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 Co_{1.043}Si_{0.957}. 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)*μ*_{B}/Co according to NPD, while the surface magnetization probed by XMCD at 3 kOe is (0.18–0.31)*μ*_{B}/Co. Both values are substantially larger than the bulk magnetization of 0.11*μ*_{B}/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.

## I. INTRODUCTION

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 P2_{1}3 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}

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 0.4*μ*_{B}/Mn atom,^{2} while the Curie temperature *T*_{c} 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., Fe* _{x}*Co

_{1−x}Si, 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 0.17

*μ*

_{B}/(Co/Fe), and the helical period between 30 and 230 nm, respectively.

^{3}Co

*Mn*

_{x}_{1−x}Si and Fe

*Mn*

_{x}_{1−x}Si 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 1.5*μ*_{B}/Mn, and a much smaller helical period (3–6 nm related to the temperature).^{5,20} FeGe is a helimagnetic material with 1.0*μ*_{B}/Fe atom and ordering temperature around 278 K and period approximately 70 nm;^{7,21–23} Co_{x}Fe_{1−x}Ge 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 (Co_{1+x}Si_{1−x}) induces magnetic ordering above a critical excess Co content of *x _{c}* = 0.028.

^{26}According to density-functional-theory (DFT) calculations, the excess Co atoms exhibit a large magnetic moment (1.7

*μ*

_{B}/Co) and also spin-polarize the surrounding Co atoms, which subsequently cause magnetic ordering in Co

_{1+x}Si

_{1−x}through a quantum phase transition above

*x*. The magnetometry measurements show magnetic transition temperatures of about 275 and 328 K for Co

_{c}_{1.029}Si

_{0.971}and Co

_{1.043}Si

_{0.957}, respectively.

^{26}Co

_{1.043}Si

_{0.957}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 Co

_{1.043}Si

_{0.957}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 Co_{1+x}Si_{1−x}. For Co_{1.043}Si_{0.957}, the theory predicts a moment of 0.18*μ*_{B}/Co, similar to that in Fe* _{x}*Co

_{1−x}Si.

^{3}In contrast, magnetometry measurements show a magnetization of 0.11

*μ*

_{B}/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 periodicity

^{27}in Co

_{1.043}Si

_{0.957}.

In this work, we report the magnetism and spin structure in single-phase B20 Co_{1.043}Si_{0.957} (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 *μ*m in the lateral direction. The measured moments, (0.3 ± 0.1)*μ*_{B}/Co at 0 kOe (NPD) and (0.18–0.31)*μ*_{B}/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.

## II. EXPERIMENTAL METHODS

The Co_{1.043}Si_{0.957} ribbons with an approximate width of 2 mm and thickness of 40 *μ*m 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 Co_{1.043}Si_{0.957} 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 10^{6} K s^{−1}. Our earlier study shows that the non-equilibrium rapid-quenching process creates Co_{1+x}Si_{1−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 Co_{1+x}Si_{1−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.

## III. RESULTS

### A. Spin structure and magnetic moment from NPD

#### 1. Crystal structure

The powder neutron diffraction pattern of Co_{1.043}Si_{0.957} 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 Co_{1.043}Si_{0.957} 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.

#### 2. Separation of magnetic and nuclear contributions of the NPD

For magnetic materials, the neutron powder diffraction with unpolarized beam contains nuclear (*I*_{n}) and magnetic (*I*_{m}) contributions, i.e., the total intensity $I(T,k\u2192)=In(T,k\u2192)+Im(T,k\u2192)$, where *T* is the temperature and $k\u2192$ 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 $I(T,k\u2192)$ summed over $k\u2192$ of different directions, but the same magnitude $|k\u2192|=2\pi d$, or $\u2211|k\u2192|=2\pi d\u2061I(T,k\u2192)$. 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,

where *A* is a common scaling factor, *F*_{n} and *F*_{m} are nuclear and magnetic diffraction amplitude, respectively, $e\u22122W(T,d)$ is the Debye–Waller factor. The nuclear contribution *I*_{n}(*T*) can be predicted if the Debye–Waller factor and *I*_{n} are known at a certain temperature since *F*_{n} 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*) = β*T*/(4*d*^{2}), 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(*I*_{n})/*T* and 1/(4*d*^{2}) yielding β = 6 × 10^{−4} Å^{2}/T. The nuclear term is obtained as $In(T,k\u2192)=In(T1,k\u2192)e\u22122\beta (T\u2212T1)/d2$ with *T*_{1} = 320 K. The magnetic contribution *I*_{m}(*T*) is then extracted by subtracting the nuclear contribution,$Im(T,k\u2192)=I(T,k\u2192)\u2212In(T,k\u2192)$. Figure 3(a) displays the magnetic contribution after the removal of the Debye–Waller factor and *M _{l}*, i.e., $\u2211|k\u2192|=2\pi d\u2061Im(T,k\u2192)Mle\u22122W(T,d)$, where

*M*is the multiplicity (number of possible $k\u2192$ of the same $|k\u2192|$) of a diffraction peak at a certain

_{l}*d*. Overall, the magnetic contribution decreases with temperature and vanishes at about 320 K, while the magnitude varies significantly between diffraction peaks of different $k\u2192$ (or

*d*).

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., $\u2211|k\u2192|=2\pi d,TIm(T,k\u2192)Mle\u22122W(T,d)$ [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

with

Here, $r\u2192$ is the position of a magnetic atomic site with spin $s\u2192(r\u2192)$; $k^=k\u2192/|k\u2192|$ is the unit vector of $k\u2192$, $f(k\u2192,r\u2192)$ is the isotope-specific form factor for each site, γ = 1.913, *r*_{0} = 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 $r\u2192=R\u2192+u\u2192$ with the lattice point vector $R\u2192$ and the relative position of an atomic site in the unit cell. The form factor $f(k\u2192,r\u2192)$ depends on atomic (and isotopic) species and $k\u2192$, but not on the lattice points, which implies $f(k\u2192,r\u2192)=f(k\u2192,u\u2192)$. A spin texture, such as a helical structure, with a propagation vector $k\u2192s$, can be considered by writing $s\u2192(r\u2192)=s\u21920(u\u2192)e\u2212ik\u2192s\u22c5R\u2192$ and $q\u2192(k\u2192,r\u2192)=q\u21920(k\u2192,u\u2192)e\u2212ik\u2192s\u22c5R\u2192$, yielding

The factor $\u2211R\u2192\u2061ei(k\u2192s+k\u2192)\u22c5R\u2192$ determines the diffraction angle where the intensity peaks appear. The factor $f\u2192s(k\u2192)\u2261\u2211u\u2192\u2061q\u21920(k\u2192,u\u2192)f(k\u2192,u\u2192)eik\u2192\u22c5u\u2192$ can be treated as the vector structure factor for magnetic scattering that becomes for Co–Si (with Co moments) $f\u2192s(k\u2192)=f(k\u2192)\u2211u\u2192\u2061q\u21920(k\u2192,u\u2192)eik\u2192\u22c5u\u2192$. Equations (3) and (5) can be used to simulate the magnetic contribution to the diffraction intensity, according to the spin arrangement $s\u2192(r\u2192)=s\u21920(u\u2192)e\u2212ik\u2192s\u22c5R\u2192$. 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 $s\u21920(u\u2192)$ and the propagation vector $k\u2192s$.

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

In the following, we examine possible spin structures allowed by the P2_{1}3 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., *k*_{s }= 0 or uniform magnetization. According to the group theory analysis,^{32} the spin structure can be described by four irreducible Γ_{1} to Γ_{4} representations of the P2_{1}3 space group, as displayed in Table I. While the one-dimensional Γ_{1} to Γ_{3} appear only once, the three-dimensional Γ_{4} appears three times. All spin structures have a zero net magnetic moment for the unit cell, meaning antiferromagnetic (AFM) order, except for Γ_{4,1}, which is ferromagnetic. More specifically, the AFM structures of Γ_{1}, Γ_{4,2}, and Γ_{4,3} are collinear, while Γ_{2} and Γ_{3} are non-collinear. Compared with the experimental observation, only the ferromagnetic spin structure Γ_{4,1} generates a magnetic contribution that matches reasonably well, while other spin structures do not [Fig. 3(b)].

. | Co1 . | Co2 . | Co3 . | Co4 . |
---|---|---|---|---|

Γ_{1} | (1, 1, 1) | (−1, −1, 1) | (−1, 1, −1) | (1, −1, −1) |

Γ_{2} | (1,$e\u2212i2\pi 3,ei2\pi 3$) | (−1,$ei\pi 3,ei2\pi 3$) | (−1,$e\u2212i2\pi 3,e\u2212i\pi 3$) | (1,$ei\pi 3,e\u2212i\pi 3$) |

Γ_{3} | (1,$ei2\pi 3,e\u2212i2\pi 3$) | (−1,$e\u2212i\pi 3,e\u2212i2\pi 3$) | (−1,$ei2\pi 3,ei\pi 3$) | (1,$e\u2212i\pi 3,ei\pi 3$) |

Γ_{4,1} | (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) |

Γ_{4,2} | (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) |

Γ_{4,3} | (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) |

. | Co1 . | Co2 . | Co3 . | Co4 . |
---|---|---|---|---|

Γ_{1} | (1, 1, 1) | (−1, −1, 1) | (−1, 1, −1) | (1, −1, −1) |

Γ_{2} | (1,$e\u2212i2\pi 3,ei2\pi 3$) | (−1,$ei\pi 3,ei2\pi 3$) | (−1,$e\u2212i2\pi 3,e\u2212i\pi 3$) | (1,$ei\pi 3,e\u2212i\pi 3$) |

Γ_{3} | (1,$ei2\pi 3,e\u2212i2\pi 3$) | (−1,$e\u2212i\pi 3,e\u2212i2\pi 3$) | (−1,$ei2\pi 3,ei\pi 3$) | (1,$e\u2212i\pi 3,ei\pi 3$) |

Γ_{4,1} | (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) |

Γ_{4,2} | (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) |

Γ_{4,3} | (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 $k\u2192s$ || [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 $k\u2192s$ || [111] allowed by the P2_{1}3 space group for Co group I. The two helical spin structures Γ_{2}(I) and Γ_{3}(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 Γ_{2}(II) and Γ_{3}(II) form helical structures as $\Gamma 2,1(II)+e\u2212i2\pi 3\Gamma 2,2(II)+e\u2212i4\pi 3\Gamma 2,3(II)$ and $\Gamma 3,1(II)+ei2\pi 3\Gamma 3,2(II)+ei4\pi 3\Gamma 3,3(II)$ 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: [$\Gamma 2(I)eiksa3$, $\Gamma 2,1(II)+e\u2212i2\pi 3\Gamma 2,2(II)+e\u2212i4\pi 3\Gamma 2,3(II)$], and [$\Gamma 3(I)eiksa3$, $\Gamma 3,1(II)+ei2\pi 3\Gamma 3,2(II)+ei4\pi 3\Gamma 3,3(II)$], for opposite chirality, respectively [Fig. 4(a)].

. | Co1 . |
---|---|

Γ_{2}(I) | (1,$e\u2212i2\pi 3,ei2\pi 3$) |

Γ_{3}(I) | (1,$ei2\pi 3,e\u2212i2\pi 3$) |

. | Co1 . |
---|---|

Γ_{2}(I) | (1,$e\u2212i2\pi 3,ei2\pi 3$) |

Γ_{3}(I) | (1,$ei2\pi 3,e\u2212i2\pi 3$) |

. | Co2 . | Co3 . | Co4 . |
---|---|---|---|

Γ_{2,1}(II) | (1,0,0) | (0,$e\u2212i2\pi 3$,0) | (0,0,$ei2\pi 3$) |

Γ_{2,2}(II) | (0,1,0) | (0,0,$e\u2212i2\pi 3$) | ($ei2\pi 3$,0,0) |

Γ_{2,3}(II) | (0,0,1) | ($e\u2212i2\pi 3$,0,0) | (0,$ei2\pi 3$,0) |

Γ_{3,1}(II) | (1,0,0) | (0,$ei2\pi 3$,0) | (0,0,$e\u2212i2\pi 3$) |

Γ_{3,2}(II) | (0,1,0) | (0,0,$ei2\pi 3$) | ($e\u2212i2\pi 3$,0,0) |

Γ_{3,3}(II) | (0,0,1) | ($ei2\pi 3$,0,0) | (0,$e\u2212i2\pi 3$,0) |

. | Co2 . | Co3 . | Co4 . |
---|---|---|---|

Γ_{2,1}(II) | (1,0,0) | (0,$e\u2212i2\pi 3$,0) | (0,0,$ei2\pi 3$) |

Γ_{2,2}(II) | (0,1,0) | (0,0,$e\u2212i2\pi 3$) | ($ei2\pi 3$,0,0) |

Γ_{2,3}(II) | (0,0,1) | ($e\u2212i2\pi 3$,0,0) | (0,$ei2\pi 3$,0) |

Γ_{3,1}(II) | (1,0,0) | (0,$ei2\pi 3$,0) | (0,0,$e\u2212i2\pi 3$) |

Γ_{3,2}(II) | (0,1,0) | (0,0,$ei2\pi 3$) | ($e\u2212i2\pi 3$,0,0) |

Γ_{3,3}(II) | (0,0,1) | ($ei2\pi 3$,0,0) | (0,$e\u2212i2\pi 3$,0) |

Another propagation direction that leads to helical spins is $k\u2192s$ || [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 $k\u2192s$ || [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: $[\Gamma 1,2(I)+i\Gamma 1,3(I)$, $\Gamma 1,2(II)+i\Gamma 1,3(II)]$, and $[\Gamma 1,2(I)\u2212i\Gamma 1,3(I)$, $\Gamma 1,2(II)\u2212i\Gamma 1,3(II)]$ for opposite chirality [Fig. 4(b)].

. | Co1 (Co3) . | Co2 (Co4) . |
---|---|---|

$\Gamma 1,2$ | (0,1,0) | (0, $e\u2212ika/2,$0) |

$\Gamma 1,3$ | (0,0,1) | (0,0, $e\u2212ika/2$) |

$\Gamma 2,2$ | (0,1,0) | (0, $\u2212e\u2212ika/2,$0) |

$\Gamma 2,3$ | (0,0,1) | (0,0, $\u2212e\u2212ika/2$) |

. | Co1 (Co3) . | Co2 (Co4) . |
---|---|---|

$\Gamma 1,2$ | (0,1,0) | (0, $e\u2212ika/2,$0) |

$\Gamma 1,3$ | (0,0,1) | (0,0, $e\u2212ika/2$) |

$\Gamma 2,2$ | (0,1,0) | (0, $\u2212e\u2212ika/2,$0) |

$\Gamma 2,3$ | (0,0,1) | (0,0, $\u2212e\u2212ika/2$) |

According to Eq. (5), the magnitude of the magnetic contribution depends on the vector structure factor $f\u2192s\u2261\u2211u\u2192\u2061q\u21920(k\u2192,u\u2192)f(k\u2192,u\u2192)eik\u2192\u22c5u\u2192$ but not on $k\u2192s$ 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 Γ_{4}(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 MnSi^{17} and Fe_{x}Co_{1−x}Ge,^{8} has a propagation vector $k\u2192s$ along the threefold rotation symmetry axis [111]. Helical propagation vectors along [111] and [100] have been reported in B20 FeGe^{34} 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

where $fn(k\u2192,r\u2192)$ 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 Γ_{4,1} 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 *μ*_{B}/Co, where the magnetic contribution is an average for T < 320 K.

### B. Surface magnetization probed by XMCD spectroscopy

XAS spectra near the Co *L*_{3,2} 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 $(\mu +)$ and antiparallel $(\mu \u2212)$ to the applied magnetic field. The x-ray magnetic circular dichroism (XMCD) signal is obtained from the difference between the corresponding blue $(\mu \u2212)$ and red $(\mu +)$ curves and is shown for different temperatures in Fig. 5(b). The integrals of both XAS and XMCD spectra, $r=\u222bL3+L2(\mu ++\mu \u2212)d\omega $, $q=\u222bL3+L2(\mu +\u2212\mu \u2212)d\omega $, and $p=\u222bL3(\mu +\u2212\mu \u2212)d\omega $, were used to quantify the orbital moment ($m0$) and spin moment ($ms$) using the sum rule by the following: $mo=\u22124q3rnh$ and $ms=\u22126p\u22124qrnh$, where *n*_{h} is the hole density per Co atom.^{23,24} Orbital and spin moments and average moment *m _{j}* =

*m*

_{o }+

*m*

_{s}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 0.31

*μ*

_{B}/Co [Fig. 5(d)], which agrees with the NPD values of (0.3 ± 0.1)

*μ*

_{B}/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.

## IV. DISCUSSION

The (0.3 ± 0.1)*μ*_{B}/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)*μ*_{B}/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 (0.18*μ*_{B}/Co)^{26} and differ substantially from magnetometry, which measures the bulk net magnetization (0.11*μ*_{B}/Co at 2 K and 0.07*μ*_{B}/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/cm^{3} 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 $k\u2192s$ is included in the factor $\u2211R\u2192\u2061ei(k\u2192s+k\u2192)\u22c5R\u2192$, which determines the diffraction angle. In other words, the helical period may be extracted from the positions of the satellite peaks $k\u2192\xb1k\u2192s$ in neutron diffraction. Unfortunately, broadening effects due to chemical disorder and multiple values of $|k\u2192\xb1k\u2192s|$ smear the satellite peaks, which prevent us from identifying satellite peaks in the powder neutron diffraction patterns.

## V. CONCLUSION

In conclusion, the results of NPD measurements on Co_{1.043}Si_{0.957} are consistent with the helical spin structure. The magnitude of the magnetic moment extracted from NPD (0.3 ± 0.1*μ*_{B}/Co) coincides with theoretical estimates (0.18*μ*_{B}/Co) and the surface magnetization retrieved from XMCD spectroscopy (0.18–0.31*μ*_{B}/Co). All these values are substantially larger than the magnetometry value (0.11*μ*_{B}/Co), and the data reflect the evolution of a coplanar helical spin structure into a noncoplanar conical spin structure with nonzero magnetization.

## SUPPLEMENTARY MATERIAL

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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflict of interest to disclose.

## DATA AVAILABILITY

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