the interstitial region was set to zero before applying the force theorem. A detailed description can be found in Ref. Ležaić et al., 2013. When calculating the spin stiffness we omitted the energy correction due to SOC, because it proved small in tests and thus we can restrict ourselves to the use of the generalized Bloch theorem.Sandratskii (1991)

iii.3 Dzyaloshinskii-Moriya interaction

The effective -vector is determined treating SOC in first-order perturbation theory on top of flat homogeneous spin-spiral solutions used to determine . The DM energy is given byHeide et al. (2009); Zimmermann et al. (2014)

(23)

The occupation numbers are given by the Fermi function , which introduces a broadening of the occupation around the Fermi energy by the temperature . They depend on the wave vector through the unperturbed (i.e., without SOC) eigenvalue spectrum . The change of the eigenvalue spectrum

(24)

due to SOC described by the Hamiltonian , depends additionally on the rotation axis . The unitary transformation directs the flat spin spiral of the unperturbed state rotating around the -axis to the global spin-rotation axis and denotes the spin-spiral eigenstates of the unperturbed Hamiltonian. The summation in Eq. (23) runs over all states characterized by the Bloch vector and band index . Due to the finite number of -points (512 -points in the whole 2D unit cell) the effect of the broadening temperature will be a subject of study in Sec. IV.2, which allows an estimation for the qualitative reliability of our results.

We analyzed , the change of the DM energy for a set of -vectors that point along the chain direction (i.e., the -axis) but vary in length, as well as two different rotation axes oriented along - and -direction ( and , see Fig. 1 and Eq. (10)) to determine independently the two non-vanishing components of the -vector (the third component vanishes due to symmetry, as already discussed in Sec. II.1). In the micromagnetic limit, i.e., in the limit of long-period spirals, Eq. (9) is applicable. Therefore, if the spin-orbit interaction is included, the DM energy is expected to change linearly with the length of the wave vector in the vicinity of the collinear spin alignment. Consequently we evaluate the effective -vector as the slope of the energy change with respect to in the limit .

As outlined in Ref. Zimmermann et al., 2014, the spin-orbit coupling operator can be safely approximated by an atom-by-atom superposition of SOC operators limited to the muffin-tin spheres of the atoms, i.e.,

(25)

where is the spin-orbit strength related to the spherical muffin-tin potential , , , and . references the center and is the radius of the th muffin-tin sphere, with running over all atoms in the unit cell. The atom-by-atom analysis is supported by the observation that for small . We observed for example in case of the Rashba effect that 90% of the Rashba strength is produced by the wave function occupying a volume in the vicinity of the nucleus given by a radius of only about 10% (0.25 a.u.) of the muffin-tin radius.Bihlmayer et al. (2006) We expect an analogous behavior for the DMI. Thus, according to Eqs. (24) and (25) also is atom dependent and the DM energy is a result of atom-by-atom contributions , at least in first-order perturbation theory that we discuss here throughout the paper. The linear fit of to at the vicinity of a high symmetry point in the Brillouin zone of propagation vectors gives then the decomposition of the -vector into contributions , which satisfy

(26)

For the interpretation of the atom dependent spiralization we refer to the discussion of the Fert-Levy model in Sec. IV.2 and in Appendix D.

Since the structure of the unit cell setup in our ab initio calculation is inversion symmetric, the contributions of the DMI to the total energy cancel when all atoms are taken into account. Thus, we manually break the inversion symmetry by considering only the energy differences due to SOC from the atoms that are placed in the upper half of the unit cell.

iii.4 Magnetic anisotropy

For the magnetic anisotropy energy the force theorem of AndersenMackintosh and Andersen (1980) is applied, now in order to extract energy differences between collinear systems with magnetizations pointing in different directions. Starting point for the force theorem are self-consistent calculations including SOC, for which the magnetic moments point along the direction. For each system we evaluate the total energy for several directions of the magnetic moments collinearly aligned within the plane and the plane. Out of the obtained energy landscape one is able to extract , , and , the principal axes of the anisotropy tensor, (cf. Eq. (2)).

Iv Results and Discussion

iv.1 Spin stiffness

Fig. 3: (color online) Determination of the spin stiffness: In the left panel the total energies relative to their respective lowest energy, , are shown as functions of the length of the wave vector, , for Mn, Fe, and Co chains (magenta circles, red diamonds, and blue triangles, respectively). All systems show a collinear ground state, i.e., a ferromagnetic () ground state for the Co and Fe chains and an antiferromagnetic ( in units of ) ground state for the Mn chains. The right panel shows the energy as function of in the linear regime with the corresponding linear fits. The slope represents the spin stiffness, . Note that for the Mn chain we consider the antiferromagnetic ordering vector, meaning that leads to the AFM spin alignment. The relative error due to the linear regression is in the order of 5% to 8% for the shown data range.

The results of the spin-spiral energy for all three investigated systems as function of the wave vector along the one-dimensional Brillouin zone are summarized in Fig. 3. For Co and Fe chains the minimal energy is found for the ferromagnetic state, i.e., the state with wave vector , whereas the Mn chains align in the antiferromagnetic order. According to the micromagnetic model in Sec. II.3 (see Eq. (9)) and the discussions in Sec. II.4 we expect in the long-wavelength limit a linear relationship between the exchange energy and the squared inverse wavelength, which is realized by these systems for a large fraction of the Brillouin zone (40%) and shown in the right panel of Fig. 3 with the resulting fit. The slope gives the spin stiffness . For the Mn system it is smallest () and rises when going to Fe () and Co (). The results are also collected in Table 1. A small spin stiffness is favorable for the stabilization of a chiral spin spiral and in this respect the Mn chain is the most favorable system.

iv.2 The Dzyaloshinskii-vector

Fig. 4: (color online) The upper panel displays for Mn/Pt(664) the Dzyaloshinskii-Moriya energy (see Eq. (23)) in the vicinity of the AFM state as function of the inverse wave length for flat homogeneous spirals rotating in the plane (magenta squares and solid line) and in the plane (magenta spheres and dashed line) for a temperature broadening of . The slopes give the values for the components of the -vector (see Eq. (2)) which are shown in the lower panel as function of a broadening temperature. Since for (dotted vertical line) the values are converged and the error bars are still reasonably small (for this system as well as for the other two the relative error due to the linear regression is in the order of 5% to 10%.), the corresponding values are considered in the following.

Fig. 4 displays the DM energies per chain atom and the and components of the spiralization vector for the Mn chains. In the upper panel we present as function of the inverse wavelength, , for clockwise rotating (negative values of ) homogeneous spin spirals of two rotational directions and . Analogously to the discussion of the spin stiffness, we utilize the micromagnetic model and expect a linear behavior of for corresponding wave vectors in the vicinity of high-symmetry points in the one-dimensional Brillouin zone, and . Indeed we find a linear behavior for wave vectors covering 10% of the Brillouin zone measured from the antiferromagnetic state at for . However, for we notice a periodic modulation on top of the linear behavior. Such oscillations can occur due to finite numerical resolutions, e.g., due to finite sampling of the Brillouin zone.Zimmermann et al. (2014) In the lower panel of Fig. 4 we analyze the effect of the electronic Fermi surface broadening temperature, , on the obtained slopes that correspond to the -vector components and the loss of linear behavior reflected in the error bars. When is decreased, the values of the slopes and thus those of the -vector components converge while at the same time the error bars are increasing. In the following, the values corresponding to (see dotted vertical black line in the lower panel of Fig. 4) are used and can be found in Table 1. Among the three systems the resulting -vectors show remarkable differences in direction and strength. Therefore, we investigate its origin in more detail in the next paragraph.

Fig. 5: (color online) The atom resolved contributions to the -vectors are shown (a) for the Mn chain, (b) for the Fe chain, and (c) for the Co chain, extracted from the performed ab initio calculations, as well as (d) for a TM chain when applying the Fert-Levy model (see Appendix D). For each of the four cases, these contributions are depicted twice. In the left image, a cross section of the step-edge structure is shown with these vectors , for convenience, located at the corresponding atom (in fact, these vectors act only on the TM atoms within the chain, represented by the light blue circles). In the right image, they are given with respect to the same origin. In addition, the resulting -vector, i.e., the sum over atoms , , is printed in boldface.

For the three investigated systems, a more detailed study of the atom-resolved contributions to the -vectors is given in Figs. 5(a)-5(c). These atom-resolved contributions, i.e., for the atom with label , are obtained by switching on the SOC contribution for atom only. For each system they are plotted as vector with and components twice, () with respect to atom in the step-edge structure in the left part of each panel and () with respect to the same origin in the right part of each panel, where in addition their sum, the -vector, is shown as bold arrow. At first, we realize that for the Mn and Fe chain both -vectors point into very similar directions. Although all three -vectors point towards the upper step-edge, the direction of the -vector of the Co chain is quite different from those of the Mn and of the Fe chain. The lengths of the -vectors for Fe and Co chains are quite similar, but about only half as large as for the Mn chain. In general, the contribution of the atom itself is nearly negligible. The largest contributions come from atoms that are located next to the chain, albeit some contributions from some farther atoms can play a role as it is the case for Fe. A dominant contribution comes from the nearest-neighbor Pt atom at the upper terrace. For all systems they are of similar size, but for Pt next to Co, points in a direction different to the Mn or Fe case (cf. discussion at the end of this section). For the Co and Fe systems, each Pt atom with a dominant contribution has a vicinal atom with a vector of opposite sign and similar size. The dominant term for the Mn system, the nearest-neighbor atom at the upper terrace, has no counteracting contribution and consequently leads overall to a larger size of the Dzyaloshinskii-vector . This analysis shows that, although the atom resolved contributions might have large values themselves, the sum of all contributions can still lead to a rather moderate -vector due to mutual compensation.

In Fig. 5(d) we show an attempt to describe the regarded structure in terms of the model proposed by Fert and Levy,Fert and Levy (1980); Levy and Fert (1981) where the DM energy is given as sum over two distinct magnetic atoms within the chain interacting with the substrate atom (see Appendix D for details). Within this model, the direction of the atom-resolved -vectors is predefined to be perpendicular to the connection of the center of atom and the chain axis and perpendicular to the chain direction. This is in good agreement with the directions for the -vectors for the Mn and the Fe chains (cf. Figs. 5(a) and 5(b) with Fig. 5(d)), while it cannot be used to explain the directions for the -vectors for the Co chain (cf. Fig. 5(c) with Fig. 5(d)).

We next discuss the strength of the -vectors. Applying the Fert-Levy model to a periodic infinite chain, one finds that vanishes in the limit , while its derivative and thus the -vector, diverges. Therefore, this model is not applicable in this limit and the introduction of corrections attenuating or truncating the interaction between atoms in the infinitely long periodic chain after a certain interaction range, e.g., due to the lack of phase coherence or the presence of disorder will resolve this problem. Here, however, we avoid this singularity by evaluating the strength of for a finite wave vector , which corresponds to and thus matches in length with a -vector used in the presented ab initio calculations, see leftmost data points in upper panel of Fig. 4. The resulting strengths of the -vectors decrease with distance to the chain (see Fig. 5(d)). The same behavior is also found for the three investigated chains, albeit the length of the vectors cannot be explained by the distance to the chain only. In Appendix D we furthermore show that the strength decays with distance much faster when the magnetic moments of the atoms within the chain show a AFM short-range order, as compared to a FM short range order in the same chain. This observation, however, cannot be extracted from the ab initio results, e.g., when comparing the Mn chain (see Fig. 5(a)) to the Fe or the Co chain (see Figs. 5(b) and 5(c)). In conclusion, with regard to the structure of an infinite chain of magnetic atoms, we find that the model of Fert and Levy does not capture the diverse behavior of the three considered chains and we advise the application of this model to chains with some precaution. A more thorough investigation of the predictive power of the Fert-Levy model with respect to films and heterostructures would be interesting.

Finally, we provide some arguments why the directions of from Pt atoms next to Co contributing to the total DMI vector are so different as compared to those next to Mn or Fe. From a simple tight-binding model that we developed in Ref. Kashid et al., 2014, we identified spin-flip transitions between occupied and unoccupied states as the relevant process for a non-vanishing DMI. For the Mn chain, the spin-up (spin-down) channels are entirely occupied (unoccupied) and all transitions yield a contribution to the DMI. Going now to Co, some spin-down states become occupied and transitions into these states do not contribute anymore to the DMI. Since the remaining empty states exhibit particular orbital characters, the vector may well be rotated as compared to Mn. The situation for Fe is similar to Mn: most of the spin-down states are still unoccupied. Of course, a quantitative analyze requires many more details, such as bandwidths, the nature of the chemical bond etc., but this goes beyond the scope of this paper.

iv.3 The anisotropy tensor

Fig. 6: (color online) Magnetic anisotropy energy for Mn/Pt(664). The energy is plotted for magnetic moments pointing along directions discretized by the angles and relative to the orientation in direction. The symbols represent ab initio calculated energy differences, whereas the fit functions correspond to Eq. (27). In the case of (solid red line) the plane and in the case of (dashed blue line) the plane is sampled.

Following the findings of Sec. IV.1 we investigate the magnetic anisotropy tensor for the ferromagnetic order for the Co and Fe chains, and for the antiferromagnetic order for the Mn chains. The two required data sets for the latter system are shown in Fig. 6. The fit functions represent the leading order term and have the form

(27)

with the energy offset , the polar angle as argument, the azimuth angle as parameter (see the inset in Fig. 6), and fit parameters and . The mirror plane perpendicular to the chain direction is reflected by the fact that , leading to a symmetric function with respect to . The resulting hard, medium, and easy axes for all three systems are summarized in Table 1. In the following we use the easy axis as energy offset.

With respect to the resulting principal components the Mn system appears to be the most promising candidate for a non-collinear ground state. The anisotropy energies for the medium and the hard axis are the smallest compared to those of the other two systems. In addition, the easy axis points along the chain direction which is of relevance for the following reason: The only spin-orbit driven spin spiral that the -vector (perpendicular to the chain axis) can stabilize are of cycloidal character, meaning that the spiral rotation plane always contains the direction along the chain. Thus, the rotation over the easy axis is achieved automatically, regardless of the rotation axis. For the Co chains the easy axis is at about tilted towards the upper terrace, which is in satisfying agreement with other experimentalGambardella et al. (2002) and theoretical findings.Komelj et al. (2002); Shick et al. (2004); Újfalussy et al. (2004) The easy axis of the system containing the Fe chains is directed in approximately the same direction as the one for the Co chains. A remarkable finding for the Fe system is the strength as well as the orientation of the hard axis. It not only exhibits the largest value among all three systems, but also is oriented along the chain direction. Therefore, it shows the most unfavorable setup for a cycloidal spiral to appear since a rotation over the hard axis would be unavoidable.

iv.4 Magnetic ground states

Fig. 7: (color online) In the upper panels we show the values of the functions and for the appearance of homogeneous and inhomogeneous spin spirals (cf. Eqs. (15) and (19), respectively) in (a) Mn, (b) Fe, and (c) Co chains as functions of , the direction of the rotation axis, see Fig. 1. For each system the inset shows the relative orientation of the -vector with respect to the principal axes of the anisotropy tensor. In the lower panels the corresponding parameters , , and are plotted. In the last two systems the critical threshold of is missed by more than one order of magnitude. For the Mn chains, however, both criteria are fulfilled, and their respective curves reach their maxima for . This can be seen as a compromise between finding the largest DMI contribution (dashed green line) and having the smallest anisotropy energy (dashed brown line).
TM
Mn   
Fe   
Co
Table 1: Collected results for the three investigated TM chains on Pt(664) step-edges: Spin stiffness , absolute value of the Dzyaloshinskii-vector and its orientation , principal components of the anisotropy tensor (cf. Eq. (2)), , , , and , the orientation of the principal axis corresponding to , the rotation angle , as well as the spin magnetic moment and orbital magnetic moment of the TM atom for the case that the spin-quantization axis points along the easy axis. All angles are measured with respect to the -axis, see Fig. 1 and the insets of Figs. 7(a), 7(b), and 7(c). indicates where the function gets maximal, representing the planar inhomogeneous spin-spiral of lowest energy among all spirals. Only for the Mn chains this energy is lower than the one for the collinear state (i.e., criterion (19) is satisfied) and a spin-spiral state is formed as ground state. For the Fe and Co chains the collinear state always remains lower in energy. Note that , the anisotropy along the chain, is the easy axis for the Mn chains but the hard axis for the Fe system. , , and can be expressed in units directly compatible to the micromagnetic equation (1) dividing the parameter values by .

In the previous Secs. IV.1, IV.2, and IV.3 we extracted the parameters that now can be used to evaluate the criteria for the appearance of homogeneous and inhomogeneous spin spirals (see Eqs. (15) and (19), respectively) and their respective properties.4 Those criteria depend on the spin stiffness as well as , the projection of the Dzyaloshinskii-vector onto the rotation axis , and , and , the two principal axes of the anisotropy tensor that describe spins rotating in the plane perpendicular to the rotation axis (see Fig. 1). Evaluating Eqs. (15) and (19) the resulting magnetic ground state is determined by the functions and and whether their value exceeds the critical threshold of 1 for at least one rotation direction, described by the rotation angle . As we can see in Fig. 7 the Mn chains indeed fulfill both criteria when the direction of the spin-rotation axis, around which the magnetic moments of the spiral rotate, is in the regime between about . The maximum values of and are obtained for , which at the same time represents the minimum of the total energy, i.e., the magnetic ground state. This rotation angle can be understood as a compromise between the optimal DMI contribution (, rotation axis parallel to -vector) and the minimal MAE barriers (, rotation axis along , the hard axis). Since is positive for , the obtained magnetic structure is a left-rotating spiral, which modulates the otherwise antiferromagnetic order. As the magnetic anisotropy within the plane (see Fig. 6, dashed blue curve) is small, the findings for homogeneous and inhomogeneous spirals are quite similar. For the same set of parameters, Eqs. (14) and (17) lead to a spiral length of for a homogeneous spin spiral and for an inhomogeneous spiral. Both values correspond to a period length of about 60 atoms along the chain, which is equivalent to an average rotation angle of between neighboring Mn atoms and a -vector of , measured from the AFM alignment. These large period lengths justify in retrospect our ansatz of a micromagnetic model. Employing Eqs. (14) and (16) we find an averaged energy gain of per chain atom () for the homogeneous spin spiral and per chain atom () in the case of an inhomogeneous spin spiral.

In contrast to the analysis of the Mn chains, the obtained parameters for the Fe and Co chains confirm a ferromagnetic ground state, which is in line with previous studies.Baud et al. (2006a, b); Honolka et al. (2009a) For one part the resulting DMI is not large enough to change the collinear order favored by the spin stiffness, which we trace back to oppositely directed atom-resolved contributions to the -vector of the Pt atoms nearby the chain (cf. Figs. 5(b) and 5(c)). On the other hand, the magnetic anisotropy causes energy barriers that prevent the system from forming a non-collinear ground state. Especially for the case of Fe chains the formation of a spin spiral turns out to be energetically unfavorable, as the spin moments would have to rotate over the hard axis, as mentioned in Sec. IV.3.

We conclude the investigation on the magnetic ground state with a brief discussion on the possibility of finding non-planar spin spirals. For systems with orthorhombic anisotropy, phase diagrams are knownHeide et al. (2011) that take such three-dimensional non-collinear spin structures into account. However, to make our parameters match the Ansatz made in Ref. Heide et al., 2011 one has to assume that the -vector is oriented along one of the two principal axes of the anisotropy vector, or . This is to some extent only reasonable for the Fe chains where the angle between easy axis direction and -vector is (see insets in Fig. 7). In addition this system is the best candidate for a three-dimensional spiral since a rotation over the unfavorable hard axis is avoided, so that we restrict our analysis onto the Fe chains only. Following the notation of Ref. Heide et al., 2011 we arrive at and , when the -vector is assumed to point along the easy axis direction. Thus, we miss the critical regime of by a factor of 3, and this pair of parameters distinctly lies in the collinear region (cf. Fig. 3(b) in Ref. Heide et al., 2011).

iv.5 Formation of domain walls

Although for Fe and Co chains the DMI is not strong enough to introduce a chiral magnetic ground state its presence can influence the formation of domain walls.Heide et al. (2008) We follow the analysis of chiral domain walls put forward by Dzyaloshinskii,Dzyaloshinskii (1965) but apply this analysis to the ferromagnetic Fe and Co chains. Once again the starting point is the energy functional as given in Eq. (1), now with the boundary condition

(28)

By this constraint a rotation by is forced to take place spreading within the infinite chain. A distinction is made between a Bloch wall (helical rotation) and a Néel wall (cycloidal rotation). For both cases a characteristic width of the planar domain wall can be defined byHeide et al. (2008)

(29)

where represents the anisotropy energy for magnetic moments that point perpendicular to the easy axis direction within the spin rotation plane. The expression for the minimal energy readsHeide et al. (2008)

(30)

where for the Néel wall is equal to the expression in Eq. (10), but vanishes for a Bloch wall (). Thus, only for a Néel wall a preference in the rotation direction is expected and Néel walls can be realized even if the MAE favors a Bloch wall. Note, that the energetically favored rotational sense of the domain wall is accounted for by the minus sign and the absolute value of the second term in Eq. (30).

The resulting domain wall energies as well as the predicted wall widths for Fe and Co chains are listed in Table 2. Since for both systems the easy-axis direction is perpendicular to the chain direction, the rotation axis is fixed by Eq. (28) and the chain direction. If the easy axis points along the chain direction, is a compromise between magnetic anisotropy and DMI as it was the case for the ground-state analysis. In such a case no Bloch wall can be established.

For the Fe chains, a Bloch wall is energetically more favorable than the Néel wall even when the DMI contribution is taken into account, so that we do not expect a preference in the rotational sense for the domain walls for this system. One reason is that a Néel wall forces a rotation of the spins over the chain direction that is the hard axis of this system. Furthermore, the rotation plane is predefined by the easy axis direction. Since the -vector is oriented nearly within this plane, the projection to the rotation axis is relatively small.

For the Co chains, the energy of the Bloch wall, , is already by more than 6 meV higher in energy than the corresponding Néel wall, , where the DMI contribution is neglected. When in Eq. (30) the DMI contribution is taken into account the preference of a Néel wall is even higher. This gain in energy is achieved only for a right-rotating domain wall. This is because the rotation angle that describes a rotation plane perpendicular to the easy axis leads to a negative , see Fig. 7(c). Such a spin-orbit driven preference of a particular rotational sense of the Néel wall should be observable in an experiment.

TM
Fe 17.49 2.94 29.06 (30.82) 1.67
Co 28.26 2.42 17.73 (21.68) 3.15
Table 2: The domain wall energies for a Bloch wall and a Néel wall, and respectively, as well as the corresponding wall widths, and , are listed. Due to the DMI, the Néel wall always exhibits a certain rotational sense that lowers the energy with respect to its value without taking the DMI into account (). For the Fe chains the Bloch wall is energetically always more favorable, for the Co chains the Néel wall is preferred.

V Summary and outlook

Fig. 8: (color online) Schematic visualization of the energetically preferred rotational sense for (a) the ground-state of the Mn chain (left-rotating spin-spiral) and (b) the domain wall for the Co chain (right-rotating Néel wall).

Density-functional theory (DFT) calculations were performed to study the magnetic interactions in Mn, Fe, and Co chains at Pt step-edges. These calculations allow to extract parameters for a micromagnetic model that takes into account the spin-stiffness constant, , the magnetic anisotropy tensor, , and the -vector, which arises from the Dzyaloshinskii-Moriya interaction (DMI). Using this model, the magnetic ground state for the three investigated systems is determined employing two different instability criteria for the appearance of spin-orbit driven non-collinear structures, one for the homogeneous and one for the inhomogeneous spin spiral. The main results are listed in Table 3.

Our results predict a spiral magnetic ground state for the Mn chains, that modulates the antiferromagnetic order with a period length of about or 50-60 atoms along the chain. These findings establish Mn as a promising candidate for experimental research groups to investigate the DMI in 1D systems. A new aspect of this system, different to the systems studied in the literature, is the non-trivial direction of the -vector, that is not fully determined by symmetry. As a result the spiral rotates in a plane that is tilted by about towards the upper terrace (see Fig. 8(a)). For the Fe and Co chains we conclude that the formation of a non-collinear spiral magnetic structure is unlikely. For one part, this is due to magnetic anisotropies that are larger compared to the Mn chains. For the other part, their -vectors are too small to overcome these anisotropy barriers. A detailed atom-resolved analysis of this quantity showed that their moderate strengths are due to compensation of the atomic contributions. For Co, the results are consistent with recent findings for the Co zigzag chain on Ir(001) (51),Dupé et al. (2015) for which also no spiraling solution was observed. On the other hand, the Fe/Pt step-edge behaves different to the biatomic Fe chain Ir(001),Menzel et al. (2012) for which a spiral with a short period pitch was observed.

The calculated directions and strengths of the -vectors for the different chains were compared to those that result from the model of Fert and Levy. It appears that this model reproduces to some extent the directions of the -vectors of the Mn and the Fe chains. For the Co chain, however, it fails to describe the direction of correctly. We noticed that the model of Fert and Levy may be used with some precaution at least for one-dimensional chains as the micromagnetic DM vector diverges for infinite chains in the limit of long wavelengths. A more thorough investigation of the predictive power of the Fert-Levy model with respect to films and heterostructures would be interesting.

Furthermore, an analysis of planar domain wall structures for the Fe and the Co chains was presented. It appears that in the Fe system a Bloch wall is energetically more favorable. Since this type of domain wall is by symmetry not affected by the DMI, a preferred sense in the rotation direction is not expected. In contrast, the Co chains form a Néel wall, which shows a homochiral preference in the wall rotation that is caused by the DMI (see Fig. 8(b)).

We encourage experimental groups to verify our findings for the Mn chains in terms of the magnetic ground state. Furthermore, a statistically preferred rotational sense of domain walls in the Co chains should be observable in experiment. This could add a substantial aspect to the understanding of magnetism in low-dimensional systems and could provide some insight into the consequences of surface and interface roughness on the DMI. Previous investigations revealed a strong dependence of the MAE on the number of transition-metal strands in the chain Gambardella et al. (2004); Baud et al. (2006b); Honolka et al. (2009b). For example, a strong softening of the MAE was observed for Fe double-chains Honolka et al. (2009b), i.e., magnetic parameters may be tuned as functions of the number of strands to meet the criterion for a chiral ground state.

In this paper we focused exclusively on infinite periodic chains. Here we comment briefly on the magnetism for chains of finite lengths. We may discuss the finiteness in terms of a boundary effect, which are strongest where the chain terminates and whose effects decay away from the boundary into the chain. This affects shorter chains stronger than larger chains. Thus, in the center of larger chains we expect the same behavior as for periodic chains. In general, due to the finiteness of the chain three additional factors may play a role: (i) Atoms in a finite chain lose the mirror symmetry in a plane normal to the chain direction. (ii) Thus, edge effects of finite chains result in non-vanishing components of -vectors along the chain direction. Although the remaining non-vanishing component is small when averaged across the finite chain, locally we expect an additional tilt of the magnetic moments and subsequently an additional energy gain. This supports the formation of a chiral magnetic ground state in a finite chain over an infinite one, similar to the surface twist in films of B20 alloys that stabilize the skyrmions phase in films over B20 bulk alloys.Rybakov et al. (2013) Even if we assume that the electronic structure at the boundary of the finite chain does not change and all microscopic magnetic parameters remain unchanged, the micromagnetic DMI experience a change due to symmetry and this additional tilt of the magnetization has been investigated by S. Rohart and A. ThiavilleRohart and Thiaville (2013) but not for chains but for nanostructures. (iii) The change of the electronic structure at the boundary of the chain is an additional factor. Actually we investigated this for finite clustersBauer et al. () and it might be an important effect. Then all micromagnetic parameters change, but most affected are the DMI and MAE. This can modify the threshold for the occurrence of a chiral magnetism in the chain. If the chain length becomes below two times the length scale, where the electronic structure is modified due to the presence of the finiteness of the chain, nothing can be said about the magnetic property of the short chain. Additional ab initio studies are required.

TM GS (no SOC) GS (with SOC) easy axis DW type
Mn AFM -SS chain
Fe FM FM chain BW
Co FM FM chain r-NW
Table 3: Summary of the outcome of the paper: Whereas in the absence of SOC only collinear ground-states (GS) occur (FM and AFM), the Mn chain forms a left-rotating homochiral spin spiral (-SS) when SOC is taken into account. The easy axis can point along the chain (Mn) or perpendicular to it (Fe and Co). For the definition of see Fig. 7 or Eq. (4). The analysis of the domain wall (DM) type reveals that the Fe chain prefers a Bloch wall (BW) whereas for the Co chain a right-rotating Néel wall (r-NW) is energetically favored.

On the methodological side we showed that for a spin-lattice model of classical spins on a Bravais lattice including Heisenberg and Dzyaloshinskii-Moriya interaction the homogeneous spin-spiral is an exact solution if the rotation vector of the spin-spiral points either parallel or antiparallel to the -vector, representing a solution of two different chiralities. This has important consequences since the spin-spiral state is a state that is frequently employed in the first-principles context using density functional theory. One consequence is that the slope and the curvature of the spiral energy as function of the wave vector as calculated in density functional theory provides directly the spin stiffness and the spiralization that enter a material specific micromagnetic model.

Vi Acknowledgment

We would like to thank Albert Fert, Miriam Hinzen, Daniel A. Klüppelberg and Christoph Melcher for fruitful suggestions and stimulating discussions during the course of this work. We gratefully acknowledge computing time on the JUROPA supercomputer provided by the Jülich Supercomputing Centre (JSC). B.S. acknowledges funding by the HGF-YIG Programme VH-NG-717 (Functional Nanoscale Structure and Probe Simulation Laboratory, Funsilab). B.Z. and S.B. acknowledge funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 665095 (MAGicSky).

Appendix A Relation between micromagnetic and spin-lattice model

A natural starting point for a multiscale analysis of a complex magnetic structure is the spin-lattice Hamiltonian

(31)

where is the exchange integral between atoms at sites and , is the Dzyaloshinskii vector and is the microscopic on-site anisotropy tensor. If these parameters are determined from first principles, one refers to a realistic spin-lattice model. Assuming lattice periodicity it follows that , , and for all sites , and considering that the exchange interaction and the DMI are even and odd functions, respectively, with respect to inversion symmetry. In this appendix we relate these parameters to the micromagnetic parameters of model (1).

If we assume that the magnetic structure is slowly varying along the chain, meaning that the magnetic moments rotate on a length scale that is much larger than the interatomic distance, then it is certainly possible to choose for the magnetization direction a continuous normalized function with , such that , where denotes the spacing between the lattice points along the -axis. If we assume further that does not vary much on a length scale at which the interactions and are relevant, then the interactions can be considered local over that length scale which is consistent with the formulation of the interactions in the micromagnetic energy functional (1). Under these conditions one can Taylor expand around . The energy expression (31) treated within the lowest relevant order reads then

(32)

For the exchange term we make explicitly use of the normalization as , and .

Reminding that the distance between atoms at site and is given by , replacing by in (32) in the limit of small changes, the energy functional of spin-model (31) approaches the energy functional of micromagnetic model (1), , with parameters , and as summarized in Eq. (22).

Appendix B Spin-spiral solution of spin-lattice model with Dzyaloshinskii-Moriya interaction

From the view-point of first-principles calculations of a magnetic crystalline solid, the planar helical or cycloidal spin-spiral represents an interesting magnetic state, because in the absence of spin-orbit coupling the spin-spiral state can be calculated by partitioning a solid into the same (chemical) unit cell that is used for non-magnetic or ferromagnetic calculations. This becomes possible by employing the generalized Bloch theorem Sandratskii (1991) and holds true for any arbitrary wave vector taken from the Brillouin zone (BZ) of wave vectors. It is a major concept to make such first-principles calculations feasible.

In this Appendix B we show that the planar homogeneous spin-spiral state of wave vector , whose rotation axis points parallel or antiparallel to the Dzyaloshinskii-Moriya vector, is a stationary solution, and for a particular wave vector , the spin-spiral state is also the energy minimizer of the spin-lattice model (31) for a periodic solid, when the magnetic anisotropy term is ignored. It is known that the spin-spiral state is the stationary solution of a classical Heisenberg model on the Bravais lattice.Yoshimori (1959); Villain (1959); Kaplan (1959, 2009) Here we show that the solution holds true also for the Heisenberg exchange plus DMI. In difference to the Heisenberg exchange only, where the energy is isotropic with respect to the rotation directions of the spirals, the DMI lowers the rotational symmetry, and selects spirals whose rotation directions are parallel and antiparallel to Dzyaloshinskii-Moriya vector of mode .

In the following we assume a crystalline solid with lattice periodicity and restrict ourselves for simplicity to one atom per unit cell. We neglect the single-site anisotropy tensor in (31). The spin-model (31) on the Bravais lattice is then replaced by the quadratic form

(33)

with prefactor 1/2 preventing a double counting of terms and the exchange tensor

(34)

and . The lattice periodicity implies . The aim is to find the set of spins , with , that minimizes subject to the constraints that the length of spins are on sphere of radius and remain unchanged at all sites ,

(35)

Luttinger and Tisza Luttinger and Tisza (1946); Luttinger (1951) realized that the minimization of a quadratic form under strong constraints can be replaced by a much simpler problem of minimizing the energy (33) subject to the weak constraint

(36)

where is the number of lattice sites. This is a necessary condition and becomes sufficient if the solution also fulfills the strong constraint, as given in Eq. (35).

To take advantage of the translational symmetry of the crystalline solid, we transform the spin at lattice site with the lattice vector into momentum space

(37)

Without loss of generality we assume here , but the derivations hold correct also for one- and two-dimensional lattices. Since is a three-tuple of real numbers, it holds that . With this definition, the quadratic form (33) and the weak constraint (36) can be expressed in momentum space as