Skyrmion Lattice in a Chiral Magnet

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Science  13 Feb 2009:
Vol. 323, Issue 5916, pp. 915-919
DOI: 10.1126/science.1166767

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Skyrmions represent topologically stable field configurations with particle-like properties. We used neutron scattering to observe the spontaneous formation of a two-dimensional lattice of skyrmion lines, a type of magnetic vortex, in the chiral itinerant-electron magnet MnSi. The skyrmion lattice stabilizes at the border between paramagnetism and long-range helimagnetic order perpendicular to a small applied magnetic field regardless of the direction of the magnetic field relative to the atomic lattice. Our study experimentally establishes magnetic materials lacking inversion symmetry as an arena for new forms of crystalline order composed of topologically stable spin states.

The formation of most crystals is related to three main aspects. First is the interplay of local repulsion and long-range attraction of atoms leading to an instability of the liquid with correlations that are completely isotropic and without preferred direction. Second, three particle collisions lower the energy and dominate the formation of the crystal out of the isotropic density fluctuations. Third, atoms are quantized with an integer number of them in a unit cell.

For the spin structures in magnetic materials, it was believed that some or all of the three above-mentioned mechanisms could not take place. Consider, for instance, the formation of a magnetically ordered state out of a paramagnetic metal. The underlying atomic lattice or the Fermi surface strongly breaks the rotational and translational invariance. Furthermore, for magnetic fluctuations in the paramagnet, three particle collisions are forbidden by time-reversal symmetry. Lastly, the question arises as to which quantized entities may play the role of the atoms in a solid. Possible candidates are topologically stable objects such as skyrmions, hedgehogs, or merons, which have received great theoretical interest ranging from magnetic monopoles in particle physics (1, 2) to the emergence of gauge theories in condensed matter (3). In analogy to the Abrikosov vortex lattice in superconductors, these objects may be considered as the building blocks of novel forms of electronic order akin to crystal structures. However, the experimental evidence for these building blocks is scarce.

We report the observation of the formation of a magnetic structure with hexagonal symmetry perpendicular to a small applied magnetic field in the cubic B20 compound MnSi. We show that this structure can approximately be visualized by a simple superposition of three helical states in the presence of a uniform field, where the superimposed state with the lowest energy can be viewed as a lattice of antiskyrmion lines, that is, magnetic vortices for which the magnetization in the center is antiparallel to the applied field. All three mechanisms mentioned above play an important role in explaining the magnetic structure. The lack of space-inversion symmetry in the atomic crystal of MnSi results in weak spin-orbit coupling that generates slow rotations of all magnetic structures such that they decouple from the underlying atomic lattice very efficiently. Further, in the presence of an external magnetic field, which breaks time-reversal symmetry, three-particle interactions of the magnetic excitations do occur. Lastly, for the magnetic state that emerges under these conditions skyrmion lines, that is, certain topologically protected knots in the magnetic structure, take over the role of the atoms in usual crystals.

At ambient pressure and zero applied magnetic field, MnSi develops helical magnetic order below a transition temperature Tc = 29.5 K that is the result of three hierarchical energy scales. The strongest scale is ferromagnetic exchange favoring a uniform spin polarization (spin alignment). The lack of inversion symmetry of the cubic B20 crystal structure results in chiral spin-orbit interactions, which may be described by the rotationally invariant Dzyaloshinsky Moriya (DM) interaction. The ferromagnetic exchange together with the chiral spin-orbit coupling lead to a rotation of the spins with a periodicity λh ≈ 190 Å that is large compared with the lattice constant, a ≈ 4.56 Å. This large separation of length scales implies an efficient decoupling of the magnetic and atomic structures. Therefore, the alignment of the helical spin spiral along the cubic space diagonal 〈111〉 is weak and is only fourth power in the small spin-orbit coupling. These crystalline field interactions, which break the rotational symmetry, are by far the weakest scale in the system.

Our study was partly inspired by recent work on the pressure dependence of the properties of MnSi (46). As shown in Fig. 1A, well below Tc an applied magnetic field, B, unpins the helical order and aligns its wave vector Q parallel to the field, QB, for a field exceeding Bc1 ≈ 0.1 T (79). This state is referred to as the conical phase. For a magnetic field exceeding Bc2 ≈ 0.6 T, the effects of the DM interaction are suppressed, giving way to a spin-aligned (ferromagnetic) state. For temperatures just below Tc, an additional phase, referred to as the A phase, is stabilized in a finite field interval shown for B ∥ 〈100〉 in Fig. 1 (7, 8). It had been believed that the A phase was explained by a single-Q helix, where Q is perpendicular to the applied field (8, 9). The specific heat exhibits a tiny peak at the border of the A phase, whereas the susceptibility discontinuously assumes a lower value (10). When taken together with the discontinuous change of the scattering intensity seen in neutron scattering, this suggests that the A phase represents a distinct magnetic phase, where the transition from the conical order to the A phase is discontinuous (first order).

Fig. 1.

(A) Magnetic phase diagram of MnSi. For B = 0, helimagnetic order develops below Tc = 29.5 K. Under magnetic field, the helical order unpins and aligns along the field above Bc1; above Bc2, the helical modulation collapses. In the conical phase, the helix is aligned parallel to the magnetic field. The transition fields shown here have been inferred from the AC susceptibility, where the DC and AC fields were parallel to 〈100〉 (10). (B) Neutron scattering setup used in our study; the applied magnetic field B was parallel to the incident neutron beam.

Because Q tends to align parallel to an applied magnetic field, neutron scattering as a function of B has been reported for setups in which the magnetic field was perpendicular to the incident neutron beam (11). In contrast, we chose the incident neutron beam to be parallel to the applied magnetic field (Fig. 1B). Two samples were studied. Sample 1 refers to a disk of 19-mm diameter and a thickness of 3 mm, where the vector normal to the disc was slightly misaligned by 11° with respect to a 〈110〉 axis. Sample 2 is a small parallelepiped, with dimensions 1.5 mm by 1.5 mm by 14 mm, where a 〈110〉 axis corresponded to the long axis. To search for higher-order peaks and double scattering, we increased the neutron flux in our measurements of sample 1, accepting a larger beam divergence. In contrast, the beam divergence was reduced for sample 2 to improve the resolution in rocking scans. All data at finite magnetic field were measured after zero-field cooling to the desired temperature, followed by a field ramp to the desired field value. However, the results for the A phase were identical when recorded after field cooling. For further details of the experimental setup, we refer to (11).

Figure 2 shows typical data recorded, where the spot sizes represent the resolution limit. All data shown represent sums over rocking scans of typically ±8° (11). Figure 2, A to C, shows data for sample 1, whereas Fig. 2, D to F, shows data for sample 2. Figure 2A shows the scattering intensity of sample 1 in a zero-field-cooled state at a temperature of 27 K for a 〈110〉 scattering plane, and the 〈110〉 axis is indicated in the figure. The pattern is consistent with previously published data and helical magnetic order along 〈111〉. Figure 2B shows the intensity pattern of sample 1 in the A phase. Six spots emerge on a regular hexagon.

Fig. 2.

Typical neutron small angle scattering intensities; note that the color scale is logarithmic to make weak features visible. Data represent the sum over rocking scans with respect to the vertical axis through the sample. (A) to (C) show data for sample 1 and (D) to (F) for sample 2. Backgrounds measured above Tc and B = 0 were subtracted in all panels except for (A) (light blue square). Spots are labeled for reference; for the intensity of these spots as a function of rocking angle, see (11). (A) Helical order in sample 1 in the zero-field-cooled state at T = 27 K and B = 0. (B) Sixfold intensity pattern in the A phase in sample 1; same orientation as in (A); T = 26.45 K, B = 0.164 T. (C) Sixfold intensity pattern in the A phase for random orientation of sample 1 (see text for details); T = 26.77 K, B = 0.164 T. (D) Helical order in sample 2 in the zero-field-cooled state at T = 16 K and B = 0. (E) A phase in sample 2, same orientation as in (D); T = 27.7 K, B = 0.162 T. (F) A phase as measured in conventional setup [compare Fig. 1A in (11)], where data in all other panels were measured in the configuration shown in Fig. 1B; T = 27.7 K, B = 0.190 T. A small residual intensity due to the conical phase is observed (spots 9 and 10), whereas spots 6 and 8 correspond to those in (E).

We tested the variation of the intensity pattern on the orientation of the field relative to the crystal axes in both samples. The field was always parallel to the incident beam, whereas the sample was rotated for a large number of different orientations. Typical data are shown in Fig. 2C for sample 1, where the sample was rotated with respect to the vertical axis into a random position. For both samples and all crystal orientations, the scattering pattern always exhibited the sixfold symmetry. In case the scattering plane contained a 〈110〉 direction, two of the peaks of the sixfold pattern coincided with this direction. As for Fig. 2C, the scattering plane did not contain a 〈110〉 direction. For sample 2, the intensities along the vertical direction, which coincided with the 〈110〉 direction, were systematically weaker. This may be explained by the demagnetizing fields caused by the large aspect ratio, which implies that part of the scattering intensity was not captured in the rocking scans [see also (11)]. The main result of our study is that, for all orientations of the magnetic field with respect to the atomic lattice, six Bragg reflections are observed on a regular hexagon that is strictly perpendicular to the magnetic field.

We performed rocking scans to test whether the A phase has long-range order. Typical data are presented in (11). In the helical state, the half-width of the rocking scans corresponded to a magnetic mosaicity ηm ≈ 3.5° consistent with previous work and long-range order (12, 13). Remarkably, in the A phase the half-width of the rocking scans corresponded to a reduced magnetic mosaicity ηm ≈ 1.75°, implying an even longer correlation length of at least χ ≈ 5500 Å, when allowing for demagnetizing fields (11).

To test for consistency with previous work, we also measured the emergence of the A phase as a function of temperature for magnetic field perpendicular to the neutron beam, where the vertical axis was the same 〈110〉 axis as before and the low-symmetry horizontal axis containing spots 6 and 8 in Fig. 2F was perpendicular to the magnetic field and incident neutron beam. Data were recorded after (i) zero-field-cooling the sample to a temperature well below Tc, (ii) increasing the magnetic field to 0.19 T, and (iii) measuring the neutron scattering pattern for selected increasing temperatures (Fig. 2F shows data for T = 27.7 K). Well below Tc we first observe the two spots parallel to the field direction labeled 9 and 10, characteristic of the conical state. When entering the A phase, the intensity of the spots of the conical phase becomes very weak but does not vanish, whereas strong scattering intensity appears in the perpendicular direction (spots 6 and 8). This is consistent with previous work and may signal a phase coexistence, as expected of a weak first-order transition with possible extra effects of the demagnetizing fields added.

The key results of our neutron scattering data may be summarized as follows: (i) the helical wave vector aligns perpendicular to the applied magnetic field; (ii) the fundamental symmetry of the intensity pattern is sixfold, suggesting a multi-Q structure; and (iii) the A phase stabilizes in a magnetic field strength of order Bc2/2. Moreover, the pattern aligns very weakly with respect to the 〈110〉 orientation. We can readily account for these features in the framework of standard Landau-Ginzburg theory in the mean-field approximation by taking fluctuations into account. Near Tc the Ginzburg-Landau energy functional can be written as (14, 15) Math(1) Math

The first and second terms represent the usual quadratic contribution with the conventional gradient term; the third term, the Dzyaloshinsky-Moriya interaction; and the last term, the coupling to an external magnetic field B. The quartic term accounts in lowest order for the effects of mode-mode interactions and stabilizes the magnetic order. We neglect higher-order spin-orbit coupling terms describing anisotropy effects (14, 15). The free energy is given by exp(–G) = ∫DM exp(–F[M]) (throughout the paper, we use a dimensionless free energy). Within mean-field approximation, G(B) is equal to minF[M], and one minimizes F with respect to the spin structure M(r).

To explain the A phase, we evoke strong analogies with the crystal formation of ordinary solids out of the liquid state. The latter is in most cases driven by the cubic interactions of density waves (16), which in momentum space can be written as Math

The ordered state can gain energy from this term only when three ordering vectors of the crystal structure add up to zero. Accordingly, in many cases (exceptions can arise only for strong first-order transitions) the ordered phase, which forms first out of a liquid state, is of body-centered cubic symmetry (16), which is the crystal structure with the largest number of such triples of reciprocal lattice vectors.

In the presence of a finite uniform component of the magnetization, Mf, a similar mechanism can also occur in MnSi. From the quartic term in Eq. 1, we obtain terms that are effectively cubic in the modulated moment amplitudes Math(2) where mq is the Fourier transform of M(r). As in the case of an ordinary crystal, one can gain energy from this term for a structure with three Q vectors adding up to zero. These vectors have a fixed modulus determined by the interplay of the two gradient terms in Eq. 1. Therefore, these three vectors have relative angles of 120° (Fig. 3A) and define a plane characterized by a normal vector, . By symmetry, the energy change is proportional to Mf· , and therefore the three Q vectors must be perpendicular with respect to the external magnetic field. Our qualitative arguments already explain the two main experimental observations in the A phase: The Bragg spots are located in the plane perpendicular to B and display a sixfold symmetry (because both Q and –Q give a Bragg reflection) independent of the orientation of the underlying lattice. We therefore suggest that the A phase is a chiral spin crystal, the A crystal, approximately characterized by the magnetization Math(3) where Math is the magnetization of a single chiral helix with amplitude A, wave vector Qi, and two unit vectors, ni1 and ni2, orthogonal to each other and to Qi. All three helices have the same chirality; that is, all Qi·(ni1 × ni2) have the same sign. More precisely, one has to add further higher-order Fourier components to Eq. 3 when minimizing F[M]. However, these terms remain small close to Tc. The relative shifts, Δri, of the helices, which we calculate theoretically, determine whether the A crystal can be described as a lattice of skyrmions (see below).

Fig. 3.

(A) Depiction of the hexagonal basis vectors of the crystalline spin order in the A phase. (B) Theoretical phase diagram as a function of magnetic field B/B0 with Embedded Image and the parameter t = r0J/D2 – 1, which is roughly proportional to TTc. We use the model parameter γ = JD/U = 5 and a momentum-space cutoff k < 40D/J. Smaller values of γ increase the A phase regime. For most values of field, the A phase is either metastable or stable, but at low fields below the dotted line it becomes unstable. Above and to the right of the red dashed line, the fluctuation correction to the size of the order parameter becomes larger than 20%, and our theoretical analysis becomes uncontrolled. Therefore in the shaded gray region, we have reliably established stability of the A phase within our model [see (11) for details]. (Inset) Energy difference between A phase and conical phase as a function of field for the same parameters and t = –3.5, both in the mean-field approximation and with fluctuation corrections. Fluctuations stabilize the A phase at intermediate fields. (C) Real space depiction of the spin arrangement in the A phase in the x-y plane. Note that this spin arrangement is translation-invariant along the z axis, which is parallel to the magnetic field. (D) Skyrmion density per unit cell area as calculated for the A phase as shown in (C). The integrated skyrmion density per unit cell is finite, Φ = –1. The arrows represent the magnetization component perpendicular to the line of sight.

One also has to take into account that an external magnetic field favors helices with Q vectors parallel to B, because this way the spins may easily tilt parallel to the field to form a conical structure. Within mean-field theory, this conical phase always has the lowest energy (11). However, in the parameter range, where the A phase occurs experimentally, that is, close to Tc, at intermediate magnetic fields (Fig. 1A), the energy difference between the two phases becomes very small as shown in the Fig. 3B inset. The origin of the energy minimum of the A crystal for moderate magnetic field can be traced back to the size of the modulations of the magnetization amplitude, |M(r)|, which is minimal close to B ≈ 0.4Bc2 (11). In our mean-field Landau-Ginzburg theory, the A crystal thus appears as a metastable phase, which becomes extremely close in energy to the conical phase for intermediate fields B ≈ 0.4Bc2.

It turns out that, when we consider thermal fluctuations around the mean-field solution, these stabilize the A crystal. To show this, we consider the leading correction to mean-field theory arising from Gaussian fluctuations Math(4) where M0 is the mean-field spin configuration for either the A phase or the conical phase. To make Eq. 4 well defined, one has to specify a cutoff scheme for short length scales. We use a cutoff in momentum space, k <2π/a, where a is the lattice spacing of the MnSi crystal. Because of the long pitch of the helix, it turns out that most contributions arise from fluctuations on short length scales with the exception of temperatures extremely close to Tc [see (11) for a detailed discussion], but both short-range and long-range fluctuations favor the A crystal for intermediate magnetic fields. As shown in the Fig. 3B inset, the fluctuations indeed stabilize the A crystal. A typical phase diagram resulting from Eq. 4 is shown in Fig. 3B. The theoretical phase diagram catches the main characteristics of the experimental phase diagram. The A crystal is stable at intermediate fields not too far from Tc. When interpreting the theoretical result, one has to take into account that Eq. 4 is only valid for small fluctuations and therefore cannot be applied too close to Tc. Indeed it is expected (17, 18) that fluctuations ultimately drive the transition first order and that such strong fluctuations will substantially shift the transition line to the paramagnet. We estimate the strength of fluctuations by calculating the leading correction to the order parameter for both the conical phase and the A crystal (11). In the shaded area of Fig. 3B, these corrections are small (less than 20%), which justifies the use of Eq. 4.

As can be seen from Fig. 3C, the magnetic structure of the A crystal obtained by minimizing F[M] is characterized by a pattern of magnetic vortices. To elucidate their nature, we compute the skyrmion density given by (19): Math(5) where x and y are the coordinates perpendicular to B and n = M(r)/|M(r)| is the orientation of the magnetization. ϕ is a measure of the winding of the magnetization profile. If ϕ integrates to 1 or –1, a topologically stable knot exists in the magnetization. As shown in Fig. 3D, the skyrmion density is finite and oscillates between positive and negative as compared with the normal helical or conical phases, where it is zero. Moreover, the skyrmion number Φ = ∫ϕ(r)d2r per two-dimensional unit cell is quantized and adds up to –1. Taken together this implies that the A crystal can be interpreted as a crystal made out of quantized objects, the skyrmion lines, with a magnetization at their core that is antiparallel to the applied magnetic field and M0.

Because of its symmetry and the cubic interactions, the A crystal has to be separated by two first-order phase transitions both from the conical and the paramagnetic phases. This is consistent with the experimental observations. Two additional features in the scattering patterns that account for less than 1% of the total integrated scattering intensity are, first, weak continuous streaks of intensity emerging radially outward from the six main spots and, second, the coexistence of conical order and the spin crystal. Both may be the result of weak heterogeneities resulting from these generic first-order boundaries of the A crystal, possibly in combination with demagnetizing fields.

Many years ago in a seminal study, Bogdanov and collaborators used a mean-field model to predict the existence of skyrmion lattices for anisotropic noncentrosymmetric magnetic materials under the application of a magnetic field (20, 21). The authors also pointed out that, within their mean-field theory for cubic materials such as MnSi, the skyrmion lattice would always be metastable. Moreover, in the absence of a magnetic field, it has been shown theoretically that certain crystalline spin structures can be stabilized by long-range interactions or an additional phenomenological parameter (2224). In contrast to these predictions, we find in this study that it is sufficient to include the effects of Gaussian thermal fluctuations to stabilize skyrmion lattices in a magnetic field in cubic materials.

It is instructive to search for analogies of the A crystal in other condensed matter systems. Because it is a multi-Q structure, we note that previously known multi-Q structures, for example, in the rare earths, involve large values of Q and exhibit very strong pinning to the atomic lattice (25, 26), whereas for MnSi we observe that the sixfold pattern of the A crystal exists independently of the underlying lattice and Q is quite tiny. Although flux lines in superconductors and the magnetic skyrmion lines observed are topologically completely different objects, there is nevertheless an intimate similarity of the Abrikosov lattice of superconducting flux lines and the hexagonal symmetry of the A crystal (20, 21). Moreover, the A crystal is characterized by broken translation symmetry in the plane perpendicular to B only. Therefore the A phase is similar to the chiral columnar phase of liquid crystals (16, 27). Further, the spin structure of the A crystal is topologically equivalent to theoretical predictions of the spin structure of the ferromagnetic quantum Hall state near 1/2 filling (28), where, however, the underlying energetics is completely different. Lastly, individual magnetic vortices attract also great interest as a micromagnetic phenomenon, which arises when conventional domain walls in soft ferromagnets are made to meet (29).

The skyrmion lattice in the chiral magnet MnSi reported here represents an example where an electronic liquid forms a spin crystal made from topologically nontrivial entities. This provides a glimpse of the large variety of magnetic states that may be expected from the particle-like magnetic objects currently discussed in the literature.

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