Introduction

A skyrmion is a topologically protected nano-size spin texture found in several magnets1,2,3,4,5. Due to its unusual spin structure, many intriguing behaviors such as topological Hall effects, current-driven motion and multiferroic behavior have been observed6. Although there is a huge potential to design novel functional materials by exploiting these unique electromagnetic properties, skyrmion engineering or skyrmion-crystal engineering is yet to be established. Here, the key issue is how to manipulate the size and helicity of skyrmions. Regarding this problem, a recent experiment for the representative skyrmion system Mn1−xFexGe has shown that we can tune the skyrmion size and helicity by changing the carrier density7,8.

The Hamiltonian which determines the nature of skyrmions is

where M is the magnetization per volume, D is the Dzyaloshinskii-Moriya (DM) interaction coefficient and J is the ferromagnetic exchange coupling, respectively. For materials design of skyrmion crystals, we need to know the precise value of D/J. However, non-empirical evaluation of these parameters in the classical continuum model has been a difficult challenge, since it requires an elaborate multi-scale approach spanning the quantum to classical regime. For the insulating skyrmion system Cu2OSeO39, there is a work in which the spin Hamiltonian (1) was derived from first principles10. By comparing the total energy of various magnetic states, they determined the value of D/J and succeeded in reproducing the experimentally measured skyrmion size. However, the guiding principle to control the values of D/J is yet to be obtained.

On the other hand, for metallic systems, the situation is different. There are several studies for the estimate of D11,12,13,14,15,16,17,18,19,20. Among them, recently, one of the present authors (NN) and his collaborators have shown that the DM interaction in the two-band model drastically changes when the band anticrossing point resides near the Fermi level20. The story is analogous to that of anomalous Hall conductivity, in which the band anticrossings act as magnetic monopoles in momentum space21. In Ref. 19, a Berry phase expression for the DM interaction has also been formulated. These studies stimulate us to explore a fascinating possibility of controlling the DM interaction in metallic systems by manipulating the electronic structure. Indeed, the fact that not only the size but also the helicity of skyrmions in Mn1−xFexGe changes as a function of x indicates that we have a good chance to control the value of D.

In this paper, we show a quantitative analysis of the DM interaction in the metallic helimagnet, Mn1−xFexGe, based on ab initio density-functional theory (DFT) calculation. From the obtained band structure, we evaluate the off-diagonal spin susceptibility which is a direct measure of the DM interaction. We find that the sign change of D observed in the experiment for Mn1−xFexGe is successfully reproduced. The carrier-density dependence of D can be systematically understood in terms of the distribution of band anti-crossing points in the electronic structure. We demonstrate that the sign and the size of D can be controlled as a function of the carrier density or the size of the exchange splitting. There is also an interesting possibility to induce gigantic anisotropy in D by applying a strain to the system.

Results

DM interaction in the continuum model

Let us first look at the second term in the Hamiltonian (1). This indicates that q-linear term in the spin susceptibility, χαβ, should be proportional to the DM interaction coefficient, D. Therefore, to estimate D in the continuum limit from the DFT calculation, we compute the long-wave length limit of the spin susceptibility, that is,

Here, (α, β, γ) = (x, y, z), (y, z, x), or (z, x, y) and corresponds to the coefficient for Mα∂Mγ/∂β in Eq. (1). Since we consider the skyrmions in the x-y plane under the total magnetic moment along the z-axis, hereafter we focus on and . In Eq. (2), we use the non-interacting spin susceptibility defined as

where, σ is the Pauli matrix and G0 is the non-interacting Green’s function in the orbital basis. Using this non-interacting spin susceptibility, we can write as with

where is the eigenvector of the Kohn-Sham Hamiltonian with the eigenvalue of εnk. Hence, we can discuss the DM interaction in terms of the band structure. Although there is a sophisticated approach to compute D19, we employ the current simple approach to explore various parameters and materials. Furthermore, this approach is appropriate to obtain a guiding principle for controlling D as discussed below.

Ab initio band structure

Figure 1(a) shows the DFT band structure of FeGe (black solid lines). Here, we include the spin-orbit couplings and assume the ferromagnetic moment along the z axis. The calculated local magnetic moment is 1.18 μB per Fe atom, which is consistent with the experiments22,23 and previous calculations24. The x- and y-component of this local magnetic moment is only about 1% of the z-component indicating that the DM interaction is sufficiently small so that Eq. (2) holds in good accuracy. Using this electronic structure, we construct the tight-binding model made of Fe 3d and Ge 4p Wannier orbitals to reproduce the band structure below the Fermi level as shown in red broken lines. The densities of states for up spin (red line) and down spin (blue line) are also shown in Fig. 1(a). As can be seen, there is a large exchange splitting, Δ. According to the energy difference of up and down spins for the Fe 3d orbitals, we obtain Δ = 1.17 eV. In Fig. 1(b), the obtained tight-binding band structure around the Fermi level is illustrated with colors representing the weight of the up spin. Since we consider ferromagnetic electronic structure, each band can be basically characterized as either up-spin or down-spin band as shown in Fig. 1(b). In addition, due to the spin-orbit couplings, there are several anticrossing points where complex spin texture emerges. In the followings, we use this tight-binding model for all the calculations.

Figure 1
figure 1

Band structure of FeGe.

(a) Comparison between DFT band structure (black solid lines) and tight-binding band structure (red broken lines). Densities of states for up spin (red line) and down spin (blue line) are also shown. (b) Detailed band structure around the Fermi level with colors representing the weight of up spin; that is, red (blue) lines correspond to up-spin (down-spin) bands. The Fermi level is set to zero.

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Ab initio evaluation of the DM interaction

Let us start with the simple two-band model in two dimensions considered in Ref. 20. The Hamiltonian is represented by a 2 × 2 matrix,

We assume that the band dispersion is linear in the kx-ky plane and dispersionless in the kz direction. We introduce m to open a gap at the band crossing point as shown in Fig. 2(a). The static spin susceptibility can be calculated analytically and the result is

Figure 2
figure 2

Schematic pictures of the DM interaction around the band anticrossing points.

(a) Band structure of the two-band model defined in Eq. (5) and (b) the chemical potential dependence of the DM interaction corresponding to this band structure. When two anticrossing points with different energies and spin textures with opposite chiralities reside close to each other, the peak structure appears as shown in (c).

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where μ is the chemical potential and . It is interesting to note that is negative for μ < −m and positive for μ > m as shown Fig. 2(b). If we reverse the spin texture by modifying the Hamiltonian as σy → −σy, becomes positive for μ < −m and negative for μ > m. This result suggests that the position of the Fermi level and the spin texture around the anticrossing point are crucial to determine the sign of D.

In real materials, the situation is not so simple as that of this two-band model. The anticrossing points form complex surfaces in a four-dimensional space spanned by the energy and the wave number and the electronic states around neighboring anticrossing points can hybridize with each other. In Fig. 2(c), as a representative case, we schematically show how changes as a function of the chemical potential when two anticrossing points with different energies and spin textures with opposite chiralities reside close to each other. Thus, when many anticrossings are densely clustered around the Fermi level, the sign and the size of should change drastically. In fact, the band structure of FeGe has many anticrossing points around the Fermi level (see Fig. 1). To visualize the distribution of the anticrossing points as a function of energy, in Fig. 3, we plot a partial density of states (DOS) where the up-spin weight, w↑, satisfies 0.05 < w↑ < 0.95. We can see that there are two peaks around μ = 0 eV and −0.5 eV, where is expected to change significantly.

Figure 3
figure 3

Distribution of anticrossing points.

Partial density of states where the up-spin weight, w↑, satisfies 0.05 < w↑ < 0.95 (red line). Total density of states is also plotted (black line).

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To analyze the contribution of each anticrossing point, next, we focus on the anticrossing points around the M (0 1/2 1/2) point. In Fig. 4(a), there are two anticrossing points between Y and M points and between M and Z points. In kx = 0 plane, such anticrossing points continuously form a closed loop around the M point. In Fig. 4(b,c), we show the k-dependence of when the chemical potential is below and above the anticrossing points shown in black lines in Fig. 4(a), respectively. We see that the texture around the M point drastically changes when the chemical potential sweeps across the anticrossing points, which is consistent with the simple two-band calculation20. Note that around the other Fermi surfaces such as the one between Γ-Y line are not negligible although the spin mixture is not so significant. This is because the effect of spin mixing extends away from anticrossing points; that is, has non-negligible values typically up to 0.2–0.3 eV away from anticrossing points. For comparison, in Fig. 4(d,e), we show the Berry curvature, Ωz(k), which is the origin of intrinsic anomalous Hall conductivity (AHC)21,25. Ωz(k) is defined as

Figure 4
figure 4

Contribution of anticrossing points to the DM interaction.

(a) Band structure around the anticrossing points and k-dependence of at two different chemical potentials (b) μ = −0.38 eV and (c) μ = −0.34 eV at T = 100 K. Fermi surface is also shown in black lines. For comparison, the Berry curvatures at (d) μ = −0.38 eV and (e) μ = −0.34 eV are plotted. (f) at μ = −0.38 eV as a function of number of (k, n) points around the anticrossing points considered in Eq. (4). Dotted line represents the value for the full summation.

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where vx and vy are velocity operators. Since the spin mixing is important for both cases, there are common regions where the contributions to the DM interaction and AHC are large. However, in the Berry curvature, the summation in Eq. (7) is restricted to n ≠ n′ while it is not in Eq. (4). As a result, only the restricted region is important for AHC, which is in sharp contrast to . To check how large region around the anticrossing points contributes to , we calculate with restricted summation in Eq. (4). As shown in Fig. 4(f), we find that the contribution from 10%–20% of the total (k, n) points around the band anticrossing points determines up to 70%–80% of the total .

Figure 5 shows the resulting at T = 300 K as a function of the carrier density together with the AHC, σxy. In the calculation, we use the rigid band approximation. The relation between the carrier density, n and the chemical potential, μ, is shown in the inset of Fig. 5. At μ = −0.506 eV, the number of hole is 1.0 per Fe atom, which corresponds to the carrier density in MnGe. In Fig. 5, we find that shows clear sign change from FeGe to MnGe , which is consistent with the experimental sign change of skyrmion helicity7,8 and is in sharp contrast to σxy. The positive (negative) hump structure in around μ ~ 0.4 (0.2) eV originates from the peak structure around μ ~ 0.5 (0.0) eV in Fig. 3, respectively. If we assume that contributions from μ ~ 0(−0.5) eV is negative (positive), we can understand why changes its sign around μ ~ 0.3 eV. As for σxy, the calculated value of σxy in MnGe is larger than that in FeGe and there is no sign change. This behavior including size and sign agrees well with the experimental anomalous Hall contribution to σxy for MnGe26 and for Mn1−xFexGe27.

Figure 5
figure 5

Carrier density dependence of the DM interaction.

DM interaction coefficients, and and the AHC, σxy as a function of the carrier density, n. The inset shows the relation between the chemical potential μ and carrier density n. We use the rigid band approximation starting from the electronic structure of FeGe (n = 0.0, μ = 0.0 eV). n = −1.0 (μ = −0.506 eV) corresponds to the carrier density of MnGe.

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Strain-induced huge anisotropy of the DM interaction

According to the above discussion, there are several ways to change the size and sign of D. As we have seen above, D can be efficiently controlled by carrier doping. We can also exploit the temperature dependence of the exchange splitting. When the relative position of up- and down- spin band changes, the distribution of anticrossing points in the band structure also changes, which will have a direct impact on D. As an example, we will show later the change in of FeGe as a function of the moment per Fe atom. This mechanism can be related to the temperature dependence of the magnetic moment and skyrmion size in MnGe26. Another interesting possibility is to make use of the strain effect. If we apply a strain to the system, the symmetry of the electronic structure can be lowered and the distribution of the anticrossing points will change drastically. This effect is expected to be prominent especially when D changes its sign. Figure 6(a) shows the calculated at n = −0.45 for which we apply the uniaxial strain along the y direction. We find that the difference between and actually enhances particularly by the elongation along the y axis; is about 40% (400%) larger than at +2% (+5%) strain. The carrier density dependence of the anisotropy, , for fixed strain of +5% is shown in Fig. 6(b). We can see that the anisotropy becomes large particularly around the region of sign change.

Figure 6
figure 6

Strain dependence of the DM interaction.

(a) at n = −0.45 as a function of uniaxial strain along the y direction. (b) Anisotropy of the DM interaction, at +5% strain along the y direction as a function of the carrier density.

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Discussion

In the present calculation, we employed the rigid band approximation. To examine its validity, we have performed a calculation for the other end material MnGe and doped negative carriers by the rigid band approximation. As shown in Fig. 7, we have obtained a qualitatively similar result in that is positive (negative) for the end material MnGe (FeGe). Thus the result that for Mn1−xFexGe changes its sign between x = 0 and 1 should be robust, even when we go beyond the rigid band approximation.

Figure 7
figure 7

Carrier density dependence of the DM interaction starting from MnGe electronic structure.

Carrier density dependence of the DM interaction coefficients, and . We use the electronic structure of MnGe (n = 0.0) for the same crystal structure as that of FeGe and employ the rigid band approximation. n = 1.0 (μ = 0.832 eV) corresponds to the carrier density of FeGe.

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Regarding the crystal structure, it has been known that the magnetic moment for the optimized structure is much smaller than the experimental value within the local density approximation28. On the other hand, for the experimental structure, the size of magnetic moment is similar to that in the experiment (~1μB). In fact, within our calculations, the magnetic moment in FeGe decreases with decreasing the lattice constant as shown in Fig. 8(a), which is consistent with previous studies24,29. As a result, exhibits non-trivial magnetic moment dependence as shown in Fig. 8(b). Since the magnetic moment dependence of D is also an important issue, let us next discuss how the change in the lattice structure or the magnetic moment affect D in Mn1−xFexGe. In Fig. 8(c), we plot the lattice constant dependence of , together with the distribution of band anti-crossings in Fig. 8(d,e) for lattice constants a = 0.99a0 and 1.01a0, where a0 is the experimental lattice constant. As can be seen, for larger a and magnetic moment, the energy difference of two peaks in Fig. 3 becomes larger (Fig. 8(e)). Consequently, the density at which changes its sign (nc) becomes larger (Fig. 8(c)). However, the qualitative feature of is robust against the change in a. Therefore, our calculation successfully explains why is positive (negative) for MnGe (FeGe) and provides a useful guideline for materials design of skyrmion crystal in metallic helimagnets.

Figure 8
figure 8

Lattice constant dependence of the DM interaction.

(a) Lattice constant dependence of the magnetic moment on each Fe atom for FeGe and (b) corresponding magnetic moment dependence of . (c) with several lattice constants, a/a0 = 0.98, 0.99, 1.00 and 1.01. Distribution of anticrossing points at (d) a/a0 = 0.99 and (e) a/a0 = 1.01.

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Finally, let us compare the quantitative values of with experiments. For this purpose, we should rescale using the exchange splitting as 20. Using Δ = 1.17 eV for FeGe and Δ = 2.14 eV for MnGe based on the calculation shown in Fig. 7, we can obtain D = −10.1 meVÅ for FeGe and D = 107 meVÅ for MnGe. The experimental values of D can be estimated using J and the helical period, λ as D = 4πJ/λ. Assuming that J scales to TN and using the values of J = 52 meVÅ2 and TN = 30 K for MnSi18,30, the experimental values of D for FeGe and MnGe are −8.7 meVÅ and 124 meVÅ, respectively, which are in good agreement with our results (see Table I).

Table 1 List of parameters for FeGe and MnGe.
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Method

Crystal structure

In the calculations, experimental values are used for the crystallographic parameters31. For the pressure and strain calculations, internal coordinates of the atoms are fixed and only the lattice parameters are modified. To symmetrize and , we use two different internal coordinates of atoms in the (4a) position, that is, (x, x, x) where xFe,Mn = 0.135 and xGe = 0.842 and the one with its 90-degree rotation along the z-axis and take the average. Note that our structure is right-handed according to Ref. 7. In the experiment, observed skyrmions on the right-handed crystal structure in FeGe (MnGe) are anticlockwise (clockwise), indicating that Dx,y < 0 (Dx,y > 0).

DFT calculations

To evaluate in FeGe, we perform the electronic structure calculation within the generalized-gradient approximation (GGA)32 based on the density functional theory33. We use ultrasoft pseudopotentials34 and plain-wave basis sets to describe the charge densities and wave functions with cutoff energies of 40Ry and 500Ry, respectively. We use 8 × 8 × 8 k-point mesh. With including the spin-orbit couplings and assuming the ferromagnetic moment along the z axis, we obtain non-collinear magnetic structure. Using this electronic structure, we calculate Wannier functions for Fe 3d and Ge 4p orbitals using wannier90 code35,36,37. Based on the Wannier functions, we construct a tight-binding model on the restricted Hilbert space and calculate χαβ using 64 × 64 × 64 k-point mesh at T = 300 K. The AHC is also calculated using the Wannier interpolation technique with 200 × 200 × 200 k-point mesh38.

Additional Information

How to cite this article: Koretsune, T. et al. Control of Dzyaloshinskii-Moriya interaction in Mn1–xFexGe: a first-principles study. Sci. Rep. 5, 13302; doi: 10.1038/srep13302 (2015).