Coherent spin rotation-induced zero thermal expansion in MnCoSi-based spiral magnets

Materials exhibiting zero thermal expansion (ZTE), namely, volume invariance during temperature change, can resist thermal shock and are highly desired in modern industries as high-precision components. However, pure ZTE materials are rare, especially those that are metallic. Here, we report the discovery of a pure metallic ZTE material: an orthorhombic Mn1-xNixCoSi spiral magnet. The introduction of Ni can efficiently enhance the ferromagnetic exchange interaction and construct the transition from a spiral magnetic state to a ferromagnetic-like state in MnCoSi-based alloys. Systematic in situ neutron powder diffraction revealed a new cycloidal spiral magnetic structure in bc plane at ground state which would transform to the helical spiral in the ab plane with increasing temperature. Combined with Lorentz transmission electron microscopy techniques, the cycloidal and helical spin order coherently rotated at varying periods along the c axis during the magnetic transition. This spin rotation drove the continuous movement of the coupled crystalline lattice and induced a large negative thermal expansion along the a axis, eventually leading to a wide-temperature ZTE effect. Our work not only introduces a new ZTE alloy but also presents a new mechanism by which to discover or design ZTE magnets.


Introduction
It is well known that the inherent anharmonicity of phonon vibrations triggers the volume expansion of most solids upon heating. However, when the crystalline lattice couples with ferroelectricity, magnetism and charge transfer, anomalous behaviour during large temperature fluctuations, namely, negative thermal expansion (NTE) behaviour, may be realized. [1][2][3][4] materials combined with positive thermal expansion (PTE) materials can reduce the overall coefficient of thermal expansion (CTE) and lead to overall zero thermal expansion (ZTE) composites, which are of great importance in industrial applications as structural components, electronic devices and high-precision instruments. 1,[5][6][7] Unfortunately, such ZTE composite materials have high internal stresses that can cause microcracking during thermal cycling, which significantly diminishes their mechanical performance and lifetime. This problem can be overcome if ZTE materials are single-phase ones with homogeneous internal structure, especially in metallic form. In addition to Invar alloys, 8 a small number of such single-phase ZTE alloys or compounds have been discovered so far, such as Mn1-xCoxB, 9 YbGaGe, 10 LaFe10.6Si2. 4,11 Ho2Fe16Cr, 12 ErFe10V1.6Mo0.4 13 and RFe2-based compounds. [14][15][16] The most found of these materials are magnetic alloys, for which the ZTE behaviour is intimately associated with spontaneous magnetic ordering, known as the magnetovolume effect (MVE). The MVE originating from spin-lattice coupling may weaken or even compensate for anharmonic lattice variations and lead to abnormal thermal expansion behaviour near the magnetic ordering temperature. 17,18 Here, we present a new class of ZTE materials, MnCoSibased metamagnetic alloys, consisting a homogeneous phase. The ternary equiatomic MnCoSi alloy crystallizes with an orthorhombic structure from the honeycomb hexagonal structure after 4 experiencing a martensitic transition at high temperature ( Fig. 1a and Fig. S1). The shortest Co-Si bonds yield wrinkled eight-membered rings, in which Mn-Mn zig-zag chains are embedded.
The interconnected Co-Si contacts along the [100] direction form the basic rigid skeleton. Due to the critical nearest Mn-Mn separation, MnCoSi alloy possesses a ground state of nonlinear antiferromagnetism (AFM) and a magnetic-field-induced magnetoelastic transition, 19,20 during which the large inverse magnetocaloric effect and giant magnetostrictive effect is realized. [21][22][23] In this study, we find that the helical magnetic structure of MnCoSi, as is widely recognized, would transit to a cycloidal spiral AFM at low temperature. Moreover, we report that when the MnCoSi system is tuned by minimal Ni introduction, an ultralow CTE over a wide temperature range can be achieved. The combination of X-ray diffraction (XRD), neutron powder diffraction (NPD) and Lorentz transmission electron microscopy (TEM) techniques reveals a new mechanism underlying ZTE: coherent spin rotation of the spiral magnetic structures.

Sample synthesis
Polycrystalline Mn1-xNixCoSi (x = 0, 0.010, 0.015, 0.017, 0.020 and 0.025) samples were prepared by arc melting the appropriate amounts of high-purity raw materials under a purified argon atmosphere three times. Then, the as-cast samples were sealed in evacuated quartz ampoules and annealed at 1123 K for 60 h before slowly cooling to room temperature over 72 h. The slow-cooling treatment guaranteed magnetic homogeneity. 24

Magnetization, XRD and NPD characterization
The magnetic properties were characterized by a superconducting quantum interference device (SQUID, Quantum Design MPMS XL7) with the reciprocating sample option. The temperature dependence of the powder XRD (Rigaku, Smartlab) was collected using a lowtemperature chamber. For each measurement at a specified temperature, the powder sample was maintained for 20 min to reach heat equilibrium. In situ variable-temperature NPD measurements (λ=1.622 Å) in the heating process were carried out on the Wombat beamline at the OPAL facility of the Australian Nuclear Science and Technology Organization (ANSTO).
Structural refinements of the XRD and NPD patterns were performed using the Rietveld refinement method, and the irreducible representation analysis of the magnetic structure was carried out using the BASIREPS programme, both implemented in the FULLPROF package. 25,26 Lorentz TEM measurement The thin plates for Lorentz TEM observations were prepared by focused ion beam. The temperature dependence of the magnetic domain structures was observed in a JEOL-dedicated 6 Lorentz TEM (JEOL2100F) equipped with liquid-nitrogen holders. To determine the in-plane spin distribution of the magnetic texture, three sets of images with under-, over-, and just (or zero) focal lengths were recorded by a charge coupled device (CCD) camera and then the highresolution in-plane magnetization distribution map was obtained using commercial software QPt on the basis of the transport-of-intensity equation (TIE) equation. The orientation of the inplane magnetization was depicted based on the color wheel. The crystalline orientation for the thin plate was checked by selected-area electron diffraction (SAED).

Results
According to the atomic occupancy rules in MnCoSi alloy, 23 when Ni atom nominally substitutes Mn atom, the Ni with more valence electrons preferably occupies Co site and then partial Co atom occupies Mn site. Thus, the occupation formula of the Mn1-xNixCoSi system should be written as (Mn1-xCox)(Co1-xNix)Si. The atomic occupation can be also confirmed by NPD refinement as shown in Fig. S6. Based on the atomic occupation, the lattice parameters and unit-cell volume of the Mn1-xNixCoSi system are obtained from the refinement of XRD patterns, as shown in Fig. 2. For further details of the refinement, see Fig. S2 and Table S1.
With decreasing temperature, the lattice parameters b and c typically decrease, while the lattice parameter a shows a NTE effect. Moreover, with the introduction of Ni, the expansion of the lattice parameter a upon cooling shifts to a low temperature and changes dramatically. As a result, the effect of the shrinkage of b and c on the unit-cell volume is compensated for; thus, ZTE behaviour is realized in Ni-containing samples, as presented in Fig. 2d. The reliability of anisotropic thermal expansion and ZTE can be also confirmed from NPD results (Fig. S7).
Notably, stoichiometric MnCoSi exhibits a linear PTE in the studied temperature range with a 7 slight inflection at approximately 230 K. When Ni substitutes minimal Mn, the samples, such as contents of x = 0.020 and 0.025 exhibit ultralow CTEs (αl = 6.9×10 -7 and 1.3×10 -7 K -1 , respectively) over a wide temperature range (10-190 K and 10-170 K, respectively) after experiencing a normal PTE and a weak NTE, respectively, which is about one order smaller than the Invar alloy of Fe65Ni35 (αl = 1.4×10 -6 K -1 ). The calculated CTEs and corresponding working temperature ranges are listed in Table S2 in the Supplementary Materials. Rietveld refinement of XRD patterns of the Mn1-xNixCoSi system. The linear CTEs of Mn1-xNixCoSi system were calculated by αl=1/3dV/(V0dT). The ZTE working temperature range is indicated in (d). 8 As mentioned before, the thermal expansion properties of magnetic materials can be affected by MVE. The temperature dependence of magnetization (M-T) curve during zero-field cooling (ZFC) and field cooling (FC) processes in Fig. 3a and Fig. S3 shows that the weak magnetization of nonlinear AFM increases slowly with increasing temperature; this behaviour is interrupted by the advent of paramagnetism (PM) in stoichiometric MnCoSi alloy. Hence, the long-range magnetic order cannot be maintained above T0 ~ 387 K, where T0 represents the order to disorder transition temperature. It is widely reported that the introduction of external elements can effectively tune the magnetic state of TMnX (T=transition metal, X=Si or Ge) alloys. 19,27 In this work, the M-T curves of the studied Mn1-xNixCoSi system indicate that minimal Ni additions strengthen the ferromagnetic (FM) interaction, leading to a rise in the hidden thermal-induced magnetic transition, which is similar to that of Fe-substituted MnCoSi alloys. 28 For the x = 0.020 sample, the AFM state smoothly transitions to the FM-like state, which is accompanied by a relatively large increase in magnetization. The magnetic transition temperature Tt, defined as the inflection point in the M-T curve, is presented in Table S2 in the Supplementary Materials and gradually decreases with increasing Ni content. Additionally, the establishment of FM coupling can also be examined by the magnetization behaviour. As shown by the room-temperature magnetization curves in Fig. 3b, the metastable nonlinear AFM can be easily destroyed by applying a magnetic field. A second-order and nonhysteretic metamagnetic transition displaying a sharp increase in magnetization is clearly seen in stoichiometric MnCoSi and tends to saturate at Bsat ~ 3.0 T. With increasing Ni content, Bsat decreases (Table S2) To further determine the evolution of the magnetic structure, the temperature-dependent NPD of the Mn1-xNixCoSi system was performed. As the NPD patterns of studied x = 0.000, 0.015 and 0.020 samples shown in Fig. 4 and Fig. S4, with decreasing the temperature, the additional peaks at the low diffraction angles corresponding to the magnetic satellite reflections gradually appears and then splits or merges, manifesting the possible AFM order. Specifically, the refinement of isothermal NPD data at selected temperatures are also shown. At 450 K, only the nuclear scattering of orthorhombic space-group is indexed due to the sample in the disorder paramagnetic state. When the sample enters into the spin ordered state below T0, the magnetic reflections can be indexed by the nonlinear magnetic structure. When the temperature further decreases to be lower than 190 K for sample x = 0.020, magnetic diffraction peaks of (101) -, (- Assisted by symmetry arguments, 29 the best fit model indicates that ordered and equal moments are detected on Mn atoms (~3 μB) or Co atoms (~0.6 μB) (Fig. S8) at 3 K, for which the cycloidal spiral magnetic arrangement lying in bc plane achieves an incommensurate propagation vector k = (0, 0, kc) for x = 0.020 sample shown in Fig. 5a-5b. This spiral magnetic structure is different from the helical magnetic structure in the literature (two NPD refinements are presented and discussed in Fig. S5), 20 which is also reported by O. Baumfeld before. 30 11 Specifically, although all the atoms occupy the same crystal site (Wyckoff position 4c (x, 1/4, z)), the wave vector group splits the magnetic Mn and Co positions into four magnetic spirals with identical k values: <Mn1, Mn3>, <Mn2, Mn4>, <Co1, Co3> and <Co2, Co4> and the magnetic spin in each cycloidal spiral rolls with a fixed angle along c axis. As the main carrier of the magnetic moment, two magnetic spirals of Mn atoms exhibit obvious phase differences, indicating different spin orientations in the x = 0.020 sample at ground state. When the temperature increases, the four groups of cycloidal AFM would transform to four groups of helical magnetic structures at about 190 K for x = 0.020 sample. As illustrated in Fig. 5d, the spin in each helix of the helical magnetic structure rotates in ab plane by a certain angle in going from layer to layer along the c axis. Therefore, the envelop of the projection of magnetic moments in bc plane is sinusoidally modulated shown in Fig. 5c. At 300 K, the two Mn helices almost rotate synchronously and show a small phase difference for x = 0.020 sample.   (Fig. 6a), indicating a decrease in the angle between the adjacent spins (θMn1-Mn3/θMn2-Mn4) in a cycloidal or helical magnetic chain and, correspondingly, an elongation of the magnetic spiral period. The temperature-dependent θMn1-Mn3 shows more obvious variation in Ni-containing samples, and the angle θMn1-Mn3 is further reduced (i.e., 23° at 300 K for the x = 0.020 sample). Moreover, the phase analysis, as shown in Fig. 6b, indicates that the angle between the nearest atoms of two Mn cycloidal or helical spirals (θMn1-Mn2) also decreases (i.e., 19° at 300 K for the x = 0.020 sample).
Consequently, the spins of all Mn atoms tend to continuously rotate towards the parallel arrangement of b axis, as shown in Fig. 6c-d; thus, a FM-like spin configuration is expected in Ni-containing samples. Mn magnetic helices of cell a×b×3c for x = 0.020 sample viewed from the c axes at 190 K, 250 K and 300 K, respectively. The angles between different Mn spins are defined and given.
In addition, Lorentz TEM is also widely employed to investigate the real-space imaging of spiral magnetic structures. 31,32 We then imaged the magnetic domain structures using Lorentz TEM under zero magnetic field between 140 K and 300 K in x = 0.020 sample, as shown in Fig.   7 (details are shown in Fig. S9). The studied thin specimen is near the [1-10] zone axis 14 orientation confirmed by the SAED in the inset. At 293 K, the uniform and nano-sized fine magnetic patterns with bright and dark contrast are repeatedly arranged perpendicular to the c axis. Based on the over-and under-focused Lorentz TEM images (Fig. S9), a TIE was adopted to characterize the spin textures of the magnetic patterns. The yellow and blue straight stripe pairs reflect the regions with opposite in-plane magnetic inductions, as indicated by the colour wheel. Together with line profile of the alternative contrast intensity, the nearly sinusoidal manner indicates the spin order is probably spiral or fan-like. It is worth mentioning that only in-plane component of moments is presented by the Lorentz TEM. Therefore, the real 3D magnetic structures of this thin MnCoSi specimen should be resolved systematically by in-situ Lorentz TEM in the future. The fine stripe-type magnetic domain can be observed during the studied temperature ranges of 140-293 K. With increasing the temperature, the width of the stripe increases. Additionally, spiral magnetic structure which contains a higher harmonic modulation of magnetic order clearly provides a pair of diffraction spots along the c axis close to (000) diffraction spots of SAED, as shown in the inset of Fig. 7b. By quantitatively analysis of these satellite spots, the magnetic spiral period can be calculated in Fig. 7b. When the temperature increases, the period gradually increases and accelerates at about 220 K, which is consistent with the NPD data and further validates the coherent rotation of the magnetic spins.

Discussion
The near-ZTE behaviour was mentioned or observed in MnCoSi-based alloys, 20,33 in which the origin of the effect was absent. The conventional mechanism underlying ZTE or NTE in magnetic alloys, such as La(Fe,Si)13 alloys and RFe2-based compounds, 11,[14][15][16]34,35 originates from either a magnetic disorder to order transition or a large change in the magnetic moment.
Here, to quantitatively uncover the contribution of magnetism to the thermal expansion behaviour of Mn1-xNixCoSi system, the spontaneous volume magnetostriction ωm of sample x = 0.020 is calculated by ωm = ωexp -ωnm, in which the ωexp is obtained from the experimental XRD results and ωnm is fitted from the nonmagnetic phase based on the Debye-Grüneisen model, 36,37 as shown in Fig. 8a. Combined with the M-T curve (Fig. 8b), the sample displays a linear PTE behaviour in the PM region. When the sample starts to enter the ordered FM-like state at below 410 K, the experimental ωexp slightly deviates from the fitted ωnm due to the MVE.
With further decreasing the temperature, a smooth magnetic transition from FM-like state to the spiral AFM state is observed, during which the thermal expansion behaviour is significantly affected. Specifically, the negative role of magnetic ωm gently exceeds or completely 16 counteracts the contribution from the lattice variation, which results in a weak NTE and a widetemperature ZTE. Moreover, it is interesting to find the ωm and the angles between Mn spins exhibit a similar temperature dependent behaviour (Fig. 8c), which indicates an intimate relationship between the anomalous thermal expansion and spin rotation of the helical magnetic structure in MnCoSi-based alloys.
Notably, thermally-induced coherent spin rotation is also observed in stoichiometric MnCoSi. During heating, the angles θMn1-Mn3 and θMn1-Mn2 decrease from 70° and 61° to 38° and 32°, respectively. Then, the rotation is forced to cease by the disordered PM state. Therefore, this weak and partial spin rotation brings about only a small fluctuation in the thermal expansion behaviour of stoichiometric MnCoSi (shown in Fig. 2d), for which a PTE is observed over the entire temperature range. The unusual magnetic tricritical behaviour of MnCoSi results in flexible tunability of the magnetic state. 20 It is widely reported that the magnetic state of this Mn-based orthorhombic alloy with space group Pnma is extremely sensitive to the Mn-Mn distance d1. 19,38 In this work, the introduction of Ni atoms can produce "chemical pressure" on the crystal lattice and change d1. As shown in Fig. 8d, d1 increases with increasing Ni content; correspondingly, nonlinear AFM tends to be FM. Therefore, the enhanced FM interaction leads to an obvious transition from the cycloidal or helical AFM state to a FM-like state, during which the magnetic spins further coherently rotate to b axis. Due to the robust magnetoelastic coupling, 20,33 this strong rotation gives rise to the giant spontaneous magnetostriction, particularly, the sharp contraction of a axis and leads to the emergence of the anomalous NTE or ZTE behaviour in the homogeneous phase. In addition, our results suggest that an appropriate internal or external stimulus, such as doping with elements, introducing vacancies or applying 17 hydrostatic pressure or a magnetic field can strengthen the FM interaction and establish this spiral AFM-FM-type transition and then ZTE would be induced in MnCoSi-based alloys.
It is worth mentioning that the change of magnetic structures from cycloidal to helical spiral cannot be obviously revealed in the evolution of lattice parameters and magnetic properties of polycrystalline samples. While due to the intimate relations between magnetic state and Mn-Mn separation, 20 the evident step change of d1 (Fig. 8d) can be observed which corresponds to the change of spiral AFM structure in MnCoSi-based alloy. Additionally, owing to the distinct easy-magnetization plane of cycloidal (bc plane) and helical (ab plane) spirals, the magnetic structures can be effectively distinguished by magnetic characterization of MnCoSi single crystal. Mn-Mn distance d1 obtained from Rietveld refinement of NPD patterns. d1 is indicated in the inset and the magnetic structure variation temperature is highlighted.

Conclusions
In summary, a wide-temperature ZTE effect and a new cycloidal spiral AFM structure were found in orthorhombic Mn1-xNixCoSi alloys. Systematic magnetic measurements and in situ XRD, NPD and Lorentz TEM characterization indicated that the introduction of Ni can enhance the FM interaction and induce a transition from the spiral AFM state to a FM-like state.
During this transition, the spin lying in the bc or ab plane rotates uniformly and leads to drastic changes in the lattice parameters due to magnetoelastic coupling, which results in ZTE behaviour. Moreover, this new mechanism sheds light on magnetic materials that possess this spiral AFM-FM-type transition, and ZTE or NTE materials may be discovered or designed.