Neutron powder diffraction (NPD) and x-ray magnetic circular dichroism (XMCD) spectroscopy are employed to investigate the magnetism and spin structure in single-phase B20 Co1.043Si0.957. The magnetic contributions to the NPD data measured in zero fields are consistent with the helical order among the allowed spin structures derived from group theory. The magnitude of the magnetic moment is (0.3 ± 0.1)μB/Co according to NPD, while the surface magnetization probed by XMCD at 3 kOe is (0.18–0.31)μB/Co. Both values are substantially larger than the bulk magnetization of 0.11μB/Co determined from magnetometry at 70 kOe and 2 K. These experimental data indicate the formation of a helical spin phase and the associated conical states in high magnetic fields.

The large atomic radius of metalloids, i.e., Si and Ge suppresses the electron transfer between transition metal (M) atoms and enlarges both local density of states and spin–orbit coupling due to hybridization between M (Mn, Fe, and Co) and Si/Ge states,1 typically leading to B20 crystal structures.2–8 As illustrated in Fig. 1(a), the B20 structure can be viewed as a rock salt structure with a dimerization-type distortion involving the displacement of M and Si/Ge atoms along the [111] direction,2 resulting in a crystal structure of non-centrosymmetric P213 space group with chirality, which induces the Dzyaloshinskii–Moriya (DM) exchange interaction,9,10 favoring helical spin structures. The chiral structure of B20 materials also leads to a topological electronic structure with chiral fermions that produce transport properties of topological features.11–15 In the helical phase, the spin directions rotate while propagating along the symmetry axis, thereby resulting in short-range ferromagnetic and long-range antiferromagnetic alignments.3–5,16 Helical spin structures with multiple propagation directions lead to the skyrmions spin texture.17 

FIG. 1.

(a) Crystal structure of B20 Co–Si. Co atoms are indexed for describing spin structures. (b) Powder neutron diffraction pattern of Co–Si at 320 K and the fitting using the B20 structure.

FIG. 1.

(a) Crystal structure of B20 Co–Si. Co atoms are indexed for describing spin structures. (b) Powder neutron diffraction pattern of Co–Si at 320 K and the fitting using the B20 structure.

Close modal

The magnetic properties of B20 materials are sensitive to atomic species of the M site. As one of the most studied B20 materials,2,16–18 MnSi exhibits a helical spin structure propagating along the [111] direction with a period around 19 nm;17 the magnetic moment is 0.4μB/Mn atom,2 while the Curie temperature Tc is only 29.5 K.17 In contrast, CoSi and FeSi are diamagnetic and paramagnetic, respectively, due to the smaller atomic radius of Si.4 Interestingly, solid solutions of CoSi and FeSi, i.e., FexCo1−xSi, have helical spin structures for 0.2 < x < 0.95,3,4,19 with the maximum magnetic ordering temperature about 59 K,4 maximum moment 0.17μB/(Co/Fe), and the helical period between 30 and 230 nm, respectively.3 CoxMn1−xSi and FexMn1−xSi are helimagnetic for x < 0.08 and x < 0.19, respectively.3,6

Replacing Si with Ge in MnSi leads to MnGe, another helimagnetic material with a B20 structure, a much higher ordering temperature (170 K), a much higher magnetic moment of 1.5μB/Mn, and a much smaller helical period (3–6 nm related to the temperature).5,20 FeGe is a helimagnetic material with 1.0μB/Fe atom and ordering temperature around 278 K and period approximately 70 nm;7,21–23 CoxFe1−xGe are also helimagnetic for x ≤ 0.8, whose helical period strongly depends on x.8,24 The formation of a helical phase in Fe–Ge is not restricted to B20 structures but also occurs with similar periodicity in amorphous materials with a short-range order.25 

Besides the atomic species on the M site and that on the Si/Ge site, relative stoichiometry between the M and Si/Ge sites also has a significant impact on the magnetic properties. Replacing a small portion of Si with Co (Co1+xSi1−x) induces magnetic ordering above a critical excess Co content of xc = 0.028.26 According to density-functional-theory (DFT) calculations, the excess Co atoms exhibit a large magnetic moment (1.7μB/Co) and also spin-polarize the surrounding Co atoms, which subsequently cause magnetic ordering in Co1+xSi1−x through a quantum phase transition above xc. The magnetometry measurements show magnetic transition temperatures of about 275 and 328 K for Co1.029Si0.971 and Co1.043Si0.957, respectively.26 Co1.043Si0.957 reveals further helimagnetic order and skyrmion lattices with a helical period of 17 nm at 300 K.26 The small skyrmion dimension at high temperatures (λ ∼ 17 nm) makes Co1.043Si0.957 an intriguing system from the viewpoint of understanding its spin structures and exploring skyrmions for practical room-temperature applications.

On the other hand, the question remains for the average Co magnetic moment in Co1+xSi1−x. For Co1.043Si0.957, the theory predicts a moment of 0.18μB/Co, similar to that in FexCo1−xSi.3 In contrast, magnetometry measurements show a magnetization of 0.11μB/Co at 70 kOe and 2 K, which still increases with the field.26 The slow saturation suggests a robust conical state, which is related to the small helical periodicity27 in Co1.043Si0.957.

In this work, we report the magnetism and spin structure in single-phase B20 Co1.043Si0.957 (here, referred to as Co–Si) using neutron powder diffraction (NPD) and x-ray magnetic circular dichroism (XMCD) spectroscopy. NPD allows for probing both the spin structure and the magnetic moment, once the magnetic and nuclear contributions are separated and the spin structure is taken into account. XMCD spectroscopy with a total electron yield quantifies the normal moments near the surface (2−5 nm)28 averaged over 100 μm in the lateral direction. The measured moments, (0.3 ± 0.1)μB/Co at 0 kOe (NPD) and (0.18–0.31)μB/Co at 3 kOe (XMCD), are both substantially larger than the high-field magnetometry value, confirming the emergence of high-field conical states in Co–Si.

The Co1.043Si0.957 ribbons with an approximate width of 2 mm and thickness of 40 μm were synthesized via melt-spinning.26 For this, high-purity Co and Si with appropriate amounts were melted using a conventional arc-melting process to prepare the Co1.043Si0.957 alloy. The arc-melted alloy was then re-melted to a molten state in a quartz tube and subsequently ejected onto the surface of a water-cooled copper wheel rotating with a speed of 15 m/s to form the ribbons. The cooling rate is of order 106 K s−1. Our earlier study shows that the non-equilibrium rapid-quenching process creates Co1+xSi1−x alloys with a maximum Co solubility of x = 0.043. The ribbons were mechanically grounded into powders suitable for neutron powder diffraction (NPD) measurements. The composition of Co1+xSi1−x (x = 0.043) was measured by energy dispersive x-ray spectroscopy (EDS). Rietveld refinement of XRD patterns and a linear increase of lattice constant on increasing Co content confirm that the excess Co atoms replace Si and our alloy is a substitutional alloy of the solid-solution type.26 Temperature-dependent high-resolution NPD measurements were carried out without an external magnetic field on the time-of-flight (TOF) powder diffractometer, POWGEN. POWGEN is a third-generation powder diffractometer and has the highest resolution to probe large unit cells in the powder suite at the Spallation Neutron Source Oak Ridge National Laboratory. The data were collected with neutrons with a central wavelength of 2.665 Å. A cryofurnace was used as the sample environment to cover the temperature region between 20 and 360 K. The sample mass was 3.53 g. The neutron diffraction patterns were analyzed using the Rietveld method, and the refinement of the structure was carried out using the FullProf program.29 X-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD) spectroscopy were performed by detecting the total electron yield at beamline 4.0.2 at the Advanced Light Source, Berkeley, CA, which performs high-resolution spectroscopy using circularly and linearly polarized x rays. For these measurements, the ribbon sample was thinned into a thin film with a thickness ≈ 100 nm using an ion-milling process at the Molecular Foundry, Berkeley, CA.

1. Crystal structure

The powder neutron diffraction pattern of Co1.043Si0.957 at 320 K is displayed in Fig. 1(b) as a function of the spacing between diffraction planes d (called d-spacing). Taking advantage of the substantial contrast of the neutron scattering length of Co (2.49 fm) and Si (4.15 fm), the experimental spectrum can be fit using the B20 structure and the Co1.043Si0.957 composition with a random distribution of Co on Si sites by varying the lattice constants and atomic positions using the software FullProf.29 All peaks are identified with the Miller indices (h, k, l), as marked in Fig. 1(b), except for the two peaks from the aluminum container. With increasing temperature, the diffraction peaks shift toward a larger d direction, indicating thermal expansion [Figs. 2(a)2(c)]. The lattice constant is extracted using structural refinement and displayed in Fig. 2(d) as a function of temperature. Above 200 K, the thermal expansion is essentially linear with a coefficient of approximately 1.1 × 10−5/K.

FIG. 2.

Neutron diffraction patterns at various temperatures for (a) (004), (b) (123), and (c) (012) peaks. (d) Temperature dependence of the lattice constants.

FIG. 2.

Neutron diffraction patterns at various temperatures for (a) (004), (b) (123), and (c) (012) peaks. (d) Temperature dependence of the lattice constants.

Close modal

2. Separation of magnetic and nuclear contributions of the NPD

For magnetic materials, the neutron powder diffraction with unpolarized beam contains nuclear (In) and magnetic (Im) contributions, i.e., the total intensity I(T,k)=In(T,k)+Im(T,k), where T is the temperature and k is the wave-vector transfer, which is along the direction normal to the diffraction planes that can be expressed in reciprocal space vectors.30 Notice that the experimental data in Fig. 1(b) and Figs. 2(a)2(c) correspond to I(T,k) summed over k of different directions, but the same magnitude |k|=2πd, or |k|=2πdI(T,k). Typically, the nuclear contribution decreases gradually with temperature due to the fluctuation of atomic positions at high temperatures. The magnetic contribution, on the other hand, is expected to vanish at the magnetic ordering temperature. Our observations in Figs. 2(a)2(c) reveal a rapidly decreasing diffraction intensity at low temperatures, which becomes nearly constant at around 320 K, consistent with the magnetic transition occurring at around 320 K.26 

To obtain insight into the spin structure from the magnetic diffraction, one needs to isolate the nuclear (n) and magnetic (m) contributions,

In(T,k)=AFn2(k)e2W(T,d)
(1)
Im(T,k)=AFm2(k)e2W(T,d)
(2)

where A is a common scaling factor, Fn and Fm are nuclear and magnetic diffraction amplitude, respectively, e2W(T,d) is the Debye–Waller factor. The nuclear contribution In(T) can be predicted if the Debye–Waller factor and In are known at a certain temperature since Fn is typically temperature independent as long as the structural symmetry is preserved. Lacking magnetic order at 320 K,26 the magnetic diffraction intensity is expected to be minimal. The function W(T,d) can be written as W(T,d) = βT/(4d2), which represents the effect of thermal fluctuation of atomic positions.31 One can fit the peak intensity as a function of d at 320 K using the linear relation between ln(In)/T and 1/(4d2) yielding β = 6 × 10−4 Å2/T. The nuclear term is obtained as In(T,k)=In(T1,k)e2β(TT1)/d2 with T1 = 320 K. The magnetic contribution Im(T) is then extracted by subtracting the nuclear contribution,Im(T,k)=I(T,k)In(T,k). Figure 3(a) displays the magnetic contribution after the removal of the Debye–Waller factor and Ml, i.e., |k|=2πdIm(T,k)Mle2W(T,d), where Ml is the multiplicity (number of possible k of the same |k|) of a diffraction peak at a certain d. Overall, the magnetic contribution decreases with temperature and vanishes at about 320 K, while the magnitude varies significantly between diffraction peaks of different k (or d).

FIG. 3.

(a) Experimental magnetic contribution as a function of temperature obtained by removing the nuclear contribution, multiplicity, and the Debye–Waller factor (see the text) from the measured intensity (peak area). (b) Experiment value of the magnetic contribution as a function of d calculated by summing the data in (a) over temperature. Also plotted are the best fit of the theoretical magnetic contribution AFm2(k) to the experimental values by varying the factor A for different spin structures. Dashed lines are guidelines for the eyes. The “error bars” of the theoretical contributions correspond to the variation when the spins are along different directions.

FIG. 3.

(a) Experimental magnetic contribution as a function of temperature obtained by removing the nuclear contribution, multiplicity, and the Debye–Waller factor (see the text) from the measured intensity (peak area). (b) Experiment value of the magnetic contribution as a function of d calculated by summing the data in (a) over temperature. Also plotted are the best fit of the theoretical magnetic contribution AFm2(k) to the experimental values by varying the factor A for different spin structures. Dashed lines are guidelines for the eyes. The “error bars” of the theoretical contributions correspond to the variation when the spins are along different directions.

Close modal

To highlight the dependence of the magnetic contribution on d, data in Fig. 3(a) were summed over T and scaled with the peak multiplicity, i.e., |k|=2πd,TIm(T,k)Mle2W(T,d) [Fig. 3(b)]. This dependence of the magnetic contribution on d is the key to analyzing the underlying spin structure.

3. Relation between spin structure and magnetic contribution

Neutron diffraction probes the arrangement of spins or spin texture. The extraction of spin structure hinges on the analysis of magnetic diffraction intensity (magnetic contribution) from the coherent scattering of the spins. The intensity of the magnetic contribution can be expressed as

Fm2(k)=12(γr0g)2|rq(k,r)f(k,r)exp(ikr)|2
(3)

with

q(k,r)s(r)k^[k^s(r)].
(4)

Here, r is the position of a magnetic atomic site with spin s(r); k^=k/|k| is the unit vector of k, f(k,r) is the isotope-specific form factor for each site, γ = 1.913, r0 = 2.818 × 10−15 m is the classical electron radius, g 2 is the Landé factor. Multiple magnetic atomic sites within the unit cell can be described by defining r=R+u with the lattice point vector R and the relative position of an atomic site in the unit cell. The form factor f(k,r) depends on atomic (and isotopic) species and k, but not on the lattice points, which implies f(k,r)=f(k,u). A spin texture, such as a helical structure, with a propagation vector ks, can be considered by writing s(r)=s0(u)eiksR and q(k,r)=q0(k,u)eiksR, yielding

rq(k,r)f(k,r)eikr=uq0(k,u)f(k,u)eikuRei(ks+k)R
(5)

The factor Rei(ks+k)R determines the diffraction angle where the intensity peaks appear. The factor fs(k)uq0(k,u)f(k,u)eiku can be treated as the vector structure factor for magnetic scattering that becomes for Co–Si (with Co moments) fs(k)=f(k)uq0(k,u)eiku. Equations (3) and (5) can be used to simulate the magnetic contribution to the diffraction intensity, according to the spin arrangement s(r)=s0(u)eiksR. As shown in Fig. 1(a), there are four Co sites in the B20 unit cell. While the site Co1 is on the [111] axis, the other three sites, Co2, Co3, and Co4, are related with the threefold rotation along the [111] axis. The spin structure is described by the spin vectors on these Co sites s0(u) and the propagation vector ks.

4. Group theory analysis of spin structures allowed by the B20 crystal structure

In the following, we examine possible spin structures allowed by the P213 space group of the B20 structure using group theory.32 The magnetic diffraction contribution simulated according to these spin structures will be compared with the experimental observation of Co–Si to determine the most likely spin structures.

First, we consider the case with no superstructure, i.e., ks = 0 or uniform magnetization. According to the group theory analysis,32 the spin structure can be described by four irreducible Γ1 to Γ4 representations of the P213 space group, as displayed in Table I. While the one-dimensional Γ1 to Γ3 appear only once, the three-dimensional Γ4 appears three times. All spin structures have a zero net magnetic moment for the unit cell, meaning antiferromagnetic (AFM) order, except for Γ4,1, which is ferromagnetic. More specifically, the AFM structures of Γ1, Γ4,2, and Γ4,3 are collinear, while Γ2 and Γ3 are non-collinear. Compared with the experimental observation, only the ferromagnetic spin structure Γ4,1 generates a magnetic contribution that matches reasonably well, while other spin structures do not [Fig. 3(b)].

TABLE I.

Directions of the magnetic moments (basis vectors) on different Co atoms [see fig. 1(a)] for the allowed spin structures when there is no magnetic superstructure (ks = 0).

Co1Co2Co3Co4
Γ1 (1, 1, 1) (−1, −1, 1) (−1, 1, −1) (1, −1, −1) 
Γ2 (1,ei2π3,ei2π3(−1,eiπ3,ei2π3(−1,ei2π3,eiπ3(1,eiπ3,eiπ3
Γ3 (1,ei2π3,ei2π3(−1,eiπ3,ei2π3(−1,ei2π3,eiπ3(1,eiπ3,eiπ3
Γ4,1 (1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
(1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
(1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
((1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
Γ4,2 (1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
(−1, 0, 0)
(0, 1, 0)
(0, 0, −1) 
(1, 0, 0)
(0, −1, 0)
(0, 0, −1) 
(−1, 0, 0)
(0, −1, 0)
(0, 0, 1) 
Γ4,3 (1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
(1, 0, 0)
(0, −1, 0)
(0, 0, −1) 
(−1, 0, 0)
(0, −1, 0)
(0, 0, −1) 
(−1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
Co1Co2Co3Co4
Γ1 (1, 1, 1) (−1, −1, 1) (−1, 1, −1) (1, −1, −1) 
Γ2 (1,ei2π3,ei2π3(−1,eiπ3,ei2π3(−1,ei2π3,eiπ3(1,eiπ3,eiπ3
Γ3 (1,ei2π3,ei2π3(−1,eiπ3,ei2π3(−1,ei2π3,eiπ3(1,eiπ3,eiπ3
Γ4,1 (1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
(1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
(1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
((1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
Γ4,2 (1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
(−1, 0, 0)
(0, 1, 0)
(0, 0, −1) 
(1, 0, 0)
(0, −1, 0)
(0, 0, −1) 
(−1, 0, 0)
(0, −1, 0)
(0, 0, 1) 
Γ4,3 (1, 0, 0)
(0, 1, 0)
(0, 0, 1) 
(1, 0, 0)
(0, −1, 0)
(0, 0, −1) 
(−1, 0, 0)
(0, −1, 0)
(0, 0, −1) 
(−1, 0, 0)
(0, 1, 0)
(0, 0, 1) 

With a propagation vector ks || [111], the arrangement Co magnetic moment can be divided into two groups, Co1 in group I and Co2–Co4 for group II. Table II shows the spin structure of the spin helix for ks || [111] allowed by the P213 space group for Co group I. The two helical spin structures Γ2(I) and Γ3(I) of opposite chirality are one-dimensional. Both spins are perpendicular to the [111] axis and rotate along the axis from the unit cell to the unit cell (see Fig. S1 in the supplementary material). Table III shows the spin structure for Co group II, where Γ2(II) and Γ3(II) form helical structures as Γ2,1(II)+ei2π3Γ2,2(II)+ei4π3Γ2,3(II) and Γ3,1(II)+ei2π3Γ3,2(II)+ei4π3Γ3,3(II) for Co group II, respectively. The additional restriction from the exchange interaction, which is not considered in the group theory analysis, connects the spins of the two groups. For the B20 material, the short-range exchange interaction is generally ferromagnetic,3,8,24,26 meaning that the magnetic moments are almost perfectly parallel within one unit cell. Under this constriction, two helical structures are formed, including all the Co atoms: [Γ2(I)eiksa3, Γ2,1(II)+ei2π3Γ2,2(II)+ei4π3Γ2,3(II)], and [Γ3(I)eiksa3, Γ3,1(II)+ei2π3Γ3,2(II)+ei4π3Γ3,3(II)], for opposite chirality, respectively [Fig. 4(a)].

FIG. 4.

Schematic illustration of helical spin structures for the propagation vector (a) ks || [111] and (b) ks || [100].

FIG. 4.

Schematic illustration of helical spin structures for the propagation vector (a) ks || [111] and (b) ks || [100].

Close modal
TABLE II.

Directions of the magnetic moments on group I Co atoms for the allowed spin structures when the propagation vector ks is along the [111] direction.

Co1
Γ2(I) (1,ei2π3,ei2π3
Γ3(I) (1,ei2π3,ei2π3
Co1
Γ2(I) (1,ei2π3,ei2π3
Γ3(I) (1,ei2π3,ei2π3
TABLE III.

Directions of the magnetic moments on group II Co atoms for the allowed spin structures when the propagation vector ks is along the [111] direction.

Co2Co3Co4
Γ2,1(II) (1,0,0) (0,ei2π3,0) (0,0,ei2π3
Γ2,2(II) (0,1,0) (0,0,ei2π3(ei2π3,0,0) 
Γ2,3(II) (0,0,1) (ei2π3,0,0) (0,ei2π3,0) 
Γ3,1(II) (1,0,0) (0,ei2π3,0) (0,0,ei2π3
Γ3,2(II) (0,1,0) (0,0,ei2π3(ei2π3,0,0) 
Γ3,3(II) (0,0,1) (ei2π3,0,0) (0,ei2π3,0) 
Co2Co3Co4
Γ2,1(II) (1,0,0) (0,ei2π3,0) (0,0,ei2π3
Γ2,2(II) (0,1,0) (0,0,ei2π3(ei2π3,0,0) 
Γ2,3(II) (0,0,1) (ei2π3,0,0) (0,ei2π3,0) 
Γ3,1(II) (1,0,0) (0,ei2π3,0) (0,0,ei2π3
Γ3,2(II) (0,1,0) (0,0,ei2π3(ei2π3,0,0) 
Γ3,3(II) (0,0,1) (ei2π3,0,0) (0,ei2π3,0) 

Another propagation direction that leads to helical spins is ks || [100]. There are also two groups of Co atoms: group I (Co1 and Co2) and group II (Co3 and Co4). Table IV shows the allowed spin structures for groups I and II with ks || [100]. Again, helical structures are formed within both atomic groups. The restriction of exchange interaction connects the two groups and results in the helical structures including all Co atoms: [Γ1,2(I)+iΓ1,3(I), Γ1,2(II)+iΓ1,3(II)], and [Γ1,2(I)iΓ1,3(I), Γ1,2(II)iΓ1,3(II)] for opposite chirality [Fig. 4(b)].

TABLE IV.

Directions of the magnetic moments on group I (and II) Co atoms for the allowed spin structures when the propagation vector k is along the [100] direction.

Co1 (Co3)Co2 (Co4)
Γ1,2 (0,1,0) (0, eika/2,0) 
Γ1,3 (0,0,1) (0,0, eika/2
Γ2,2 (0,1,0) (0, eika/2,0) 
Γ2,3 (0,0,1) (0,0, eika/2
Co1 (Co3)Co2 (Co4)
Γ1,2 (0,1,0) (0, eika/2,0) 
Γ1,3 (0,0,1) (0,0, eika/2
Γ2,2 (0,1,0) (0, eika/2,0) 
Γ2,3 (0,0,1) (0,0, eika/2

According to Eq. (5), the magnitude of the magnetic contribution depends on the vector structure factor fsuq0(k,u)f(k,u)eiku but not on ks which determines the diffraction angles, as long as the helical period is much larger than one unit cell. In other words, except for the diffraction angles, the magnetic contributions of the helical spin structure and the ferromagnetic spin structure Γ4(1) are expected to be nearly the same [Fig. 3(b)]. Therefore, the helical spins, whose periods are much larger than the unit cell, also match the experimentally observed magnetic contribution.

The result that the zero-field helical propagation vector is along either the [111] or the [001] directions [see the supplementary material) is consistent with the micromagnetic analysis in the earlier work.33 Experimentally, the helical spin structure of B20 magnets, such as MnSi17 and FexCo1−xGe,8 has a propagation vector ks along the threefold rotation symmetry axis [111]. Helical propagation vectors along [111] and [100] have been reported in B20 FeGe34 and MnGe.5 

5. Magnitude of the magnetic moment

Once the magnetic and nucleation contributions to the NPD are separated and the spin structure is known, one can estimate the magnitude of the spin by comparing the magnetic contribution and the nuclear contribution, which can be written as

Fn2=|rfn(k,r)exp(ikr)|2
(6)

where fn(k,r) is the isotope-specific nuclear structure form factor. Note that this is typically not the case for small angle neutron diffraction that only measures the magnetic contribution.22 According to Eqs. (5) and (6), the ratio of the magnetic to the nuclear contribution is, for the ferromagnetic spin structure Γ4,1 and the helical spin structure, solely determined by the magnitude of the spins (magnetic moments),30,35 which is found to be 0.3 ± 0.1 μB/Co, where the magnetic contribution is an average for T < 320 K.

XAS spectra near the Co L3,2 edges (760–830 eV) were measured at different temperatures from 100 to 300 K in an external magnetic field (±3 kOe) applied normal to the sample surface. To exclude possible degradation of the free-standing film under the beam, observed in previous experiments, the following temperature series were used: 300, 200, 100, 150, and 250 K. Figure 5(a) shows the room-temperature XAS spectra for the photon angular momentum parallel (μ+) and antiparallel (μ) to the applied magnetic field. The x-ray magnetic circular dichroism (XMCD) signal is obtained from the difference between the corresponding blue (μ) and red (μ+) curves and is shown for different temperatures in Fig. 5(b). The integrals of both XAS and XMCD spectra, r=L3+L2(μ++μ)dω, q=L3+L2(μ+μ)dω, and p=L3(μ+μ)dω, were used to quantify the orbital moment (m0) and spin moment (ms) using the sum rule by the following: mo=4q3rnh and ms=6p4qrnh, where nh is the hole density per Co atom.23,24 Orbital and spin moments and average moment mj = mo + ms are shown as a function of temperature in Figs. 5(c) and 5(d), respectively. The spin and, hence, total magnetic moment increase linearly with temperature from 0.18 to 0.31μB/Co [Fig. 5(d)], which agrees with the NPD values of (0.3 ± 0.1)μB/Co. The unusual increase of the magnetic moment with temperature (Fig. 5) mimicks the trend of peak broadening in NPD (see Fig. S2 in the supplementary material). It is unclear whether this trend is related to the temperature-dependent transition from helical to skyrmion spin structures around room temperature,26 an interesting aspect that calls for future studies.

FIG. 5.

(a) X-ray absorption spectra (XAS) measured with left and right circular polarized light at 300 K near Co L3,2 edges (2p → 3d) and (b) XMCD spectra temperatures for Co1.043Si0.957. The dashed (black) curves in (a) and (b) correspond to the integrals of room-temperature XAS (r) and XMCD (q) spectra and p is an integral point used for determining the spin moment (ms). (c) Spin and orbital (mo) moments at different temperatures. (d) Total magnetic moment mj = mo + ms.

FIG. 5.

(a) X-ray absorption spectra (XAS) measured with left and right circular polarized light at 300 K near Co L3,2 edges (2p → 3d) and (b) XMCD spectra temperatures for Co1.043Si0.957. The dashed (black) curves in (a) and (b) correspond to the integrals of room-temperature XAS (r) and XMCD (q) spectra and p is an integral point used for determining the spin moment (ms). (c) Spin and orbital (mo) moments at different temperatures. (d) Total magnetic moment mj = mo + ms.

Close modal

The (0.3 ± 0.1)μB/Co value extracted from zero-field NPD corresponds to the magnetic moment after accounting for the spin alignment in the magnetic structure. The XMCD values in the range of (0.18–0.31)μB/Co are most likely only a small portion of the helical period due to surface sensitivity with a 2–5 nm probing depth. In addition, the weaker coupling of spins at the surface due to a smaller coordination number implies a stronger response to external magnetic fields (larger normal moment), which increases with temperature. Both NPD and XMCD values coincide with density functional theory (0.18μB/Co)26 and differ substantially from magnetometry, which measures the bulk net magnetization (0.11μB/Co at 2 K and 0.07μB/Co at 300 K).26 In fact, magnetometry shows that the field-dependent magnetization curves at 2 and 300 K do not saturate in fields up to 70 kOe.26 This is inconsistent with the ferromagnetic ordering since the incomplete saturation cannot be explained by the small magnetocrystalline anisotropies (1.8 and 0.043 Merg/cm3 at 2 and 300 K, respectively).26 Instead, slow saturation originates from helimagnetism in the Co–Si alloy, as indicated by DC susceptibility and Lorentz transmission electron microscopy, and the transformation from the helical to conical phase at a high magnetic field.

Correlating the experimental with modeled magnetic contributions suggests that both ferromagnetic and helical spin structures (with a period exceeding the size of the unit cell) are most likely [Fig. 3(b)]. According to Eq. (5), the information of the helical period or ks is included in the factor Rei(ks+k)R, which determines the diffraction angle. In other words, the helical period may be extracted from the positions of the satellite peaks k±ks in neutron diffraction. Unfortunately, broadening effects due to chemical disorder and multiple values of |k±ks| smear the satellite peaks, which prevent us from identifying satellite peaks in the powder neutron diffraction patterns.

In conclusion, the results of NPD measurements on Co1.043Si0.957 are consistent with the helical spin structure. The magnitude of the magnetic moment extracted from NPD (0.3 ± 0.1μB/Co) coincides with theoretical estimates (0.18μB/Co) and the surface magnetization retrieved from XMCD spectroscopy (0.18–0.31μB/Co). All these values are substantially larger than the magnetometry value (0.11μB/Co), and the data reflect the evolution of a coplanar helical spin structure into a noncoplanar conical spin structure with nonzero magnetization.

See the supplementary material for a symmetry analysis of the spin structure by the group theory and a peak boarding analysis according to neutron powder diffraction.

This work was supported by NSF-DMREF SusChEM No. 1729288 (sample fabrication and neutron diffraction studies), NSF-EPSCoR EQUATE OIA-2044049 (theoretical analysis), Nebraska EPSCoR FIRST OIA-1557417 (x-ray absorption spectroscopy), and the Nebraska Center for Materials and Nanoscience. We thank Qiang Zhang from the Oak Ridge National Laboratory for his help on neutron measurement. This research used resources at the Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory, the Advanced Light Source, and the Molecular Foundry, a DOE Office of Science User Facility under Contract No. DE-AC02-05CH11231. This work was performed in part in the Nebraska Nanoscale Facility and Nebraska Center for Materials and Nanoscience, which are supported by the National Science Foundation under Award No. ECCS: 2025298, and the Nebraska Research Initiative.

The authors have no conflict of interest to disclose.

The data that support the findings of this study are available within the article and its supplementary material.

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Supplementary Material