Thermodynamic phase transitions of nematic order in magnetic helices

Abstract

A nematic phase lacks translation order but has orientational order. Nematic phases have been discovered in a variety of systems, including liquid crystals, correlated materials, and superconductors. Here, we report on a magnetic nematic phase, where the basis components are composed of magnetic helices. We directly probed the order parameters associated with the magnetic helices using resonant soft x-ray scattering and find two distinct nematic phases with complex spatiotemporal signatures. Using x-ray correlation spectroscopy, we find that near the phase boundary between the two nematic phases, fluctuations coexist on multiple disparate timescales. Our micromagnetic simulations and density functional theory calculations show that the fluctuations occur concomitantly with a reorientation of the magnetic helices, indicating spontaneous symmetry breaking and the emergence of additional degrees of freedom. Our results provide a framework for characterizing exotic phases that can be extended to a broad class of physical systems.

The helimagnetic phase in amorphous FeGe shows surprising similarities to the nematic phase commonly found in liquid crystals.

INTRODUCTION

Nematic order describes a system that has orientational order, but lacks translational order (1), while also being defined by the nature of its phase transitions. For instance, in a layered nematic system, the transition is mediated by unbinding of topological defect pairs between layers (2). The origin of the nematic phase lies in the research field of liquid crystals. However, the classes of materials and the phenomena associated with nematic phases have turned out to be quite general, spanning bacterial suspensions (3), synthetic active matter (4), and polymer networks (5). Nematicity can be understood by its order parameter(s). In liquid crystals, the order parameter is based on the molecular alignment known as the director, which has no head-tail distinction. In the nematic phase, the molecules have a defined orientation but have no translational symmetry (Fig. 1A). Similar to the liquid crystals, in a spin nematic, the spin symmetry of the system is broken without the breaking of time reversal symmetry (Fig. 1B) (6–9). Nematicity has also been discovered in unconventional superconductors, where the order parameter is described by the anisotropic electronic wavefunction (Fig. 1C) as revealed by its electrical transport properties (10, 11). Electronic nematicity in quantum materials is considered to be a previously unknown state of matter due to its lowering of rotational symmetry and correlations of itinerant electrons that lead to anisotropic magnetoresistance (12). The nematic state we propose has an order parameter where the director is the helical propagation vector (Fig. 1D) with orientational anisotropy and, as we show, exhibits spontaneous fluctuations.

Fig. 1. Nematic phases in condensed matter systems.

(A) The prototypical nematic phase found in liquid crystals (LC) have a preferential orientation but are translationally isotropic. (B) Spin nematics break symmetry in spin space while preserving time-reversal symmetry and are typically classified by a nematic tensor order parameter. (C) In superconductivity (SC), there is spontaneous breaking of the rotational symmetry of the superconducting gap. (D) The nematic phase with the basis of magnetic helices presented in this work preserves translational symmetry while breaking rotational symmetry in coordinate space. (E) Experimental geometry used for resonant soft x-ray scattering measurements. Spin and superconducting nematic diagrams adapted from (53) and (54), respectively.

Magnetic spin textures offer a fertile playground for studying topological phases and their transitions, offering a diverse selection of different phases, experimentally measurable length scales, and a coupling to external stimuli. A canonical example is the B20 crystalline polymorph of FeGe, whose ground state is helical with an ordering temperature T_C = 280 K with a pitch = ∼70 nm. B20 FeGe has generated considerable interest over the past decade due to the discovery that it can host a skyrmion crystal close to room temperature (13, 14), stabilized by the inversion symmetry breaking of the atomic unit cell. It is also known from small-angle neutron scattering that the helical propagation axis reorients from <100> to <111> at ≈ 210 K on cooling (15). The dynamics of helimagnetic phases in FeGe has been shown to exist far below T_C due to dislocation and associated collective motions (16). More recently, however, it has been found that a well-defined crystal structure is not a requirement to host helimagnetic and skyrmionic spin textures (17). This opens the possibility to tailor different phases by controlling the atomic composition and measure phase transitions with reduced constraints to the underlying crystalline environment (18, 19).

For this study, we use a thin film of amorphous FeGe with the stoichiometry Fe₅₁Ge₄₉ (a-Fe₅₁Ge₄₉). To thoroughly understand the thermodynamic transitions of the nematic phase in a magnetic system, we performed static and dynamic resonant soft x-ray scattering (RSXS) and x-ray photon correlation spectroscopy (XPCS) measurements. Additionally, we combined micromagnetic simulations and density functional theory (DFT) calculations with our experimental findings to further investigate the degrees of freedom of the order parameters. We found that a helical magnetic texture stabilizes in a-Fe₅₁Ge₄₉, which can be described as chiral nematic (N*), with two distinct thermodynamic phases exhibiting characteristic spatiotemporal correlations. We identify the two phases by their distinct correlation length behavior with a low-temperature phase, $N_{{\mathrm{\xi }}_{>}}^{*}$ , possessing a larger correlation length and a high-temperature phase, $N_{{\mathrm{\xi }}_{<}}^{*}$ , possessing a shorter correlation length. Just before the $N_{{\mathrm{\xi }}_{<}}^{*}$ phase, we found that the system becomes unstable (shown by shaded region in Fig. 2A).

Fig. 2. Static scattering of a-Fe₅₁Ge₄₉.

(A) Magnetization versus temperature with an applied in-plane field of 1000 Oe. The Curie temperature (T_C) was found to be 298 K from analysis of M(H,T) and χ(T) up to 400 K. The $N_{{\mathrm{\xi }}_{>}}^{*}$ to $N_{{\mathrm{\xi }}_{<}}^{*}$ transition temperature (T^∗) is ∼151.5 K, determined by the fit in (F). (B and C) Quantification of the static scattering patterns. Exemplary averages are between the temperature range of 125 K (top) and 162.5 K (bottom). The white arrow in the insets shows the averaging directions. Solid lines correspond to (B) Lorentzian and (C) pseudo-Voigt fits to the data for the radial-q and azimuthal-ϕ averages, respectively. (D) Average helix periodicity extracted from the peak position in (B). (E) The inverse correlation length, κ, extracted from the full width at half maximum (FWHM) in (B) showing the translational order parameter and a crossover from power law (blue line) to exponential (red line) functional form. (F) FWHM, δϕ, of the azimuthal distribution in (C) showing the orientational order parameter. Solid line in (F) shows a fit to Eq. 1.

We studied second and sub-nanosecond timescale temporal properties and showed simultaneous existence of fluctuations at both scales. We found that the onset of the fluctuations coincides with a plausible reorientation of the helical propagation direction from being purely in-plane to canted out-of-plane. The degree of freedom change from planar confinement for the helix propagation vector to a three-dimensional (3D) orientation results in the availability of an additional degree of freedom: the cant angle of the helix propagation vector. With Fig. 1D as a reference, the cant angle of the helix propagation vector would represent a tilting of the magnetic directors to be out of the page.

Within this article, we refer to the phases using standard nomenclature used to describe liquid crystals, which include rod-like directors describing the molecular orientation and have broadly been extended to other physical systems. We would like to point out, however, that the magnetic helices do not preserve inversion symmetry. Despite this distinction, all elastic properties are identical to more traditional nematics (20). Additionally, an asterisk (*) is used to emphasize that the magnetic helix is a chiral entity.

RESULTS

The temperature dependence of the magnetization for a-Fe₅₁Ge₄₉ is shown in Fig. 2A, where the highlighted regions correspond to the temperature range measured by RSXS in this study, $T$* marks the $N_{{\mathrm{\xi }}_{>}}^{*}$-$N_{{\mathrm{\xi }}_{<}}^{*}$ transition temperature, and T_C is the Curie temperature. In Fig. 2 (B and C), we show temperature dependence of the radial and azimuthal averages from the recorded scattering patterns (insets) and corresponding fits to Lorentzian and pseudo-Voigt functions, respectively (solid lines are fits). The average helical periodicity of the system shows an interesting temperature dependence, as shown in Fig. 2D. At lower temperatures, the periodicity is stable with a value around 135 nm. When the temperature is increased above ∼144 K, there is a turning point, and the periodicity decreases more rapidly. This decrease in periodicity is linear with increasing temperature until the signal is no longer observed at T ∼ 163 K. It is important to note that this disappearance of the x-ray scattering is ∼100 K below T_C, where there is an appreciable in-plane net magnetization, as seen in Fig. 2A. This result indicates that for $T$* ≤ T ≤ T_C, either the magnetic helical texture orients itself in such a way that the periodic structure is not observable in transmission scattering geometry or regions between the magnetic helices are weakly ferromagnetic.

To quantify the positional order of the system, the inverse correlation length, κ, was extracted (Fig. 2E) from the full width at half maximum (FWHM) of the radial averages. Below the periodicity inflection point, κ grows algebraically (i.e., power law) with an initial correlation length of ∼500 nm. At the inflection point, there is a crossover from an algebraic scaling to an exponential. Note that the crossover takes place within the shaded region close to the phase boundary where the phase instability exists. The different functional scaling with temperature of the correlation length within different temperature windows indicates a multi-step order-disorder transition.

From the azimuthal averages (Fig. 2C), we obtain the orientational order by determining the angular spread δϕ, or FWHM of the distribution shown in Fig. 2F. In contrast to the translational order, δϕ shows a slow increase until ∼150 K. At this point, δϕ sharply increases and then plateaus around 155 K. The solid line in Fig. 2F is a fit to the following expression

$$\mathrm{\delta }\mathrm{\varphi }\propto e^{-T_{\mathrm{R}}^{-1/2}}$$

where T_R ∝ T − $T$* . This functional scaling of the broadening of the azimuthal x-ray scattering peak is in remarkable agreement with Ostlund and Halperin’s prediction near a nematic phase boundary (20), suggesting that the $N_{{\mathrm{\xi }}_{<}}^{*}$ phase maintains nematic character. The fit to Eq. 1 evaluates our transition temperature, $T$* = 151.5 K. We note here once again the important distinction that we are relating the distribution of magnetic spin helices to that of the directrix in a conventional nematic. The distinct changes in the azimuthal correlation, with translation correlation being very short, confirm that the system has entered a different phase, $N_{{\mathrm{\xi }}_{<}}^{*}$ , where the orientational order is destroyed slightly, as evidenced by the peak broadening, but is still preserved, since the twofold symmetric scattering pattern indicates that an anisotropy persists.

In addition to these static scattering signatures of the phase transition, unique temporal correlations are also present upon approaching the $N_{{\mathrm{\xi }}_{<}}^{*}$ phase, as evidenced by our time dynamic studies. Using the technique of x-ray photon correlation spectroscopy (XPCS), we observe a change from a small spatial volume of fluctuations at low temperatures to an increasing volume of fluctuations at an overall slower timescale as the $N_{{\mathrm{\xi }}_{<}}^{*}$ phase is approached from below. XPCS is a coherent x-ray scattering–based technique to study spontaneous fluctuations at the nanoscale (21, 22). Coherent x-ray scattering gives rise to speckles due to the interference of scattered wavefronts that are randomly phase shifted by the morphology of the sample. The speckle patterns obtained are the exact instantaneous Fourier representation of the sample inhomogeneities. For a static sample, the speckle pattern remains the same, and hence, any two speckle pattern collected at two different times are fully correlated with each other. For a nonstatic system, the speckle pattern changes, and hence, the correlation decays over the correlation time. By measuring the decorrelation, we obtain the intermediate scattering function S(q, t). In XPCS, a time series of images is collected at a particular temperature on the scattering peak associated with the length scale of the helical period. This time series of scattering patterns is then processed by calculating the autocorrelation at a particular temperature, producing a g₂ curve that encapsulates the dynamics present over the timescale measured. By performing XPCS as a function of several temperatures near T_C, detailed information about the temporal behavior is obtained.

A typical speckle pattern is shown in the inset of Fig. 3A, and solid lines show experimental g₂ autocorrelation values. Note that all of the g₂ data are preprocessed by subtracting unity and dividing by the first g₂ value. In our g₂ autocorrelation calculation, a region containing the entire speckle pattern of a magnetic peak is chosen so that the dynamics measured are a statistical ensemble of the fluctuations present at the length scale of the helical periodicities. It is clear from our g₂ data that there is not a single characteristic timescale representative of the system. Each temperature shows dynamics at a faster timescale with decorrelations on the order of seconds followed by additional dynamics on a much slower timescale.

Fig. 3. Dynamic correlations of a-Fe₅₁Ge₄₉.

(A) X-ray photon correlation spectroscopy (XPCS) of a coherent scattering pattern (inset) showing complex dynamics with multiple relevant timescales. The solid lines are calculated autocorrelation values, and the dashed lines are fits to Eq. 2. The characteristic relaxation time of the exponential term (B), the stretching exponent (C), and the constant offset (D) extracted from (A) as a function of temperature from 100 to 149 K measured on heating after cooling in zero field. The blue and red lines in (B) correspond to linear and exponential trends, respectively, and are guides to the eye.

These multi-timescale dynamics exclude the possibility of analysis by the common treatment of fitting a single stretched exponential function of Kohlrausch-Williams-Watts (KWW) form (22, 23). In addition to this complicated temporal structure, the timescale of the slower dynamics is prohibitively long to adequately sample experimentally and fit the data to a sum of multiple exponential functions. Instead, we fit our data with a single stretched exponential function modified by a linear offset, given by the following equation

$$g_2\propto e^{-{(t/\mathrm{\tau })}^{\mathrm{\gamma }}}+\mathit{mt}+g_2(\infty )$$

Here, τ is the relaxation time, γ is the stretching exponent, and g₂(∞) is a constant. This modified exponential includes only a single additional variable, m, with linear time dependence, which provides a simple fit to allow long timescale fluctuations and enables the possibility of extracting the fast timescale dynamics. Fits to Eq. 2 are shown by the dashed lines in Fig. 3A. Extracted fit parameters are given in Fig. 3 (B to D). Fitted values for m are provided in the Supplementary Materials for completeness but are presently used to isolate the fast timescale dynamics. The relaxation time τ describes the characteristic timescale of the fast fluctuations, and the stretching exponent represents the type of dynamics present. g₂(∞) is a measure of the spatial volume of fluctuations over the measurement time due to the g₂ autocorrelation function including the entire magnetic scattering peak and therefore representing a statistical ensemble of all dynamics present at the length scale associated with the magnetic helices. Therefore, fluctuations exist in the experimentally probed timescale but can be combined with other static scattering present and only show up as a small change in the overall decorrelation. Our analysis indicates that for T ∼ 144 K, all the extracted parameters show a change in slope, indicating that this temperature is where the ground state becomes unstable. In the low-temperature region (T < 144 K), the characteristic relaxation time exhibits a relatively fast relaxation rate that is on the order of seconds. This coincides with a superdiffusive stretching exponent (γ > 1), typical for systems with jammed dynamics or heterogeneous local rearrangements (23–25), and a g₂(∞) value close to one. This would indicate that the system is mostly static with only a small portion of the sample fluctuating, consistent with reported entropically driven small movements of topological defects in the helical phase of crystalline B20 FeGe and its dynamics well below the helical ordering temperature on the timescale of seconds to minutes (16).

As the temperature is increased above 144 K, which is the onset of the instability region, the characteristic faster fluctuations rapidly slow down. The stretching exponent goes from faster than exponential (superdiffusive, γ > 1) to slower than exponential (subdiffusive, γ < 1), and a sharp decrease in g₂(∞) is observed. This result, perhaps unintuitive, means that initially a small amount of the sample fluctuates with a single timescale, which then transitions to having a broader distribution of fluctuation timescales but on average has a slower timescale. With increasing temperature, our signal is increasingly dominated by collective slower motion that covers a larger sample volume. Thus, although faster fluctuations are still present, most of the fluctuations being measured are slower and that overshadows the faster fluctuations in the g₂ calculation. This is further bolstered by our nanosecond fluctuation measurements to be discussed below.

To investigate if faster timescale fluctuations are present, we performed a two-pulse–based x-ray correlation experiment (26–28) at the Linearly Coherent Light Source (LCLS), in which the speckle visibility of two integrated images is calculated on an individual photon level (29). In a two-pulse XPCS experiment, the X-ray Free Electron Laser (XFEL) at LCLS produces two time-delayed pulses, with each pulse having its own speckle pattern. The time delays are in the nanosecond time regime. The sum of the two speckle patterns is collected. For the static case, the speckle patterns are on top of each other and show a maximum contrast. If fluctuations are present, then the speckles move and the contrast decreases. A measure of the contrast therefore gives a measure of the fluctuations in the system. A statistical ensemble of sparse detector images (Fig. 4A) is converted to discrete photon maps (Fig. 4B) to fit the probability distribution of k photon-arrival events that are used to calculate the contrast (Fig. 4C). The speckle contrast extracted from a two-pulse measurement is normalized by a single-pulse measurement such that a contrast value close to one is a static measurement. Contrast deviations below one therefore represent a lower visibility and measure fluctuations on the timescale defined by the delay between the two pulses. Note that as $T$* is approached, the scattering intensity is greatly reduced, as shown in Fig. 4D by the average number of photons in a single detector image, ⟨k⟩. Insets are the integration of 300 detector images at 142 K, showing well-developed speckle (left), and at 155 K, showing loss of scattering intensity (right).

Fig. 4. Sub-nanosecond fluctuations near the phase transition at T^∗ = 151.5 K.

The contrast value was extracted from a statistical ensemble of sparse detector images (A) converted to discrete photon maps (B) to measure the probability distribution of k photon-arrival events given the average intensity, $\overline{k}$ , of the photon map (C) at a particular two-pulse delay time. As the phase transition occurs, the order decreases, resulting in a reduction in scattering intensity (D). Insets are the accumulation of 300 sparse detector images showing speckle (142 K) and Poissonian noise (155 K). (E) Just below (blue) and above (orange) the onset of the region of instability, decays in contrast show the existence of nanosecond dynamics. (F) While measuring with a fixed delay time of 700 ps, the enhancement of magnetic fluctuations is observed just before T* is reached. The solid line is a guide to the eye. Because of the decrease in scattering intensity, the contrast at T ≥ 150 K shows a contrast value returning back to one. Potential dynamics as measured with increased scattering intensity given by dashed line.

Error bars presented in Fig. 4 (E and F) represent the standard error calculated from the statistical fits and are not related to the signal-to-noise ratio associated with the scattered intensity resulting from speckle. Just below and above the temperature where the slow timescale fluctuations set in, the contrast was measured between 700 ps and 28 ns (Fig. 4E). Despite the sparse number of delay times measured, it is evident that nanosecond fluctuations are present and a decay in the measured contrast is observed. More interestingly, a measurement at a single timescale of 700 ps as a function of increasing temperature (Fig. 4F) through the region of instability shows an enhancement of fluctuations just before the ${\mathrm{N}}_{{\mathrm{\xi }}_{<}}^{*}$ phase is reached. Increasing the temperature even further through $T$* brings the contrast back to a value close to one, a result of the decrease in scattering intensity and minimal deviation compared to a measurement taken with a single pulse incident. It is likely that faster fluctuations are present, which we were not able to capture. A possible decay curve obtained by fitting the data points with an exponential for the fast fluctuation is shown as a dashed line in Fig. 4F as $T$* is approached.

To complement our experimental observations, we calculate, as described in Materials and Methods, the magneto-structural anisotropy energy (MSAE) surfaces of three amorphous FeGe structures and report them in Fig. 5 (A to C). This calculation is the amorphous analog of magneto-crystalline anisotropy energy in crystalline systems and is vital in determining the orientation of magnetic helices (30). MSAE is found to reach as high as 9.42 meV (equivalent to 109 K), and the average highest MSAE for all three structures is 6.92 meV (equivalent to 80 K). This indicates that thermally induced switching of the magnetization axis could occur beyond 80 K, which is in line with our experimental observation of 144 K, where the periodicity has an inflection point and the fluctuations set in. We expect this calculation to generally describe the range of relevant structures and interactions in our system. However, to fully capture the complexity of our amorphous system, a more comprehensive study would be required.

Fig. 5. Computational support of helical reorientation.

(A to C) MSAE surfaces are plotted for three amorphous structures prepared as described in Materials and Methods. (D) Micromagnetic simulation showing the low-temperature phase where the helical propagation direction (Q-vector) is purely in-plane. (E) Simulation in the same region after the reorientation begins, showing a decrease in periodicity and an inhomogeneous rotation of the Q-vector to be both in-plane (solid box) and canted out-of-plane (dashed box). Rendered regions in (D) and (E) represent a volume of ∼500 nm by 1 μm by 80 nm.

Thermal reorientation of the helical propagation direction is reported to exist in the B20 crystalline form of FeGe, where the helix switches from the <111> orientation to <100> at 279 K (15). In contrast, we believe that the reorientation in our system occurs over a broad temperature range and the helical propagation direction (Q-vector) is not a coherent reorientation, as evidenced by our micromagnetic simulations (Fig. 5, D and E) discussed below.

To simulate the effect of temperature on the magnetic structure using micromagnetic simulations, we varied the ratio of the exchange stiffness to the Dzyaloshinskii-Moriya interaction (DMI) and matched the periodicity of the relaxed state to experimental scattering patterns (see the Supplementary Materials). It is important to note that raising temperature alone in the simulations was not sufficient to reproduce the periodicity change observed in our experimental data. In the simulations, when exchange stiffness is reduced while keeping DMI constant (equivalent to raising temperature), regions with in-plane (solid box in Fig. 5E) and others with canted out-of-plane (dashed box in Fig. 5E) Q-vectors are observed, indicating an inhomogeneous rotation of the Q-vector. This inhomogeneity is a manifestation of the topological constraint of the system and results in the nucleation of defects in the 80-nm thickness (out-of-plane direction) of the sample. An example of such a defect is shown in Fig. 5E between the two highlighted regions with different Q-vectors. This reorientation would also reduce the out-of-plane magnetic contrast when averaged along the film thickness despite retaining a helical spin structure, as observed in the scattering data.

DISCUSSION

Our study has identified a magnetic system with chiral nematic ( $N$ *) character, where the role of the director is replaced by magnetic spin helices. The thermodynamic order-disorder transition of $N$* can be distinguished into two distinct phases, with a low-temperature $N_{{\mathrm{\xi }}_{>}}^{*}$ phase continuously transforming into a high-temperature $N_{{\mathrm{\xi }}_{<}}^{*}$ phase with distinct spatiotemporal correlations. The $N_{{\mathrm{\xi }}_{>}}^{*}$ phase is characterized by a well-ordered helical orientation and pitch, with dynamics consisting of only a small spatial volume of the sample. Despite having a well-ordered Q-vector, this phase should be distinguished from a smectic phase, which gives a resolution-limited diffraction peak that is indicative of a long-range 1D order. The radial broadness of the scattering peak is well above the resolution limit of the measurement, meaning that there is an appreciable number of defects present even at the lowest temperatures measured.

When the temperature is raised to just below the transition temperature T*, there is a turning point in the helical periodicity, a crossover in the functional scaling of the inverse correlation length, and a striking change in the dynamics that are present. This marks a region of instability between the phases (identified by the hatched region in Figs. 2 to 4). As illuminated by the micromagnetic simulations and DFT calculations, there is a crossover in the dimensionality of the system. At low temperatures, the Q-vector direction lies within a plane and periodicity is rather well defined as evidenced by the relatively sharp Lorentzian profile of the radial scattering peaks. At higher temperatures, we see broadening of the peaks, which means that there is additional freedom of the helix to change its pitch and direction of propagation. In addition, the Q-vector loses the planar orientational confinement. As shown by our simulation induced by topological detects along the depth of the sample, the helices can orient in 3D space. These changes show that there is spontaneous symmetry breaking and a crossover in the degrees of freedom is observed, allowing for the rich dynamics measured.

Deep in the $N_{{\mathrm{\xi }}_{>}}^{*}$ phase, the topological defects introduce several local minima in the energy landscape. There are only a few configurations that are accessible via thermal excitations on an experimentally feasible timescale. The dynamics that we measured are likely the motion of individual topological defects, such as dislocations that have a high energy density that are not in equilibrium even at the lowest temperatures (16, 31); the motion is heavily constrained. Superposed on these localized dynamics is the equilibrium global motion of the helix. These multi-time and space scale events continue and merge toward a coherent event as the $N_{{\mathrm{\xi }}_{<}}^{*}$ phase is approached.

It is worth noting that qualitative descriptions of noncollinear spin textures in connection to liquid crystals (e.g., smectic and nematic) have been previously reported (32). The present work not only provides a theoretical framework but also contains key distinctions and interpretations that are unique and are worth emphasizing. The nematic phase transitions discussed presently are thermally induced and do not include any transitions that result from an externally applied magnetic field. This means that there is no net magnetization in the helical directors, as would be the case in a conical state. Reorientation of the helical Q-vectors has been discussed in both field-induced conical (9, 33, 34) and zero applied magnetic field helical phases in terms of temperature-dependent anisotropic exchange interaction (AEI) and magnetocrystalline energy but is always confined to crystallographic orientations. Investigation of a helical reorientation transition in an amorphous material has not been reported previously. As discussed above, we understand the crossover in correlation length scaling and onset of dynamics around T* to be a result of additional degrees of freedom that become available as the planar confinement of the helical Q-vectors is lost and transitions between two nematic phases. The experimental demonstration that strict crystalline symmetry constraints are not a requirement for obtaining nontrivial spin textures is not only interesting from a theoretical perspective and motivates the need to revisit theoretical models of topological spin textures, but it will also be important for applications. This amorphous helimagnetic texture is highly attractive for incorporation into spintronic technologies without relying on high-quality crystals and exemplifies a gap in the scientific understanding of noncrystalline systems.

There are interesting questions that arise on the role of shape anisotropy on nematicity. We observe that just before the scattering signal is lost, the helical periodicity coincides with both the correlation length and the thickness of the film (80 nm), indicating that the canting is limited by the film thickness. Additional experiments where the film thickness is varied or interaction strengths are tailored and tuned by varying the Fe concentration could reveal previously undiscovered critical scaling and potentially discover nontrivial phases. (19). This could enable measurements that couple to external stimuli, such as quantum oscillations in an applied magnetic field, of critical points near smectic, nematic, and isotropic phase boundaries in a Fermionic liquid crystal–type system.

The results presented in this article can find relevance to nematic phases of many other systems. For example, it has been long predicted that a chiral nematic phase can disorder by breaking apart into groups of size dictated by the correlation length of dislocation defects (2). It is worthwhile to draw parallels between a helical nematic that we describe and a spin nematic. For example, in a spin nematic, frustration and transition from a uniaxial to biaxial exchange interaction are two important constituents (8). In case of amorphous FeGe sample, randomized DMI and dimensionality crossover can play similar roles. It is conceivable that our discovered exotic nematic phase with a magnetic helix basis may have interesting optical and transport properties with potential applications in microelectronics and spintronics. This work helps understand the thermal stability and impact of fluctuations for confined helical systems, such as those proposed in recent spintronic applications (35, 36).

Identification and investigation of nematicity in quantum materials is challenging. In particular, dynamical measurements that characterize the fluctuations are essential in understanding phase transitions (37). For example, in the case of superconductivity, the identification of the nematic phase is largely reliant on electron transport. While transport measurements are sensitive probe, explicit determination of spatiotemporal distributions and correlations of the order parameter(s) are inaccessible. This warrants the need for nematic systems that can be measured with coherent x-ray scattering since those distributions become available and can be combined with other spectroscopic measurements to provide multimodal characterization (38). In addition to this sensitivity, with the increasing coherent flux in the upcoming diffraction-limited light sources, coherent x-ray–based measurements are becoming more precise. This precision allows for further probing into the nematic and other exotic nontrivial phases and accessibility to faster timescales, which will provide deep insight into the interplay of fluctuation and phases.

MATERIALS AND METHODS

Films of a-Fe₅₁Ge₄₉ were grown on Si₃N₄ membranes by DC magnetron co-sputtering as previously reported (19). Static and dynamic RSXS measurements were performed at Cosmic-Scattering beamline 7.0.1.1 at the Advanced Light Source. A transmission scattering geometry was used wherein a linearly horizontal polarized x-ray beam tuned to the Fe L₃ edge was incident on the sample. A schematic of the experimental geometry is shown in the Supplementary Materials. This measurement is strictly sensitive to the out-of-plane component of the magnetization. The scattered signal was collected by a Lawrence Berkeley National Laboratory fast charge-coupled device (LBNL-FCCD) camera placed 284 mm downstream of the sample (39). For the static measurements, the sample was first cooled to 100 K, well below the helical ordering temperature, and subsequently heated until the scattering signal disappeared. For dynamic measurements, we used x-ray photon correlation spectroscopy (XPCS). The exposure time was fixed at 200 ms for all measurements with a CCD readout time of 53 ms. A 7-μm pinhole was used to define a coherent incident beam and placed upstream of the sample. To probe faster timescale fluctuations, we performed a two-pulse x-ray correlation (27) experiment at the ChemRIXS endstation (40) at the LCLS at Stanford Linear Accelerator Lab (SLAC). A transmission geometry was used with linearly horizontal polarized x-rays tuned to the Fe L₃ edge, and the pulse separation was varied between 700 ps and 28 ns. The speckle contrast was extracted from a statistical ensemble of sparse CCD images with a fixed pulse separation. Preliminary filtering of the data ensured that the two pulses were within a 20% tolerance and all contrast measurements were normalized by a single pulse incident on the sample to eliminate the possibility of measuring dynamics that are not inherent to the sample. Sample dynamics were then extracted by tracking the speckle contrast change at different temperatures in the transition. Micromagnetic simulations were performed using MuMax³ (41) with a volume of 512 by 512 by 20 cells and a cell size of 4 nm with in-plane periodic boundary conditions. To model the magnetic structure at different temperatures, a starting state was obtained by initializing with a random magnetization and subsequently minimizing the global energy. This state was then relaxed with a reduced exchange stiffness since, as the transition temperature is approached from below, the exchange stiffness weakens (42) and is the predominant energy term for the stabilization of the magnetic helices. To obtain representative amorphous structures, we used ab initio molecular dynamics (AIMD) simulations with the canonical ensemble (NVT) as implemented in Vienna Ab Initio Software Package (VASP) (43) with the projector augmented-wave potentials (44). We used the “melt-quench” methodology to generate amorphous snapshots, which was previously demonstrated on several systems (18) and found to agree with experimental measurements of the structural and local electronic properties (45–50). We constructed a cell of 96 atoms (Fe_0.5Ge_0.5), which were first randomly distributed in a cubic simulation cell using Packmol (51). We obtained the stable liquid phase by equilibrating the pressure through a series of AIMD simulations at 3000 K, concurrently rescaling the unit cell between each AIMD simulation until it reached an internal pressure of 0 bar. Following this, we collected three snapshots at regular intervals after performing a 10-ps production run. We quenched these three snapshots following a stepped cooling scheme with 400-fs cooling and 1-ps isothermal steps. Final structures were optimized using VASP with an energy cutoff of 860 eV and Γ point only sampling for the Brillouin zone. All structural relaxations were performed until the Hellmann-Feynman force on each atom was less than 0.002 eV/Å and included spin-orbit coupling (SOC) self-consistently as implemented in VASP. We used the local density approximation (LDA) (52) for the final relaxations as this gave better agreement with the experimentally reported magnetic moments of 0.75 μB (17, 45). We calculated the MSAE surfaces by varying the spin quantization axes over 50 points with SOC and comparing the total energies with respect to the lowest total energy.

Supplementary Materials

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Supplementary Text

Figs. S1 to S6