Frozen Magnon Calculations Beyond the Long Wavelength ...

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Frozen Magnon Calculations Beyond the Long Wavelength Approximation A. Jacobsson1 ,∗ C. Etz1 , M. Leˇzai´c3 , B. Sanyal2 , and S. Bl¨ ugel3

arXiv:1702.00599v2 [cond-mat.mtrl-sci] 5 Feb 2017

1

Department of Engineering Sciences and Mathematics, Lule˚ a University of Technology, Lule˚ a, Sweden 2 Department of Physics and Astronomy, Uppsala University, Box 516, 75120 Uppsala, Sweden 3 Peter Gr¨ unberg Institut and Institute for Advanced Simulation, Forschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, Germany The magnetic force theorem provides convenient ways to study exchange interactions in magnetic systems since it gives access to total energy differences between different magnetic states from nonself-consistent density functional theory calculations. This enables the efficient mapping of the total energy of the system as a function of the directions of the atomic moments in a Heisenberg model. However it is well known that short range interactions in itinerant magnetic systems are poorly described with the conventional use of the theorem and numerous strategies have been developed over the years to overcome this deficiency. In this study it is shown that utilizing pre-converged constraining fields in non-self-consistent spin-spiral total energy calculations gives results that match self-consistent total energy calculations for bcc Fe, fcc Ni and FeCo, while calculations without the fields fail to provide a good match for both fcc Ni and FeCo.

I.

INTRODUCTION

In the view of solid-state applications, it is of utmost importance to be able to accurately calculate magnetic exchange interactions and predict the correct critical temperatures for all kinds of materials in different geometries (from bulk, to thin films, to nano structures). Furthermore for a fundamental understanding of the materials magnetic response to different stimuli, as well as for a multitude of applications in solid state physics, it is necessary to properly describe spin excitations: collective spin excitations (spin waves) or single particle excitations (Stoner excitations). Building on the formulation of Andersen’s Force theorem1–3 the works by Liechtenstein et al.4 and Oswald et al.5 introduced the magnetic force theorem (MFT). The general formulation by Liechtenstein et al.4,6 is one of the most widely used method for the determination of inter-atomic exchange interactions since it it may be applied to any collinear magnetic configuration. The great utility of the theorem for the electronic structure community is due to the fact that the exchange interactions are determined from non-self consistent calculations that are orders of magnitude faster than most self-consistent approaches. Developments in time dependent density functional theory (TD-DFT) and many body perturbation theory have made it possible to access the full dynamical magnetic susceptibility7–10 , which give detailed information of both magnon and Stoner excitations, the latter not being accessible by the different versions of MFT calculations that operate in the adiabatic approximation. However, by a multi-code and multi-scale approach using ab initio, atomistic spin-dynamics and Monte Carlo calculations, magnon excitations and magnetic phase transitions may be accurately simulated for larger systems11–13 . The MFT relies on the assumption of small changes in the magnetisation and charge density. In addition, usually the internal magnetic fields are varied in the calculations rather than the direction of the moments.

Short wave-length excitations are therefore determined less accurately and the calculations are said to be performed in the long wave-length approximation (LWA) (for a more extensive discussion see the work of Antropov et al.14,15 ). Various strategies to avoid the LWA have been suggested over the years14–17 . In particular, developments by Patrick Bruno using ”constrained” density functional theory introduced by Dederichs et al.18 , included constraining magnetic fields acting on each atomic site. These fields reinforce the directions of the small transverse spin displacements used to probe the exchange interactions. It was suggested that neglecting the constraining fields one obtains a set of ”bare” exchange parameters which differ from the real exchange parameters. The inclusion of the constraining fields leads to a ’renormalisation’ of the spin-wave spectrum and the Curie temperature. The work by Katsnelson et al.17 showed that while the renormalisation suggested by Bruno gives better thermodynamic properties, the improvement in the magnon dispersion is questionable since the renormalisation should be small in every case where the adiabatic approximation is valid. The MFT is usually applied in the frozen magnon method19 , where exchange interactions are determined from the energy differences or evaluation of the spintorque for different spin spiral configurations. It has been noted that in the absence of constraining fields (converged in self-consistent spin spiral calculations), a missmatch is introduced between the desired and the resulting directions of the magnetic moments, thus leading to a significant un-physical spin-torque.20,21 In analogy with Bruno’s correction this can be solved by adding an additional term that includes the constraining fields to the potential, a procedure referred to as a corrected frozen magnon method in the continuation of the text. It should right away be noted that the constraining fields are actually equal to the Heisenberg exchange fields that act on the magnetic moments. It is therefore quite possible to derive the exchange parameters directly from the self-consistently converged constraining fields. This we

2 do for comparison with a simple real space version of the method of Grotheer et al.20,21 . As a practical scheme for the estimation of exchange parameters this method is usually preferable to the frozen magnon procedures due to better scaling and the avoidance of a frozen potential. A possible exception could be for systems with strong relativistic effects, where spin-orbit coupling can be introduced through perturbation theory.22,23 .

II. A.

THEORY

The frozen magnon methods

The constraining fields BC R are the Lagrange multipliers necessary to perform the energy minimisation under the constraint of the specific magnetisation density.24 In the APW-based methods the crystals are divided into spherical muffin-tin regions around the atomic sites and interstitial regions between the sites. The constraining fields are introduced in the Kohn-Sham equation (1) as uniform vector fields in the muffin-tins in addition to the magnetic field BXC (r) generated by the exchangecorrelation potential (eq. (2)). They are converged together with the rest of the potential such that the components of the total integrated magnetic moments perpendicular to the chosen direction of the moments are zero. The procedure still allows for intra-atomic noncollinearity. When calculating the total energy of a cone-spin-spiral it is usually necessary to add constraining magnetic fields to each magnetic sub-lattice in order to preserve the coneangle during the self consistency cycles. Exceptions are systems with the cone-spin-spiral ground state25 since the constraining fields are equal to the transverse part of the Heisenberg exchange fields. Non-zero transverse exchange fields imply that the system is not in equilibrium. 

~2 2 ∇ + Vef f (r) + σ · Bef f (r) − ν − 2m



XC Bef f (r) = BC (r) R+B

ψν (r) = 0 (1)

(2)

In the frozen magnon methods a number of reference Γpoint spin configurations are converged self-consistently. Configurations where there is a non-zero cone angle of the magnetic moment on either a single magnetic sub-lattice or a pair of magnetic sub-lattices are considered. These are only collinear if all magnetic sub-lattices are tilted with the same angle. In some previous publications,13,26 these reference states were obtained by rotations of the potential of a collinear state. Since only the potential in the muffin-tins were rotated this left a discontinuity in the magnetisation density at the border between the muffin-tins and the interstitial region. It was therefore

found that better agreement with self-consistent calculations was achieved by setting the interstitial magnetisation to zero.13 The procedure followed in this work makes it possible to keep a non-zero interstitial magnetisation since we don’t perform any rotations of the potential. On the other hand, it leads to a moderate increase in computational cost for systems with more than one magnetic sublattice since several self-consistent non-collinear calculations have to be carried out. Once the potentials for the reference configurations are obtained, non-self consistent total energy calculations are done for a set of spin-spiral wave vectors using the converged Vef f (r) and BXC (r) from the reference configurations and BC (r) set to zero. The total energy differences ∆E(q) are approximated as the difference in the sums of eigenvalues through the MFT:6 ∆E(q) ≈

X ν

ν (q) −

X

ν (0)

(3)

ν

In the last step, exchange parameters JRR0 of a Heisenberg Hamiltonian are obtained through a least square fitting procedure:26

∆E(q) =

1 X JRR0 (eR (q) · eR0 (q) − eR (0) · eR0 (0)) 2 0 R6=R

(4) Here the direction of the magnetic moments of a sublattice with a cone-angle θ is given by the expression: eR (q) = sinθcos(q · R)x + sinθsin(q · R)y + cos(θ)z (5) In the corrected frozen magnon method an extra calculation step is done compared to the conventional method. Vef f (r) is kept fixed and BC (r) is converged separately for each spin-spiral wave vector. In the final total energy calculations, Vef f (r) and BXC (r) are kept fixed as in the conventional method but now the pre-converged BC R are added for each one-shot calculation of the sums of eigenvalues. There are slight numerical differences between the constraining fields obtained in this way compared to fields obtained fully self-consistently. However, this procedure is much faster than a fully self-consistent calculation and no significant difference was obtained with respect to the total energy differences. A noteworthy difference between the two frozen magnon methods is that while eR is roughly parallel to BXC (r) in both cases – only in the corrected method is eR also guarantied to be the direction of the integrated moment after a single iteration of the electronic structure code. For materials and magnetic states – where the self-consistently converged magnetic fields fullfil BC ∼ BXC (r) – the correction to the conventional R frozen magnon method can be expected to be important.

3 The constraining field BC (r) compensates for the spin-torques the system experiences when it is forced to assume magnetic configurations different from the ground state structure. The correction will thus tend to be more relevant for itinerant magnetic systems with small moments, materials with strong exchange interactions or magnetic configurations far from the the ground state. B.

Exchange parameters from a transverse field

In order to obtain exchange parameters from selfconsistent quantities one could perform fully self consistent spin-spiral total energy calculations and extract the exchange parameters from eq. (4). This is however computationally expensive even for systems that can be described with a small unit-cell if we want to use an accurate all-electron code. A faster procedure – that still is fully self-consistent – was worked out by Grotheer et al.20,21 which exploits the fact that the constraining fields are equal to the transverse part of the Heisenberg exchange fields. The exchange field acting on P a magnetic moment mR0 is given by the sum: m1 0 R JRR0 eR . R Once the constraining fields and magnetic moments are converged for a number of magnetic configurations we can therefore solve the following system of equations for the exchange parameters: C BC R0 · BR0 =

1 X JRR0 eR · BC R0 mR0

(6)

R

Besides the difference in starting point (real space vs. reciprocal space) to previous studies, a further difference is the choice of the constraining field BC R as a free variable in our approach, while Grotheer et al.20,21 consider the cone-angle as a free-variable. For convenience our procedure in setting up and solving eq. (6) is called the transverse field method in the following. It has a better scaling with respect to the size of the system than the frozen magnon method since the number of determined variables is equal to the number of magnetic atoms for each calculation, while the latter method determines only a single variable in each calculation. While the constraining fields in the transverse field method are converged self-consistently which computationally is more demanding compared to non-self consistent methods, this is compensated well by the fact that the constraining fields are robust quantities that are usually much faster to converge with respect to the computational parameters than small total energy differences.27 C.

Computational Details

The main results presented in this paper are obtained with the full potential APW+lo Elk code28 . The computational details for the three systems bcc Fe, fcc Ni and

FeCo are the same, except for the number of k-points. The local spin density approximation (LSDA) functional by Perdew and Wang29 is used throughout this work. For the band energy calculations a 31 × 31 × 31 k-point mesh is used for fcc Ni and FeCo while a 41 × 41 × 41 k-point mesh is used for bcc Fe. The constraining fields are obtained using a 15 × 15 × 15 k-point mesh for fcc Ni and FeCo, while a 21 × 21 × 21 k-point mesh is used for bcc Fe. The muffin-tin radius was set to 1.23 ˚ A for all atoms. The angular cutoff for the APW functions, for the muffin-tin potential and the density was set to 9. The maximum length of the {G + k}-vectors was regulated by fixing its product with the average muffin-tin radius to 9. The maximum length of the {G}-vectors describing the interstitial potential and the density was set to 9.5 ˚ A−1 . We utilise unrestricted intra-atomic noncollinearity in the calculations. The experimental lattice constants of 2.87 ˚ A, 3.50 ˚ A and 2.83 ˚ A were considered for bcc Fe, fcc Ni and FeCo respectively. For the calculations of exchange parameters 100 spin-spirals were considered with randomly generated wave-vectors for bcc Fe, while 90 spin-spirals were considered for fcc Ni and FeCo. Cone-angles of 0.25 rad on one or two magnetic sub-lattices were used for all the results in this paper, while angles up to 0.5 rad were considered to ensure that our results did not depend sensitively on the chosen angles. The constraining fields were considered converged when the change of the constraining fields between two iterations was less than 0.5 %. For FeCo we also calculate the exchange parameters using the Korringa-KohnRostocker (KKR) method and the MFT method as developed by Lichtenstein et al.6 and implemented in the M¨ unich SPR-KKR band structure program package30 . The computational parameters were checked for convergence.

III.

RESULTS

In order to evaluate the accuracy of the spin-spiral total energy differences using the MFT with and without the use of constraining fields, spin-spiral total energies calculated fully self-consistently were used as a benchmark. Three different situations were considered for the calculation of total energies: (i) non-self-consistent calculations without any constraining fields (referred in the following as nsc and marked with green triangles); (ii) non-self-consistent calculation with constraining fields (referred to as nsc-field, blue diamonds) and (iii) self-consistent calculations (referred to as sc-field, red squares). Only for bcc Fe and fcc Ni are the spin-spiral dispersions related to the magnon dispersion through a scaling factor since they – unlike FeCo – have a single magnetic sub-lattice.

4 A.

Bcc Fe

We calculated the spin-spiral energy dispersion in bcc Fe along the H–Γ–P direction (fig. 1) for all the three cases mentioned before: nsc, npc-field and sc-field. The magnetic field Bef f (r) is aligned closer to the z-axis when the constraining fields are neglected and thus corresponds to a situation of smaller cone-angles. This results in lower total energy differences with respect to q = 0 and is why the nsc-field energies lie above the nsc case. The behaviour is consistent with the results of previous studies where quantities such as exchange parameters, magnon energies and the TC are increasing when the MFT calculations are corrected for the LWA.14–17 Indeed we also obtain a higher TC for bcc Fe with the corrected frozen magnon method than with the conventional one as can be seen in table I. FIG. 2. (color online) Exchange parameters for bcc Fe. The blue diamonds and green triangles represent exchange parameters calculated non-self-consistently with- (blue) and without (green) constraining fields. The black circles represent exchange parameters calculated with the transverse field method.

FIG. 1. (color online) Spin-spiral dispersion in bcc Fe. The blue diamonds and green triangles represent non-selfconsistent total energies calculated with- (blue) or without (green) constraining fields. The red squares represent selfconsistent total energies.

It is worth noting that we have intra atomic collinearity in our non-self consistent calculations for bcc Fe (and for fcc Ni) since we have only a single magnetic sublattice in this case and therefore have a collinear Γ-point reference state. In a previous study,15 it was shown that suppressing intra-atomic non-collinearity in bcc Fe gives rise to an increase of the calculated TC when derived from self-consistent spin spiral total energies. This observation is consistent with our results since it suggests that the non-self consistent calculations – where we have intra-atomic collinearity – should get higher total energy differences compared to the self consistent calculations. This might explain why the agreement with self-consistent spin-spiral total energies decreases somewhat with the introduction of the fields in the non-selfconsistent calculations, since the error cancelation is lost between the neglect of the fields (that lowers the effective cone-angles and thus energies) and the imposition of intra-atomic collinearity (that increases the energies of the spin-spirals).

The most pronounced difference in the exchange parameters is the larger nearest neighbour interaction of the nsc-field case compared to the nsc case (fig. 2). This difference is of similar size as the one obtained in previous studies between methods that worked in the LWA and those that went beyond using the linear response theory14 or self-consistent spin-spirals.15 Similarly the error cancelation can explain the excellent match obtained between self-consistently converged exchange parameters and the exchange parameters from previous studies using the MFT31–33 . In fact an almost perfect match is obtained with the results of Bergqvist31 . It should be noted however that bcc Fe is not very well described in a strict Heisenberg model. The exchange parameters are configuration-dependent and specifically the ratio between the dominating nearest- and second nearest neighbour interactions is known to depend sensitively on the cone-angle of the spin-spirals.34 Some differences between the results of various calculations available in the literature can be expected for that reason. To avoid the configuration-dependence a more complex Hamiltonian has to be considered.35

B.

Fcc Ni

Introducing the pre-converged constraining fields into the non-self consistent calculations gives results that match excellently with self-consistent results while the conventional frozen magnon method produces a dispersion curve that is significantly lower (fig. 3). This means that correcting for the LWA removes almost entirely the

5 TABLE I. (color online)The Curie temperatures (TC ) are listed in K and the theoretical results are all obtained in the mean-field approximation that typically overestimates the TC by ∼ 15 %. The exchange parameters of ref. [36] are obtained using the MFT approach of Liechtenstein et al.6 in the LWA. nsc nsc-field t-field ref. [36] Exp. Bcc Fe 1067 1143 1077 1050 1043 Fcc Ni 351 572 533 406 627 FeCo 1853 2261 1934 -

large differences between non-self consistent and selfconsistent results. As a consequence, the exchange parameters of the corrected magnon method correspond much closer to the exchange parameters obtained by the transverse field method, shown in fig. 4. We note that there is a substantial difference between the size of the nearest neighbour interaction obtained in our study when derived from the transverse field or the corrected frozen magnon method and previous self-consistent results.15 . We also get substantially weaker nearest neighbour interaction compared to results of linear response theory.14 It is however clear from fig. 3 that the energy differences that enter into the corrected frozen magnon method are close to the self-consistent results, witch indicates that we are doing a correct mapping of the self-consistent total energy landscape as calculated by our full potential APW+lo code. This is further reinforced by the close agreement between the exchange parameters calculated self-consistently from the transverse field and the ones obtained from the corrected frozen magnon method. We also note that our results for the TC give a decent match previous time-dependent density functional simulations36 .

FIG. 4. (color online) Exchange parameters in fcc Ni. The blue diamonds and green triangles represent exchange parameters calculated non-self-consistently with- (blue) or without (green) constraining fields. The black circles represent exchange parameters calculated with the transverse field method.

MFT calculations that operate in the LWA and employ a LSDA functional.14–16,19,37 The transverse field method and corrected frozen magnon method give a much improved prediction of the TC , as can be seen in table I, but still fall significantly short of the experimental TC . This further strengthens the case that the LSDA functionals underestimate the exchange interactions or that non-adiabatic processes and longitudinal fluctuations not captured by the Heisenberg model also play an important role in determining the TC in fcc Ni.36,38

C.

FIG. 3. (color online) Spin-spiral dispersion in fcc Ni. The blue diamonds and green triangles represent non-selfconsistent total energies calculated with- (blue) or without (green) constraining fields. The red squares represent selfconsistent total energies.

The TC of fcc Ni is well known to be underestimated by

B2 Structured FeCo

For FeCo two different cases of magnetic configurations were considered for the dispersion: (i) moments are only titled on one magnetic sub lattice (either Fe or Co and presented in figs. 5 and 6), while the orientations of the moments on the other sub-lattice are kept fixed and (ii) moments are tilted simultaneously on both magnetic sublattices (Fe and Co and presented in fig. 7). The nsc-field dispersion curves correspond considerably more accurately to the sc-field case than the nsc for FeCo. The difference becomes especially pronounced qualitatively when only the Co (fig. 5) or the Fe (fig. 6) sub-lattice is tilted even though it needs to be pointed out that the energy scale is much smaller in this case compared to the case where both magnetic sub-lattices are tilted. The energy differences of the spin spirals in figs. 5 and 6 do not depend on the inter-lattice exchange parameters since the angles between atomic moments belonging to different sub-lattices are constant. Hence we

6

FIG. 5. (color online) Spin-spiral dispersion curves in FeCo, when only the magnetic moments on the Co magnetic sublattice are tilted, while the orientations of the moments on the Fe sub-lattice are kept fixed. The blue diamonds and green triangles represent non-self-consistent total energies calculated with- (blue) or without (green) constraining fields. The red squares represent self-consistent total energies. Negative values occurs here since the gamma point is a non-collinear state.

FIG. 6. (color online) Same as in fig. 5, but with the magnetic moments on the Fe magnetic sub-lattice tilted. Negative values occurs also here since the gamma point is a non-collinear state.

can expect significant differences in the intra-lattice exchange parameters. Indeed the nsc-field intra-lattice parameters, shown in figs. 9 and 10, correspond much closer to the exchange parameters obtained by the transverse field method than the nsc-exchange parameters obtained by the conventional MFT calculations. Also the nearest neighbour Fe-Co interaction is significantly stronger when obtained by the methods that goes beyond the LWA. Exchange parameters were also obtained using a Korringa-Kohn-Rostocker (KKR) method employing the method of Lichtenstein et al.6 and found to be similar to the nsc calculations ((the purple vs the green symbols in

FIG. 7. (color online) Same as in figs. 5 and 6, when the moments on both Fe and Co sub-lattice are tilted. Only positive values occurs here since the gamma point is the collinear ground-state

FIG. 8. (color online) Exchange interactions between between the two magnetic sub-lattices (Fe and Co). The blue diamonds and green triangles represent exchange parameters calculated non-self-consistently with- (blue) or without (green) constraining fields. The black circles and purple squares represent exchange parameters calculated with the transverse field method (black) and the method of Lichtenstein et al.6 (purple).

figs. 8, 9 and 10. Also previous KKR calculations by MacLaren et al.39 match our KKR results well. This shows that the conventional MFT calculations fail to describe the self-consistent energy landscape of FeCo due to the LWA regardless of the specifics of the implementation. In this case there is no experimental value to compare to since FeCo undergoes a structural phase transition before losing the magnetic ordering.

7

FIG. 9. (color online) Exchange interactions with the same conventions as in fig. 8 but now within the Co sub-lattice.

FIG. 10. (color online) Exchange interactions with the same conventions as in fig. 8 but now within the Fe magnetic sublattice.

IV.

DISCUSSION AND CONCLUSIONS

Much of the discussions regarding the accuracy of the exchange parameters determined with the MFT has revolved around the results for bcc Fe. This is a material where conventional MFT calculations in the LWA and self-consistent total energy calculations agree very well and where the introduction of any correction scheme therefore has little room to improve the results. This excellent agreement can be understood to be partly due to the error cancelation between the neglect of the constraining fields and the assumption of intra-atomic collinearity in the calculations. The exchange parameters of the transverse field method therefore agree very well with the results of calculations using the MFT in the LWA, while the corrected frozen magnon method gives slightly different quantitative predictions for the ex-

change parameters. But for any material and state where BC (r) ∼ XC B (r), significant differences can be expected between non-self-consistent and self-consistent results. This is the case for the spin-spiral total energy calculations for fcc Ni and FeCo, where the conventional use of the MFT doesn’t result in a correct description of the total energy as a function of the directions of the moments. The neglect of the contributions of the constraining fields to the potential is crucial for this discrepancy as we can see from the comparison between our two different frozen magnon methods. We can see that going beyond the LWA through the use of the corrected frozen magnon- or transverse field method does result in a critical temperature that is significantly closer to experiments for fcc Ni using LSDA. Since the theoretical temperature is still too low this means that either the LSDA itself fails to give an accurate description of fcc Ni or a more complex treatment beyond the Heisenberg model is needed. This finding is consistent with previous studies that go beyond the LWA. Besides turning to more accurate functionals, it would be interesting to see wether the theoretical predictions could be made more accurate by introducing longitudinal fluctuations of the magnetic moments into the model. The corrected frozen magnon method gives overall an accurate description of the self-consistent energy landscape of all the three materials covered in this study while the conventional method only works well for bcc Fe. This indicates that non-self consistent spin-spiral calculations are not in general the most efficient approach for the purpose of parameterising the total energy of noncollinear states in itinerant magnetic systems, since the constraining fields that remove the differences to selfconsistent total energies need to contain information of the parameterisation itself. However the procedure can still make sense if the constraining fields are not fully selfconsistently converged. In our corrected frozen magnon method the potential is kept fixed while the constraining fields are converged. This results in computational costs for the corrected frozen magnon method comparable with the transverse field method for the small magnetic systems considered in this study, while the transverse field method scales better and therefore is usually clearly favourable to use for larger magnetic systems. The formalism of the transverse field method is simple to implement in any code that properly handles noncollinear magnetism. It can be used with or without the generalised Bloch theorem. The advantage of using the spin-spiral formalism is the possibility of calculating interactions between pairs of atoms that are not both contained in the unit cell. The method has several clear advantages over the conventional frozen magnon method besides the improved scaling and avoidance of the LWA. It is reasonable to expect the method to produce more accurate results than non-self-consistent approaches for systems where inter-atomic non-collinearity results in sizeable intra-atomic non-collinearity. Further-

8 more more accurate results can be expected for systems with large induced moments, that depends on the directions of the inducing moments. Simple rotations of the potentials at the sites of the magnetic sub-lattices are in these cases not likely to capture the intricate changes of the self-consistent potentials and corresponding total energies.

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ACKNOWLEDGEMENTS

The authors wish to thank Prof. Olle Eriksson, Dr. Yaroslav Kvashnin, Dr. Lars Nordstr¨om, Dr Phivos Mavropoulos and Dr. Gustav Bihlmayer for valuable discussions. This work was partly supported by the Young Investigators Group Programme of the Helmholtz Association, Germany, contract VHNG-409 and by the Kempe Foundations, Sweden. We gratefully acknowledge the support of J¨ ulich Supercomputing Centre in allocating resources to project is JIFF3800 and the National Supercomputer Centre in Sweden for the allocation of computing time for SNIC 2015/16-30 and SNIC 2016/1-44 .

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