Complex magnetism of B20-MnGe: from spin-spirals, hedgehogs to monopoles

B20 compounds are the playground for various non-trivial magnetic textures such as skyrmions, which are topologically protected states. Recent measurements on B20-MnGe indicate no clear consensus on its magnetic behavior, which is characterized by the presence of either spin-spirals or 3-dimensional objects interpreted to be a cubic lattice of hedgehogs and anti-hedgehogs. Utilizing a massively parallel linear scaling all-electron density functional algorithm, we find from full first-principles simulations on cells containing thousands of atoms that upon increase of the compound volume, the state with lowest energy switches across different magnetic phases: ferromagnetic, spin-spiral, hedgehog and monopole.


I. INTRODUCTION
Most of the current research activity dealing with magnetism in B20 compounds is closely connected to the field of skyrmionics. Skyrmions are two-dimensional nontrivial magnetization solitons 1,2 , i.e. two-dimensional magnetic structures localized in space, of topological nature 3 , which have particle-like properties. Such magnetic objects are heavily prospected with the aim of establishing them as possible information-carrying particles that are small-sized and stable up to room temperature [4][5][6] . This motivated numerous studies on cubic B20-type compounds with broken lattice inversion symmetry, where skyrmion phases have been observed for the first time experimentally 7 . B20-MnGe, in particular, is the subject of extensive investigations [8][9][10][11][12][13][14] . While there is a consensus on the importance of both long-range magnetic interactions leading to frustration interplaying with the spinorbit coupling driven Dzyaloshinskii-Moriya interaction (DMI) 15,16 , there is no convincing explanation on what is observed experimentally. After the observation of a large topological Hall effect in B20-MnGe 8 , it was found in a recent transmission electron microscopy study that 3dimensional (3D) magnetic objects exist in B20-MnGe 11 . The authors came to the conclusion that their data indicate a cubic lattice of skyrmionic hedgehogs and antihedgehogs with a periodicity of about 3-6 nm which is stable up to a temperature of 170 K and a magnetic field of 12 T 17 . While the hedgehog-antihedgehog lattice state certainly constitutes the most interesting nontrivial magnetic texture in B20-MnGe, there are also reports that the magnetic ground state in this system is actually a helical spiral 18 which was observed up to a temperature of 170 K by powder neutron diffraction, too 19 .
In this article, we present results on B20-MnGe both from a magnetic model approach and large-scale allelectron calculations for a supercell whose extent compares to the periodicity of the experimentally observed helical textures. After discussing the electronic properties and analyzing the magnetic interactions obtained from the ferromagnetic unit cell, we explore various potential magnetic textures obtained from the selfconsistent large-scale ab-initio simulations. We discovered that the experimentally proposed hedgehog structures are marginally higher in energy than the ferromagnetic state around the experimental lattice parameters. Increasing the volume of the cell can lead to a stabilization of single spirals and of hedgehog lattice states.

II. METHODS
All of the results presented below are obtained with Density Functional Theory (DFT). We utilize the codes juKKR 20,21 and KKRnano 22 which are both based on the KKR Green function formalism. While juKKR is used to extract magnetic model quantities by infinitesimal rotations of the magnetic moments in the ferromagnetic 8-atomic unit cell [23][24][25] , KKRnano was especially designed to perform large-scale electronic structure calculations and allows us to perform self-consistent all-electron calculations for supercells that contain a few thousand atoms 22 .
In section III, we review the basic electronic and magnetic properties of the primitive cell of B20-MnGe. To choose the appropriate exchange and correlation functional to utilize for the large supercells, we extract the theoretical equilibrium lattice parameter within the scalar-relativistic approximation as implemented in KKRnano.
As exchange-correlation functionals, we choose the local spin density approximation (LSDA) according to the scheme of Vosko, Wilk and Nusair 26 and the generalized gradient approximation as given in PBEsol 27 . A grid of 14 × 14 × 14 k-points in combination with a broadening temperature of 800 K is used.
In section IV we use the juKKR code including spinorbit coupling to extract the isotropic exchange interactions (J ij ) and the Dzyaloshinskii-Moriya interactions ( D ij ) between the Mn atoms in B20-MnGe. The calculations are performed for a = 4.76Å, i.e. the theoretical lattice parameter obtained with PBEsol and larger lattice parameters, with a broadening temperature of T = 800 K. The k-point mesh was increased up to 60 × 60 × 60 points. In order to also investigate the impact of a temperature-driven small shift of atomic positions, as reported from a neutron diffraction study 19 , the interaction parameters are determined for the experimentally obtained positions of the Mn atoms with the atomic positioning parameter u Mn = 0.135 and a slightly smaller (larger) value of u Mn = 0.125 (u Mn = 0.145). When the Mn atoms are moved, the distance between Mn and Ge atoms in the crystal structure changes. Additionally, we use an inhouse Monte-Carlo code that only considers the isotropic exchange interactions to determine the Curie temperature of the system 28,29 .
For the large-scale calculations with KKRnano in section V we use supercells built of 6 × 6 × 6 unit cells so that 1728 atoms are treated. PBEsol is used as exchangecorrelation functional and only a single k-point, i.e. the Γ-point, is included. The Green function is truncated beyond a distance of 2a. The magnetic states are imposed on the system by forcing the atomic exchange-correlation B-fields to point into specific directions.

III. BASIC PROPERTIES OF B20-MNGE
B20-MnGe orders in a cubic structure that is described by the P 2 1 3 space group. This space group is noncentrosymmetric, which means that there is no lattice inversion symmetry. The eight atoms in the primitive cell are located at the Wyckoff positions (u, u, u), , where u is a constant value that is determined for each atom type. We choose these parameters as u Mn = 0.135 and u Ge = 0.8435 which is in good agreement with experimental findings 19 .
In Figure 1 the density of states obtained with KKRnano is shown for the ferromagnetic 8-atomic unit cell with the lattice parameter set to 4.795Å. One recognizes the large spin-splitting characterizing Mn atoms, which gives rise to a magnetic moment of roughly 2 µ B /f.u.. Interestingly, the Ge states do not contribute significantly to the density of states at and around the Fermi level.
By varying the lattice constant, we recover the pressure-induced magnetic transition from a high-spin state (HS) to a low-spin (LS) state (see Figure 2) predicted by Rößler 30 and confirmed experimentally by Deutsch et al. 31 . The latter reported additionally that the helical ordering in the material collapses above an applied pressure of 10 GPa. In our analysis, the equilibrium lattice constant obtained by LDA, 4.65Å does not coincide with that obtained by PBEsol, 4.76Å. The latter is closer to the experimental value, 4.8Å 32 . The calculation also sheds light on the behavior of the magnetic moment under variation of the lattice constant. As can be seen in the lower part of Figure 2, the magnetic moment of each Mn atom becomes larger with increasing lattice constant. The main difference between LDA and PBEsol is the location of the crossover region between the LS and HS state in which the moments increase abruptly and the system goes into the HS state. For PBEsol it is found around 4.65Å while it is slightly below 4.7Å for LDA. Furthermore, it is remarkable that the magnetic moments per Mn atom differ a lot for the equilibrium lattice constant of LDA and PBEsol. Here, LDA predicts a magnetic moment/f.u. that is a bit larger than 1 µ B /f.u., where instead PBEsol prefers a value of almost exactly 2 µ B /f.u.. Experimentally, a magnetic moment between 1.6 µ B /f.u. and 2.3 µ B /f.u. was measured 18 . A closer look reveals that there are actually two parabolalike energy curves for each functional. One describes the total energy of the system in the HS state (solid line) while the other does the same for the LS state (dashed line). At the transition point the two curves intersect and the two states are degenerate. To summarize, a HS/LS-state transition is predicted with both LDA and PBEsol, where PBEsol correctly finds the ground state to be the HS state while LDA does not. In combination with the fact that PBEsol also gives a more realistic estimate of the equilibrium lattice constant, we restrict ourselves to PBEsol as exchangecorrelation functional in the following.

IV. MAGNETIC EXCHANGE INTERACTIONS
The most common theoretical explanation of helical magnetic structures in the B20 materials is given by the competition of the symmetric exchange interactions and the DMI 33 . A detailed account of the magnetic exchange interactions is given in Figure 3, where both the atomistic model parameters J ij and | D ij | are depicted for u Mn = 0.125, u Mn = 0.135 and u Mn = 0.145. Those parameters are associated with the following extended Heisenberg model where m i corresponds to the unit vector of the magnetic moment of Mn at site i. Since the Ge atoms carry a small induced magnetic moment of only 0.1µ B /f.u., their explicit contribution can be neglected in our magnetic model analysis. For all three u-parameters the first nearest-neighbor interaction is strongly positive which means that ferromagnetic coupling is preferred here. However, the alternating sign of the second nearestneighbor isotropic exchange interactions indicates magnetic frustration. The absolute values of D ij are two orders of magnitude smaller than the values of J ij . Interestingly, a different choice of the structural parameter u has a strong influence on their behavior between the second and fifth shell. When the Mn atoms are moved, the distance between Mn and Ge atoms in the crystal structure changes. This potentially gives rise to increased hybridization between both atom types which might be the reason for the changes of the DM interaction.
Since the magnetic behavior of MnGe is often described in terms of a micromagnetic model (see e.g. Ref. 3 ) where the energy reads it is interesting to evaluate the exchange stiffness, A, and the DM spiralization D from their respective interatomic counterparts, J ij and D ij by summation over individual shells. (see the thorough derivation in appendix A). The helical period of a magnetic structure is then proportional to A/D and given by the provision where q is the wave number of the spin spiral.
Because of the long-range oscillations observed for the J ij and D ij , neither the spin stiffness A nor antisymmetric exchange D converge to a constant value, when shells up to 3a are taken into account. It is therefore not possible to estimate the helical pitch λ by eq. (3). Note that in any case, estimating the pitch in this manner is only valid, if the magnetic texture is created by the competition of spin-stiffness and DM spiralization in the micromagnetic model. If frustration plays a crucial role 34 , it is no longer applicable. While the absolute values of D ij are very similar for the first shell as function of the u-parameter, the values of D differ if only the first shell is taken into account. This can be attributed to the scalar product in eq. (A9) which makes D dependent not only on the absolute value of D ij but also on its direction with respect to the corresponding atomic bond vectors. In general, it can be observed that a change of the u Mnparameter results in a global shift of D. Its value can be increased by decreasing u Mn .
Using Monte-Carlo simulations, it is interesting to evaluate the Curie temperature T C and its dependence on the size of shells of interatomic magnetic interactions (see Figure 4). We find that although there is a dependence on the number of shells included in the simulations, in contrast to the micromagnetic quantities A and D, T C is more or less converged, when interactions up to 2a are taken into account. However, this converged value of T C ≈ 300 K lies considerably above the experimental value T exp. C ≈ 170 K 9 . Another structural parameter whose effect on the magnetic model parameters can be studied is the lattice constant. Our results are shown in Figure 5, where the atomistic and micromagnetic parameters for a = 4.85, 4.90, 5.00 and 5.10Å are visualized. All of these lie above the equilibrium lattice constant. By increasing a, and therefore the inter-atomic distances, the first nearestneighbor isotropic exchange can be reduced. The J ijcouplings between the neighbors that lay further apart are less affected. The behavior of the absolute values of D ij differs between the two smaller (a = 4.85, 4.90 A) and the two larger (a = 5.00, 5.10Å) lattice constants. For the smaller lattice constants the first nearestneighbor contribution dominates. Interestingly, the be-havior changes for the larger lattice constants. Here, the most significant D ij values are found for | r j − r i | = 1.0 a. Similar to the data obtained with varied u-parameter in Figure 3, the spin stiffness parameter A cannot be converged for any data set. Nevertheless, it is interesting that A becomes increasingly negative with larger lattice constants. A comparison with eq. (A7) shows that a negative spin stiffness A < 0 would favor a magnetic configuration where q = 0. This could be a hint to the existence of a helical magnetic texture. Similarly, to the spin stiffness tensor, the micromagnetic D does not converge either. However, it is interesting to see that its sign differs between different lattice parameters if we consider only the first shell for a moment. This indicates that there is also a sign change in the components of D ij . Having the parameters calculated, it is in principle straightforward to use them to explore the magnetic textures characterizing B20-MnGe using the corresponding Heisenberg Hamiltonian in eq. (1). Although the DM spiralization obtained from our scheme is in reasonable agreement with previous ab-initio simulations 13,14 , it cannot lead to an accurate description of the magnetic states. The lack of convergence with respect to the interatomic distance hinders the use of such basic mod- els. Interestingly, Koretsune et al. 16 obtain a much larger DM spiralization (107 meV/Å) utilizing a tight-binding model. This leads to a reasonable pitch of the magnetic texture (≈ 3nm). Using our parameters in Atomistic Spin Dynamics 35 , we found 2D metastable textures but no stable 3D textures. Besides the non-convergence of the parameters, this might indicate that interactions beyond the bilinear magnetic interactions play an important role. Naturally, the first-principles based simulations contain all the detailed electronic information required for a realistic description of the magnetic states.

V. LARGE-SCALE ELECTRONIC STRUCTURE CALCULATIONS WITH KKRNANO
In this section we present the large-scale DFT results that we obtained with KKRnano for B20-MnGe. Besides the previously suggested non-trivial magnetic states: the helical spiral (1Q state) and the hedgehog lattice (3Q state), we explored the possibility of stabilizing the Bloch point (BP) state.
In our study, the lattice constant is varied and the total energies corresponding to the three states are tracked with respect to the ferromagnetic phase. As mentioned in the introduction, the main reason that motivates the usage of KKRnano in conjunction with B20-MnGe is that Tanigaki et al. reported on the existence of the 3Q hedgehog lattice state in this material 11 . Findings by Kanazawa et al. suggest that this lattice is set up by a superposition of three orthogonal helical structures also referred to as 3Q state 36 . Here, the local magnetization is determined by the provision M ( r) =   sin qy + cos qz sin qz + cos qx sin qx + cos qy where q = 2π λ is the wavenumber given in terms of the helical wavelength λ and x, y and z are the spatial coordinates within the unit cell. Note, that M ( r) is not normalized. Equation (4) describes an alternating pattern of hedgehog and anti-hedgehog textures. An illustration of an anti-hedgehog is given in Figure 6 b). Following the micromagnetic description, singularities in the magnetization are expected within the magnetic texture 37 . Our ab initio simulations indicate, however, that all atomic magnetic moments are finite, although our method does not prevent the occurrence of a fully quenched magneti- zation density within or in between atoms. In contrast to other systems exhibiting a similar magnetic phase, the rather short helical wavelength of 3-6 nm in B20-MnGe allows one to perform density functional theory (DFT) calculations with KKRnano.
Other experimental works 18,19 suggest that in B20-MnGe a helical spin spiral forms along the (001) direction where the magnetization is described by the relation In the following, we refer to this as the 1Q state (see Figure 6 a)) Based on our findings in section IV, where we encounter a DM spiralization that does not seem to be larger than 10 meVÅ, we also consider a magnetic configuration which can exist without a large DM spiralization but could yield transmission electron microscopy stripe contrasts similar to the 3Q state. An obvious candidate for this is a Bloch point texture which can be conveniently defined by means of the four spherical parameters φ, θ, Φ and Θ. The parameters φ and θ designate the position of an individual atom in the unit cell which is described by the common polar and azimuthal angle φ = arctan (y/x) (6) and θ = arccos z Usually, the atomic positions are given in the Cartesian coordinates x, y and z. In the definition above, we define the origin of the coordinate system, i.e. the tuple (x = 0, y = 0, z = 0), to be at the center of the unit cell. In this frame of reference, all atoms that lay in an x-yplane that intersects with the center are described by θ = π/2. The orientation of the individual atomic magnetic moments for a BP texture is then defined by the polar angle and the azimuthal angle where the angles designating the atomic position enter as arguments. The phase factor is set to φ 1 = π, hence all magnetic moments point at the origin of the coordinate system. An illustration of this configuration is given in Figure 6 c).
Since the lattice parameter of a material can be modified via strain that originates from the manufacturing process of the sample, we investigated the dependence of B20-MnGe's magnetic properties as function of volume. Such a dependence is depicted in the upper part of Figure 7, where the total energy is evaluated for FM, 1Q, 3Q and BP states. The FM state constitutes the ground state, when the experimental lattice constant is assumed. However, 1Q and 3Q states are energetically not far from the FM state (within less than 10 meV/f.u.). When we further increase the lattice constant the picture changes. A crucial transition point is found around a = 5.0Å, where by imposing the 1Q or 3Q state the energy can be made lower than that of the ferromagnetic state. In general, for a > 5.0Å 1Q and 3Q states are favored over the ferromagnetic one. The BP state is energetically not preferred for any lattice constant except for the rather large a = 5.2Å.
In the lower part of Figure 7 the evolution of the magnetic moment with varying lattice constant is tracked. The resulting magnetic moment for the experimental lattice constant nicely falls on top of the magnetic moment of approximately 2µ B /f.u. which is reported experimentally 18 . The HS/LS transition that is already shown in Figure 2 is recognizable between a = 4.60 and a = 4.70Å. Furthermore, the magnetic moment increases, when the lattice constant is increased. This is a common behavior which is often observed in metallic systems. For larger lattice constants the magnetic moments of the different magnetic textures differ more than for the smaller lattice constants.

VI. CONCLUSIONS
We performed full ab initio calculations in order to investigate the magnetic properties of B20-MnGe. The basic properties of B20-MnGe are discussed, in particular the pressure-induced transition from a high-spin to a low-spin state. Subsequently, the magnetic exchange interactions that were extracted with the KKR method by means of infinitesimal rotations are presented in Section IV. Finally, the energetic behaviour of the magnetic 1Q (helial spiral), 3Q (hedgehog-antihedgehog lattice) and BP (Bloch point) state is analyzed by means of largescale electronic structure calculations with KKRnano.
Our first-principles based simulations indicate that neither the experimentally reported 1Q nor the 3Q texture is predicted to be the ground state when assuming the equilibrium lattice constant. However, by increasing the lattice constant both textures can be made energetically preferable. A possible explanation for this is the firstnearest neighbor isotropic magnetic exchange coupling which, as shown in Figure 5, is lowered for larger lattice constants. Experimentally, it is not possible to have the ideal lattice structures considered in our study. Thus various effects can affect the experimental observations and the related interpretations. For instance, impurities in the sample can potentially exert chemical pressure, which leads to spatial expansion of the lattice structure (see the example of Co-doped B20-FeGe 38 ).
Regarding the BP texture, it should be stressed that in KKRnano periodic boundary conditions are assumed and therefore the magnetic spins at the edges of each periodic image are aligned in a non-favorable anti-ferromagnetic way. Thus there is a large energy-penalty to pay. Interestingly, by increasing the lattice parameter, this energy cost decreases which can again be related to the reduction of the first-nearest neighbor isotropic exchange interaction upon increase of the lattice parameter.
theory. The latter is widely used in the skyrmion community and we adopt it to complement our toolbox for the investigation of the magnetic properties of B20-MnGe. We exemplify the connection between atomistic and continuum model by considering a helical spin spiral that points along the z-axis (c.f. 1Q state in eq. (5)) and is described by the wave vector q = 0, 0, q , i.e. the magnetic moments rotate within the x-y-plane and the wave vector points along the z-axis. The magnetization of each atom i is then given by where z i denotes the z-coordinate of the respective atomic site. It can be shown that such a magnetic structure interpolates smoothly between the discrete lattice and the continuum limit. We define the Heisenberg energy with isotropic exchange interaction and DM interaction as Insertion of eq. (A1) and usage of addition theorems leads to where we used the translational invariance of J ij and N is the number of atoms. For the helical spiral defined in eq. (A1), only the z-component D z ij of D ij needs to be considered.
The micromagnetic energy reads where A is the so-called spin stiffness, D the DM spiralization 3 and V Ω the volume of the unit cell. Insertion of the magnetization of the helical spiral given by eq. (A1) yields E micro,1Q = 1 V Ω dV Aq 2 sin 2 (qz) + cos 2 (qz) + D (cos (qz)ê x − sin (qz)ê y ) · ∂ ∂z sin (qz)ê x + ∂ ∂z cos (qz)ê y = 1 V Ω dV Aq 2 + Dq cos 2 (qz) + sin 2 (qz) The wave number q will take the value which minimizes E micro,1Q and we can thus impose the condition which gives us a provision on how the wave number q depends on the magnitude of DM spiralization and spin stiffness.
The atomistic and the micromagnetic model are connected in the limit q → 0, i.e. for a helical spiral that extends over multiple unit cells. Equation (A3) can then be simplified to Thus in this limit, it is possible to derive the micromagnetic parameters A and D from the atomistic parameters J ij and D ij which can be obtained from a KKR calculation by following the procedure described in [23][24][25] . The term E 0 determines the ferromagnetic reference energy. The exchange stiffness A describes the increase in energy if a spin spiral is assumed instead of the ferromagnet. The micromagnetic DMI D can lower the energy if the product of D and q is negative and can thus make the spin spiral configuration the energetically preferred state.
In general, A and D are 3 × 3 tensors that we denote with A and D. For B20 compounds this simplifies to diagonal matrices due to symmetry arguments and we Here, the summation is performed over all shells s, so that symmetrically equivalent parameters are omitted. It should be noted that from eq. (A9) it follows that D vanishes for D n ⊥ R n and is largest for D n R n .